Solid geometry

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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)

Bicone

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)

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.

Zone

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.

Insphere

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)

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)

Hemisphere

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).

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

Circumsphere

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.

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..

Hosohedron

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.

Dihedron

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)..

Superellipsoid

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 3.2.2.2(9)in 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)..

Spheroid

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)...

Flattening

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).

Ellipticity

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).

Ellipsoid

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.

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..

Sphericon

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.

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..

Tube

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).

Torus

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.

Elliptic torus

A surface of revolution which is generalization of the ring torus. It is produced by rotating an ellipse having horizontal semi-axis , vertical semi-axis , embedded in the -plane, and located a distance away from the -axis about the -axis. It is given by the parametric equations(1)(2)(3)for .This gives first fundamental form coefficientsof(4)(5)(6)second fundamental form coefficients of(7)(8)(9)The Gaussian curvature and meancurvature are(10)(11)By Pappus's centroid theorems, the surface area and volume are(12)(13)(14)(15)where is a complete elliptic integral of the first kind and(16)is the eccentricity of the ellipse cross section.

Toroid

A surface of revolution obtained by rotating a closed plane curve about an axis parallel to the plane which does not intersect the curve. The simplest toroid is the torus. The word is also used to refer to a toroidal polyhedron (Gardner 1975).

South pole

The south pole is the point on a sphere with minimum -coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is defined by the rotation axis, with the south pole given by the point in the southern hemisphere that is farthest from the equator (i.e., midplane of the sphere).The coordinate in spherical coordinates is measured from the north pole and takes on the value at the south pole.

Vertex figure

The vertex figure at a vertex of a polygon is the line segment joining the midpoints of the two adjacent sides meeting at . For a regular -gon with side length , the length of the vertex figure isThe vertex figure at a vertex of a polyhedron is the polygon whose sides are the vertex figures of the faces surrounding . The faces that join at a polyhedron vertex form a solid angle whose section by the plane is the vertex figure, as illustrated above for one vertex of the cube.The vertex figures of the Platonic solids yield the polyhedra (with holes centered on the centroids of the original faces) have convex hulls illustrated above and summarized in the following table.polyhedronconvex hull of vertex figurescubecuboctahedrondodecahedronicosidodecahedronicosahedronicosidodecahedronoctahedroncuboctahedrontetrahedronoctahedronThe illustrations above show the Archimedean solids, their vertex figures, and the solids obtained by taking..

Toric section

A toric section is a curve obtained by slicing a torus (generally a horn torus) with a plane. A spiric section is a special case of a toric section in which the slicing plane is perpendicular to both the midplane of the torus and to the plane .Consider a torus with tube radius . For a cutting plane parallel to the -plane, the toric section is either a single circle (for ) or two concentric circles (for ). For planes containing the z-axis, the section is two equal circles.Toric sections at oblique angles can be more complicated, passing from a crescent shape, through a U-shape, and into two disconnected kidney-shaped curves.

Unistable polyhedron

A uniform-density polyhedral solid is unistable (also called monostable) if it is stable on exactly one face (Croft et al. 1991, p. 61). For example, the 19-faced polyhedron illustrated above is unistable.Whether unistability is possible with fewer faces is an unsolvedproblem.Various turtles, such as the Indian star tortoise, have unistable shapes (Rehmeyer 2007).

Hyperbolic octahedron

The hyperbolic octahedron is a hyperbolic version of the Euclidean octahedron, which is a special case of the astroidal ellipsoid with .It is given by the parametric equations(1)(2)(3)for and .It is an algebraic surface of degree 18 withcomplicated terms. However, it has the simple Cartesian equation(4)where is taken to mean . Cross sections through the , , or planes are therefore astroids.The first fundamental form coefficientsare(5)(6)(7)the second fundamental form coefficientsare(8)(9)(10)The area element is(11)giving the surface area as(12)The volume is given by(13)an exact expression for which is apparently not known.The Gaussian curvature is(14)while the mean curvature is given by a complicatedexpression.

Oblate spheroid geodesic

The geodesic on an oblate spheroid can be computed analytically, although the resulting expression is much more unwieldy than for a simple sphere. A spheroid with equatorial radius and polar radius can be specified parametrically by(1)(2)(3)where . Using the second partial derivatives(4)(5)(6)(7)(8)(9)gives the geodesics functions as(10)(11)(12)(13)(14)(15)where(16)is the ellipticity.Since and and are explicit functions of only, we can use the special form of the geodesic equation(17)(18)(19)where is a constant depending on the starting and ending points. Integrating gives(20)where(21)(22) is an elliptic integral of the first kind with parameter , and is an elliptic integral of the third kind.Geodesics other than meridians of an oblate spheroid undulate between two parallels with latitudes equidistant from the equator. Using the Weierstrass sigma function and Weierstrass zeta function, the geodesic on the oblate spheroid..

Osculating sphere

The center of any sphere which has a contact of (at least) first-order with a curve at a point lies in the normal plane to at . The center of any sphere which has a contact of (at least) second-order with at point , where the curvature , lies on the polar axis of corresponding to . All these spheres intersect the osculating plane of at along a circle of curvature at . The osculating sphere has centerwhere is the unit normal vector, is the unit binormal vector, is the radius of curvature, and is the torsion, and radiusand has contact of (at least) third order with .

Wedge

The term "wedge" has a number of meanings in mathematics. It is sometimes used as another name for the caret symbol, as well as being the notation () for logical AND.In solid geometry, a wedge is a right triangular prism turned so that it rests on one of its lateral rectangular faces (left figure). Harris and Stocker (1998) define a more general type of oblique wedge in which the top edge is symmetrically shortened, causing the end triangles to slant obliquely (right figure).For an oblique wedge of base lengths and , top edge length , and height (right figure), the volume of the wedge is(1)(2)In the case of a right wedge , this simplifies to(3)The geometric centroid is located at a height(4)above the base, which simplifies to for a right wedge .

Spherical shell

A spherical shell is a generalization of an annulus to three dimensions. A spherical shell is therefore the region between two concentric spheres of differing radii.The spherical shell is implemented in the Wolfram Language as SphericalShell[x, y, z, b, a].

Sphere

A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius") from a given point (the "center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes.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). As a result, geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the..

Cylinder

The term "cylinder" has a number of related meanings. In its most general usage, the word "cylinder" refers to a solid bounded by a closed generalized cylinder (a.k.a. cylindrical surface) and two parallel planes (Kern and Bland 1948, p. 32; Harris and Stocker 1998, p. 102). A cylinder of this sort having a polygonal base is therefore a prism (Zwillinger 1995, p. 308). Harris and Stocker (1998, p. 103) use the term "general cylinder" to refer to the solid bounded a closed generalized cylinder.Unfortunately, the term "cylinder" is commonly used not only to refer to the solid bounded by a cylindrical surface, but to the cylindrical surface itself (Zwillinger 1995, p. 311). To make matters worse, according to topologists, a cylindrical surface is not even a true surface, but rather a so-called surface with boundary (Henle 1994, pp. 110 and 129).As if this were..

