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

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

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

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

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

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

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

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

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

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

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

A cylindric section is the intersection of a plane with a right circular cylinder. It is a circle (if the plane is at a right angle to the axis), an ellipse, or, if the plane is parallel to the axis, a single line (if the plane is tangent to the cylinder), pair of parallel lines bounding an infinite rectangle (if the plane cuts the cylinder), or no intersection at all (if the plane misses the cylinder entirely; Hilbert and Cohn-Vossen 1999, pp. 7-8).

A spheric section is the curve formed by the intersection of a plane with a sphere. Excluding the degenerate cases of the plane tangent to the sphere or the plane not intersecting the sphere, all spheric sections are circles.A spheric section that does not contain a diameter of the sphere is known as a small circle, while a spheric section containing a diameter is known as a great circle.

The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola (Hilbert and Cohn-Vossen 1999, p. 8). The curve produced by a plane intersecting both nappes is a hyperbola (Hilbert and Cohn-Vossen 1999, pp. 8-9).The ellipse and hyperbolaare known as central conics.Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newton was then able to derive the shape of orbits mathematically using calculus, under..

The (not necessarily regular) tetrahedron of least volume circumscribed around a convex body with volume is not known. If is a parallelepiped, then the smallest-volume tetrahedron containing it has volume 9/2. It is conjectured that this is the worst possible fit for the general problem, but this remains unproved.

The mean triangle area of a triangle picked at random inside a unit cube is , with variance .The distribution of areas, illustrated above, is apparently not known exactly.The probability that a random triangle in a cube is obtuse is approximately .

Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. random tetrahedra can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 4].Pick four points on a sphere. What is the probability that the tetrahedron having these points as polyhedron vertices contains the center of the sphere? In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the two-dimensional case, pick two points. In order for the third to form a triangle containing the center, it must lie in the quadrant bisected by a line segment passing through the center of the circle and the bisector of the two points. This happens for one quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one octant, or 1/8.Pick four points at random on the surface of a unit sphereusing(1)(2)(3)with..

Given a simplex of unit content in Euclidean -space, pick points uniformly and independently at random, and denote the expected content of their convex hull by . Exact values are known only for and 2.(1)(2)(Buchta 1984, 1986), giving the first few values 0, 1/3, 1/2, 3/5, 2/3, 5/7, ...(OEIS A026741 and A026741).(3)(4)where is a harmonic number (Buchta 1984, 1986), giving the first few values 0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, ... (OEIS A093762 and A093763).Not much is known about , although(5)(Buchta 1983, 1986) and(6)(Buchta 1986).Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of points chosen at random in a three-dimensional simplex for arbitrary .

Let two points and be picked randomly from a unit -dimensional hypercube. The expected distance between the points , i.e., the mean line segment length, is then(1)This multiple integral has been evaluated analytically only for small values of . The case corresponds to the line line picking between two random points in the interval .The first few values for are given in the following table.OEIS1--0.3333333333...2A0915050.5214054331...3A0730120.6617071822...4A1039830.7776656535...5A1039840.8785309152...6A1039850.9689420830...7A1039861.0515838734...8A1039871.1281653402...The function satisfies(2)(Anderssen et al. 1976), plotted above together with the actual values.M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the -dimensional integral to an integral over a 1-dimensional integrand such that(3)The first few values are(4)(5)(6)(7)In the limit as , these..

A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called small circles. A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. 220-221).The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle (geodesic) distance between two points located at latitude and longitude of and on a sphere of radius , convert spherical coordinates to Cartesian coordinates using(1)(Note that the latitude is related to the colatitude of spherical coordinates by , so the conversion to Cartesian coordinates replaces and by and , respectively.) Now find the angle between and using the dot product,(2)(3)(4)The great circle distance is then(5)For the Earth, the equatorial radius is km, or 3963 (statute) miles. Unfortunately, the flattening..

A convex polyhedron is defined as the set ofsolutions to a system of linear inequalities(i.e., a matrix inequality), where is a real matrix and is a real -vector. Given and , vertex enumeration is the determination of the polyhedron's polyhedron vertices.

