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Teardrop curve

A plane curve given by the parametric equations(1)(2)The plots above show curves for values of from 0 to 7.The teardrop curve has area(3)


In the triangle illustrated above, side subtends angle . More generally, given a geometric object in the plane and a point , let be the angle from one edge of to the other with vertex at . Then is said to subtend an angle from .

Full angle

A full angle, also called a complete angle, round angle, or perigon, is an angle equal to radians corresponding to the central angle of an entire circle.Four right angles or two straightangles equal one full angle.

Angle standard position

An angle drawn on the coordinate plane is said to be in standard position if its initial side lies on the positive x-axis so that its vertex coincides with the origin and its rotation is in the counterclockwise direction.In the above image, the angle is in standard position due to the locations of its vertex and its initial side and because of the direction of its rotation.

Right angle

A right angle is an angle equal to half the angle from one end of a line segment to the other. A right angle is radians or . A triangle containing a right angle is called a right triangle. However, a triangle cannot contain more than one right angle, since the sum of the two right angles plus the third angle would exceed the total possessed by a triangle.The patterns of cracks observed in mud that has been dried by the sun form curves that often intersect in right angles (Williams 1979, p. 45; Steinhaus 1999, p. 88; Pearce 1990, p. 12).

Exterior angle bisector

The exterior angle bisectors (Johnson 1929, p. 149), also called the external angle bisectors (Kimberling 1998, pp. 18-19), of a triangle are the lines bisecting the angles formed by the sides of the triangles and their extensions, as illustrated above.Note that the exterior angle bisectors therefore bisect the supplementaryangles of the interior angles, not the entire exterior angles.There are therefore three pairs of oppositely oriented exterior angle bisectors. The exterior angle bisectors intersect pairwise in the so-called excenters , , and . These are the centers of the excircles, i.e., the three circles that are externally tangent to the sides of the triangle (or their extensions).The points determined on opposite sides of a triangle by an angle bisector from each vertex lie on a straight line if either (1) all or (2) one out of the three bisectors is an external angle bisector (Johnson 1929, p. 149; Honsberger..

Exterior angle

An exterior angle of a polygon is the angle formed externally between two adjacent sides. It is therefore equal to , where is the corresponding internal angle between two adjacent sides (Zwillinger 1995, p. 270).Consider the angles formed between a side of a polygon and the extension of an adjacent side. Since there are two directions in which a side can be extended, there are two such angles at each vertex. However, since corresponding angles are opposite, they are also equal.Confusingly, a bisector of an angle is known as an exterior angle bisector, while a bisector of an angle (which is simply a line oriented in the opposite direction as the interior angle bisector) is not given any special name.The sum of the angles in a convex polygon is equal to radians (), since this corresponds to one complete rotation of the polygon...

Angle bisector

The (interior) bisector of an angle, also called the internal angle bisector (Kimberling 1998, pp. 11-12), is the line or line segment that divides the angle into two equal parts.The angle bisectors meet at the incenter , which has trilinear coordinates 1:1:1.The length of the bisector of angle in the above triangle is given bywhere and .The points , , and have trilinear coordinates , , and , respectively, and form the vertices of the incentral triangle.

Direction cosine

Let be the angle between and , the angle between and , and the angle between and . Then the direction cosines are equivalent to the coordinates of a unit vector ,(1)(2)(3)From these definitions, it follows that(4)To find the Jacobian when performing integrals overdirection cosines, use(5)(6)(7)The Jacobian is(8)Using(9)(10)(11)(12)so(13)(14)(15)(16)Direction cosines can also be defined between two sets of Cartesiancoordinates,(17)(18)(19)(20)(21)(22)(23)(24)(25)Projections of the unprimed coordinates onto the primed coordinates yield(26)(27)(28)(29)(30)(31)and(32)(33)(34)(35)(36)(37)Projections of the primed coordinates onto the unprimed coordinates yield(38)(39)(40)(41)(42)(43)and(44)(45)(46)Using the orthogonality of the coordinate system, it must be true that(47)(48)giving the identities(49)for and , and(50)for . These two identities may be combined into the single identity(51)where is the..


Given two intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is called the angle between them. The term "plane angle" is sometimes used to distinguish angles in a plane from solid angles measured in space (International Standards Organization 1982, p. 5).The term "angle" can also be applied to the rotational offset between intersecting planes about their common line of intersection, in which case the angle is called the dihedral angle of the planes.Angles are usually measured in degrees (denoted ), radians (denoted rad, or without a unit), or sometimes gradians (denoted grad).The concept of an angle can be generalized from the circle to the sphere, in which case it is known as solid angle. The fraction of a sphere subtended by an object (its solid angle) is measured in steradians, with the entire sphere..


The radian is a unit of angular measure defined such that an angle of one radian subtended from the center of a unit circle produces an arc with arc length 1.A full angle is therefore radians, so there are per radians, equal to or 57./radian. Similarly, a right angle is radians and a straight angle is radians.Radians are the most useful angular measure in calculus because they allow derivative and integral identities to be written in simple terms, e.g.,for measured in radians.Unless stated otherwise, all angular quantities considered in this work are assumed to be specified in radians.

Generalized cone

A ruled surface is called a generalized cone if it can be parameterized by , where is a fixed point which can be regarded as the vertex of the cone. A generalized cone is a regular surface wherever . The above surface is a generalized cone over a cardioid. A generalized cone is a flat surface, and is sometimes called "conical surface."

Elliptic cone

A cone with elliptical cross section. The parametric equations for an elliptic cone of height , semimajor axis , and semiminor axis are(1)(2)(3)where and .The elliptic cone is a quadratic ruledsurface, and has volume(4)The coefficients of the first fundamental form(5)(6)(7)second fundamental form coefficients(8)(9)(10)The lateral surface area can then be calculated as(11)(12)(13)where is a complete elliptic integral of the second kind and assuming .The Gaussian curvature is(14)and the mean curvature is(15)

Conical frustum

A conical frustum is a frustum created by slicing the top off a cone (with the cut made parallel to the base). For a right circular cone, let be the slant height and and the base and top radii. Then(1)The surface area, not including the top and bottomcircles, is(2)(3)The volume of the frustum is given by(4)But(5)so(6)(7)(8)This formula can be generalized to any pyramid by letting be the base areas of the top and bottom of the frustum. Then the volume can be written as(9)The area-weighted integral of over the frustum is(10)(11)so the geometric centroid is located alongthe z-axis at a height(12)(13)(Eshbach 1975, p. 453; Beyer 1987, p. 133; Harris and Stocker 1998, p. 105). The special case of the cone is given by taking , yielding .

Cone net

The mapping of a grid of regularly ruled squares onto a cone with no overlap or misalignment. Cone nets are possible for vertex angles of , , and , where the dark edges in the upper diagrams above are joined. Beautiful photographs of cone net models (lower diagrams above) are presented in Steinhaus (1999). The transformation from a point in the grid plane to a point on the cone is given by(1)(2)(3)where , 1/2, or 3/4 is the fraction of a circle forming the base, and(4)(5)(6)


Two cones placed base-to-base.The bicone with base radius and half-height has surface area and volume(1)(2)The centroid is at the origin, and the inertia tensor about the centroid is given by(3)

Nielsen's spiral

Nielsen's spiral, also called the sici spiral (von Seggern 1993) is the spiralwith parametric equations(1)(2)where is the cosine integral and is the sine integral.The curvature is given by(3)and the arc length measured from by(4)

Fermat's spiral

Fermat's spiral, also known as the parabolic spiral, is an Archimedean spiral with having polar equation(1)This curve was discussed by Fermat in 1636 (MacTutor Archive). For any given positive value of , there are two corresponding values of of opposite signs. The left plot above shows(2)only, while the right plot shows equation (1) in red and(3)in blue. Taking both signs, the resulting spiral is symmetrical about the origin.The curvature and arc lengthof the positive branch of Fermat's spiral are(4)(5)(6)where is a hypergeometric function and is an incomplete beta function.

Epispiral inverse curve

The inverse curve of the epispiralwith inversion center at the origin and inversion radius is the rose


The epispiral is a plane curve with polar equationThere are sections if is odd and if is even.A slightly more symmetric version considers instead

Theodorus spiral

The Theodorus spiral is a discrete spiral formed by connecting the ends of radial spokes corresponding to the hypotenuses of a sequence of adjoining right triangles. The initial spoke is of length , the next spoke is of length , etc., and each segment of the spiral (corresponding to the outer leg of a triangle) has unit length. It is also known as the square root spiral, Einstein spiral, Pythagorean spiral, or--to contrast it with certain continuous analogs--the discrete spiral of Theodorus.The slope of a continuous analog of the discrete Theodorus spiral due to Davis (1993) at the point is sometimes known as Theodorus's constant.

Logarithmic spiral pedal curve

The pedal curve of a logarithmicspiral with parametric equation(1)(2)for a pedal point at the pole is an identical logarithmicspiral(3)(4)so(5)

Logarithmic spiral inverse curve

The inverse curve of the logarithmicspiralwith inversion center at the origin and inversion radius is the logarithmic spiral

Logarithmic spiral evolute

For a logarithmic spiral given parametricallyas(1)(2)evolute is given by(3)(4)As first shown by Johann Bernoulli, the evolute of a logarithmic spiral is therefore another logarithmic spiral, having and ,In some cases, the evolute is identical to the original,as can be demonstrated by making the substitution to the new variable(5)Then the above equations become(6)(7)(8)(9)which are equivalent to the form of the original equation if(10)(11)(12)where only solutions with the minus sign in exist. Solving gives the values summarized in the following table.10.2744106319...20.1642700512...30.1218322508...40.0984064967...50.0832810611...60.0725974881...70.0645958183...80.0583494073...90.0533203211...100.0491732529...

Cotes' spiral

A spiral that gives the solution to the central orbitproblem under a radial force law(1)where is a positive constant. There are three solution regimes,(2)where and are constants,(3)(4)and is the specific angular momentum (Whittaker 1944, p. 83). The case gives an epispiral, while leads to a hyperbolic spiral.

Logarithmic spiral catacaustic

The catacaustic of a logarithmic spiral, where the origin is taken as the radiant point, is another logarithmic spiral. For an original spiral with parametric equations(1)(2)the catacaustic with radiant point at the originis(3)(4)

Logarithmic spiral

The logarithmic spiral is a spiral whose polarequation is given by(1)where is the distance from the origin, is the angle from the x-axis, and and are arbitrary constants. The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira mirabilis. It can be expressed parametrically as(2)(3)This spiral is related to Fibonacci numbers, the golden ratio, and the golden rectangle, and is sometimes called the golden spiral.The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3).The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw..

Sinusoidal spiral pedal curve

The pedal curve of a sinusoidalspiralwith pedal point at the center is another sinusoidalspiral with polar equationA few examples are illustrated above.

Sinusoidal spiral inverse curve

The inverse curve of a sinusoidalspiralwith inversion center at the origin and inversion radius is another sinusoidal spiral

Steiner's theorem

The most common statement known as Steiner's theorem (Casey 1893, p. 329) states that the Pascal lines of the hexagons 123456, 143652, and 163254 formed by interchanging the vertices at positions 2, 4, and 6 are concurrent (where the numbers denote the order in which the vertices of the hexagon are taken). The 20 points of concurrence so generated are known as Steiner points.Another theorem due to Steiner lets lines and join a variable point on a conic section to two fixed points on the same conic section. Then and are projectively related.A third "Steiner's theorem" states that if two opposite edges of a tetrahedron move on two fixed skew lines in any way whatsoever but remain fixed in length, then the volume of the tetrahedron remains constant (Altshiller-Court 1979, p. 87)...