Cone

A (finite, circular) conical surface is a ruled surface created by fixing one end of a line segment at a point (known as the vertex or apex of the cone) and sweeping the other around the circumference of a fixed circle (known as the base). When the vertex lies above the center of the base (i.e., the angle formed by the vertex, base center, and any base radius is a right angle), the cone is known as a right cone; otherwise, the cone is termed "oblique." When the base is taken as an ellipse instead of a circle, the cone is called an elliptic cone.In discussions of conic sections, the word "cone" is commonly taken to mean "double cone," i.e., two (possibly infinitely extending) cones placed apex to apex. The infinite double cone is a quadratic surface, and each single cone is called a "nappe." The hyperbola can then be defined as the intersection of a plane with both nappes of the double cone.As can be seen from the above,..

Wythoff symbol

A symbol consisting of three rational numbers that can be used to describe uniform polyhedra based on how a point in a spherical triangle can be selected so as to trace the vertices of regular polygonal faces. For example, the Wythoff symbol for the tetrahedron is . There are four types of Wythoff symbols, , , and , and one exceptional symbol, (which is used for the great dirhombicosidodecahedron).The meaning of the bars may be summarized as follows (Wenninger 1989, p. 10; Messer 2002). Consider a spherical triangle whose angles are , , and . 1. : is a special point within that traces snub polyhedra by even reflections. 2. (or ): is the vertex . 3. (or ): lies on the arc and the bisector of the opposite angle . 4. (or any permutation of the three letters): is the incenter of the triangle . Some special cases in terms of Schläfli symbolsare(1)(2)(3)(4)(5)(6)..

Schl&auml;fli function

The function giving the volume of the spherical quadrectangulartetrahedron:(1)where(2)and(3)

North pole

The north pole is the point on a sphere with maximum -coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is defined by the rotation axis, with the north pole given by the point in the northern hemisphere that is farthest from the equator (i.e., midplane of the sphere).The coordinate in spherical coordinates is measured from the north pole.

Bang's theorem

There are least two Bang's theorems, one concerning tetrahedra (Bang 1897), and theother with widths of convex domains (Bang 1951).The theorem of Bang (1897) states that the lines drawn to the polyhedron vertices of a face of a tetrahedron from the point of contact of the face with the insphere form three angles at the point of contact which are the same three angles in each face.The theorem of Bang (1951) states that if a convex domain is covered by a collection of strips, then the sum of the widths of the strips is at least , where is the width of the narrowest strip which covers .

Oloid

Let two disks of radius intersect one another perpendicularly and have a diameter in common. If the distance between the centers of the disks is times their radius, then the distance from the center of gravity remains constant and so the object, known as a "two circle roller," rolls smoothly (Nishihara).If the distance of two centers of disk is equal to the radius, then the convex hull produces another figure that rolls smoothly and is known as the oloid (Schatz 1975, p. 122; Nishihara), illustrated above. The oloid is an octic surface (Trott 2004, pp. 1194-1196).For circles of radii , the surface area of the resulting oloid is(the same as that of a sphere with radius ), but no closed form is apparently known for the enclosed volume.

Cube dissection

A cube can be divided into subcubes for only , 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and (OEIS A014544; Hadwiger 1946; Scott 1947; Gardner 1992, p. 297). This sequence provides the solution to the so-called Hadwiger problem, which asks for the largest number of subcubes (not necessarily different) into which a cube cannot be divided by plane cuts, and has the answer 47 (Gardner 1992, pp. 297-298).If and are in the sequence, so is , since -dissecting one cube in an -dissection gives an -dissection. The numbers 1, 8, 20, 38, 49, 51, 54 are in the sequence because of dissections corresponding to the equations(1)(2)(3)(4)(5)(6)(7)Combining these facts gives the remaining terms in the sequence, and all numbers , and it has been shown that no other numbers occur.It is not possible to cut a cube into subcubes that are all different sizes (Gardner 1961, p. 208; Gardner 1992, p. 298).The seven pieces used to construct..

Trapezohedron

An -gonal trapezohedron, also called an antidipyramid, antibipyramid, or deltohedron (not to be confused with a deltahedron), is a dual polyhedra of an -antiprism. Unfortunately, the name for these solids is not particularly well chosen since their faces are not trapezoids but rather kites. The trapezohedra are isohedra.The 3-trapezohedron (trigonal trapezohedron) is a rhombohedron having all six faces congruent. A special case is the cube (oriented along a space diagonal), corresponding to the dual of the equilateral 3-antiprism (i.e., the octahedron).A 4-trapezohedron (tetragonal trapezohedron) appears in the upper left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).The trapezohedra generated by taking the duals of the equilateral antiprisms have side length , half-heights (half the peak-to-peak distance) , surface areas , and volumes..

Escher's solid

"Escher's solid" is the solid illustrated on the right pedestal in M. C. Escher's Waterfall woodcut (Bool et al. 1982, p. 323). It is obtained by augmenting a rhombic dodecahedron until incident edges become parallel, corresponding to augmentation height of for a rhombic dodecahedron with unit edge lengths.It is the first rhombic dodecahedron stellation and is a space-filling polyhedron. Its convex hull is a cuboctahedron.It is implemented in the Wolfram Languageas PolyhedronData["EschersSolid"].It has edge lengths(1)(2)surface area and volume(3)(4)and moment of inertia tensor(5)The skeleton of Escher's solid is the graph of the disdyakis dodecahedron.Escher's solid can also be viewed as a polyhedron compound of three dipyramids (nonregular octahedra) with edges of length 2 and ...

Pentagonal dipyramid

The pentagonal dipyramid is one of the convex deltahedra, and Johnson solid . It is also the dual polyhedron of the pentagonal prism and is an isohedron.It is implemented in the Wolfram Language as PolyhedronData["Dipyramid", 5].A pentagonal dipyramid appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).For a pentagonal dipyramid having a base with unit edge lengths, the circumradiusof the base pentagon is(1)In order for the top and bottom edges to also be of unit length, the polyhedron must be of height(2)The ratio of is therefore given by(3)where is the golden ratio.The surface area and volume of a unit pentagonal dipyramid are(4)(5)

Elongated square dipyramid

The elongated square dipyramid with unit edge lengths is Johnson Solid .An elongated square dipyramid (having a central ribbon composed of rectangles instead of squares) appears in the top center as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

D&uuml;rer's solid

Dürer's solid, also known as the truncated triangular trapezohedron, is the 8-faced solid depicted in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989), the same engraving in which Dürer's magic square appears, which depicts a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Although Dürer does not specify how his solid is constructed, Schreiber (1999) has noted that it appears to consist of a distorted cube which is first stretched to give rhombic faces with angles of , and then truncated on top and bottom to yield bounding triangular faces whose vertices lie on the circumsphere of the azimuthal cube vertices.It is implemented in the Wolfram Languageas PolyhedronData["DuererSolid"].The skeleton of Dürer's solid is the Dürer graph (i.e., generalized Petersen graph ).Starting..

Stella octangula

The stella octangula is a polyhedron compound composed of a tetrahedron and its dual (a second tetrahedron rotated with respect to the first). The stella octangula is also (incorrectly) called the stellated tetrahedron, and is the only stellation of the octahedron. A wireframe version of the stella octangula is sometimes known as the merkaba and imbued with mystic properties.The name "stella octangula" is due to Kepler (1611), but the solid was known earlier to many others, including Pacioli (1509), who called it the "octaedron elevatum," and Jamnitzer (1568); see Cromwell (1997, pp. 124 and 152).It is implemented in the Wolfram Languageas PolyhedronData["StellaOctangula"].A stella octangula can be inscribed in a cube, deltoidal icositetrahedron, pentagonal icositetrahedron, rhombic dodecahedron, small triakis octahedron, and tetrakis hexahedron, (E. Weisstein, Dec. 24-25,..