Given four points chosen at random inside a unit cube, the average volume of the tetrahedron determined by these points is given by(1)where the polyhedron vertices are located at where , ..., 4, and the (signed) volume is given by the determinant(2)The integral is extremely difficult to compute, but the analytic result for the mean tetrahedron volume is(3)(OEIS A093524; Zinani 2003). Note that the result quoted in the reply to Seidov (2000) actually refers to the average volume for tetrahedron tetrahedron picking.

Cube point picking is the three-dimensional case of hypercubepoint picking.The average distance from a point picked at random inside a unitcube to the center is given by(1)(2)(3)(4)Similarly, the average distance from a point picked at random to a fixed corner is given by(5)(6)(7)(8)(9)where is the -box integral.The average distance from the center of a unit cube to a given face is(10)(11)(12)(13)(OEIS A097047).

Consider the distribution of distances between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form(1)The first even raw moments for , 2, 4, ... are 1, 2/3, 11/18, 211/315, 187/225, 11798/10395, ....

Instead of picking two points from the interior of the cube, instead pick two points on different faces of the unit cube. In this case, the average distance between the points is(1)(OEIS A093066; Borwein and Bailey 2003, p. 26;Borwein et al. 2004, pp. 66-67). Interestingly,(2)as apparently first noted by M. Trott (pers. comm., Mar. 21, 2008).The two integrals above can be written in terms of sums as(3)(4)(Borwein et al. 2004, p. 67), where however appears to be classically divergent and perhaps must be interpreted in some regularized sense.Consider a line whose endpoints are picked at random on opposite sides of the unit cube. The probability density function for the length of this line is given by(5)(Mathai 1999; after simplification). The mean length is(6)(7)The first even raw moments for , 2, 4, ... are 1, 4/3, 167/90, 284/105, 931/225, 9868/1485, ....Consider a line whose endpoints are picked at random..

The average distance between two points chosen at random inside a unit cube (the case of hypercube line picking), sometimes known as the Robbins constant, is(1)(2)(3)(OEIS A073012; Robbins 1978, Le Lionnais 1983).The probability function as a function of line length, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form(4)The first even raw moments for , 2, ... are 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 and A160694).Pick points on a cube, and space them as far apart as possible. The best value known for the minimum straight line distance between any two points is given in the following table. 51.118033988749861.0606601482100718190.86602540378463100.74999998333331110.70961617562351120.70710678118660130.70710678118660140.70710678118660150.625..

To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates and from uniform distributions and , since the area element is a function of , and hence points picked in this way will be "bunched" near the poles (left figure above). random points can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n].To obtain points such that any small area on the sphere is expected to contain the same number of points (right figure above), choose and to be random variates on . Then(1)(2)gives the spherical coordinates for a set of points which are uniformly distributed over . This works since the differential element of solid angle is given by(3)The distribution of polar angles can be found from(4)by taking the derivative of (2) with respect to to get , solving (2) for , and plugging the results back in to (4) with to obtain the distribution(5)Similarly, we can pick to be uniformly..

Sphere line picking is the selection of pairs of points corresponding to vertices of a line segment with endpoints on the surface of a sphere. random line segments can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 2].Pick two points at random on a unit sphere. The first one can be placed at the north pole, i.e., assigned the coordinate (0, 0, 1), without loss of generality. The second point is then chosen at random using sphere point picking, and so can be assigned coordinates(1)(2)(3)with and . The distance between first and second points is then(4)and solving for gives(5)Now the probability function for distance is then given by(6)(Solomon 1978, p. 163), since and . Here, .Therefore, somewhat surprisingly, large distances are the most common, contrary to most people's intuition. A plot of 15 random lines is shown above. The raw moments are(7)giving the first few as(8)(9)(10)(11)(OEIS..

The Robbins constant is the mean line segment length, i.e., the expected distance between two points chosen at random in cube line picking, namely(1)(2)(3)(OEIS A073012; Robbins 1978, Le Lionnais 1983).

A golden rhombohedron is a rhombohedron whose faces consist of congruent golden rhombi. Golden rhombohedra are therefore special cases of a trigonal trapezohedron as well as zonohedra.There are two distinct golden rhombohedra: the acute golden rhombohedron and obtuse golden rhombohedron. Both are built from six golden rhombi and comprise two of the five golden isozonohedra. These polyhedra are implemented in the Wolfram Language as PolyhedronData["AcuteGoldenRhombohedron"] and PolyhedronData["ObtuseGoldenRhombohedron"], respectively.The acute and obtuse golden rhombohedra with edge length both have surface area(1)and have volumes(2)(3)respectively.