Lemoine hexagon

The Lemoine hexagon is a cyclic hexagon with vertices given by the six concyclic intersections of the parallels of a reference triangle through its symmedian point . The circumcircle of the Lemoine hexagon is therefore the first Lemoine circle. There are two definitions of the hexagon that differ based on the order in which the vertices are connected.The first definition is the closed self-intersecting hexagon in which alternate sides , , and pass through the symmedian point (left figure). The second definition (Casey 1888, p. 180) is the hexagon formed by the convex hull of the first definition, i.e., the hexagon (right figure).The sides of this hexagon have the property that, in addition to , , and , the remaining sides , , and are antiparallel to , , and , respectively.For the self-intersecting Lemoine hexagon, the perimeter and area are(1)(2)and for the simple hexagon, they are given by(3)(4)(Casey 1888, p. 188), where is the..

Fuhrmann's theorem

Let the opposite sides of a convex cyclic hexagon be , , , , , and , and let the polygon diagonals , , and be so chosen that , , and have no common polygon vertex (and likewise for , , and ), thenThis is an extension of Ptolemy's theorem tothe hexagon.

Pascal's theorem

The dual of Brianchon's theorem (Casey 1888, p. 146), discovered by B. Pascal in 1640 when he was just 16 years old (Leibniz 1640; Wells 1986, p. 69). It states that, given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line.In 1847, Möbius (1885) published the following generalization of Pascal's theorem: if all intersection points (except possibly one) of the lines prolonging two opposite sides of a -gon inscribed in a conic section are collinear, then the same is true for the remaining point.

Schoch line

In the arbelos, consider the semicircles and with centers and passing through . The Apollonius circle of , and the large semicircle of the arbelos is an Archimedean circle . This circle has radius(as it must), and centerThe line perpendicular to and passing through the center of is called the Schoch line.Now let and be two semicircles through with radii proportional to and respectively. The circle tangent to and with its center on the Schoch line is an Archimedean circle. These circles are called Woo circles.Let be the radical axis of the great semicircle of the arbelos and . From a point on consider the tangents to the circle on diameter . The circle with center on the Schoch line and tangent to these tangents is a Woo circle (Okumura and Watanabe 2004).An applet for investigating Woo circles and Schoch lines has been prepared by Schoch (2005)...

Bankoff circle

The circle through the cusp of the arbelos and the tangent points of the first Pappus circle, which is congruent to the two Archimedes' circles. If and , then the radius of the Bankoff circle is

Archimedes' circles

Draw the perpendicular line from the intersection of the two small semicircles in the arbelos. The two circles and tangent to this line, the large semicircle, and each of the two semicircles are then congruent and known as Archimedes' circles.For an arbelos with outer semicircle of unit radius and parameter , Archimedes' circles have radii(1)and centers(2)(3)Circles that are constructed in a natural way using an arbelos and are congruent to Archimedes' circles are known as Archimedean circles.

Archimedean circle

An Archimedean circle is a circle defined in the arbelos in a natural way and congruent to Archimedes' circles, i.e., having radiusfor an arbelos with outer semicircle of unit radius and parameter .


The term "arbelos" means shoemaker's knife in Greek, and this term is applied to the shaded area in the above figure which resembles the blade of a knife used by ancient cobblers (Gardner 1979). Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. The position of the central notch is arbitrary and can be located anywhere along the diameter.The arbelos satisfies a number of unexpected identities (Gardner 1979, Schoch). 1. Call the diameters of the left and right semicircles and , respectively, so the diameter of the enclosing semicircle is 1. Then the arc length along the bottom of the arbelos is(1)so the arc length along the enclosing semicircle is the same as the arc length along the two smaller semicircles. 2. Draw the perpendicular from the tangent of the two semicircles to the edge of the large circle. Then the area of the arbelos is the same as the area of the circle with..

Polygon inscribing

Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these triangles is the same independent of the polygon vertex chosen (Johnson 1929, p. 193).If a triangle is inscribed in a circle, another circle inside the triangle, a square inside the circle, another circle inside the square, and so on. Then the equation relating the inradius and circumradius of a regular polygon,(1)gives the ratio of the radii of the final to initial circles as(2)Numerically,(3)(OEIS A085365), where is the corresponding constant for polygon circumscribing. This constant is termed the Kepler-Bouwkamp constant by Finch (2003). Kasner and Newman's (1989) assertion that is incorrect, as is the value of 0.8700... given by Prudnikov et al. (1986, p. 757)...

Polygon circumscribing

Circumscribe a triangle about a circle, another circle around the triangle, a square outside the circle, another circle outside the square, and so on. The circumradius and inradius for an -gon are then related by(1)so an infinitely nested set of circumscribed polygons and circles has(2)(3)(4)Kasner and Newman (1989) and Haber (1964) state that , but this is incorrect, and the actual answer is(5)(OEIS A051762).By writing(6)it is possible to expand the series about infinity, change the order of summation, do the sum symbolically, and obtain the quickly converging series(7)where is the Riemann zeta function.Bouwkamp (1965) produced the following infinite productformulas for the constant,(8)(9)(10)where is the sinc function (cf. Prudnikov et al. 1986, p. 757), is the Riemann zeta function, and is the Dirichlet lambda function. Bouwkamp (1965) also produced the formula with accelerated convergence(11)where(12)(cited in Pickover..


A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior.The study of triangles is sometimes known as triangle geometry, and is a rich area of geometry filled with beautiful results and unexpected connections. In 1816, while studying the Brocard points of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?" (Wells 1991, p. 21).It is..

Heronian triangle

A Heronian triangle is a triangle having rational side lengths and rational area. The triangles are so named because such triangles are related to Heron's formula(1)giving a triangle area in terms of its side lengths , , and semiperimeter . Finding a Heronian triangle is therefore equivalent to solving the Diophantine equation(2)The complete set of solutions for integer Heronian triangles (the three side lengths and area can be multiplied by their least common multiple to make them all integers) were found by Euler (Buchholz 1992; Dickson 2005, p. 193), and parametric versions were given by Brahmagupta and Carmichael (1952) as(3)(4)(5)(6)(7)This produces one member of each similarity class of Heronian triangles for any integers , , and such that , , and (Buchholz 1992).The first few integer Heronian triangles sorted by increasing maximal side lengths, are ((3, 4, 5), (5, 5, 6), (5, 5, 8), (6, 8, 10), (10, 10, 12), (5, 12, 13), (10, 13,..

Prince rupert's cube

Prince Rupert's cube is the largest cube that can be made to pass through a given cube. In other words, the cube having a side length equal to the side length of the largest hole of a square cross section that can be cut through a unit cube without splitting it into two pieces.Prince Rupert's cube cuts a hole of the shape indicated in the above illustration (Wells 1991). Curiously, it is slightly larger than the original cube, with side length (OEIS A093577). Any cube this size or smaller can be made to pass through the original cube.

Hyperbolic map

A linear transformation is hyperbolic if none of its eigenvalues has modulus 1. This means that can be written as a direct sum of two -invariant subspaces and (where stands for stable and for unstable) such that there exist constants , , and with(1)(2)for , 1, ....

Galilean transformation

A transformation from one reference frame to another moving with a constant velocity with respect to the first for classical motion. However, special relativity shows that the transformation must be modified to the Lorentz transformation for relativistic motion. The forward Galilean transformation isand the inverse transformation is

Rotation operator

The rotation operator can be derived from examining an infinitesimalrotationwhere is the time derivative, is the angular velocity, and is the cross product operator.


The turning of an object or coordinate system by an angle about a fixed point. A rotation is an orientation-preserving orthogonal transformation. Euler's rotation theorem states that an arbitrary rotation can be parameterized using three parameters. These parameters are commonly taken as the Euler angles. Rotations can be implemented using rotation matrices.Rotation in the plane can be concisely described in the complex plane using multiplication of complex numbers with unit modulus such that the resulting angle is given by . For example, multiplication by represents a rotation to the right by and by represents rotation to the left by . So starting with and rotating left twice gives , which is the same as rotating right twice, , and . For multiplication by multiples of , the possible positions are then concisely represented by , , , and .The rotation symmetry operation for rotation by is denoted "." For periodic arrangements of points..

Rodrigues' rotation formula

Rodrigues' rotation formula gives an efficient method for computing the rotation matrix corresponding to a rotation by an angle about a fixed axis specified by the unit vector . Then is given by(1)(2)(3)where is the identity matrixand denotes the antisymmetric matrix with entries(4)Note that the entries in this matrix are defined analogously to the differentialmatrix representation of the curl operator.Note that(5)so applying the rotation matrix given by Rodrigues' formula to any point on the rotation axis returns the same point.

Infinitesimal rotation

An infinitesimal transformation of a vector is given by(1)where the matrix is infinitesimal and is the identity matrix. (Note that the infinitesimal transformation may not correspond to an inversion, since inversion is a discontinuous process.) The commutativity of infinitesimal transformations and is established by the equivalence of(2)(3)(4)(5)Now let(6)The inverse is then , since(7)(8)(9)Since we are defining our infinitesimal transformation to be a rotation, orthogonalityof rotation matrices requires that(10)but(11)(12)(13)so and the infinitesimal rotation is antisymmetric. It must therefore have a matrix of the form(14)The differential change in a vector upon application of the rotation matrix is then(15)Writing in matrix form,(16)(17)(18)(19)Therefore,(20)where(21)The total rotation observed in the stationary frame will be a sum of the rotational velocity and the velocity in the rotating frame. However,..

Improper rotation

The symmetry operation corresponding to a rotation followed by an inversion operation, also called a rotoinversion. This operation is denoted for an improper rotation by , so the crystallography restriction gives only , , , , for crystals. The mirror plane symmetry operation is , etc., which is equivalent to .

Euler parameters

The four parameters , , , and describing a finite rotation about an arbitrary axis. The Euler parameters are defined by(1)(2)(3)where is the unit normal vector, and are a quaternion in scalar-vector representation(4)Because Euler's rotation theorem states that an arbitrary rotation may be described by only three parameters, a relationship must exist between these four quantities(5)(6)(Goldstein 1980, p. 153). The rotation angle is then related to the Euler parameters by(7)(8)(9)and(10)The Euler parameters may be given in terms of the Eulerangles by(11)(12)(13)(14)(Goldstein 1980, p. 155).Using the Euler parameters, the rotation formulabecomes(15)and the rotation matrix becomes(16)where the elements of the matrix are(17)Here, Einstein summation has been used, is the Kronecker delta, and is the permutation symbol. Written out explicitly, the matrix elements are(18)(19)(20)(21)(22)(23)(24)(25)(26)..

Euler angles

According to Euler's rotation theorem, any rotation may be described using three angles. If the rotations are written in terms of rotation matrices , , and , then a general rotation can be written as(1)The three angles giving the three rotation matrices are called Euler angles. There are several conventions for Euler angles, depending on the axes about which the rotations are carried out. Write the matrix as(2)The so-called "-convention," illustrated above, is the most common definition. In this convention, the rotation given by Euler angles , where 1. the first rotation is by an angle about the z-axis using , 2. the second rotation is by an angle about the former x-axis (now ) using , and 3. the third rotation is by an angle about the former z-axis (now ) using . Note, however, that several notational conventions for the angles are in common use. Goldstein (1980, pp. 145-148) and Landau and Lifschitz (1976) use , Tuma (1974) says is..


The surface area of a spherical segment. Call the radius of the sphere , the upper and lower radii and , respectively, and the height of the spherical segment . The zone is a surface of revolution about the z-axis, so the surface area is given by(1)In the -plane, the equation of the zone is simply that of a circle,(2)so(3)(4)and(5)(6)(7)(8)This result is somewhat surprising since it depends only on the height ofthe zone, not its vertical position with respect to the sphere.

Solid angle

The solid angle subtended by a surface is defined as the surface area of a unit sphere covered by the surface's projection onto the sphere. This can be written as(1)where is a unit vector from the origin, is the differential area of a surface patch, and is the distance from the origin to the patch. Written in spherical coordinates with the colatitude (polar angle) and for the longitude (azimuth), this becomes(2)Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is steradians.To see how the solid angle of simple geometric shapes can be computed explicitly, consider the solid angle subtended by one face of a cube of side length centered at the origin. Since the cube is symmetrical and has six sides, one side obviously subtends steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using(3)(4)and(5)(6)Considering the top face of the cube, which is located at and has sides..