Great dirhombicosidodecahedron

The uniform polyhedron whose dual is the great dirhombicosidodecacron. This polyhedron is exceptional because it cannot be derived from Schwarz triangles and because it is the only uniform polyhedron with more than six polygons surrounding each polyhedron vertex (four squares alternating with two triangles and two pentagrams). This unique polyhedron has features in common with both snub forms and hemipolyhedra, and its octagrammic faces pass through the origin.It has pseudo-Wythoff symbol . Its faces are , and its circumradius for unit edge length isThe great dirhombicosidodecahedron appears on the cover of issue 4, volume 3 of TheMathematica Journal.

Great rhombic triacontahedron

The great rhombic triacontahedron, also called the great stellated triacontahedron, is a zonohedron which is the dual of the great icosidodecahedron and Wenninger model . It is one of the rhombic triacontahedron stellations.It appears together with an isometric projection of the 5-hypercube on the cover (and p. 103) of Coxeter's well-known book on polytopes (Coxeter 1973).The great rhombic triacontahedron can be constructed by expanding the size of the faces of a rhombic triacontahedron by a factor of , where is the golden ratio (Kabai 2002, p. 183) and keeping the pieces illustrated in the above stellation diagram.

Small dodecicosahedron

The uniform polyhedron whose dual polyhedron is the small dodecicosacron. It has Wythoff symbol . Its faces are . Its circumradius for unit edge lengths is

Truncated tetrahedron

The Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedTetrahedron"].The dual of the truncated tetrahedron is the triakis tetrahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances from the center of the solid to the centroids of the triangular and hexagonal faces are given by(4)(5)The surface area and volumeare(6)(7)

Snub cube

The snub cube, also called the cubus simus (Kepler 1619, Weissbach and Martini 2002) or snub cuboctahedron, is an Archimedean solid having 38 faces (32 triangular and 6 square), 60 edges, and 24 vertices. It is a chiral solid, and hence has two enantiomorphous forms known as laevo (left-handed) and dextro (right-handed).It is Archimedean solid , uniform polyhedron , and Wenninger model . It has Schläfli symbol and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["SnubCube"].Surprisingly, the tribonacci constant is intimately related to the metric properties of the snub cube.It can be constructed by snubification of a unit cube with outward offset(1)(2)and twist angle(3)(4)(5)(6)Here, the notation indicates a polynomial root and is the tribonacci constant.An attractive dual of the two enantiomers superposed on one another is illustrated above.Its dual polyhedron is the pentagonalicositetrahedron.The..

Pentagonal icositetrahedron

The pentagonal icositetrahedron is the 24-faced dual polyhedron of the snub cube and Wenninger dual . The mineral cuprite () forms in pentagonal icositetrahedral crystals (Steinhaus 1999, pp. 207 and 209).Because it is the dual of the chiral snub cube, the pentagonal icositetrahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right). An attractive dual of the two enantiomers superposed on one another is illustrated above.A cube, octahedron, and stella octangula can all be inscribed on the vertices of the pentagonal icositetrahedron (E. Weisstein, Dec. 25, 2009).Surprisingly, the tribonacci constant is intimately related to the metric properties of the pentagonal icositetrahedron cube.Its irregular pentagonal faces have vertex angles of(1)(2)(3)(four times) and(4)(5)(6)(once), where is a polynomial root and is the tribonacci constant.The dual formed from a snub cube with..

Truncated octahedron

The truncated octahedron is the 14-faced Archimedean solid , with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol . It was called the "mecon" by Buckminster Fuller (Rawles 1997).The dual polyhedron of the truncated octahedron is the tetrakis hexahedron. The truncated octahedron has the octahedral group of symmetries. The form of the fluorite () resembles the truncated octahedron (Steinhaus 1999, pp. 207-208).It is implemented in the Wolfram Languageas PolyhedronData["TruncatedOctahedron"].The solid of edge length can be formed from an octahedron of edge length via truncation by removing six square pyramids, each with edge slant height , base on a side, and height . The height and base area of the square pyramid are then(1)(2)(3)and its volume is(4)(5)The volume of the truncated octahedron is then given bythe volume of the octahedron(6)(7)minus..

Small triakis octahedron

In general, a triakis octahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of regular octahedron. Such a solid is also known as a trisoctahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting icositetrahedron is not regular, its faces are all identical. The small triakis octahedron, called simply the triakis octahedron by Holden (1971, p. 55), is the 24-faced dual polyhedron of the truncated cube and is Wenninger dual . The addition of the word "small" is necessary to distinguish it from the great triakis octahedron, which is the dual of the stellated truncated hexahedron. The small triakis octahedron It can be constructed by augmentation of a unit edge-length octahedron by a pyramid with height .A small triakis octahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's..

Pentagonal hexecontahedron

The pentagonal hexecontahedron is the 60-faced dual polyhedron of the snub dodecahedron (Holden 1971, p. 55). It is Wenninger dual .A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the pentagonal hexecontahedron (E. Weisstein, Dec. 25-27, 2009).Its irregular pentagonal faces have vertex angles of(1)(2)(four times) and(3)(4)(once), where is a polynomial root.Because it is the dual of the chiral snub dodecahedron, the pentagonal hexecontahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right). An attractive dual of the two enantiomers superposed on one another is illustrated above.Starting with a snub dodecahedron with unit edge lengths, the edges lengths of the pentagonal hexecontahedron are given by the roots of (5)(6)which have approximate values and .The surface area and volume are both given by the roots of 12th-order..

Small rhombicuboctahedron

The small rhombicuboctahedron is the 26-faced Archimedean solid consisting of faces . Although this solid is sometimes also called the truncated icosidodecahedron, this name is inappropriate since true truncation would yield rectangular instead of square faces. It is uniform polyhedron and Wenninger model . It has Schläfli symbol r and Wythoff symbol .The solid may also be called an expanded (or cantellated) cube or octahedron sinceit may be constructed from either of these solids by the process of expansion.A small rhombicuboctahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).It is implemented in the Wolfram Languageas PolyhedronData["SmallRhombicuboctahedron"].Its dual polyhedron is the deltoidal icositetrahedron, also called the trapezoidal icositetrahedron. The inradius of the..

Truncated dodecahedron

The 32-faced Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedDodecahedron"].The dual polyhedron is the triakisicosahedron.To construct the truncated dodecahedron by truncation, note that we want the inradius of the truncated pentagon to correspond with that of the original pentagon, , of unit side length . This means that the side lengths of the decagonal faces in the truncated dodecahedron satisfy(1)giving(2)The length of the corner which is chopped off is therefore given by(3)The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(4)(5)(6)The distances from the center of the solid to the centroids of the triangular and decagonal faces are given by(7)(8)The surface area and volumeare(9)(10)..