As defined by Gray (1997, p. 201), Viviani's curve, sometimes also called Viviani's window, is the space curve giving the intersection of the cylinder of radius and center (1)and the sphere(2)with center and radius . This curve was studied by Viviani in 1692 (Teixeira 1908-1915, pp. 311-320; Struik 1988, pp. 10-11; Gray 1997, p. 201).Solving directly for and as a function of gives(3)(4)This curve is given by the parametric equations(5)(6)(7)for (Gray 1997, p. 201).From the parametric equations, it can be immediately seen that views of the curve from the front, top, and left are given by a lemniscate-like curve, circle, and parabolic segment, respectively. The lemniscate-like figure has parametric equations(8)(9)which can be written in Cartesian coordinatesas the quartic curve(10)Viviani's curve has arc length(11)where is a complete elliptic integral of the second kind.The arc length function,..

The azimuthal coordinate on the surface of a sphere ( in spherical coordinates) or on a spheroid (in prolate or oblate spheroidal coordinates). Longitude is defined such that . Lines of constant longitude are generally called meridians. The other angular coordinate on the surface of a sphere is called the latitude.The shortest distance between any two points on a sphere is the so-called great circle distance, which can be directly computed from the latitude and longitudes of two points.

What is the area of the largest square that can be inscribed on a unit cube (Trott 2004, p. 104)? The answer is 9/8, given by a square with vertices (1/4, 0, 0), (0, 1, 1/4), (3/4, 1, 1), (1, 0, 3/4), or any configuration equivalent by symmetry.In general, let be the edge of the largest -dimensional cube that fits inside an -dimensional cube, with . Then(1)(2)(3)(4)(Croft et al. 1991, p. 53). For larger , little is known.

Four circles may be drawn through an arbitrary point on a torus. The first two circles are obvious: one is in the plane of the torus and the second perpendicular to it. The third and fourth circles (which are inclined with respect to the torus) are much more unexpected and are known as the Villarceau circles (Villarceau 1848, Schmidt 1950, Coxeter 1969, Melzak 1983).To see that two additional circles exist, consider a coordinate system with origin at the center of torus, with pointing up. Specify the position of by its angle measured around the tube of the torus. Define for the circle of points farthest away from the center of the torus (i.e., the points with ), and draw the x-axis as the intersection of a plane through the z-axis and passing through with the -plane. Rotate about the y-axis by an angle , where(1)In terms of the old coordinates, the new coordinates are(2)(3)So in coordinates, equation (◇) of the torus becomes(4)Expanding the left..

Draw three circles in the plane, none of which lies completely inside another, and the common external tangent lines for each pair. Then points of intersection of the three pairs of tangent lines lie on a straight line.Monge's theorem has a three-dimensional analog which states that the apexes of the cones defined by four spheres, taken two at a time, lie in a plane (when the cones are drawn with the spheres on the same side of the apex; Wells 1991).

A line segment joining the midpointsof opposite sides of a quadrilateral or tetrahedron.Varignon's theorem states that the bimedians of a quadrilateral bisect each other (left figure). In addition, the three bimedians of a tetrahedron are concurrent and bisect each other (right figure; Altshiller-Court 1979, p. 48).

An unfolding is the cutting along edges and flattening out of a polyhedron to form a net. Determining how to unfold a polyhedron into a net is tricky. For example, cuts cannot be made along all edges that surround a face or the face will completely separate. Furthermore, for a polyhedron with no coplanar faces, at least one edge cut must be made from each vertex or else the polyhedron will not flatten. In fact, the edges that must be cut corresponds to a special kind of graph called a spanning tree of the skeleton of the polyhedron (Malkevitch).In 1987, K. Fukuda conjectured that no convex polyhedra admit a self-overlapping unfolding. The top figure above shows a counterexample to the conjecture found by M. Namiki. An unfoldable tetrahedron was also subsequently found (bottom figure above). Another nonregular convex polyhedra admitting an overlapping unfolding was found by G. Valette (shown in Buekenhout and Parker 1998).Examples..

