1.
Convex uniform honeycomb
–
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. They can be considered the three-dimensional analogue to the uniform tilings of the plane, the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. 1905, Alfredo Andreini enumerated 25 of these tessellations,1991, Norman Johnsons manuscript Uniform Polytopes identified the complete list of 28. 1994, Branko Grünbaum, in his paper Uniform tilings of 3-space, also independently enumerated all 28 and he found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991, alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time. Only 14 of the uniform polyhedra appear in these patterns. This set can be called the regular and semiregular honeycombs and it has been called the Archimedean honeycombs by analogy with the convex uniform polyhedra, commonly called Archimedean solids. Recently Conway has suggested naming the set as the Architectonic tessellations, the individual honeycombs are listed with names given to them by Norman Johnson. For cross-referencing, they are given with list indices from Andreini, Williams, Johnson, and Grünbaum. Coxeter uses δ4 for a honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb. The fundamental infinite Coxeter groups for 3-space are, The C ~3, cubic, The B ~3, alternated cubic, The A ~3 cyclic group, or, There is a correspondence between all three families. Removing one mirror from C ~3 produces B ~3 and this allows multiple constructions of the same honeycombs. If cells are colored based on positions within each Wythoff construction. In addition there are 5 special honeycombs which dont have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations, the total unique honeycombs above are 18. The total unique honeycombs above are 10. Combining these counts,18 and 10 gives us the total 28 uniform honeycombs, the regular cubic honeycomb, represented by Schläfli symbol, offers seven unique derived uniform honeycombs via truncation operations. The reflectional symmetry is the affine Coxeter group, There are four index 2 subgroups that generate alternations, and +, with the first two generated repeated forms, and the last two are nonuniform. The B ~4, group offers 11 derived forms via truncation operations, There are 3 index 2 subgroups that generate alternations, and +

2.
Cube
–
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors

3.
Cuboctahedron
–
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2

4.
Deltoidal icositetrahedron
–
In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron, the short and long edges of each kite are in the ratio 1, ≈1,1.292893. The shape is called a trapezohedron in mineral contexts, although in solid geometry that name has another meaning. The deltoidal icositetrahedron has three positions, all centered on vertices, The great triakis octahedron is a stellation of the deltoidal icositetrahedron. The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants and it can also be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a cube or octahedron, in crystallography a rotational variation is called a dyakis dodecahedron or diploid. The deltoidal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and this polyhedron is topologically related as a part of sequence of deltoidal polyhedra with face figure, and continues as tilings of the hyperbolic plane. These face-transitive figures have reflectional symmetry, deltoidal hexecontahedron Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube. The Haunter of the Dark, a story by H. P, lovecraft, whose plot involves this figure Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, deltoidal Icositetrahedron – Interactive Polyhedron model

5.
Disdyakis dodecahedron
–
In geometry, a disdyakis dodecahedron, or hexakis octahedron or kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons, more formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron. Its collective edges represent the reflection planes of the symmetry and it can also be seen in the corner and mid-edge triangulation of the regular cube and octahedron, and rhombic dodecahedron. Seen in stereographic projection the edges of the dodecahedron form 9 circles in the plane. Between a polyhedron and its dual, vertices and faces are swapped in positions, the disdyakis dodecahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron. It is a polyhedra in a sequence defined by the face configuration V4.6. 2n, with an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. Each face on these domains also corresponds to the domain of a symmetry group with order 2,3, n mirrors at each triangle face vertex. First stellation of rhombic dodecahedron Disdyakis triacontahedron Kisrhombille tiling Great rhombihexacron—A uniform dual polyhedron with the surface topology Williams. The Geometrical Foundation of Natural Structure, A Source Book of Design, the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Disdyakis dodecahedron at MathWorld

6.
Octahedron
–
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size

7.
Pentagonal icositetrahedron
–
In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron is a Catalan solid which is the dual of the snub cube. In crystallography it is called a gyroid. It has two forms, which are mirror images of each other. Denote the tribonacci constant by t, approximately 1.8393, then the pentagonal faces have four angles of cos−1 ≈114. 8° and one angle of cos−1 ≈80. 75°. The pentagon has three edges of unit length each, and two long edges of length t + 1/2 ≈1.42. The acute angle is between the two long edges and this polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations. These face-transitive figures have rotational symmetry, the pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3. n. The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube, the Geometrical Foundation of Natural Structure, A Source Book of Design

8.
Regular dodecahedron
–
In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does

9.
Regular icosahedron
–
In geometry, a regular icosahedron is a convex polyhedron with 20 faces,30 edges and 12 vertices. It is one of the five Platonic solids, and also the one with the most sides and it has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol, or sometimes by its vertex figure as 3.3.3.3.3 or 35 and it is the dual of the dodecahedron, which is represented by, having three pentagonal faces around each vertex. A regular icosahedron is a pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedrons or icosahedra. The surface area A and the volume V of a regular icosahedron of edge length a are, note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length √5 − 1/2 = 1/ϕ = ϕ −1, the 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. The locations of the vertices of a regular icosahedron can be described using spherical coordinates, if two vertices are taken to be at the north and south poles, then the other ten vertices are at latitude ±arctan ≈ ±26. 57°. These ten vertices are at evenly spaced longitudes, alternating between north and south latitudes and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, an icosahedron has 43,380 distinct nets. To color the icosahedron, such that no two adjacent faces have the color, requires at least 3 colors. A problem dating back to the ancient Greeks is to determine which of two shapes has larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere, the problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers, as it turns out, the icosahedron occupies less of the spheres volume than the dodecahedron. The following construction of the icosahedron avoids tedious computations in the number field ℚ necessary in more elementary approaches, the existence of the icosahedron amounts to the existence of six equiangular lines in ℝ3. Indeed, intersecting such a system of lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of an icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system. In order to such an equiangular system, we start with this 6 ×6 square matrix