Small rhombicosidodecahedron

The 62-faced Archimedean solid with faces . It is uniform polyhedron and Wenninger model . It has Schläfli symbol r and Wythoff symbol . The small dodecicosidodecahedron and small rhombidodecahedron are faceted versions.It is implemented in the Wolfram Languageas PolyhedronData["SmallRhombicosidodecahedron"].Its dual polyhedron is the deltoidal hexecontahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)It has surface area(4)and volume(5)

Great rhombicuboctahedron

The 26-faced Archimedean solid consisting of faces . It is sometimes (improperly) called the truncated cuboctahedron (Ball and Coxeter 1987, p. 143), and is also more properly called the rhombitruncated cuboctahedron. It is uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .The great rhombicuboctahedron is an equilateral zonohedron and the Minkowski sum of three cubes. It can be combined with cubes and truncated octahedra into a regular space-filling pattern.The small cubicuboctahedron is a facetedversion of the great rhombicuboctahedron.Its dual is the disdyakis dodecahedron, also called the hexakis octahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)(4)(5)(6)Additional quantities are(7)(8)(9)(10)(11)The distances between the solid center and centroids of the square and octagonal faces are(12)(13)The surface..

Truncated cube

The 14-faced Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedCube"].The dual polyhedron of the truncated cube is the small triakis octahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances from the center of the solid to the centroids of the triangular and octagonal faces are(4)(5)The surface area and volumeare(6)(7)

Rhombic triacontahedron

The rhombic triacontahedron is a zonohedron which is the dual polyhedron of the icosidodecahedron (Holden 1971, p. 55). It is Wenninger dual . It is composed of 30 golden rhombi joined at 32 vertices. It is a zonohedron and one of the five golden isozonohedra.The intersecting edges of the dodecahedron-icosahedron compound form the diagonals of 30 rhombi which comprise the triacontahedron. The cube 5-compound has the 30 facial planes of the rhombic triacontahedron and its interior is a rhombic triacontahedron (Wenninger 1983, p. 36; Ball and Coxeter 1987).More specifically, a tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the rhombic triacontahedron (E. Weisstein, Dec. 25-27, 2009).The rhombic triacontahedron is implemented in the WolframLanguage as PolyhedronData["RhombicTriacontahedron"].The short diagonals of the faces..

Triakis tetrahedron

In general, a triakis tetrahedron is a non-regular dodecahedron that can be constructed as a positive augmentation of a regular tetrahedron. Such a solid is also known as a tristetrahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting dodecahedron is not regular, its faces are all identical. "The" triakis tetrahedron is the dual polyhedron of the truncated tetrahedron (Holden 1971, p. 55) and Wenninger dual . It can be constructed by augmentation of a unit edge-length tetrahedron by a pyramid with height .Five tetrahedra of unit edge length (corresponding to a central tetrahedron and its regular augmentation) and one tetrahedron of edge length 5/3 can be inscribed in the vertices of the unit triakis tetrahedron, forming the configurations illustrated above.The triakis tetrahedron formed by taking the dual of a truncated tetrahedron with unit edge..

Rhombic dodecahedron

The (first) rhombic dodecahedron is the dual polyhedron of the cuboctahedron (Holden 1971, p. 55) and Wenninger dual . Its sometimes also called the rhomboidal dodecahedron (Cotton 1990), and the "first" may be included when needed to distinguish it from the Bilinski dodecahedron (Bilinski 1960, Chilton and Coxeter 1963).A rhombic dodecahedron appears in the upper right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).The rhombic dodecahedron is implemented in the WolframLanguage as PolyhedronData["RhombicDodecahedron"].The 14 vertices of the rhombic dodecahedron are joined by 12 rhombiof the dimensions shown in the figure below, where(1)(2)(3)(4)(5)The rhombic dodecahedron can be built up by a placing six cubes on the faces of a seventh, in the configuration of a metal "jack" (left figure). Joining..

Disdyakis triacontahedron

The disdyakis triacontahedron is the dual polyhedron of the Archimedean great rhombicosidodecahedron . It is also known as the hexakis icosahedron (Holden 1971, p. 55). It is Wenninger dual .A tetrahedron 10-compound, octahedron 5-compound, cube 5-compound, icosahedron, dodecahedron, and icosidodecahedron can be inscribed in the vertices of a disdyakis triacontahedron (E. Weisstein, Dec. 26-27, 2009).Starting with an Archimedean great rhombicosidodecahedron of unit edge lengths, the edge lengths of the corresponding disdyakis triacontahedron are(1)(2)(3)The corresponding midradius is(4)The surface area and volume are(5)(6)

Triakis icosahedron

The 60-faced dual polyhedron of the truncated dodecahedron (Holden 1971, p. 55) and Wenninger dual . Wenninger (1989, p. 46) calls the small triambic icosahedron the triakis octahedron.A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed on the vertices of the triakis icosahedron (E. Weisstein, Dec. 25-27, 2009).Taking the dual of a truncated dodecahedronwith unit edge lengths gives a triakis icosahedron with edge lengths(1)(2)The surface area and volumeare(3)(4)

Disdyakis dodecahedron

The disdyakis dodecahedron is the dual polyhedron of the Archimedean great rhombicuboctahedron and Wenninger dual . It is also called the hexakis octahedron (Unkelbach 1940; Holden 1971, p. 55).If the original great rhombicuboctahedronhas unit side lengths, then the resulting dual has edge lengths(1)(2)(3)The inradius is(4)Scaling the disdyakis dodecahedron so that gives a solid with surface area and volume(5)(6)

Tetrakis hexahedron

In general, a tetrakis hexahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of a cube. Such a solid is also known as a tetrahexahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting icositetrahedron is not regular, its faces are all identical. "The" tetrakis hexahedron is the 24-faced dual polyhedron of the truncated octahedron (Holden 1971, p. 55) and Wenninger dual . It can be constructed by augmentation of a unit cube by a pyramid with height 1/4.A cube, octahedron, and stella octangula can all be inscribed in the vertices of the tetrakis hexahedron (E. Weisstein, Dec. 25, 2009).The edge lengths for the tetrakis hexahedron constructed as the dual of the truncatedoctahedron with unit edge lengths are(1)(2)Normalizing so that gives a tetrakis hexahedron with surface area and volume(3)(4)..

Deltoidal icositetrahedron

The deltoidal icositetrahedron is the 24-faced dual polyhedron of the small rhombicuboctahedron and Wenninger dual . It is also called the trapezoidal icositetrahedron (Holden 1971, p. 55).A deltoidal icositetrahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).A stella octangula, attractive octahedron 4-compound (whose dual is an attractive cube 4-compound), and cube can all be inscribed in a deltoidal icositetrahedron (E. Weisstein, Dec. 24, 2009). Superposing all these solids gives the beautiful compound illustrated above.For a small rhombicuboctahedron withunit edge length, the deltoidal icositetrahedron has edge lengths(1)(2)and inradius(3)Normalizing so the smallest edge has unit edge length gives a deltoidal icositetrahedron with surface area and volume(4)(5)..

Square antiprism

The square antiprism is the antiprism with square bases whose dual is the tetragonal trapezohedron. The square antiprism has 10 faces.The square antiprism with unit edge lengths has surfacearea and volume(1)(2)

Deltoidal hexecontahedron

The deltoidal hexecontahedron is the 60-faced dual polyhedron of the small rhombicosidodecahedron . It is sometimes also called the trapezoidal hexecontahedron (Holden 1971, p. 55), strombic hexecontahedron, or dyakis hexecontahedron (Unkelbach 1940). It is Wenninger dual .A tetrahedron 10-compound, octahedron 5-compound, cube 5-compound, icosahedron, dodecahedron, and icosidodecahedron can all be inscribed in the vertices of the deltoidal hexecontahedron (E. W. Weisstein, Dec. 24-27, 2009). The resulting compound of all these inscriptable solids is also illustrated above.Starting from a small rhombicosidodecahedron of unit edge length, the edge lengths of the corresponding deltoidal hexecontahedron are(1)(2)The corresponding midradius is(3)The surface area and volume are(4)(5)..