An insphere is a sphere inscribed in a given solid. The radius of the insphere is called the inradius.Platonic solids (whose duals are themselves Platonic solids) and Archimedean duals have inspheres that touch all their faces, but Archimedean solids do not. Note that the insphere is not necessarily tangent at the centroid of the faces of a dual polyhedron, but is rather only tangent at some point lying on the face.The figures above depict the inspheres of the Platonicsolids.

Torispherical dome

A torispherical dome is the surface obtained from the intersection of a spherical cap with a tangent torus, as illustrated above. The radius of the sphere is called the "crown radius," and the radius of the torus is called the "knuckle radius." Torispherical domes are used to construct pressure vessels.Let be the distance from the center of the torus to the center of the torus tube, let be the radius of the torus tube, and let be the height from the base of the dome to the top. Then the radius of the base is given by . In addition, by elementary geometry, a torispherical dome satisfies(1)so(2)The transition from sphere to torus occurs at the critical radius(3)so the dome has equation(4)where(5)The torispherical dome has volume(6)(7)


The -hypersphere (often simply called the -sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (called by geometers the 3-sphere) to dimensions . The -sphere is therefore defined (again, to a geometer; see below) as the set of -tuples of points (, , ..., ) such that(1)where is the radius of the hypersphere.Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). A geometer would therefore regard the object described by(2)as a 2-sphere,..

Spherical wedge

The volume of a spherical wedge isThe surface area of the corresponding spherical luneis

Hemispherical function

The hemisphere function is defined as(1)Watson (1966) defines a hemispherical function as a function which satisfies the recurrence relations(2)with(3)


Half of a sphere cut by a plane passing through its center. A hemisphere of radius can be given by the usual spherical coordinates(1)(2)(3)where and . All cross sections passing through the z-axis are semicircles.The volume of the hemisphere is(4)(5)(6)The weighted mean of over the hemisphere is(7)The geometric centroid is then given by(8)(Beyer 1987).

Double sphere

The double sphere is the degenerate quartic surfaceobtained by squaring the left-hand side of the equation of a usual sphere

Dandelin spheres

The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres.The spheres can be used to show that the intersection of the plane with the cone is an ellipse. Let be a plane intersecting a right circular cone with vertex in the curve . Call the spheres tangent to the cone and the plane and , and the circles on which the spheres are tangent to the cone and . Pick a line along the cone which intersects at , at , and at . Call the points on the plane where the sphere are tangent and . Because intersecting tangents have the same length,(1)(2)Therefore,(3)which is a constant independent of , so is an ellipse with .

Spherical lune

A sliver of the surface of a sphere of radius cut out by two planes through the azimuthal axis with dihedral angle . The surface area of the lune iswhich is just the area of the sphere times . The volume of the associated spherical wedge has volume


A sphere circumscribed in a given solid. Its radius is called the circumradius. By analogy with the equation of the circumcircle, the equation for the circumsphere of the tetrahedron with polygon vertices for , ..., 4 is(1)Expanding the determinant,(2)where(3) is the determinant obtained from the matrix(4)by discarding the column (and taking a plus sign) and similarly for (this time taking the minus sign) and (again taking the plus sign)(5)(6)(7)and is given by(8)Completing the square gives(9)which is a sphere of the form(10)with circumcenter(11)(12)(13)and circumradius(14)The figures above depict the circumspheres of the Platonicsolids.


A bubble is a minimal-energy surface of the type that is formed by soap film. The simplest bubble is a single sphere, illustrated above (courtesy of J. M. Sullivan). More complicated forms occur when multiple bubbles are joined together. The simplest example is the double bubble, and beautiful configurations can form when three or more bubbles are conjoined (Sullivan).An outstanding problem involving bubbles is the determination of the arrangements of bubbles with the smallest surface area which enclose and separate given volumes in space.

Sphere with tunnel

Find the tunnel between two points and on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of radius , and with uniform density . Then the standard form Euler-Lagrange differential equation in polar coordinates is(1)along with the boundary conditions , , , and . Integrating once gives(2)But this is the equation of a hypocycloid generated by a circle of radius rolling inside the circle of radius , so the tunnel is shaped like an arc of a hypocycloid. The transit time from point to point is(3)where(4)is the surface gravity with the universal gravitational constant.

Kollros' theorem

For every ring containing spheres, there exists a ring of spheres, each touching each of the spheres, where(1)which can also be written(2)This was stated without proof by Jakob Steiner and proved by Kollros in 1938.The hexlet is a special case with . if more than one turn is allowed, then(3)where and are the numbers of turns on both necklaces before closing (M. Buffet, pers. comm., Feb. 14, 2003).

Bowl of integers

Place two solid spheres of radius 1/2 inside a hollow sphere of radius 1 so that the two smaller spheres touch each other at the center of the large sphere and are tangent to the large sphere on the extremities of one of its diameters. This arrangement is called the "bowl of integers" (Soddy 1937) since the bend of each of the infinite chain of spheres that can be packed into it such that each successive sphere is tangent to its neighbors is an integer. The first few bends are then , 2, 5, 6, 9, 11, 14, 15, 18, 21, 23, ... (OEIS A046160). The sizes and positions of the first few rings of spheres are given in the table below.100--220--3546059611071481591801021112312270, 1330143315380Spheres can also be packed along the plane tangent to the two spheres of radius 2 (Soddy 1937). The sequence of integers for can be found using the equation of five tangent spheres. Letting givesFor example, , , , , , and so on, giving the sequence , 2, 3, 11, 15, 27, 35, 47,..

Spherical segment

A spherical segment is the solid defined by cutting a sphere with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface of the spherical segment (excluding the bases) is called a zone. However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for spherical cap and "zone" for what is here called a spherical segment.Call the radius of the sphere and the height of the segment (the distance from the plane to the top of sphere) . Let the radii of the lower and upper bases be denoted and , respectively. Call the distance from the center to the start of the segment , and the height from the bottom to the top of the segment . Call the radius parallel to the segment , and the height above the center . Then ,(1)(2)(3)(4)(5)(6)Relationships among the various quantities include(7)(8)(9)(10)(11)Plugging in gives(12)(13)(14)The..

Spherical sector

A spherical sector is a solid of revolution enclosed by two radii from the center of a sphere. The spherical sector may either be "open" and have a conical hole (left figure; Beyer 1987), or may be a "closed" spherical cone (right figure; Harris and Stocker 1998). The volume of a spherical sector in either case is given bywhere is the vertical distance between where the upper and lower radii intersect the sphere and is the sphere's radius.

Spherical cap

A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment. However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for what is here called a spherical cap and "zone" for spherical segment.Let the sphere have radius , then the volume of a spherical cap of height and base radius is given by the equation of a spherical segment(1)with , giving(2)Using the Pythagorean theorem gives(3)which can be solved for as(4)so the radius of the base circle is(5)and plugging this in gives the equivalent formula(6)In terms of the so-called contact angle (the anglebetween the normal to the sphere at the bottom of the cap and the base plane)(7)(8)so(9)The geometric centroid occurs at a distance(10)above the center of the..

Reuleaux tetrahedron

The Reuleaux tetrahedron, sometimes also called the spherical tetrahedron, is the three-dimensional solid common to four spheres of equal radius placed so that the center of each sphere lies on the surface of the other three. The centers of the spheres are therefore located at the vertices of a regular tetrahedron, and the solid consists of an "inflated" tetrahedron with four curved edges.Note that the name, coined here for the first time, is based on the fact that the geometric shape is the three-dimensional analog of the Reuleaux triangle, not the fact that it has constant width. In fact, the Reuleaux tetrahedron is not a solid of constant width. However, Meißner (1911) showed how to modify the Reuleaux tetrahedron to form a surface of constant width by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. Depending on which three edge arcs are replaced (three that have a common..


A hosohedron is a regular tiling or map on a sphere composed of digons or spherical lunes, all with the same two vertices and the same vertex angles, . Its Schläfli symbol is . Its dual is the dihedron .

General prismatoid

A solid such that the area of any section parallel to and a distance from a fixed plane can be expressed asThe volume of such a solid is the same as for a prismatoid,Examples include the cone, conical frustum, cylinder, prismatoid, pyramidal frustum, sphere, spherical segment, and spheroid.


A dihedron is a regular tiling or map on a sphere composed of two regular -gons, each occupying a hemisphere and with edge lengths of on a unit sphere. Its Schläfli symbol is . Its dual is the hosohedron .

Cork plug

A cork plug is a three-dimensional solid that can stopper a square, triangular, or circular hole. There is an infinite family of such shapes.The shape with smallest volume has triangular cross sections.The plug with the largest volume is made using two cuts from the top diameter to the edge, as illustrated above. Such a plug has to obtain a square cross section. For a general such a plug of height and radius , the volume of the plug is

Slant height

The slant height of an object (such as a frustum, or pyramid) is the distance measured along a lateral face from the base to the apex along the "center" of the face. In other words, it is the altitude of the triangle comprising a lateral face (Kern and Bland 1948, p. 50).The slant height of a right circular cone is the distance from the apex to a point on the base (Kern and Bland 1948, p. 60), and is related to the height and base radius byFor a right pyramid with a regular -gonal base of side length , the slant height is given bywhere is the inradius of the base.

Truncated square pyramid

The truncated square pyramid is a special case of a pyramidal frustum for a square pyramid. Let the base and top side lengths of the truncated pyramid be and , and let the height be . Then the volume of the solid isThis formula was known to the Egyptians ca. 1850 BC. The Egyptians cannot have proved it without calculus, however, since Dehn showed in 1900 that no proof of this equation exists which does not rely on the concept of continuity (and therefore some form of integration).

Pyramidal frustum

A pyramidal frustum is a frustum made by chopping thetop off a pyramid. It is a special case of a prismatoid.For a right pyramidal frustum, let be the slant height, the height, the bottom base perimeter, the top base perimeter, the bottom area, and the top area. Then the surface area (of the sides) and volume of a pyramidal frustum are given by(1)(2)The geometric centroid of a right pyramidalfrustum occurs at a height(3)above the bottom base (Harris and Stocker 1998).The bases of a right -gonal frustum are regular polygons of side lengths and with circumradii(4)where is the side length, so the diagonal connecting corresponding vertices on top and bottom has length(5)and the edge length is(6)(7)The triangular () and square () right pyramidal frustums therefore have side surface areas(8)(9)The area of a regular -gon is(10)so the volumes of these frustums are(11)(12)..


The superellipsoid is a generalization of the ellipsoid.The version also called the superquadratic ellipsoid is defined by the equation(1)where and are the east-west and north-south exponents, respectively. The superellipsoid can be rendered in POVRay® with the command superellipsoid{ <e, n> }The generalization(2)of the surface considered by Gray (1997) might also be called a superellipsoid. This surface can be given parametrically by(3)(4)(5)for and . Some special cases of this surface are summarized in the following table.ellipsoid, sphere, Hauser's "cube"The volume of the solid with is(6)(7)As , the solid becomes a cube, so(8)as it must. This is a special case of the integral Prudnikov et al. (1986, p. 583). The cases and appear to be the only integers whose corresponding solids have simple moment of inertia tensors, given by(10)(11)..


A superegg is a solid described by the equation(1)The special case gives a spheroid.Special cases of volume are given by(2)(3)


A spheroid is an ellipsoid having two axes of equal length, making it a surface of revolution. By convention, the two distinct axis lengths are denoted and , and the spheroid is oriented so that its axis of rotational symmetric is along the -axis, giving it the parametric representation(1)(2)(3)with , and .The Cartesian equation of the spheroid is(4)If , the spheroid is called oblate (left figure). If , the spheroid is prolate (right figure). If , the spheroid degenerates to a sphere.In the above parametrization, the coefficients of the firstfundamental form are(5)(6)(7)and of the second fundamental form are(8)(9)(10)The Gaussian curvature is given by(11)the implicit Gaussian curvature by(12)and the mean curvature by(13)The surface area of a spheroid can be variously writtenas(14)(15)(16)(17)where(18)(19)and is a hypergeometric function.The volume of a spheroid can be computed from the formula for a general ellipsoid with ,(20)(21)(Beyer..

