Snub cubic antiprism

Snub cubic antiprism
File:Snub cubic antiprism.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Sniccap
Coxeter diagram	s2s4s3s ()
Elements
Cells	48 irregular tetrahedra, 12 rhombic disphenoids, 8 triangular gyroprisms, 6 square gyroprisms, 2 snub cubes
Faces	48+48+48+48 scalene triangles, 16 triangles, 12 squares
Edges	24+24+24+24+48+48
Vertices	48
Vertex figure	Triangular-pentagonal gyrowedge
Measures (as derived from unit-edge great rhombicuboctahedral prism)
Edge lengths	Diagonals of original squares (24+24+24+24): ${\sqrt {2}}\approx 1.41421$
	Edges of equilateral triangles (48): ${\sqrt {3}}\approx 1.73205$
	Edges of squares (48): ${\sqrt {2+{\sqrt {2}}}}\approx 1.84776$
Circumradius	${\sqrt {\frac {7+3{\sqrt {2}}}{2}}}\approx 2.37093$
Central density	1
Related polytopes
Army	Sniccap
Regiment	Sniccap
Dual	Pentagonal icositetrahedral antitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B3×A1)+, order 48
Convex	Yes
Nature	Tame

The snub cubic antiprism, omnisnub cubic antiprism, or sniccap, also known as the alternated great rhombicuboctahedral prism, is a convex isogonal polychoron that consists of 2 snub cubes, 6 square gyroprisms, 8 triangular gyroprisms, 12 rhombic disphenoids, and 48 irregular tetrahedra. 4 tetrahedra and one each of the other 4 types of cells join at each vertex. It can be obtained through the process of alternating the great rhombicuboctahedral prism. However, it cannot be made uniform, as it generally has 6 edge lengths, which can be minimized to no fewer than 3 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {5}}{2}}$ ≈ 1:1.11803.

Vertex coordinates

Vertex coordinates for a snub cubic antiprism, created from the vertices of a great rhombicuboctahedral prism of edge length 1, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes in the first three coordinates, of:

• $\left({\frac {1+2{\sqrt {2}}}{2}},\,{\frac {1+{\sqrt {2}}}{2}},\,{\frac {1}{2}},\,{\frac {1}{2}}\right),$
• $\left({\frac {1+{\sqrt {2}}}{2}},\,{\frac {1+2{\sqrt {2}}}{2}},\,{\frac {1}{2}},\,-{\frac {1}{2}}\right),$

A variant with uniform snub cubes as bases is given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes excluding the last coordinate of:

• $\left(c_{1},\,c_{2},\,c_{3},\,c_{4}\right),$
• $\left(c_{2},\,c_{1},\,c_{3},\,-c_{4}\right),$

where

• $c_{1}={\sqrt {{\frac {1}{12}}\left(4-{\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{17-3{\sqrt {33}}}}\right)}},$
• $c_{2}={\sqrt {{\frac {1}{12}}\left(2+{\sqrt[{3}]{17+3{\sqrt {33}}}}+{\sqrt[{3}]{17-3{\sqrt {33}}}}\right)}},$
• $c_{3}={\sqrt {{\frac {1}{12}}\left(4+{\sqrt[{3}]{199+3{\sqrt {33}}}}+{\sqrt[{3}]{199-3{\sqrt {33}}}}\right)}}.$
• $c_{4}={\sqrt {{\frac {1}{12}}\left(-2+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}\right)}}.$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes excluding the last coordinate of:

• $\left(c_{1},\,c_{2},\,c_{3},\,c_{4}\right),$
• $\left(c_{2},\,c_{1},\,c_{3},\,-c_{4}\right),$

where

• $c_{1}={\text{root}}(9472x^{8}-5504x^{6}+1136x^{4}-98x^{2}+3,6)\approx 0.3135135258027234561506493,$
• $c_{2}={\text{root}}(14208x^{8}-19840x^{6}+7432x^{4}-679x^{2}+8,7)\approx 0.6542869462841313854118897,$
• $c_{3}={\text{root}}(14208x^{8}-20800x^{6}+3640x^{4}-159x^{2}+2,8)\approx 1.1264771628934748152619404,$
• $c_{4}={\text{root}}(18944x^{8}-7936x^{6}+1120x^{4}-60x^{2}+1,7)\approx 0.3894987408692678350179804,$

which has rhombic disphenoids (via the absolute value method), or

• $\left({\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {10}}{5}},\,{\frac {\sqrt {15+10{\sqrt {2}}}}{5}},\,{\frac {\sqrt {15}}{10}}\right),$
• $\left({\frac {\sqrt {10}}{5}},\,{\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {15+10{\sqrt {2}}}}{5}},\,-{\frac {\sqrt {15}}{10}}\right),$

where the ratio of the largest edge length to the smallest edge length is lowest (via the ratio method).

External links

• Klitzing, Richard. "sniccap".

• Wikipedia Contributors. "Omnisnub cubic antiprism".