User:Harry Princeton/Catalan Solid and Stereohedron

Summary of Catalan Solid and Stereohedron results.

Catalan Solid

Dual Solids and Faces

Main article: Catalan solid § List of Catalan solids and their duals

Remaining dual Platonic solids, and dual Johnson solids allowing a midradius, are drawn below to scale with original solid edges superimposed. Solids in each group are sorted by midradius in descending order. Below each solid is its non-Catalan face:

General Dual Faces

Main article: Catalan solid § Geometry

Furthermore, we have for 3.4.5.6 that 60+90+108+120=378>360 degrees (hyperbolic face), so no face can have regular polygons of four or more different covertex types. Hence every dihedral angle, therefore midradius, thus dual face, can be explicitly computed using the above equations. Some other dual faces are drawn below with determined side lengths, angles, and incircles (even the spherical area of such faces can be computed); although they cannot form convex dual solids:

Other dual facets with determined side lengths, angles, and incircles; although they cannot form convex dual solids (same scale as above).

Surface Area, Volume, Sphericity

The surface areas, volumes, and sphericities of all Archimedean solids and Catalan solids with unit edge length (plus for some general dual faces):

Surface Area, Volume, Sphericity

Vertex	Archimedean			Catalan
	Surface	Volume	Sphericity	Surface	Volume	Sphericity
(3.4)2	9.46410	2.35702	0.904997	9.54594	2.38649	0.904700
3.62	12.12436	2.71058	0.775413	17.90977	5.72756	0.864385
34.4	19.85641	7.88948	0.965196	19.29941	7.44740	0.955601
3.43	21.46410	8.71405	0.954080	21.51345	8.75069	0.954558
4.62	26.78461	11.31371	0.909918	30.18692	14.31891	0.944652
(3.5)2	29.30598	13.83553	0.951024	30.33814	14.80021	0.960890
3.82	32.43466	13.59966	0.849494	42.69177	23.31371	0.924445
34.5	55.28674	37.61665	0.982011	55.28053	37.58842	0.981630
3.4.5.4	59.03598	41.61532	0.979237	59.76740	42.25537	0.981615
4.6.8	61.75517	41.79899	0.943166	67.42485	49.66382	0.969075
5.62	72.60725	55.28773	0.966622	75.56554	59.87641	0.979484
3.102	100.99076	85.03966	0.726012	115.56969	111.14947	0.967338
4.6.10	174.29203	206.80340	0.970313	183.19555	228.17899	0.985719
5.6.7	272.21009	415.76248	0.989639	275.83563	426.89674	0.993991
3.7.41	13499.641	145654.04	0.991692	13725.202	150553.97	0.997149

Stereohedra

Main article: Stereohedron

All possible tangential stereohedra which have inspheres of radius 1/2 are shown below. They are dual to vertex figures of honeycombs by Platonic solids, Archimedean solids, and Johnson solids.

Nondegenerate Duals

• The barn pentagonal prism is dual to two honeycombs, the gyroelongated triangular prismatic honeycomb, and the elongated triangular prismatic honeycomb.
• The pentagonal bifastigaum is the dual of the gyrated triangular prismatic honeycomb, and is topologically identical to the elongated gyrobifastigaum. In fact, the gyrobifastigaum honeycomb is degenerately self-dual.
• The prismatic decahedron is half a hexagonal prism and half a trapezo-rhombic dodecahedron and is dual to the gyroelongated-alternated-cubic and elongated-alternated-cubic honeycombs.
• Both bifastigaums in the prism category are formed by the vertex figures when rotating one prismatic slab of a k-uniform tiling with respect to the other in 3-D space.
• The skew quadrilateral and tie kite prisms are analogous in the planigon case.
• The quarter cubic honeycomb can be split into 4 sets of parallel planes with trihexagonal tiling cross-section. Such slabs can be gyrated 60 degrees with respect to one another (vertex figure is trapezo), or they can be elongated with trihexagonal prism slabs (vertex figure is elongated).

Degenerate Duals

The remaining stereohedra are degenerate duals because some vertices extend to the base of the corresponding Johnson solids (square pyramid and triangular cupola).

Strict Combinations

• Octahedron√2+Oblate is the degenerate dual of the elongated square bipyramid honeycomb, and is mentioned at the last line here. When superimposed, the edges of the dual regular octahedra meet the base edges of the triangular faces of the elongated square bipyramids. The regular octahedra have inspheres of radius 1/√6 and the oblate octahedra don't have inspheres.
• Tetrahedron·2+Antiprism is the degenerate dual of the triakis truncated tetrahedral honeycomb, and is mentioned at the last line here. It replaces each degree-4 vertex of the rhombohedral honeycomb with a regular tetrahedron. When superimposed, the edges of the dual regular tetrahedra lay outside the triangular faces of the triakis truncated tetrahedra. The regular tetrahedra also have inspheres of radius 1/√6 and the antiprisms don't have inspheres.

Oblate Pyramidille

• Twelve oblate pyramids can be inset into twelve rhombic frustums plus one rhombic dodecahedron replacing the cuboctahedral co-vertex (e.g. the oblate pyramid is truncated into the rhombic frustum).
• The slant elongated [OP] is half an oblate pyramid and half an equilateral triangular prism. This is because that the cantic cubic honeycomb can be split in into 4 sets of parallel planes whose projection is the regular hexagonal tiling, inducing wallpaper group p3m1. The cuboctahedra are split into two triangular cupolae, and we add hexagonal prism slabs between the layers.
• This is similar for the slant elongated truncated [OP], where the triangular cupolae are dissected and between layers we add sets of six triangular prisms.
• The trapezoidal pyramid is not degenerate, and it has a triangular orthobicupola at its apex. This is the result of gyrating one cantic cubic slab 60 degrees with respect to another across such a plane and gyrating the cuboctahedra into triangular orthobicupolae. This pyramid can be truncated to the trapezoidal frustum.

Augmentations

• All cuboctahedra can be degenerately dissected into eight tetrahedra and four square pyramids, and hence the rhombic dodecahedron can be augmented at non-adjacent rhombic faces. In fact, all such augmentations are either truncations of the square bipyramid or its slanted gyro version at cuboctahedral co-vertices (gyrated 90 degrees with respect to a height plane containing a square diagonal).
• The prismatic decahedron can also be augmented (detruncated) at the non-adjacent right trapezoidal faces, in the same manner from the slant elongated truncated [OP] to the slant elongated [OP]. A special case, upper+2*side' (rightmost augmentation) is the dual of the s₃{2,6,3} scaliform honeycomb.
• The triangular orthobicupola has a similar dissection, and hence the trapezo-rhombic dodecahedron can have augmentations at non-adjacent trapezoidal faces.

Miscellaneous

• The [ESB] is half a cube and half a square bipyramid.
• The prismatic nonahedron is the degenerate dual of the prismatic stacks 3s₄{4,4,2,∞} honeycomb, and it and the [ESB] have a half-variation.
• The symmetric-skew square bipyramid corresponds to the dissection of the cuboctahedra in the cantellated cubic honeycomb.
• The elongated square pyramid is half a pyramid and half a kisquadrille prism.

Architectonic and Catoptric Tessellations

Main article: Architectonic and catoptric tessellation

The image below shows the convex uniform honeycomb corresponding to the dual cell that subdivides the cube or rhombic dodecahedron, with Tomruen's version on the right. Moreover, the left image shows the Conway operations and side lengths of the cubes when the uniform honeycombs have unit edge length.

The image below shows the superposition of the architectonic cells over the catoptric cell (3D vertex figure) along with its translation cell. The vertex figures are also included for reference.