Prolate spheroid

A prolate spheroid is a spheroid that is "pointy" instead of "squashed," i.e., one for which the polar radius is greater than the equatorial radius , so (called "spindle-shaped ellipsoid" by Tietze 1965, p. 27). A symmetrical egg (i.e., with the same shape at both ends) would approximate a prolate spheroid. A prolate spheroid is a surface of revolution obtained by rotating an ellipse about its major axis (Hilbert and Cohn-Vossen 1999, p. 10), and has Cartesian equations(1)The ellipticity of the prolate spheroid is definedby(2)The surface area of a prolate spheroid can be computedas a surface of revolution about the z-axis,(3)with radius as a function of given by(4)The integrand is then(5)and the integral is given by(6)(7)Using the identity(8)gives(9)(Beyer 1987, p. 131). Note that this is the conventional form in which the surface area of a prolate spheroid is written, although it..

Oblate spheroid

A "squashed" spheroid for which the equatorial radius is greater than the polar radius , so (called an oblate ellipsoid by Tietze 1965, p. 27). An oblate spheroid is a surface of revolution obtained by rotating an ellipse about its minor axis (Hilbert and Cohn-Vossen 1999, p. 10). To first approximation, the shape assumed by a rotating fluid (including the Earth, which is "fluid" over astronomical time scales) is an oblate spheroid.For a spheroid with z-axisas the symmetry axis, the Cartesian equation is(1)The ellipticity of an oblate spheroid is definedby(2)The surface area of an oblate spheroid can be computedas a surface of revolution about the z-axis,(3)with radius as a function of given by(4)Therefore(5)(6)(7)(8)where the last step makes use of the logarithm identity(9)valid for . Re-expressing in terms of the ellipticity then gives(10)yielding the particular simple form(11)(Beyer 1987, p. 131)...


The flattening of a spheroid (also called oblateness) is denoted or (Snyder 1987, p. 13). It is defined as(1)where is the polar radius and is the equatorial radius.It is related to the ellipticity by(2)(3)(Snyder 1987, p. 13).


Given a spheroid with equatorial radius and polar radius , the ellipticity is defined by(1)It is defined analogously to eccentricity and is commonly denoted using the symbols (Snyder 1987, p. 13) or (Beyer 1987).It is related to the flattening by(2)(3)(Snyder 1987, p. 13).


The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by(1)where the semi-axes are of lengths , , and . In spherical coordinates, this becomes(2)If the lengths of two axes of an ellipsoid are the same, the figure is called a spheroid (depending on whether or , an oblate spheroid or prolate spheroid, respectively), and if all three are the same, it is a sphere. Tietze (1965, p. 28) calls the general ellipsoid a "triaxial ellipsoid."There are two families of parallel circular cross sections in every ellipsoid. However, the two coincide for spheroids (Hilbert and Cohn-Vossen 1999, pp. 17-19). If the two sets of circles are fastened together by suitably chosen slits so that they are free to rotate without sliding, the model is movable. Furthermore, the disks can always be moved into the shape of a sphere (Hilbert and Cohn-Vossen 1999, p. 18).In 1882, Staude..

Cylindrical hoof

The cylindrical hoof is a special case of the cylindrical wedge given by a wedge passing through a diameter of the base (so that ). Let the height of the wedge be and the radius of the cylinder from which it is cut be . Then plugging the points , , and into the 3-point equation for a plane gives the equation for the plane as(1)Combining with the equation of the circle that describes the curved part remaining of the cylinder (and writing ) then gives the parametric equations of the "tongue" of the wedge as(2)(3)(4)for . To examine the form of the tongue, it needs to be rotated into a convenient plane. This can be accomplished by first rotating the plane of the curve by about the x-axis using the rotation matrix and then by the angle(5)above the z-axis. The transformed plane now rests in the -plane and has parametric equations(6)(7)and is shown below. The length of the tongue (measured down its middle) is obtained by plugging into the above equation for..

Generalized cylinder

A ruled surface is called a generalized cylinder if it can be parameterized by , where is a fixed point. A generalized cylinder is a regular surface wherever . The above surface is a generalized cylinder over a cardioid. A generalized cylinder is a flat surface, and is sometimes called a "cylindrical surface" (Kern and Bland 1948, p. 32) or "cylinder surface" (Harris and Stocker 1998, p. 102).A generalized cylinder need not be closed (Kern and Bland 1948, p. 32).Kern and Bland (1948, p. 32) define a cylinder as a solid bounded by a generalized cylinder and two parallel planes. However, when used without qualification, the term "cylinder" generally refers to the particular case of a right circular cylinder.


Let a vault consist of two equal half-cylinders of radius which intersect at right angles so that the lines of their intersections (the "groins") terminate in the polyhedron vertices of a square. Two vaults placed bottom-to-top form a Steinmetz solid on two cylinders.Solving the equations(1)(2)simultaneously gives(3)(4)One quarter of the vault can therefore be described by the parametricequations(5)(6)(7)The surface area of the vault is therefore givenby(8)where is the length of a cross section at height and is the angle a point on the center of this line makes with the origin. But , so(9)and(10)(11)(12)The volume of the vault is(13)(14)The geometric centroid is(15)

Steinmetz solid

The solid common to two (or three) right circular cylinders of equal radii intersecting at right angles is called the Steinmetz solid. Two cylinders intersecting at right angles are called a bicylinder or mouhefanggai (Chinese for "two square umbrellas"), and three intersecting cylinders a tricylinder. Half of a bicylinder is called a vault.For two cylinders of radius oriented long the - and -axes gives the equations(1)(2)which can be solved for and gives the parametric equations of the edges of the solid,(3)(4)The surface area can be found as , where(5)(6)Taking the range of integration as a quarter or one face and then multiplying by 16 gives(7)The volume common to two cylinders was known to Archimedes (Heath 1953, Gardner 1962) and the Chinese mathematician Tsu Ch'ung-Chih (Kiang 1972), and does not require calculus to derive. Using calculus provides a simple derivation, however. Noting that the solid has a square cross section..

Spherical ring

A spherical ring is a sphere with a cylindrical hole cut so that the centers of the cylinder and sphere coincide, also called a napkin ring. Let the sphere have radius and the cylinder radius .From the right diagram, the surface area of the sphericalring is equal to twice that of a cylinder of half-height(1)and radius plus twice that of the zone of radius and height , giving(2)(3)Note that as illustrated above, the hole cut out consists of a cylindrical portion plus two spherical caps. The volume of the entire cylinder is(4)and the volume of the upper segment is(5)The volume removed upon drilling of a cylindricalhole is then(6)(7)where the expressions(8)(9)obtained from trigonometry have been used to re-express the result.The volume of the spherical ring itself is then givenby(10)(11)(12)By the final equation, the remaining volume of any center-drilled sphere can be calculated given only the length of the hole. In particular, if the sphere..

Elliptic cylinder

An elliptic cylinder is a cylinder with an elliptical cross section.The elliptic cylinder is a quadratic ruledsurface.The parametric equations for the laterals sides of an elliptic cylinder of height , semimajor axis , and semiminor axis are(1)(2)(3)where and .The volume of the elliptic cylinder is(4)The coefficients of the first fundamental formare(5)(6)(7)and of the second fundamental form are(8)(9)(10)The area element is(11)The Gaussian and meancurvatures are(12)(13)

Cylindrical wedge

A wedge is cut from a cylinder by slicing with a plane that intersects the base of the cylinder. The volume of a cylindrical wedge can be found by noting that the plane cutting the cylinder passes through the three points illustrated above (with ), so the three-point form of the plane gives the equation(1)(2)Solving for gives(3)Here, the value of is given by(4)(5)The volume is therefore given as an integralover rectangular areas along the x-axis,(6)Using the identities(7)(8)(9)(10)gives the equivalent alternate forms(11)(12)(Harris and Stocker 1998, p. 104). This simplifies in the case of to(13)The lateral surface areacan be found from(14)where is simply with , so(15)(16)(17)(18)(Harris and Stocker 1998, p. 104).A special case of the cylindrical wedge, also called a cylindrical hoof, is a wedge passing through a diameter of the base (so that )...

Horizontal cylindrical segment

The solid cut from a horizontal cylinder of length and radius by a single plane oriented parallel to the cylinder's axis of symmetry (i.e., a portion of a horizontal cylindrical tank which is partially filled with fluid) is called a horizontal cylindrical segment.For a cut made a height above the bottom of the horizontal cylinder (as illustrated above), the volume of the cylindrical segment is given by multiplying the area of a circular segment of height by the length of the tank ,plotted above. Note that the above equation gives , , and , as expected. Since a circular segment is the cross section of the horizontal cylindrical segment, determining the fraction of the tank that is full is equivalent to determining the fractional area of a circle covered by the circular segment.Finding the height above the bottom of a horizontal cylinder (such as a cylindrical gas tank) to which the it must be filled for it to be one quarter full is sometimes known as the..

Cylindrical segment

A cylindrical segment, sometimes also called a truncated cylinder, is the solid cutfrom a circular cylinder by two (or more) planes.If there are two cutting planes, one perpendicular to the axis of the cylinder and the other titled with respect to it, the resulting solid is known as a cylindrical wedge.If the plane is titled with respect to a circular cross section but does not cut the bottom base, the resulting cylindrical segment has one circular cap and one elliptical cap (see above figure). Consider a cylinder of radius and minimum and maximum heights and . Set up a coordinate system with lower cap in the -plane, origin at the center of the lower cap, and the -axis passing through the center of the lower cap parallel to the projection of the semimajor axis of the upper cap. Then the height of the solid at distance is given byThe volume of the cylindrical section can be obtained instantly by noting that two such sections can be fitted together to form a cylinder..


A sphericon is the solid formed from a bicone with opening angle of (and therefore with ) obtained by slicing the solid with a plane containing the rotational axes resulting in a square cross section, then rotating the two pieces by and reconnecting them. It was constructed by Israeli game and toy inventor David Hirsch who patented the shape in Israel in 1984. It was given the name "sphericon" by Colin Roberts, who independently discovered the solid in the 1960s while attempting to carve a Möbius strip without a hole in the middle out of a block of wood.The solid is not as widely known as it should be.The above net shows another way the sphericon can be constructed. In this figure radians . A sphericon has a single continuous face and rolls by wobbling along that face, resulting in straight-line motion. In addition, one sphericon can roll around another.The sphericon with radius has surface area and volume(1)(2)The centroid is at the..

Spherical cone

The surface of revolution obtained by cutting a conical "wedge" with vertex at the center of a sphere out of the sphere. It is therefore a cone plus a spherical cap, and is a degenerate case of a spherical sector. The volume of the spherical cone is(1)(Kern and Bland 1948, p. 104). The surface areaof a closed spherical sector is(2)and the geometric centroid is located at aheight(3)above the sphere's center (Harris and Stocker 1998).The inertia tensor of a uniform spherical cone of mass is given by(4)The degenerate case of gives a hemisphere with circular base, yielding(5)(6)as expected.

Harborth's tiling

A tiling consisting of a rhombus such that 17 rhombuses fit around a point and a second tile in the shape of six rhombuses stuck together. These two tiles can fill the plane in exactly four different ways. Two tiles which tile the plane in ways can be constructed using a rhombus of a shape such that pack around a point together with a complex piece made by sticking rhombuses together (Wells 1991).

Penrose tiles

The Penrose tiles are a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders). These two tiles, illustrated above, are called the "kite" and "dart," respectively. In strict Penrose tiling, the tiles must be placed in such a way that the colored markings agree; in particular, the two tiles may not be combined into a rhombus (Hurd).Two additional types of Penrose tiles known as the rhombs (of which there are two varieties: fat and skinny) and the pentacles (or which there are six type) are sometimes also defined that have slightly more complicated matching conditions (McClure 2002).In 1997, Penrose sued the Kimberly Clark Corporation over their quilted toilet paper, which allegedly resembles a Penrose aperiodic tiling (Mirsky 1997). The suit was apparently settled out of court.To see how the plane may be tiled aperiodically using the kite and dart, divide the kite into..