Cuboctahedron

A cuboctahedron, also called the heptaparallelohedron or dymaxion (the latter according to Buckminster Fuller; Rawles 1997), is Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .A cuboctahedron appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43), as well is in the mezzotint "Crystal" (Bool et al. 1982, p. 293).It is implemented in the Wolfram Languageas PolyhedronData["Cuboctahedron"].It is shown above in a number of symmetric projections.The dual polyhedron is the rhombic dodecahedron. The cuboctahedron has the octahedral group of symmetries. According to Heron, Archimedes ascribed the cuboctahedron to Plato (Heath 1981; Coxeter 1973, p. 30). The polyhedron..

Snub dodecahedron

The snub dodecahedron is an Archimedean solid consisting of 92 faces (80 triangular, 12 pentagonal), 150 edges, and 60 vertices. It is sometimes called the dodecahedron simum (Kepler 1619, Weissbach and Martini 2002) or snub icosidodecahedron. It is a chiral solid, and therefore exists in two enantiomorphous forms, commonly called laevo (left-handed) and dextro (right-handed).It is Archimedean solid , uniform polyhedron and Wenninger model . It has Schläfli symbol s and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["SnubDodecahedron"].An attractive dual of the two enantiomers superposed on one another is illustrated above.The dual polyhedron of the snub dodecahedron isthe pentagonal hexecontahedron.It can be constructed by snubification of a dodecahedron of unit edge length with outward offset(1)and twist angle(2)Here, the notation indicates a polynomial root.The inradius..

Pentakis dodecahedron

The pentakis dodecahedron is the 60-faced dual polyhedron of the truncated icosahedron (Holden 1971, p. 55). It is Wenninger dual . It can be constructed by augmentation of a unit edge-length dodecahedron by a pyramid with height .A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the pentakis dodecahedron (E. Weisstein, Dec. 25-27, 2009).Taking the dual of a truncated icosahedronwith unit edge lengths gives a pentakis dodecahedron with edge lengths(1)(2)Normalizing so that , the surface area and volume are(3)(4)

Truncated icosahedron

The truncated icosahedron is the 32-faced Archimedean solid corresponding to the facial arrangement . It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Rhodes 1996, p. 195). The truncated icosahedron has 60 vertices, and is also the structure of pure carbon known as buckyballs (a.k.a. fullerenes).The truncated icosahedron is uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedIcosahedron"].Several symmetrical projections of the truncated icosahedron are illustrated above.The dual polyhedron of the truncated icosahedron is the pentakis dodecahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances..

Rhombic triacontahedron stellations

Ede (1958) enumerates 13 basic series of stellations of the rhombic triacontahedron, the total number of which is extremely large. Pawley (1973) gave a set of restrictions upon which a complete enumeration of stellations can be achieved (Wenninger 1983, p. 36). Messer (1995) describes 227 stellations (including the original solid in the count as usual), some of which are illustrated above.The Great Stella stellation software reproduces Messer's 227 fully supported stellations. Using Miller's rules gives 358833098 stellations, 84959 of them reflexible and 358748139 of them chiral.The convex hull of the dodecadodecahedron is an icosidodecahedron and the dual of the icosidodecahedron is the rhombic triacontahedron, so the dual of the dodecadodecahedron (the medial rhombic triacontahedron) is one of the rhombic triacontahedron stellations (Wenninger 1983, p. 41). Others include the great rhombic triacontahedron,..

Great icosahedron

One of the Kepler-Poinsot solids whose dual is the great stellated dodecahedron. It is also uniform polyhedron , Wenninger model , and has Schläfli symbol and Wythoff symbol .The great icosahedron can be constructed from an icosahedron with unit edge lengths by taking the 20 sets of vertices that are mutually spaced by a distance , the golden ratio. The solid therefore consists of 20 equilateral triangles. The symmetry of their arrangement is such that the resulting solid contains 12 pentagrams.The great icosahedron can most easily be constructed by building a "squashed" dodecahedron (top right figure) from the corresponding net (top left). Then, using the net shown in the bottom left figure, build 12 pentagrammic pyramids (bottom middle figure) and affix them into the dimples (bottom right). This method of construction is given in Cundy and Rollett (1989, pp. 98-99). If the edge lengths of the dodecahedron are unity,..

Spikey

"Spikey" is the logo of Wolfram Research, makers of Mathematica and the Wolfram Language. In its original (Version 1) form, it is an augmented icosahedron with an augmentation height of , not to be confused with the great stellated dodecahedron, which is a distinct solid. This gives it 60 equilateral triangular faces, making it a deltahedron. More elaborate forms of Spikey have been used for subsequent versions of Mathematica. In particular, Spikeys for Version 2 and up are based on a hyperbolic dodecahedron with embellishments rather than an augmented icosahedron (Trott 2007, Weisstein 2009).The "classic" (Version 1) Spikey illustrated above is implemented in theWolfram Language as PolyhedronData["MathematicaPolyhedron"].The skeleton of the classic Spikey is the graph of thetriakis icosahedron.A glyph corresponding to the classic Spikey, illustrated above, is available as the character \[MathematicaIcon]..

Small stellated dodecahedron

The small stellated dodecahedron is the Kepler-Poinsot solids whose dual polyhedron is the great dodecahedron. It is also uniform polyhedron , Wenninger model , and is the first stellation of the dodecahedron (Wenninger 1989). The small stellated dodecahedron has Schläfli symbol and Wythoff symbol . It is concave, and is composed of 12 pentagrammic faces ().The small stellated dodecahedron appeared ca. 1430 as a mosaic by Paolo Uccello on the floor of San Marco cathedral, Venice (Muraro 1955). It was rediscovered by Kepler (who used th term "urchin") in his work Harmonice Mundi in 1619, and again by Poinsot in 1809.The skeleton of the small stellated dodecahedron is isomorphic to the icosahedralgraph.Schläfli (1901, p. 134) did not recognize the small stellated dodecahedron as a regular solid because it violates the polyhedral formula, instead satisfying(1)where is the number of vertices, the number of edges,..

Regular tetrahedron

The regular tetrahedron, often simply called "the" tetrahedron, is the Platonic solid with four polyhedron vertices, six polyhedron edges, and four equivalent equilateral triangular faces, . It is also uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and the Wythoff symbol is . It is an isohedron, and a special case of the general tetrahedron and the isosceles tetrahedron.The regular tetrahedron is implemented in the Wolfram Language as Tetrahedron[], and precomputed properties are available as PolyhedronData["Tetrahedron"].The tetrahedron has 7 axes of symmetry: (axes connecting vertices with the centers of the opposite faces) and (the axes connecting the midpoints of opposite sides).There are no other convex polyhedra other than the tetrahedron having four faces.The tetrahedron has two distinct nets (Buekenhout and Parker 1998). Questions of polyhedron coloring..