The midsphere is the sphere with respect to which the polyhedron vertices of a polyhedron are the inversion poles of the planes of the faces of the dual polyhedron (and vice versa), also called the intersphere, reciprocating sphere, or inversion sphere. The radius of the midsphere is called the midradius.The midsphere touches all polyhedron edges of a semiregular or regular polyhedron, as well as the edges of the dual of that solid (Cundy and Rollett 1989, p. 117). Note that the midsphere does not necessarily pass through the midpoints of the edges a polyhedron dual, but is rather only tangent to the edges at some point along their lengths.The figure above shows the Platonic solids and their duals, with the circumsphere of the solid, midsphere, and insphere of the dual superposed.

The radius of the midsphere of a polyhedron, also called the interradius. Let be a point on the original polyhedron and the corresponding point on the dual. Then because and are inverse points, the radii , , and satisfy(1)The above figure shows a plane section of a midsphere.Let be the inradius the dual polyhedron, circumradius of the original polyhedron, and the side length of the original polyhedron. For a regular polyhedron with Schläfli symbol , the dual polyhedron is . Then(2)(3)(4)Furthermore, let be the angle subtended by the polyhedron edge of an Archimedean solid. Then(5)(6)(7)so(8)(Cundy and Rollett 1989).For a Platonic or Archimedean solid, the midradius of the solid and dual can be expressed in terms of the circumradius of the solid and inradius of the dual gives(9)(10)and these radii obey(11)..

The sphere with respect to which inverse points are computed (i.e., with respect to which geometrical inversion is performed). For example, the cyclides are inversions in a sphere of tori. The center of the inversion sphere is called the inversion center, and its radius is called the inversion radius. When dual polyhedra are being considered, the inversion sphere is commonly called the midsphere (or intersphere, or reciprocating sphere).In two dimensions, the inversion sphere collapses to an inversioncircle.

Sangaku problems, often written "san gaku," are geometric problems of the type found on devotional mathematical wooden tablets ("sangaku") which were hung under the roofs of shrines or temples in Japan during two centuries of schism from the West (Fukagawa and Pedoe 1989). During the time of isolation, Japanese mathematicians developed their own "traditional mathematics," which, in the 1850s, began giving way to Western methods. There were also changes in the script in which mathematics was written and, as a result, few people now living know how to interpret the historic tablets (Kimberling).Japanese mathematicians represented in sangaku include Seki Kowa (1642-1708), Ajima Chokuen (also called Naonobu; 1732-1798), and Shoto Kenmotu (1790-1871).Sangaku problems typically involve mutually tangent circles or tangent spheres, with specific examples including the properties of the Ajima-Malfatti..

An auxiliary latitude which gives a sphere having correct distances along the meridians. It is denoted (or ) and is given by(1) is evaluated for at the north pole (), and is given by(2)(3)A series for is(4)and a series for is(5)where(6)The inverse formula is(7)

A polyhedron is rigid if it cannot be continuously deformed into another configuration. A rigid polyhedron may have two or more stable forms which cannot be continuously deformed into each other without bending or tearing (Wells 1991).A polyhedron that can change form from one stable configuration to another with only a slight transient nondestructive elastic stretch is called a multistable polyhedron (Goldberg 1978).A non-rigid polyhedron may be "shaky" (infinitesimally movable) or flexible. An example of a concave flexible polyhedron with 18 triangular faces was given by Connelly (1978), and a flexible polyhedron with only 14 triangular faces was subsequently found by Steffen (Mackenzie 1998).Jessen's orthogonal icosahedronis an example of a shaky polyhedron.

Although the rigidity theorem states that if the faces of a convex polyhedron are made of metal plates and the polyhedron edges are replaced by hinges, the polyhedron would be rigid, concave polyhedra need not be rigid. A nonrigid polyhedron may be "shaky" (infinitesimally movable) or flexible (continuously movable; Wells 1991).In 1897, Bricard constructed several self-intersecting flexible octahedra (Cromwell 1997, p. 239). Connelly (1978) found the first example of a true flexible polyhedron, consisting of 18 triangular faces (Cromwell 1997, pp. 242-244). Mason discovered a 34-sided flexible polyhedron constructed by erecting a pyramid on each face of a cube adjoined square antiprism (Cromwell 1997). Kuiper and Deligne modified Connelly's polyhedron to create a flexible polyhedron having 18 faces and 11 vertices (Cromwell 1997, p. 245), and Steffen found a flexible polyhedron with only 14 triangular..