The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...

Rounded rectangle

A rounded rectangle is the shape obtained by taking the convex hull of four equal circles of radius and placing their centers at the four corners of a rectangle with side lengths and .A filled rounded rectangle with (or ) is called a stadium.The rounded rectangle has perimeter(1)A filled rounded rectangle has area(2)For a rounded square with (center) length and width , the corner radius can be determined by measuring the generalized diameter from the edge of one rounded corner to the diagonally opposite corner. From the Pythagorean theorem, the diagonal distance from the rounded corner to the corner of the circumscribed square is(3)and the corner radius is related to the edge length of the square circumscribing the corner circle by(4)Combining gives(5)(6)


The salinon is the figure illustrated above formed from four connected semicircles. The word salinon is Greek for "salt cellar," which the figure resembles. If the radius of the large enclosing circle is and the radius of the small central circle is , then the radii of the two small side circles are .In his Book of Lemmas, Archimedes proved that the salinon has an area equal to the circle having the line segment joining the top and bottom points as its diameter (Wells 1991), namely


A lune is a plane figure bounded by two circular arcs of unequal radii, i.e., a crescent. (By contrast, a plane figure bounded by two circular arcs of equal radius is known as a lens.) For circles of radius and whose centers are separated by a distance , the area of the lune is given by(1)where(2)is the area of the triangle with side lengths , , and . The second of these can be obtained directly by subtracting the areas of the two half-lenses whose difference producing the colored region above.In each of the figures above, the area of the lune is equal to the area of the indicated triangle. Hippocrates of Chios squared the above left lune (Dunham 1990, pp. 19-20; Wells 1991, pp. 143-144), as well as two others, in the fifth century BC. Two more squarable lunes were found by T. Clausen in the 19th century (Shenitzer and Steprans 1994; Dunham 1990 attributes these discoveries to Euler in 1771). In the 20th century, N. G. Tschebatorew..

Vesica piscis

The term "vesica piscis," meaning "fish bladder" in Latin, is used for the particular symmetric lens formed by the intersection of two equal circles whose centers are offset by a distance equal to the circle radii (Pedoe 1995, p. xii). The height of the lens is given by letting in the equation for a circle-circle intersection(1)giving(2)The vesica piscis therefore has two equilateral triangles inscribed in it as illustrated above.The area of the vesica piscis is given by plugging into the circle-circle intersection area equation with ,(3)giving(4)(5)(OEIS A093731). Since each arcof the lens is precisely 1/3 of a circle, perimeter is given by(6)Renaissance artists frequently surrounded images of Jesus with the vesica piscis (Pedoe 1995, p. xii; Rawles 1997).


The perpendicular distance from an arc's midpoint to the chord across it, equal to the radius minus the apothem ,(1)For a regular polygon of side length ,(2)(3)(4)(5)(6)where is the circumradius, the inradius, is the side length, and is the number of sides.


A (general, asymmetric) lens is a lamina formed by the intersection of two offset disks of unequal radii such that the intersection is not empty, one disk does not completely enclose the other, and the centers of curvatures are on opposite sides of the lens. If the centers of curvature are on the same side, a lune results.The area of a general asymmetric lens obtained from circles of radii and and offset can be found from the formula for circle-circle intersection, namely(1)(2)Similarly, the height of such a lens is(3)(4)A symmetric lens is lens formed by the intersection of two equal disk. The area of a symmetric lens obtained from circles with radii and offset is given by(5)and the height by(6)A special type of symmetric lens is the vesica piscis (Latin for "fish bladder"), corresponding to a disk offset which is equal to the disk radii.A lens-shaped region also arises in the study of Bessel functions, is very important in the theory of..

Square quadrants

The areas of the regions illustrated above can be found from the equations(1)(2)Since we want to solve for three variables, we need a third equation. This can be taken as(3)where(4)(5)leading to(6)Combining the equations (1), (2), and (6) gives the matrix equation(7)which can be inverted to yield(8)(9)(10)

Circular segment

A portion of a disk whose upper boundary is a (circular) arc and whose lower boundary is a chord making a central angle radians (), illustrated above as the shaded region. The entire wedge-shaped area is known as a circular sector.Circular segments are implemented in the Wolfram Language as DiskSegment[x, y, r, q1, q2]. Elliptical segments are similarly implemented as DiskSegment[x, y, r1, r2, q1, q2].Let be the radius of the circle, the chord length, the arc length, the height of the arced portion, and the height of the triangular portion. Then the radius is(1)the arc length is(2)the height is(3)(4)(5)and the length of the chord is(6)(7)(8)(9)From elementary trigonometry, the angle obeys the relationships(10)(11)(12)(13)The area of the (shaded) segment is then simply given by the area of the circular sector (the entire wedge-shaped portion) minus the area of the bottom triangular portion,(14)Plugging in gives(15)(16)(17)(18)where..


Half a circle. The area of a semicircle of radius is given by(1)(2)(3)The weighted mean of is(4)(5)The semicircle is the cross section of a hemispherefor any plane through the z-axis.The perimeter of the curved boundary is given by(6)With , this gives(7)The perimeter of the semicircular lamina is then(8)The weighted value of of the semicircular curve is given by(9)(10)(11)so the geometric centroid is(12)The geometric centroid of the semicircularlamina is given by(13)(Kern and Bland 1948, p. 113).

Circular sector

A circular sector is a wedge obtained by taking a portion of a disk with central angle radians (), illustrated above as the shaded region. A sector with central angle of radians would correspond to a filled semicircle. Let be the radius of the circle, the chord length, the arc length, the sagitta (height of the arced portion), and the apothem (height of the triangular portion). Then(1)(2)(3)(4)(5)(6)(7)(8)(9)The angle obeys the relationships(10)(11)(12)(13)The area of the sector is(14)(15)(Beyer 1987). The area can also be found by direct integration as(16)It follows that the weighted mean of the is(17)(18)so the geometric centroid of the circular sectoris(19)(20)(21)(Gearhart and Schulz 1990). Checking shows that this obeys the proper limits for a semicircle () and for an isosceles triangle ()...


There are a number of meanings for the word "arc" in mathematics. In general, an arc is any smooth curve joining two points. The length of an arc is known as its arc length.In a graph, a graph arc isan ordered pair of adjacent vertices.In particular, an arc is any portion (other than the entire curve) of the circumference of a circle. An arc corresponding to the central angle is denoted . Similarly, the size of the central angle subtended by this arc (i.e., the measure of the arc) is sometimes (e.g., Rhoad et al. 1984, p. 421) but not always (e.g., Jurgensen 1963) denoted .The center of an arc is the center of the circle of whichthe arc is a part.An arc whose endpoints lie on a diameter of a circleis called a semicircle.For a circle of radius , the arc length subtended by a central angle is proportional to , and if is measured in radians, then the constant of proportionality is 1, i.e.,(1)The length of the chord connecting the arc's endpointsis(2)As..

Orthodiagonal quadrangle

An orthodiagonal quadrangle is a quadrangle whose diagonals are perpendicular to each other. If , , , and are the sides of a quadrangle, then this quadrangle is orthodiagonal iff .If is a cyclic orthodiagonal quadrangle, then the quadrangle formed by the tangents to the circumcircle through the vertices of form a bicentric quadrilateral . The circumcenters of and and the point of intersection of the diagonals of are collinear.

Pascal lines

The lines containing the three points of the intersection of the three pairs of oppositesides of a (not necessarily regular) hexagon.There are 6! (i.e., 6 factorial) possible ways of taking all polygon vertices in any order, but among these are six equivalent cyclic permutations and two possible orderings, so the total number of different hexagons (not all simple) isThere are therefore a total of 60 Pascal lines created by connecting polygonvertices in any order.The 60 Pascal lines form a very complicated pattern which can be visualized most easily in the degenerate case of a regular hexagon inscribed in a circle, as illustrated above for magnifications ranging over five powers of 2. Only 45 lines are visible in this figure since each of the three thick lines (located at angles to each other) represents a degenerate group of four Pascal lines, and six of the Pascal lines are lines at infinity (Wells 1991). The pattern for a general ellipse and hexagon..

Pappus's hexagon theorem

If , , and are three points on one line, , , and are three points on another line, and meets at , meets at , and meets at , then the three points , , and are collinear. Pappus's hexagon theorem is self-dual.The incidence graph of the configuration corresponding to the theorem is the Pappus graph.


The regular tessellation consisting of regular hexagons (i.e., a hexagonal grid).In general, the term honeycomb is used to refer to a tessellation in dimensions for . The only regular honeycomb in three dimensions is , which consists of eight cubes meeting at each polyhedron vertex. The only quasiregular honeycomb (with regular cells and semiregular vertex figures) has each polyhedron vertex surrounded by eight tetrahedra and six octahedra and is denoted .Ball and Coxeter (1987) use the term "sponge" for a solid that can be parameterized by integers , , and that satisfy the equationThe possible sponges are , , , , and .There are many semiregular honeycombs, such as , in which each polyhedron vertex consists of two octahedra and four cuboctahedra .

Hexyl triangle

Given a triangle and the excentral triangle , define the -vertex of the hexyl triangle as the point in which the perpendicular to through the excenter meets the perpendicular to through the excenter , and similarly define and . Then is known as the hexyl triangle of , and forms a hexagon with parallel sides (Kimberling 1998 pp. 79 and 172).The hexyl triangle has trilinear vertex matrix(1)where , , and (Kimberling 1998, p. 172).It has side lengths(2)(3)(4)and area(5)(6)(7)where is the area of the reference triangle, is the circumradius, and is the inradius. It therefore has the same side lengths and area as the excentral triangle.The Cevians triangles with Cevian points corresponding to Kimberling centers with , 20, 21, 27, 63, and 84 are perspective to the hexyl triangle. That anticevian triangles and antipedal triangles corresponding to Kimberling centers for , 9, 19, 40, 57, 63, 84, 610, 1712, and 2184 are also perspective to the..

Thomsen's figure

Take any triangle with polygon vertices , , and . Pick a point on the side opposite , and draw a line parallel to . Upon reaching the side at , draw the line parallel to . Continue (left figure). Then the line closes for any triangle. If is the midpoint of , then (right figure).Let be the ratio in which the sides of the reference triangle are divided i.e., , and define . Then the coordinates of the vertices of the figure are shown above.The six vertexes of Thomsen's figure lie on an ellipse having the triangle centroid as its center. The area of this ellipse iswhere is the area of the reference triangle. When (or ), the ellipse becomes the Steiner circumellipse, and when , it becomes the Steiner inellipse (M. Tarquini, pers. comm., Sep. 2, 2005).Thomsen's figure is similar to a Tucker hexagon. While Thomsen's hexagon closes after six parallels, a Tucker hexagon closes after alternately three parallels and three antiparallels...

Marion's theorem

Marion's theorem (Mathematics Teacher 1993, Maushard 1994, Morgan 1994) states that the area of the central hexagonal region determined by trisection of each side of a triangle and connecting the corresponding points with the opposite vertex is given by 1/10 the area of the original triangle.This can easily be shown using trilinear coordinates. In the above diagram, , , and, from the multisection formula, the trisection points have trilinear coordinates(1)(2)(3)(4)(5)(6)The other labeled points can then be computed as(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)Using the trilinear equation for the area of a triangle then gives the following areas of the colored triangles illustrated above in terms of the area of the original triangle.(19)(20)(21)(22)Taking the remaining red portion then gives(23)(24)as originally stated.A generalization of Marion's theorem sometimes known as Morgan's theorem was found by Ryan Morgan,..