Regular octahedron

The regular octahedron, often simply called "the" octahedron, is the Platonic solid with six polyhedron vertices, 12 polyhedron edges, and eight equivalent equilateral triangular faces, denoted . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and Wythoff symbol . The octahedron of unit side length is the antiprism of sides with height . The octahedron is also a square dipyramid with equal edge lengths.The regular octahedron is implemented in the Wolfram Language as Octahedron[], and precomputed properties are available as PolyhedronData["Octahedron"].There are 11 distinct nets for the octahedron, the same as for the cube (Buekenhout and Parker 1998). Questions of polyhedron coloring of the octahedron can be addressed using the Pólya enumeration theorem.The dual polyhedron of an octahedron with unit edge lengths is a cube with edge lengths .The illustration..

Regular icosahedron

The regular icosahedron (often simply called "the" icosahedron) is the regular polyhedron and Platonic solid illustrated above having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces, .The regular icosahedron is also uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and Wythoff symbol . Coxeter et al. (1999) have shown that there are 58 icosahedron stellations (giving a total of 59 solids when the icosahedron itself is included).The regular icosahedron is implemented in the Wolfram Language as Icosahedron[], and precomputed properties are available as PolyhedronData["Icosahedron"].Two icosahedra constructed in origami are illustrated above (Gurkewitz and Arnstein 1995, p. 53). This construction uses 30 triangle edge modules, each made from a single sheet of origami paper.Two icosahedra appears as polyhedral "stars"..

Regular dodecahedron

The regular dodecahedron, often simply called "the" dodecahedron, is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and the Wythoff symbol .The regular dodecahedron is implemented in the Wolfram Language as Dodecahedron[], and precomputed properties are available as PolyhedronData["Dodecahedron"].There are 43380 distinct nets for the regular dodecahedron, the same number as for the icosahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998). Questions of polyhedron coloring of the regular dodecahedron can be addressed using the Pólya enumeration theorem.The image above shows an origami regular dodecahedron constructed using six dodecahedron units, each consisting of a single sheet of paper (Kasahara and Takahama 1987, pp. 86-87).A..

Icosidodecahedron

In general, an icosidodecahedron is a 32-faced polyhedron. "The" icosidodecahedron is the 32-faced Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is also uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["Icosidodecahedron"].Several symmetric projections of the icosidodecahedron are illustrated above. The dual polyhedron is the rhombic triacontahedron. The polyhedron vertices of an icosidodecahedron of polyhedron edge length are , , , , , . The 30 polyhedron vertices of an octahedron 5-compound form an icosidodecahedron (Ball and Coxeter 1987). Faceted versions include the small icosihemidodecahedron and small dodecahemidodecahedron.The icosidodecahedron constructed in origami is illustrated above (Kasahara and Takahama 1987, pp. 48-49). This construction..

Great rhombicosidodecahedron

The 62-faced Archimedean solid with faces . It is also known as the rhombitruncated icosidodecahedron, and is sometimes improperly called the truncated icosidodecahedron (Ball and Coxeter 1987, p. 143), a name which is inappropriate since truncation would yield rectangular instead of square. The great rhombicosidodecahedron is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .The great rhombicosidodecahedron is an equilateral zonohedron and is the Minkowski sum of five cubes.Its dual is the disdyakis triacontahedron, also called the hexakis icosahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The great rhombicosidodecahedron has surface area(4)and volume(5)The great rhombicosidodecahedron constructed by E. K. Herrstrom in origami is illustrated above (Kasahara and Takahama 1987, pp. 46-49)...

Augmentation

Augmentation is the dual operation of truncation which replaces the faces of a polyhedron with pyramids of height (where may be positive, zero, or negative) having the face as the base (Cromwell 1997, p. 124 and 195-197). The operation is sometimes also called accretion, akisation (since it transforms a regular polygon to an -akis polyhedron, i.e., quadruples the number of faces), capping, or cumulation.B. Grünbaum used the terms elevatum and invaginatum for positive-height (outward-pointing) and negative-height (inward-pointing), respectively, pyramids used in augmentation.The term "augmented" is also sometimes used in the more general context of affixing one polyhedral cap over the face of a base solid. An example is the Johnson solid called the augmented truncated cube, for which the affixed shape is a square cupola--not a pyramid.Augmentation is implemented under the misnomer Stellate[poly,..

Latitude

The latitude of a point on a sphere is the elevation of the point from the plane of the equator. The latitude is related to the colatitude (the polar angle in spherical coordinates) by . More generally, the latitude of a point on an ellipsoid is the angle between a line perpendicular to the surface of the ellipsoid at the given point and the plane of the equator (Snyder 1987).The equator therefore has latitude , and the north and south poles have latitude , respectively. Latitude is also called geographic latitude or geodetic latitude in order to distinguish it from several subtly different varieties of authalic latitudes.The shortest distance between any two points on a sphere is the so-called great circle distance, which can be directly computed from the latitudes and longitudes of the two points...

Medial parallelogram

When a pair of non-incident edges of a tetrahedron is chosen, the midpoints of the remaining 4 edges are the vertices of a planar parallelogram. Furthermore, the area of this parallelogram determined by the edges of lengths and in the figure above is given by(Yetter 1998; Trott 2004, pp. 65-66)

Holyhedron

A holyhedron is polyhedron whose faces and holes are all finite-sided polygons and that contains at least one hole whose boundary shares no point with a face boundary. D. Wilson coined the term in 1997, although no actual holyhedron was known until 1999, when a holyhedron with faces was constructed (Vinson 2000).J. H. Conway believes that the minimal number of faces should be closer to 100, and offered a prize of divided by the number of faces for a better solution. A holyhedron with 492 faces was subsequently discovered, good for a prize of (Hatch).

Torus coloring

The number of colors sufficient for map coloring on a surface of genus is given by the Heawood conjecture,where is the floor function. The fact that (which is called the chromatic number) is also necessary was proved by Ringel and Youngs (1968) with two exceptions: the sphere (which requires the same number of colors as the plane) and the Klein bottle.A -holed torus therefore requires colors. For , 1, ..., the first few values of are 4, 7 (illustrated above, M. Malak, pers. comm., Feb. 22, 2006), 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, ... (OEIS A000934). A set of regions requiring the maximum of seven regions is shown above for a normal torusThe above figure shows the relationship between the Heawoodgraph and the 7-color torus coloring.

Polyhedron coloring

Define a valid "coloring" to occur when no two faces with a common edge share the same color. Given two colors, there is a single way to color an octahedron (Ball and Coxeter 1987, pp. 238-239). Given three colors, there is one way to color a cube (Ball and Coxeter 1987, pp. 238-239) and 144 ways to color an icosahedron (Ball and Coxeter 1987, pp. 239-242). Given four colors, there are two distinct ways to color a tetrahedron (Ball and Coxeter 1987, p. 238) and four ways to color a dodecahedron, consisting of two enantiomorphous ways (Steinhaus 1999, pp. 196-198; Ball and Coxeter 1987, p. 238). Given five colors, there are four ways to color an icosahedron. Given six colors, there are 30 ways to color a cube (Steinhaus 1999, p. 167). These values are related to the chromatic polynomial of the corresponding dual skeleton graph, which however overcounts since it does not take rotational equivalence..

