# Snub cubic antiprism

Snub cubic antiprism
File:Snub cubic antiprism.png
Rank4
TypeIsogonal
Notation
Bowers style acronymSniccap
Coxeter diagrams2s4s3s ()
Elements
Cells48 irregular tetrahedra, 12 rhombic disphenoids, 8 triangular gyroprisms, 6 square gyroprisms, 2 snub cubes
Faces48+48+48+48 scalene triangles, 16 triangles, 12 squares
Edges24+24+24+24+48+48
Vertices48
Vertex figureTriangular-pentagonal gyrowedge
Measures (as derived from unit-edge great rhombicuboctahedral prism)
Edge lengthsDiagonals of original squares (24+24+24+24): ${\displaystyle {\sqrt {2}}\approx 1.41421}$
Edges of equilateral triangles (48): ${\displaystyle {\sqrt {3}}\approx 1.73205}$
Edges of squares (48): ${\displaystyle {\sqrt {2+{\sqrt {2}}}}\approx 1.84776}$
Circumradius${\displaystyle {\sqrt {\frac {7+3{\sqrt {2}}}{2}}}\approx 2.37093}$
Central density1
Related polytopes
ArmySniccap
RegimentSniccap
DualPentagonal icositetrahedral antitegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(B3×A1)+, order 48
ConvexYes
NatureTame

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:${\displaystyle {\frac {\sqrt {5}}{2}}}$ ≈ 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:

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

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:

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

where

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

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:

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

where

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

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

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

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