Hexagon tiling

A hexagon tiling is a tiling of the planeby identical hexagons.The regular hexagon forms a regular tessellation,also called a hexagonal grid, illustrated above.There are at least three tilings of irregular hexagons,illustrated above.They are given by the following types:(1)(Gardner 1988). Note that the periodic hexagonal tessellationis a degenerate case of all three tilings with(2)and(3)Amazingly, the number of plane partitions contained in an box also gives the number of hexagon tilings by rhombi for a hexagon of side lengths , , , , , (David and Tomei 1989, Fulmek and Krattenthaler 2000). The asymptotic distribution of rhombi in a random hexagon tiling by rhombi was given by Cohn et al. (1998). A variety of enumerations for various explicit positions of rhombi are given by Fulmek and Krattenthaler (1998, 2000)...

Coin paradox

After a half rotation of the coin on the left around the central coin (of the same radius), the coin undergoes a complete rotation. In other words, a coin makes two complete rotations when rolled around the boundary of an identical coin. This fact is readily apparent in the generation of the cardioid as one disk rolling on another.


In geometry, the term "enlargement" is a synonym for expansion.In nonstandard analysis, let be a set of urelements, and let be the superstructure with individuals in : 1. , 2. , 3. . Let be a superstructure monomorphism, with and for . Then is an enlargement of provided that for each set in , there is a hyperfinite set that contains all the standard entities of .It is the case that is an enlargement of if and only if every concurrent binary relation satisfies the following: There is an element of the range of such that for every in the domain of , the pair is in the relation .

Steiner's segment problem

Given points, find the line segments with the shortest possible total length which connect the points. The segments need not necessarily be straight from one point to another.For three points, if all angles are less than , then the line segments are those connecting the three points to a central point which makes the angles , , and all . If one angle is greater that , then coincides with the offending angle.For four points, is the intersection of the two diagonals, but the required minimum segments are not necessarily these diagonals.A modified version of the problem is, given two points, to find the segments with the shortest total length connecting the points such that each branch point may be connected to only three segments. There is no general solution to this version of the problem...


The game of billiards is played on a rectangular table (known as a billiard table) upon which balls are placed. One ball (the "cue ball") is then struck with the end of a "cue" stick, causing it to bounce into other balls and reflect off the sides of the table. Real billiards can involve spinning the ball so that it does not travel in a straight line, but the mathematical study of billiards generally consists of reflections in which the reflection and incidence angles are the same. However, strange table shapes such as circles and ellipses are often considered. The popular 1959 animated short film Donald in Mathmagic Land features a tutorial by Donald Duck on how to win at billiards using the diamonds normally inscribed around the edge of a real billiard table.Many interesting problems can arise in the detailed study of billiards trajectories. For example, any smooth plane convex set has at least two double normals, so there are..

Reflection property

In the plane, the reflection property can be stated as three theorems (Ogilvy 1990, pp. 73-77): 1. The locus of the center of a variable circle, tangent to a fixed circle and passing through a fixed point inside that circle, is an ellipse. 2. If a variable circle is tangent to a fixed circle and also passes through a fixed point outside the circle, then the locus of its moving center is a hyperbola. 3. If a variable circle is tangent to a fixed straight line and also passes through a fixed point not on the line, then the locus of its moving center is a parabola. Let be a smooth regular parameterized curve in defined on an open interval , and let and be points in , where is an -dimensional projective space. Then has a reflection property with foci and if, for each point , 1. Any vector normal to the curve at lies in the vector space span of the vectors and . 2. The line normal to at bisects one of the pairs of opposite angles formed by the intersection of the lines joining..


The operation of exchanging all points of a mathematical object with their mirror images (i.e., reflections in a mirror). Objects that do not change handedness under reflection are said to be amphichiral; those that do are said to be chiral.Consider the geometry of the left figure in which a point is reflected in a mirror (blue line). Then(1)so the reflection of is given by(2)The term reflection can also refer to the reflection of a ball, ray of light, etc. off a flat surface. As shown in the right diagram above, the reflection of a points off a wall with normal vector satisfies(3)If the plane of reflection is taken as the -plane, the reflection in two- or three-dimensional space consists of making the transformation for each point. Consider an arbitrary point and a plane specified by the equation(4)This plane has normal vector(5)and the signed point-plane distance is(6)The position of the point reflected in the given plane is therefore given by(7)(8)The..

Alhazen's billiard problem

In a given circle, find an isosceles triangle whose legs pass through two given points inside the circle. This can be restated as: from two points in the plane of a circle, draw lines meeting at the point of the circumference and making equal angles with the normal vector at that point.The problem is called the billiard problem because it corresponds to finding the point on the edge of a circular "billiard" table at which a cue ball at a given point must be aimed in order to carom once off the edge of the table and strike another ball at a second given point.The problem is equivalent to the determination of the point on a spherical mirror where a ray of light will reflect in order to pass from a given source to an observer. It is also equivalent to the problem of finding, given two points and a circle such that the points are both inside or outside the circle, the ellipse whose foci are the two points and which is tangent to the given circle.The problem was..

Parabolic rotation

The map(1)(2)which leaves the parabola(3)invariant.

Ulam map

for . Fixed points occur at , 1/2, and order 2 fixed points at . The natural invariant of the map is

Twist map

A class of area-preserving maps ofthe form(1)(2)which maps circles into circles but with a twist resulting from the term.

Crossed hyperbolic rotation

Exchanges branches of the hyperbola .(1)(2)


A transformation (a.k.a., map, function) over a domain takes the elements to elements , where the range (a.k.a., image) of is defined asNote that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system, with space and objects embedded in it being viewed as fixed (a so-called alias transformation), or on the space itself relative to a fixed coordinate system (a so-called alibi transformation).Examples of transformations are summarized in the following table.TransformationCharacterizationdilationcenter of dilation, scale decrease factorexpansioncenter of expansion, scale increase factorreflectionmirror line or planerotationcenter of rotation, rotation angleshearinvariant line and shear factorstretch (1-way)invariant line and scale factorstretch (2-way)invariant lines and scale factorstranslationdisplacement..

M&ouml;bius transformation

Let and , thenis a Möbius transformation, where is the complex conjugate of . is a conformal mapping self-map of the unit disk for each , and specifically of the boundary of the unit disk to itself. The same holds for .Any conformal self-map of the unit disk to itself is a composition of a Möbius transformation with a rotation, and any conformal self-map of the unit disk can be written in the formfor some Möbius transformation and some complex number with (Krantz 1999, p. 81).

Cremona transformation

An entire Cremona transformation is a birational transformation of the plane. Cremona transformations are maps of the form(1)(2)in which and are polynomials. A quadratic Cremona transformation is always factorable.

Map class

A map from a domain is called a map of class if each component ofis of class ( or ) in , where denotes a continuous function which is differentiable times.

Continued fraction map

Min Max Re Im for , where is the floor function. The natural invariant of the map is


A bijective map between twometric spaces that preserves distances, i.e.,where is the map and is the distance function. Isometries are sometimes also called congruence transformations. Two figures that can be transformed into each other by an isometry are said to be congruent (Coxeter and Greitzer 1967, p. 80).An isometry of the plane is a linear transformation which preserves length. Isometries include rotation, translation, reflection, glides, and the identity map. Two geometric figures related by an isometry are said to be geometrically congruent (Coxeter and Greitzer 1967, p. 80).If a plane isometry has more than one fixed point, it must be either the identity transformation or a reflection. Every isometry of period two (two applications of the transformation preserving lengths in the original configuration) is either a reflection or a half-turn rotation. Every isometry in the plane is the product of at most three reflections..


A linear transformation of period two. Sincea linear transformation has the form,(1)applying the transformation a second time gives(2)For an involutory, , so(3)Since each coefficient must vanishseparately,(4)(5)(6)Equation (5) requires . Taking in turn requires that , giving , i.e., the identity map, while taking gives , so(7)which is the general form of a line involution.

Shear matrix

The shear matrix is obtained from the identity matrix by inserting at , e.g.,(1)Bolt and Hobbs (1998) define a shear matrix as a matrix(2)such that(3)(4)

Hyperbolic rotation

Also known as the a Lorentz transformation or Procrustian stretch, a hyperbolic transformation leaves each branch of the hyperbola invariant and transforms circles into ellipses with the same area.(1)(2)


An automorphism is an isomorphism of a system of objects onto itself. The term derives from the Greek prefix (auto) "self" and (morphosis) "to form" or "to shape."The automorphisms of a graph always describea group (Skiena 1990, p. 19).An automorphism of a region of the complex planeis a conformal self-map (Krantz 1999, p. 81).


A transformation in which all points along a given line remain fixed while other points are shifted parallel to by a distance proportional to their perpendicular distance from . Shearing a plane figure does not change its area. The shear can also be generalized to three dimensions, in which planes are translated instead of lines.

Appell transformation

A homographic transformation(1)(2)with substituted for according to(3)


Let be a map. Then is expansive if the statement that the distance for all implies that . Equivalently, is expansive if the orbits of two points and are never very close.

Unimodular transformation

A transformation is unimodular if the determinant of the matrix satisfiesA necessary and sufficient condition that a linear transformation transform a lattice to itself is that the transformation be unimodular.If is a complex number, then the transformationis called a unimodular if , , , and are integers with . The set of all unimodular transformations forms a group called the modular group.


A projective correlation of period two. In a polarity, is called the polar of , and the inversion pole .

Perspective collineation

A perspective collineation with center and axis is a collineation which leaves all lines through and points of invariant. Every perspective collineation is a projective collineation.

Harmonic homology

A perspective collineation with center and axis not incident is called a geometric homology. A geometric homology is said to be harmonic if the points and on a line through are harmonic conjugates with respect to and . Every perspective collineation of period two is a harmonic homology.

Geometric correlation

A point-to-line and line-to-point transformation which transforms points into lines and lines into points such that passes through iff lies on .


Expansion is an affine transformation (sometimes called an enlargement or dilation) in which the scale is increased. It is the opposite of a geometric contraction, and is also sometimes called an enlargement. A central dilation corresponds to an expansion plus a translation.Another type of expansion is the process of radially displacing the edges or faces of a polyhedron (while keeping their orientations and sizes constant) while filling in the gaps with new faces (Ball and Coxeter 1987, pp. 139-140). This procedure was devised by Stott (1910), and can be used to construct all 11 amphichiral (out of 13 total) Archimedean solids. The opposite operation of expansion (i.e., inward expansion) is called contraction. Expansion is a special case of snubification in which no twist occurs.The following table summarizes some expansions of some unit edge length Platonic and Archimedean solids, where is the displacement and is the golden ratio.base..


A similarity transformation which transforms each line to a parallel line whose length is a fixed multiple of the length of the original line. The simplest dilation is therefore a translation, and any dilation that is not merely a translation is called a central dilation. Two triangles related by a central dilation are said to be perspective triangles because the lines joining corresponding vertices concur. A dilation corresponds to an expansion plus a translation.


The tesseract is the hypercube in , also called the 8-cell or octachoron. It has the Schläfli symbol , and vertices . The figure above shows a projection of the tesseract in three-space (Gardner 1977). The tesseract is composed of 8 cubes with 3 to an edge, and therefore has 16 vertices, 32 edges, 24 squares, and 8 cubes. It is one of the six regular polychora.The tesseract has 261 distinct nets (Gardner 1966, Turney 1984-85, Tougne 1986, Buekenhout and Parker 1998).In Madeleine L'Engle's novel A Wrinkle in Time, the characters in the story travel through time and space using tesseracts. The book actually uses the idea of a tesseract to represent a fifth dimension rather than a four-dimensional object (and also uses the word "tesser" to refer to movement from one three dimensional space/world to another).In the science fiction novel Factoring Humanity by Robert J. Sawyer, a tesseract is used by humans on Earth to enter the fourth..


Move a point along a line from an initial point to a final point. It traces out a line segment . When is translated from an initial position to a final position, it traces out a parallelogram . When is translated, it traces out a parallelepiped . The generalization of to dimensions is then called a parallelotope. has vertices ands, where is a binomial coefficient and , 1, ..., (Coxeter 1973). These are also the coefficients of .


A collection of faces of an -dimensional polytope or simplicial complex, one of each dimension 0, 1, ..., , which all have a common nonempty intersection. In normal three dimensions, the flag consists of a half-plane, its bounding ray, and the ray's endpoint.