By the duality principle, for every polyhedron, there exists another polyhedron in which faces and polyhedron vertices occupy complementary locations. This polyhedron is known as the dual, or reciprocal. The process of taking the dual is also called reciprocation, or polar reciprocation. Brückner (1900) was among the first to give a precise definition of duality (Wenninger 1983, p. 1).Starting with any given polyhedron, the dual of its dual is the original polyhedron.Any polyhedron can be associated with a second (abstract, combinatorial, topological) dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Even when a pair of polyhedra cannot be obtained by reciprocation, they may be called (abstract, combinatorial, or topogical) duals of each other as long as the vertices of one correspond to the faces..

The number (the cube root of 2) which is to be constructed in the cube duplication problem. This number is not a Euclidean number although it is an algebraic number of third degree.It has decimal digits 1.25992104989... (OEIS A002580).Its continued fraction is [1, 3, 1, 5, 1, 1,4, 1, 1, 8, 1, 14, 1, ...] (OEIS A002945).

What space-filling arrangement of similar cells of equal volume has minimal surface area? This questions arises naturally in the theory of foams when the liquid content is small. Kelvin (Thomson 1887) proposed that the solution was a 14-sided truncated octahedron having a very slight curvature of the hexagonal faces.The isoperimetric quotient the uncurved truncated octahedron is given by(1)(2)(3)while Kelvin's slightly curved variant has a slightly less optimal quotient of 0.757.Despite one hundred years of failed attempts and Weyl's (1952) opinion that the curved truncated octahedron could not be improved upon, Weaire and Phelan (1994) discovered a space-filling unit cell consisting of six 14-sided polyhedra and two 12-sided polyhedra with irregular faces and only hexagonal faces remaining planar. This structure has an isoperimetric quotient of 0.765, or approximately 1.0% more than Kelvin's cell.The building for water events..

The term rectification is sometimes used to refer to the determination of the length of a curve.Rectification also refers to the operation which converts the midpoints of the edges of a regular polyhedron to the vertices of the related "rectified" polyhedron. Rectified forms are bounded by a combination of rectified cells and vertex figures. Therefore, a rectified polychoron is bounded by s and s. For example, is bounded by 600 truncated tetrahedra (truncated cells) and 120 icosahedra (vertex figures). A rectified polyhedron is indicated by prepending an "r" to the Schläfli symbol.polyhedronSchläfli symbolrectified polyhedronSchläfli symboltetrahedronoctahedronoctahedroncuboctahedroncubecuboctahedronicosahedronicosidodecahedrondodecahedronicosidodecahedron16-cell24-cellRectification of the six regular polychora gives five (not six) new polychora since the rectified..

The tetrix is the three-dimensional analog of the Sierpiński sieve illustrated above, also called the Sierpiński sponge or Sierpiński tetrahedron.The th iteration of the tetrix is implemented in the Wolfram Language as SierpinskiMesh[n, 3].Let be the number of tetrahedra, the length of a side, and the fractional volume of tetrahedra after the th iteration. Then(1)(2)(3)The capacity dimension is therefore(4)(5)so the tetrix has an integer capacity dimension (which is one less than the dimension of the three-dimensional tetrahedra from which it is built), despite the fact that it is a fractal.The following illustrations demonstrate how the dimension of the tetrix can be the same as that of the plane by showing three stages of the rotation of a tetrix, viewed along one of its edges. In the last frame, the tetrix "looks" like the two-dimensional plane. ..

The maximum number of pieces into which a cylinder can be divided by oblique cuts is given by(1)(2)(3)where is a binomial coefficient.This problem is sometimes also called cake cutting or pie cutting, and has the same solution as space division by planes. For , 2, ... cuts, the maximum number of pieces is 2, 4, 8, 15, 26, 42, ... (OEIS A000125). Unsurprisingly, the numbers of this sequence are called cake numbers.

The number of regions into which space can be divided by mutually intersecting spheres isgiving 2, 4, 8, 16, 30, 52, 84, ... (OEIS A046127) for , 2, ....