Cube

The cube is the Platonic solid (also called the regular hexahedron). It is composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and Wythoff symbol .The cube is illustrated above, together with a wireframe version and a net(top figures). The bottom figures show three symmetric projections of the cube.The cube is implemented in the Wolfram Language as Cube[], and precomputed properties are available as PolyhedronData["Cube"].There are a total of 11 distinct nets for the cube (Turney 1984-85, Buekenhout and Parker 1998, Malkevitch), illustrated above, the same number as the octahedron. Questions of polyhedron coloring of the cube can be addressed using the Pólya enumeration theorem.A cube with unit edge lengths is called a unit cube.The surface area and volume of a cube with edge..

Heronian tetrahedron

A Heronian tetrahedron, also called a perfect tetrahedron, is a (not necessarily regular) tetrahedron whose sides, face areas, and volume are all rational numbers. It therefore is a tetrahedron all of whose faces are Heronian triangles and additionally that has rational volume. (Note that the volume of a tetrahedron can be computed using the Cayley-Menger determinant.)The integer Heronian tetrahedron having smallest maximum side length is the one with edge lengths 51, 52, 53, 80, 84, 117; faces (117, 80, 53), (117, 84, 51), (80, 84, 52), (53, 51, 52); face areas 1170, 1800, 1890, 2016; and volume 18144 (Buchholz 1992; Guy 1994, p. 191). This is the only integer Heronian triangle with all side lengths less than 157.The integer Heronian tetrahedron with smallest possible surface area and volume has edges 25, 39, 56, 120, 153, and 160; areas 420, 1404, 1872, and 2688 (for a total surface area of 6384); and volume 8064 (Buchholz 1992, Peterson..

Perfect cuboid

A perfect cuboid is a cuboid having integer side lengths,integer face diagonals(1)(2)(3)and an integer space diagonal(4)The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem.No perfect cuboids are known despite an exhaustive search for all "odd sides" up to (Butler, pers. comm., Dec. 23, 2004).Solving the perfect cuboid problem is equivalent to solving the Diophantineequations(5)(6)(7)(8)A solution with integer space diagonal and two out of three face diagonals is , , and , giving , , , and , which was known to Euler. A solution giving integer space and face diagonals with only a single nonintegral polyhedron edge is , , and , giving , , , and .

Euler brick

An Euler brick is a cuboid that possesses integer edges and face diagonals(1)(2)(3)If the space diagonal is also an integer, the Euler brick is called a perfect cuboid, although no examples of perfect cuboids are currently known.The smallest Euler brick has sides and face polyhedron diagonals , , and , and was discovered by Halcke (1719; Dickson 2005, pp. 497-500). Kraitchik gave 257 cuboids with the odd edge less than 1 million (Guy 1994, p. 174). F. Helenius has compiled a list of the 5003 smallest (measured by the longest edge) Euler bricks. The first few are (240, 117, 44), (275, 252, 240), (693, 480, 140), (720, 132, 85), (792, 231, 160), ... (OEIS A031173, A031174, and A031175).Interest in this problem was high during the 18th century, and Saunderson (1740) found a parametric solution always giving Euler bricks (but not giving all possible Euler bricks), while in 1770 and 1772, Euler found at least two parametric solutions...

Volume

The volume of a solid body is the amount of "space" it occupies. Volume has units of length cubed (i.e., , , , etc.) For example, the volume of a box (cuboid) of length , width , and height is given byThe volume can also be computed for irregularly-shaped and curved solids such as the cylinder and cone. The volume of a surface of revolution is particularly simple to compute due to its symmetry.The volume of a region can be computed in the WolframLanguage using Volume[reg].The following table gives volumes for some common surfaces. Here denotes the radius, the height, and the base area, and, in the case of the torus, the distance from the torus center to the center of the tube (Beyer 1987).surfacevolumeconeconical frustumcubecylinderellipsoidoblate spheroidprolate spheroidpyramidpyramidal frustumspherespherical capspherical sectorspherical segmenttorusEven simple surfaces can display surprisingly counterintuitive properties...

Uniform polyhedron

The uniform polyhedra are polyhedra with identical polyhedron vertices. Badoureau discovered 37 nonconvex uniform polyhedra in the late nineteenth century, many previously unknown (Wenninger 1983, p. 55). The uniform polyhedra include the Platonic solids and Kepler-Poinsot solids.Coxeter et al. (1954) conjectured that there are 75 such polyhedra in which only two faces are allowed to meet at an polyhedron edge, and this was subsequently proven. The five pentagonal prisms can also be considered uniform polyhedra, bringing the total to 80. In addition, there are two other polyhedra in which four faces meet at an edge, the great complex icosidodecahedron and small complex icosidodecahedron (both of which are compounds of two uniform polyhedra).The polyhedron vertices of a uniform polyhedron all lie on a sphere whose center is their geometric centroid (Coxeter et al. 1954, Coxeter 1973, p. 44. The polyhedron vertices joined..

Spherical distance

The spherical distance between two points and on a sphere is the distance of the shortest path along the surface of the sphere (paths that cut through the interior of the sphere are not allowed) from to , which always lies along a great circle.For points and on the unit sphere, the spherical distance is given bywhere denotes a dot product.

Kulikowski's theorem

For every positive integer , there exists a sphere which has exactly lattice points on its surface. The sphere is given by the equation(1)where and are the coordinates of the center of the so-called Schinzel circle(2)and is its radius.

Sphere eversion

Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy from the standard embedding of the 2-sphere in Euclidean three-space to the mirror-reflection embedding such that at every stage in the homotopy, the sphere is being immersed in Euclidean space. This result is so counterintuitive and the proof so technical that the result remained controversial for a number of years.In 1961, Arnold Shapiro devised an explicit eversion but did not publicize it. Phillips (1966) heard of the result and, in trying to reproduce it, actually devised an independent method of his own. Yet another eversion was devised by Morin, which became the basis for the movie by Max (1977). Morin's eversion also produced explicit algebraic equations describing the process. The original method of Shapiro was subsequently published by Francis and Morin (1979).The..

Zome

A kit consisting of colored rods and slotted balls that can be used to construct three-dimensional configurations. The balls into which the rods are placed resemble an "expanded" small rhombicosidodecahedron, with the squares replaced by rectangles, as illustrated above. The expansion is chosen so that the resulting rectangles are golden rectangles.For a solid zome unit with edge lengths 1 and (where is the golden ratio), the circumradius is(1)the volume is(2)and the surface area is(3)In the zome kit, the rods come in four colors, and there are three lengths for each color, as summarized in the table below. Here, is the golden ratio.colorlengthsblueyellowredgreen

Cube dovetailing problem

Given the above figure (without looking at the figure below!), determine how to disengage the two slotted cube halves without cutting, breaking, or distorting.One possible solution is shown above; the slots are not perpendicular to one another but instead consist of parallel slotted tracks. Other solutions are also possible. For example, another construction involves two circular arcs sharing the same center (Gardner 2001, p. 117).

Ball

The -ball, denoted , is the interior of a sphere , and sometimes also called the -disk. (Although physicists often use the term "sphere" to mean the solid ball, mathematicians definitely do not!)The ball of radius centered at point is implemented in the Wolfram Language as Ball[x, y, z, r].The equation for the surface area of the -dimensional unit hypersphere gives the recurrence relation(1)Using then gives the hypercontent of the -ball of radius as(2)(Sommerville 1958, p. 136; Apostol 1974, p. 430; Conway and Sloane 1993). Strangely enough, the content reaches a maximum and then decreases towards 0 as increases. The point of maximal content of a unit -ball satisfies(3)(4)(5)where is the digamma function, is the gamma function, is the Euler-Mascheroni constant, and is a harmonic number. This equation cannot be solved analytically for , but the numerical solution to(6)is (OEIS A074455) (Wells 1986, p. 67)...