Cross polytope

The cross polytope is the regular polytope in dimensions corresponding to the convex hull of the points formed by permuting the coordinates (, 0, 0, ..., 0). A cross-polytope (also called an orthoplex) is denoted and has vertices and Schläfli symbol . The cross polytope is named because its vertices are located equidistant from the origin along the Cartesian axes in Euclidean space, which each such axis perpendicular to all others. A cross polytope is bounded by -simplexes, and is a dipyramid erected (in both directions) into the th dimension, with an -dimensional cross polytope as its base.In one dimension, the cross polytope is the line segment . In two dimensions, the cross polytope is the filled square with vertices , , , . In three dimensions, the cross polytope is the convex hull of the octahedron with vertices , , , , , . In four dimensions, the cross polytope is the 16-cell, depicted in the above figure by projecting onto one of the four mutually..


A simplex, sometimes called a hypertetrahedron (Buekenhout and Parker 1998), is the generalization of a tetrahedral region of space to dimensions. The boundary of a -simplex has 0-faces (polytope vertices), 1-faces (polytope edges), and -faces, where is a binomial coefficient. An -dimensional simplex can be denoted using the Schläfli symbol . The simplex is so-named because it represents the simplest possible polytope in any given space.The content (i.e., hypervolume) of a simplex can be computedusing the Cayley-Menger determinant.In one dimension, the simplex is the line segment . In two dimensions, the simplex is the convex hull of the equilateral triangle. In three dimensions, the simplex is the convex hull of the tetrahedron. The simplex in four dimensions (the pentatope) is a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The regular simplex in dimensions with is denoted..


The associahedron is the -dimensional generalization of the pentagon. It was discovered by Stasheff in 1963 and it is also known as the Stasheff polytope. The number of nodes in the -associahedron is equivalent to the number of binary trees with nodes, which is the Catalan number .The associahedron is the basic tool in the study of homotopy associative Hopf spaces.Loday (2004) provides the following method for associahedron construction. Take , the set of planar binary trees with leaves. Define as the number of leaves to the left of the th vertex and as the number of leaves to the right of the th vertex. For in , defineThe -associahedron is then defined as the convex hull of .The associahedron can be obtained by removing facets from the permutohedron,and is related to the cyclohedron and permutohedron...


The word polytope is used to mean a number of related, but slightly different mathematical objects. A convex polytope may be defined as the convex hull of a finite set of points (which are always bounded), or as a bounded intersection of a finite set of half-spaces. Coxeter (1973, p. 118) defines polytope as the general term of the sequence "point, line segment, polygon, polyhedron, ...," or more specifically as a finite region of -dimensional space enclosed by a finite number of hyperplanes. The special name polychoron is sometimes given to a four-dimensional polytope. However, in algebraic topology, the underlying space of a simplicial complex is sometimes called a polytope (Munkres 1991, p. 8). The word "polytope" was introduced by Alicia Boole Stott, the somewhat colorful daughter of logician George Boole (MacHale 1985).The part of the polytope that lies in one of the bounding hyperplanes is called..


The permutohedron is the -dimensional generalization of the hexagon. The -permutohedron is the convex hull of all permutations of the vector in . The number of vertices is .

Howe's theorem

Let be a primitive polytope with eight vertices. Then there is a unimodular map that maps to the polyhedron whose vertices are (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1), (1, , ), (1, , ), and (1, , ) with , , and . Furthermore, any primitive polyhedron with fewer than eight vertices can be embedded in one with eight vertices.


A zonotope is a set of points in -dimensional space constructed from vectors by taking the sum of , where each is a scalar between 0 and 1. Different choices of scalars give different points, and the zonotope is the set of all such points. Alternately it can be viewed as a Minkowski sum of line segments connecting the origin to the endpoint of each vector. It is called a zonotope because the faces parallel to each vector form a so-called zone wrapping around the polytope (Eppstein 1996).A three-dimensional zonotope is called a zonohedron.There is some confusion in the definition of zonotopes (Eppstein 1996). Wells (1991, pp. 274-275) requires the generating vectors to be in general position (all -tuples of vectors must span the whole space), so that all the faces of the zonotope are parallelotopes. Others (Bern et al. 1995; Ziegler 1995, pp. 198-208; Eppstein 1996) do not make this restriction. Coxeter (1973) starts with one definition..


The hypersine (-dimensional sine function) is a function of a vertex angle of an -dimensional parallelotope or simplex. If the content of the parallelotope is and the contents of the facets of the parallelotope that meet at vertex are , then the value of the -dimensional sine of that vertex is(1)Changing the length of an edge of the parallelotope by a factor changes the content by the same factor and the contents of all but one of the facets by the same factor. Thus, a change in edge length does not affect the value of the right-hand side, and the sine function is dependent solely on the angles between the edges of the parallelotope, not their lengths. In addition, the sines of all of the vertex angles of the parallelotope are the same, since the opposite facets have the same content, and one of each pair of opposite facets meets at each vertex. If we extend the facets at a vertex, all of the vertex angles thus formed have the same sine, as they are simply translations..

Tetracyclic plane

The set of all points that can be put into one-to-one correspondence with sets of essentially distinct values of four homogeneous coordinates , not all simultaneously zero, which are connected by the relation

Sphere embedding

A 4-sphere has positive curvature,with(1)(2)Since(3)(4)(5)To stay on the surface of the sphere,(6)(7)(8)(9)(10)(11)With the addition of the so-called expansion parameter, this is the Robertson-Walker line element.


The hypercube is a generalization of a 3-cube to dimensions, also called an -cube or measure polytope. It is a regular polytope with mutually perpendicular sides, and is therefore an orthotope. It is denoted and has Schläfli symbol .The following table summarizes the names of -dimensional hypercubes.object1line segment2square3cube4tesseractThe number of -cubes contained in an -cube can be found from the coefficients of , namely , where is a binomial coefficient. The number of nodes in the -hypercube is therefore (OEIS A000079), the number of edges is (OEIS A001787), the number of squares is (OEIS A001788), the number of cubes is (OEIS A001789), etc.The numbers of distinct nets for the -hypercube for , 2, ... are 1, 11, 261, ... (OEIS A091159; Turney 1984-85).The above figure shows a projection of the tesseract in three-space. A tesseract has 16 polytope vertices, 32 polytope edges, 24 squares, and eight cubes.The dual of the tesseract..

Hyperbolic polar sine

The hyperbolic polar sine is a function of an -dimensional simplex in hyperbolic space. It is analogous to the polar sine of an -dimensional simplex in elliptic or spherical space. If the edges between vertices and have length , the value of the hyperbolic polar sine of the -dimensional hyperbolic simplex in space with Gaussian curvature is given byThe hyperbolic polar sine is used in the generalizedlaw of sines for a hyperbolic simplex.The limit of the hyperbolic polar sine of an -dimensional hyperbolic simplex as the curvature of the space approaches zero is , where is the content of the Euclidean simplex with the same edge lengths.

Polar sine

The polar sine is a function of a vertex angle of an -dimensional parallelotope or simplex. If the content of the parallelotope is and the lengths of the edges of the parallelotope that meet at vertex are , , ..., , then the value of the polar sine of that vertex isChanging the length of an edge of the parallelotope by a factor changes the content by the same factor. Thus, a change in edge length does not affect the value of the right-hand side, and the polar sine function is dependent solely on the angles between the edges of the parallelotope, not their lengths. Also the polar sines of all of the vertex angles of the parallelotope are the same, since the right-hand side of the definition does not depend on the vertex chosen. If we extend the facets at a vertex, all of the vertex angles thus formed have the same polar sine, as they are simply translations of the vertex angles of the parallelotope.If a sphere is centered at the vertex of an -dimensional angle, the rays..

Generalized law of sines

The generalized law of sines applies to a simplex in space of any dimension with constant Gaussian curvature. Let us work up to that. Initially in two-dimensional space, we define a generalized sine function for a one-dimensional simplex (line segment) with content (length) in space of constant Gaussian curvature as(1)For particular values of , we have(2)giving(3)Thus in elliptic space (), the function is the sine function; in Euclidean space (), the function is simply the content itself; and in hyperbolic space (), the function is the hyperbolic sine function. Thus for a two-dimensional simplex with edges of length , , and , we can express the law of sines for space with any constant Gaussian curvature as(4)For Euclidean space (), equation (4) specializes to(5)For the elliptic plane or the unit sphere (), equation (4) specializes to(6)For the hyperbolic plane (), equation (4) specializes to(7)Our generalization for the two-dimensional..

Regular polychoron

The necessary condition for the polychoron to be regular (with Schläfli symbol ) and finite isSufficiency can be established by consideration ofthe six figures satisfying this condition.There are sixteen regular polychora, six of which are convex (Wells 1986, p. 68) and ten of which are stellated (Wells 1991, p. 209). The regular convex polychora have four principal types of symmetry axes, and the projections into three-spaces orthogonal to these may be called the "canonical" projections.Of the six regular convex polychora, five are typically regarded as being analogous to the Platonic solids: the 4-simplex (a hyper-tetrahedron), the 4-cross polytope (a hyper-octahedron), the 4-cube (a hyper-cube), the 600-cell (a hyper-icosahedron), and the 120-cell (a hyper-dodecahedron). The 24-cell, however, has no perfect analogy in higher or lower spaces. The pentatope and 24-cell are self-dual, the 16-cell..


The pentatope is the simplest regular figure in four dimensions, representing the four-dimensional analog of the solid tetrahedron. It is also called the 5-cell, since it consists of five vertices, or pentachoron. The pentatope is the four-dimensional simplex, and can be viewed as a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The pentatope has Schläfli symbol .It is one of the six regular polychora.The skeleton of the pentatope is isomorphic to the complete graph , known as the pentatope graph.The pentatope is self-dual, has five three-dimensional facets (each the shape of a tetrahedron), 10 ridges (faces), 10 edges, and five vertices. In the above figure, the pentatope is shown projected onto one of the four mutually perpendicular three-spaces within the four-space obtained by dropping one of the four vertex components (R. Towle)...

Solid of revolution

To find the volume of a solid of revolution by adding up a sequence of thin cylindrical shells, consider a region bounded above by , below by , on the left by the line , and on the right by the line . When the region is rotated about the z-axis, the resulting volume is given byThe following table gives the volumes of various solidsof revolution computed using the method of cylinders.solidvolumecone0conical frustum0cylinder0oblate spheroidprolate spheroidspheretorusspherical segmenttorispherical dome0To find the volume of a solid of revolution by adding up a sequence of thin flat washers, consider a region bounded on the left by , on the right by , on the bottom by the line , and on the top by the line . When the region is rotated about the z-axis, the resulting volume isThe following table gives the volumes of various solids of revolution computed using the method of washers.solidvolumebarrel (elliptical)0barrel (parabolic)0cone0conical frustum0cylinder0oblate..

Gabriel's horn

Gabriel's horn, also called Torricelli's trumpet, is the surface of revolution of the function about the x-axis for . It is therefore given by parametric equations(1)(2)(3)The surprising thing about this surface is that it (taking for convenience here) has finite volume(4)(5)(6)but infinite surface area,since(7)(8)(9)(10)(11)(12)This leads to the paradoxical consequence that while Gabriel's horn can be filled up with cubic units of paint, an infinite number of square units of paint are needed to cover its surface!The coefficients of the first fundamental formare,(13)(14)(15)and of the second fundamental form are(16)(17)(18)The Gaussian and meancurvatures are(19)(20)The Gaussian curvature can be expressed implicitly as(21)


The funnel surface is a regular surface and surface of revolution defined by the Cartesian equation(1)and the parametric equations(2)(3)(4)for and . The coefficients of the first fundamental form are(5)(6)(7)the coefficients of the second fundamentalform are(8)(9)(10)the area element is(11)and the Gaussian and mean curvatures are(12)(13)The Gaussian curvature can be given implicitly as(14)Both the surface area and volumeof the solid are infinite.


