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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)...

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.

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..

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).

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)..

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 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..

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..

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).

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..

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..

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.

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 .

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)..

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

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..

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).

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 .

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..

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" 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 ...

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)

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).

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..

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,..

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.

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.

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

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)

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..

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..

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..

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..

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..

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..

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)..

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)

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..

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)

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..

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..

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..

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)

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)

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)

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)..

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)..

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)

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)..

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..

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..

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)

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..

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,..

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" 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]..

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,..

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..

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..

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"..

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..

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..

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 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,..

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.

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.

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)

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..

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).

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..

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..

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..

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 .

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...

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..

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.

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

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).

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).

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..

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,..

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..

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 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..

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..

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..

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..

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