Tangent spheres

Any four mutually tangent spheres determine six points of tangency. A pair of tangencies is said to be opposite if the two spheres determining are distinct from the two spheres determining . The six tangencies are therefore grouped into three opposite pairs corresponding to the three ways of partitioning four spheres into two pairs. These three pairs of opposite tangencies are coincident (Altshiller-Court 1979, p. 231; Eppstein 2001).A special case of tangent spheres is given by Soddy's hexlet, which consists of a chain of six spheres externally tangent to two mutually tangent spheres and internally tangent to a circumsphere. The bends of the circles in the chain obey the relationship(1)A Sangaku problem from 1798 asks to distribute 30 identical spheres of radius such that they are tangent to a single central sphere of radius and to four other small spheres. This can be accomplished (left figure) by placing the spheres at the vertices..

Hexlet

Consider two mutually tangent (externally) spheres and together with a larger sphere inside which and are internally tangent. Then construct a chain of spheres each tangent externally to , and internally to (so that encloses the chain as well as the two original spheres). Surprisingly, every such chain closes into a "necklace" after six spheres, regardless of where the first sphere is placed.This beautiful and amazing result due to Soddy (1937) is a special case of Kollros' theorem. It can be demonstrated using inversion of six identical spheres around an equal center sphere, all of which are sandwiched between two planes (Wells 1991, pp. 120 and 232). This result was given in a Sangaku problem from Kanagawa Prefecture in 1822, more than a century before it was published by Soddy (Rothman 1998).Moreover, the centers of the six spheres in the necklace and their six points of contact all lie in a plane. Furthermore, there are two..

Parametric latitude

An auxiliary latitude also called the reduced latitude and denoted or . It gives the latitude on a sphere of radius for which the parallel has the same radius as the parallel of geodetic latitude and the ellipsoid through a given point. It is given by(1)In series form,(2)where(3)

Cube duplication

Cube duplication, also called the Delian problem, is one of the geometric problems of antiquity which asks, given the length of an edge of a cube, that a second cube be constructed having double the volume of the first. The only tools allowed for the construction are the classic (unmarked) straightedge and compass.The problem appears in a Greek legend which tells how the Athenians, suffering under a plague, sought guidance from the Oracle at Delos as to how the gods could be appeased and the plague ended. The Oracle advised doubling the size of the altar to the god Apollo. The Athenians therefore built a new alter twice as big as the original in each direction and, like the original, cubical in shape (Wells, 1986, p. 33). However, as the Oracle (notorious for ambiguity and double-speaking in his prophecies) had advised doubling the size (i.e., volume), not linear dimension (i.e., scale), the new altar was actually eight times as big as the old..

Hexagonal close packing

In hexagonal close packing, layers of spheres are packed so that spheres in alternating layers overlie one another. As in cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives Johnson solid known as the triangular orthobicupola (Steinhaus 1999, pp. 203-205; Wells 1991, p. 237).Hexagonal close packing must give the same packing density as cubic close packing, since sliding one sheet of spheres cannot affect the volume they occupy. To verify this, construct a three-dimensional diagram containing a hexagonal unit cell with three layers (Steinhaus 1999, pp. 203-204). Both the top and the bottom contain six -spheres and one hemisphere. The total number of spheres in these two rows is therefore(1)The volume of spheres in the middle row cannot be simply computed using geometry. However,..

Ellipsoid packing

Bezdek and Kuperberg (1991) have constructed packings of identical ellipsoidsof densities arbitrarily close to(OEIS A093824), greater than the maximum density of (OEIS A093825) that is possible for sphere packing (Sloane 1998), as established by proof of the Kepler conjecture. Furthermore, J. Wills has modified the ellipsoid packing to yield an even higher density of (Bezdek and Kuperberg 1991).Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.

Sphere packing

Define the packing density of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. It was hypothesized by Kepler in 1611 that close packing (cubic or hexagonal, which have equivalent packing densities) is the densest possible, and this assertion is known as the Kepler conjecture. The problem of finding the densest packing of spheres (not necessarily periodic) is therefore known as the Kepler problem, where(OEIS A093825; Steinhaus 1999, p. 202;Wells 1986, p. 29; Wells 1991, p. 237).In 1831, Gauss managed to prove that the face-centered cubic is the densest lattice packing in three dimensions (Conway and Sloane 1993, p. 9), but the general conjecture remained open for many decades.While the Kepler conjecture is intuitively obvious, the proof remained..

Cubic close packing

There are three types of cubic lattices corresponding to three types of cubic close packing, as summarized in the following table. Now that the Kepler conjecture has been established, hexagonal close packing and face-centered cubic close packing, both of which have packing density of , are known to be the densest possible packings of equal spheres.lattice typebasis vectorspacking densitysimple cubic (SC), , face-centered cubic (FCC), , body-centered cubic (BCC), , Simple cubic packing consists of placing spheres centered on integer coordinates in Cartesian space.Arranging layers of close-packed spheres such that the spheres of every third layer overlay one another gives face-centered cubic packing. To see where the name comes from, consider packing six spheres together in the shape of an equilateral triangle and place another sphere on top to create a triangular pyramid. Now create another such grouping of seven spheres and place..

Local density

Let each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces. Call the resulting polyhedron the local cell. Then the local density is given byWhen the local cell is a regular dodecahedron,then

Random close packing

The concept of "random close packing" was shown by Torquato et al. (2000) to be mathematically ill-defined idea that is better replaced by the notion of "maximally random jammed."Random close packing of spheres in three dimensions gives a packing density of only (Jaeger and Nagel 1992), significantly smaller than the optimal packing density for cubic or hexagonal close packing of 0.74048.Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.

Net

The word net has several meanings in mathematics. It refers to a plane diagram in which the polyhedron edges of a polyhedron are shown, a point set satisfying certain uniformity of distribution conditions, and a topological generalization of a sequence.The net of a polyhedron is also known as a development, pattern, or planar net (Buekenhout and Parker 1998). The illustrations above show polyhedron nets for the cube and tetrahedron.In his classic Treatise on Measurement with the Compass and Ruler, Dürer(1525) made one of the first presentations of a net (Livio 2002, p. 138).The net of a polyhedron must in general also specify which edges are to be joined since there might be ambiguity as to which of several possible polyhedra a net might fold into. For simple symmetrical polyhedra, the folding procedure can only be done one way, so edges need not be labeled. However, for the net shown above, two different solids can be constructed from..

Polyhedron

The word polyhedron has slightly different meanings in geometry and algebraic geometry. In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. The word derives from the Greek poly (many) plus the Indo-European hedron (seat). A polyhedron is the three-dimensional version of the more general polytope (in the geometric sense), which can be defined in arbitrary dimension. The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons").The term "polyhedron" is used somewhat differently in algebraic topology, where it is defined as a space that can be built from such "building blocks" as line segments, triangles, tetrahedra, and their higher dimensional analogs by "gluing them together" along their faces (Munkres 1993, p. 2). More specifically, it can be defined as the underlying space..