The surface of revolution of the parabola which is the shape used in the reflectors of automobile headlights (Steinhaus 1999, p. 242; Hilbert and Cohn-Vossen 1999). It is a quadratic surface which can be specified by the Cartesian equation(1)The paraboloid which has radius at height is then given parametrically by(2)(3)(4)where , .The coefficients of the first fundamental formare given by(5)(6)(7)and the second fundamental form coefficientsare(8)(9)(10)The area element is then(11)giving surface area(12)(13)The Gaussian curvature is given by(14)and the mean curvature(15)The volume of the paraboloid of height is then(16)(17)The weighted mean of over the paraboloid is(18)(19)The geometric centroid is then given by(20)(Beyer 1987).

Perpendicular foot

The perpendicular foot, also called the foot of an altitude, is the point on the leg opposite a given vertex of a triangle at which the perpendicular passing through that vertex intersects the side. The length of the line segment from the vertex to the perpendicular foot is called the altitude of the triangle.When a line is drawn from a point to a plane,its intersection with the plane is known as the foot.

Hjelmslev's theorem

When all the points on one line are related by an isometry to all points on another, the midpoints of the segments are either distinct and collinear or coincident.

Three conics theorem

If three conics pass through two given points and , then the lines joining the other two intersections of each pair of conics are concurrent at a point (Evelyn 1974, p. 15). The converse states that if two conics and meet at four points , , , and , and if and are chords of and , respectively, which meet on , then the six points lie on a conic. The dual of the theorem states that if three conics share two common tangents, then their remaining pairs of common tangents intersect at three collinear points.If the points and are taken as the points at infinity, then the theorem reduces to the theorem that radical lines of three circles are concurrent in a point known as the radical center (Evelyn 1974, p. 15).If two of the points and are taken as the points at infinity, then the theorem becomes that if two circles and pass through two points and on a conic , then the lines determined by the pair of intersections of each circle with the conic are parallel (Evelyn..

Tangent line

A straight line is tangent to a given curve at a point on the curve if the line passes through the point on the curve and has slope , where is the derivative of . This line is called a tangent line, or sometimes simply a tangent.

Kosnita theorem

The lines joining the vertices , , and of a given triangle with the circumcenters of the triangles , , and (where is the circumcenter of ), respectively, are concurrent. Their point of concurrence is known as the Kosnita point.


Two or more lines are said to be concurrent if they intersect in a single point. Two lines concur if their trilinear coordinates satisfy(1)Three lines concur if their trilinearcoordinates satisfy(2)(3)(4)in which case the point is(5)Three lines(6)(7)(8)are concurrent if their coefficients satisfy(9)

Parallel lines

Two lines in two-dimensional Euclidean space aresaid to be parallel if they do not intersect.In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. Therefore, parallel lines in three-space lie in a single plane (Kern and Blank 1948, p. 9). Lines in three-space which are not parallel but do not intersect are called skew lines.Two trilinear lines(1)(2)are parallel if(3)(Kimberling 1998, p. 29).


A bitangent is a line that is tangentto a curve at two distinct points.Aa general plane quartic curve has 28 bitangents in the complex projective plane. However, as shown by Plücker (1839), the number of real bitangents of a quartic must be 28, 16, or a number less than 9. Plücker (Plücker 1839, Gray 1982) constructed the first as(correcting the typo of for ) for small and positive. Without mentioning its origin or significance, this curve with is termed the ampersand curve by Cundy and Rowlett (1989, p. 72).As noted by Gray (1982), "the 28 bitangents became, and remain, a topic of delight."Trott (1997) subsequently gave the beautiful symmetric quartic curve with 28 real bitangentswhich is illustrated above.

Sylvester's line problem

Sylvester's line problem, known as the Sylvester-Gallai theorem in proved form, states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single line. This problem was proposed by Sylvester (1893), who asked readers to "Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line."Woodall (1893) published a four-line "solution," but an editorial comment following his result pointed out two holes in the argument and sketched another line of enquiry, which is characterized as "equally incomplete, but may be worth notice." However, no correct proof was published at the time (Croft et al. 1991, p. 159), but the problem was revived by Erdős (1943) and correctly solved by Grünwald..

Four conics theorem

If two intersections of each pair of three conics , , and lie on a conic , then the lines joining the other two intersections of each pair are concurrent (Evelyn et al. 1974, pp. 23 and 25).The dual theorem states that if two common tangents of each pair of three conics touch a fourth conic, then the remaining common tangents of each pair intersect in three collinear points (Evelyn et al. 1974, pp. 24-25).

Semilatus rectum

The chord through a focus parallel to the conic section directrix of a conic section is called the latus rectum, and half this length is called the semilatus rectum (Coxeter 1969). "Semilatus rectum" is a compound of the Latin semi-, meaning half, latus, meaning 'side,' and rectum, meaning 'straight.'For an ellipse, the semilatus rectum is the distance measured from a focus such that(1)where and are the apoapsis and periapsis, and is the ellipse's eccentricity. Plugging in for and then gives(2)so(3)For a parabola,(4)where is the distance between the focus and vertex (or directrix).

Holditch's theorem

Let a chord of constant length be slid around a smooth, closed, convex curve , and choose a point on the chord which divides it into segments of lengths and . This point will trace out a new closed curve , as illustrated above. Provided certain conditions are met, the area between and is given by , as first shown by Holditch in 1858.The Holditch curve for a circle of radius is another circle which, from the theorem, has radius

Equireciprocal point

is an equireciprocal point if, for every chord of a curve , satisfiesfor some constant . The foci of an ellipse are equichordal points.

Equichordal point

An equichordal point is a point for which all the chords of a curve passing through are of the same length. In other words, is an equichordal point if, for every chord of length of the curve , satisfies(1)A function satisfying(2)corresponds to a curve with equichordal point (0, 0) and chord length defined by letting be the polar equation of the half-curve for and then superimposing the polar equation over the same range. The curves illustrated above correspond to polar equations of the form(3)for various values of .Although it long remained an outstanding problem (the equichordal point problem), it is now known that a planar convex region can not have two equichordal points (Rychlik 1997).


In plane geometry, a chord is the line segment joining two points on a curve. The term is often used to describe a line segment whose ends lie on a circle.The term is also used in graph theory, where a cycle chord of a graph cycle is an edge not in whose endpoints lie in .In the above figure, is the radius of the circle, is the chord length, is called the apothem, and the sagitta. The shaded region in the left figure is called a circular sector, and the shaded region in the right figure is called a circular segment.There are a number of interesting theorems about chords of circles. All angles inscribed in a circle and subtended by the same chord are equal. The converse is also true: The locus of all points from which a given segment subtends equal angles is a circle.In the left figure above,(1)(Jurgensen 1963, p. 345). In the right figure above,(2)which is a statement of the fact that the circle power is independent of the choice of the line (Coxeter 1969, p. 81;..

Centrosymmetric set

A convex set is centro-symmetric, sometimes also called centrally symmetric, if it has a center that bisects every chord of through .


Given a circle, the apothem is the perpendicular distance from the midpoint of a chord to the circle's center. It is also equal to the radius minus the sagitta ,For a regular polygon, the apothem simply is the distance from the center to a side, i.e., the inradius of the polygon.

Pappus's centroid theorem

The first theorem of Pappus states that the surface area of a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance traveled by the curve's geometric centroid ,(Kern and Bland 1948, pp. 110-111). The following table summarizes the surface areas calculated using Pappus's centroid theorem for various surfaces of revolution.solidgenerating curveconeinclined line segmentcylinderparallel line segmentspheresemicircleSimilarly, the second theorem of Pappus states that the volume of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area of the lamina and the distance traveled by the lamina's geometric centroid ,(Kern and Bland 1948, pp. 110-111). The following table summarizes the surface areas and volumes calculated using Pappus's centroid theorem..

Harmonic parameter

The harmonic parameter of a polyhedron is the weighted mean of the distances from a fixed interior point to the faces, where the weights are the areas of the faces, i.e.,(1)This parameter generalizes the identity(2)where is the volume, is the inradius, and is the surface area, which is valid only for symmetrical solids, to(3)The harmonic parameter is independent of the choice of interior point (Fjelstad and Ginchev 2003). In addition, it can be defined not only for polyhedron, but any -dimensional solids that have -dimensional content and -dimensional content .Expressing the area and perimeter of a lamina in terms of gives the identity(4)The following table summarizes the harmonic parameter for a few common laminas. Here, is the inradius of a given lamina, and and are the side lengths of a rectangle.laminacirclerectanglesquaretriangleExpressing and for a solid in terms of then gives the identity(5)The following table summarizes the harmonic..


A tube of radius of a set is the set of points at a distance from . In particular, if is a regular space curve whose curvature does not vanish, then the normal vector and binormal vector are always perpendicular to , and the circle is perpendicular to at . So as the circle moves around , it traces out a tube, provided the tube radius is small enough so that the tube is not self-intersecting. A formula for the tube around a curve is therefore given byfor over the range of the curve and . The illustrations above show tubes corresponding to a circle, helix, and two torus knots.The surface generated by constructing a tube around a circleis known as a torus.

Standard tori

One of the three classes of tori illustrated above andgiven by the parametric equations(1)(2)(3)The three different classes of standard tori arise from the three possible relative sizes of and . corresponds to the ring torus shown above, corresponds to a spindle torus which touches itself at the point (0, 0, 0), and corresponds to a self-intersecting horn torus (Pinkall 1986). If no specification is made, "torus" is taken to mean ring torus.The standard tori and their inversions are cyclides.

Spiric section

The equation of the curve of intersection of a torus with a plane perpendicular to both the midplane of the torus and to the plane . (The general intersection of a torus with a plane is called a toric section). Let the tube of a torus have radius , let its midplane lie in the plane, and let the center of the tube lie at a distance from the origin. Now cut the torus with the plane . The equation of the torus with gives the equation(1)(2)(3)The above plots show a series of spiric sections for the ring torus, horn torus, and spindle torus, respectively. When , the curve consists of two circles of radius whose centers are at and . If , the curve consists of one point (the origin), while if , no point lies on the curve.The spiric extensions are an extension of the conic sections constructed by Menaechmus around 150 BC by cutting a cone by a plane, and were first considered around 50 AD by the Greek mathematician Perseus (MacTutor).If , then (3) simplifies to(4)which is the..

Spindle torus

One of the three standard tori given by the parametricequations(1)(2)(3)with . The exterior surface is called an apple surface and the interior of a lemon surface. The above left figure shows a spindle torus, the middle a cutaway, and the right figure shows a cross section of the spindle torus through the -plane. The inversion of a spindle torus is a spindle cyclide (or parabolic spindle cyclide).

Horn torus

One of the three standard tori given by the parametricequations(1)(2)(3)corresponding to the torus with .It has coefficients of the first fundamentalform given by(4)(5)(6)and of the second fundamental form givenby(7)(8)(9)The area element is(10)and the surface area and volumeare(11)(12)The geometric centroid is at , and the moment of inertia tensor for a solid torus is given by(13)for a uniform density torus of mass .The inversion of a horn torus is a horn cyclide. The above figures show a horn torus (left), a cutaway (middle), and a cross section of the horn torus through the -plane (right).


An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an "anchor ring." It can be constructed from a rectangle by gluing both pairs of opposite edges together with no twists (right figure; Gardner 1971, pp. 15-17; Gray 1997, pp. 323-324). The usual torus embedded in three-dimensional space is shaped like a donut, but the concept of the torus is extremely useful in higher dimensional space as well.In general, tori can also have multiple holes, with the term -torus used for a torus with holes. The special case of a 2-torus is sometimes called the double torus, the 3-torus is called the triple torus, and the usual single-holed torus is then simple called "the" or "a" torus.A second definition for -tori relates to dimensionality. In one dimension, a line bends into..

Ring torus

One of the three standard tori given by the parametricequations(1)(2)(3)with . This is the torus which is generally meant when the term "torus" is used without qualification. The inversion of a ring torus is a ring cyclide if the inversion center does not lie on the torus and a parabolic ring cyclide if it does. The above left figure shows a ring torus, the middle a cutaway, and the right figure shows a cross section of the ring torus through the -plane.

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