The word "median" has several different meanings in mathematics all related to the "middle" of mathematical objects.The statistical median is an order statistic that gives the "middle" value of a sample. More specifically, it is the value such that an equal number of samples are less than and greater than the value (for an odd sample size), or the average of the two central values (for an even sample size). The Wolfram Language function Median[list] can be used to find the statistical median of the elements in a list.A triangle median is the Cevian from one of its vertices to the midpoint of the opposite side. The medians intersect in a point known as the triangle centroid that is sometimes also called the median point.Similarly, a tetrahedron median is a line joining a vertex of a tetrahedron to the geometric centroid of the opposite face.The median of a trapezoid is the line segment determinedby the midpoints of..

The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and iff vertex is incident upon edge (Skiena 1990, p. 135). However, some authors define the incidence matrix to be the transpose of this, with a column for each vertex and a row for each edge. The physicist Kirchhoff (1847) was the first to define the incidence matrix.The incidence matrix of a graph (using the first definition) can be computed in the Wolfram Language using IncidenceMatrix[g]. Precomputed incidence matrices for a many named graphs are given in the Wolfram Language by GraphData[graph, "IncidenceMatrix"].The incidence matrix of a graph and adjacency matrix of its line graph are related by(1)where is the identity matrix (Skiena 1990, p. 136).For a -D polytope , the incidence matrix is defined by(2)The th row shows which s surround , and the th column shows which s bound . Incidence matrices are also..

Origami is the Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual "units," each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be constructed, especially using modular origami. The images above show a number of modular polyhedral origami, together with an animated crane constructed in the Wolfram Language by L. Zamiatina.To distinguish the two directions in which paper can be folded, the notations illustrated above are conventionally used in origami. A "mountain fold" is a fold in which a peak is formed, whereas a "valley fold" is a fold forming..

In 1611, Kepler proposed that close packing (either cubic or hexagonal close packing, both of which have maximum densities of ) is the densest possible sphere packing, and this assertion is known as the Kepler conjecture. Finding the densest (not necessarily periodic) packing of spheres is known as the Kepler problem.Buckminster Fuller (1975) claimed to have a proof, but it was really a description of face-centered cubic packing, not a proof of its optimality (Sloane 1998). A second putative proof of the Kepler conjecture was put forward by W.-Y. Hsiang (Cipra 1991, Hsiang 1992, 1993, Cipra 1993), but was subsequently determined to be flawed (Conway et al. 1994, Hales 1994, Sloane 1998). According to J. H. Conway, nobody who has read Hsiang's proof has any doubts about its validity: it is nonsense.Soon thereafter, Hales (1997a) published a detailed plan describing how the Kepler conjecture might be proved using a significantly..

There are a number of algebraic equations known as the icosahedral equation, all of which derive from the projective geometry of the icosahedron. Consider an icosahedron centered , oriented with -axis along a fivefold () rotational symmetry axis, and with one of the top five edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest icosahedral equation is defined by projecting the vertices of the icosahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not the height above..

The tetrahedral equation, by way of analogy with the icosahedral equation, is a set of related equations derived from the projective geometry of the octahedron. Consider a tetrahedron centered , oriented with -axis along a fourfold () rotational symmetry axis, and with one of the top three edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest tetrahedral equation is defined by projecting the vertices of the tetrahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not..

The octahedral equation, by way of analogy with the icosahedral equation, is a set of related equations derived from the projective geometry of the octahedron. Consider an octahedron centered , oriented with -axis along a fourfold () rotational symmetry axis, and with one of the top four edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest octahedral equation is defined by projecting the vertices of the octahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not the..

There are several meanings of the word content in mathematics.The content of a polytope or other -dimensional object is its generalized volume (i.e., its "hypervolume"). Just as a three-dimensional object has volume, surface area, and generalized diameter, an -dimensional object has "measures" of order 1, 2, ..., . The content of a region can be computed in the Wolfram Language using RegionMeasure[reg].The content of an integer polynomial , denoted , is the largest integer such that also has integer coefficients. Gauss's lemma for contents states that if and are two polynomials with integer coefficients, then (Séroul 2000, p. 287).For a general univariate polynomial , the Wolfram Language command FactorTermsList[poly, x] returns a list of three elements, the first being the integer content , the second being the polynomial content, i.e., a primitive (with respect to all variables) polynomial that..

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