Snub tetracontoctachoron

(Redirected from Snub icositetrachoron)
Snub tetracontoctachoron
Rank4
TypeIsogonal
Notation
Bowers style acronymSnoc
Coxeter diagrams3s4s3s ()
Elements
Cells576 phyllic disphenoids, 192 triangular gyroprisms, 48 snub cubes
Faces384 triangles, 1152+1152 scalene triangles, 144 squares
Edges288+576+576+1152
Vertices576
Vertex figurePolyhedron with 2 pentagons, 2 tetragons, and 4 triangles
Measures (as derived from unit-edge great prismatotetracontoctachoron)
Edge lengthsEdges from diagonals of original squares (288+576): ${\displaystyle {\sqrt {2}}\approx 1.41421}$
Edges of equilateral triangles (1152): ${\displaystyle {\sqrt {3}}\approx 1.73205}$
Edges of squares (576): ${\displaystyle {\sqrt {2+{\sqrt {2}}}}\approx 1.84776}$
Circumradius${\displaystyle {\sqrt {14+9{\sqrt {2}}}}\approx 5.16991}$
Central density1
Related polytopes
ArmySnoc
RegimentSnoc
DualEnneahedral pentacosiheptacontahexachoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(F4×2)+, order 1152
ConvexYes
NatureTame

The snub tetracontoctachoron, or snoc, also commonly called the snub icositetrachoron or omnisnub 24-cell, is a convex isogonal polychoron that consists of 48 snub cubes, 192 triangular gyroprisms, and 576 phyllic disphenoids. 2 snub cubes, 2 triangular gyroprisms, and 4 disphenoids join at each vertex. It can be obtained through the process of alternating the great prismatotetracontoctachoron. However, it cannot be made uniform, as it generally has 4 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:${\displaystyle {\frac {\sqrt {4+2{\sqrt {2{\sqrt {2}}-2}}}}{2}}}$ ≈ 1:1.20627.

This polychoron generally has a doubled symmetry. A variant without this extended symmetry also exists, known as the snub icositetrachoron.

Vertex coordinates

Vertex coordinates for a snub tetracontoctachoron, created from the vertices of a great prismatotetracontoctachoron 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 of:

• ${\displaystyle \left({\frac {5+3{\sqrt {2}}}{2}},\,{\frac {1+2{\sqrt {2}}}{2}},\,{\frac {1+{\sqrt {2}}}{2}},\,{\frac {1}{2}}\right),}$
• ${\displaystyle \left(3{\frac {1+{\sqrt {2}}}{2}},\,{\frac {3+2{\sqrt {2}}}{2}},\,{\frac {3+{\sqrt {2}}}{2}},\,{\frac {1}{2}}\right),}$
• ${\displaystyle \left({\frac {4+3{\sqrt {2}}}{2}},\,(1+{\sqrt {2}}),\,{\frac {2+{\sqrt {2}}}{2}},\,1\right).}$

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 of:

• ${\displaystyle \left({\frac {\sqrt {4-2{\sqrt {2}}}}{4}},\,{\frac {\sqrt {4+2{\sqrt {2}}}}{4}},\,\pm {\frac {2+{\sqrt {4+2{\sqrt {2}}}}}{4}},\,{\frac {2+2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{4}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {2}}{4}},\,{\frac {{\sqrt {2}}+{\sqrt {8+4{\sqrt {2}}}}}{4}},\,{\frac {2+{\sqrt {2}}+{\sqrt {8+4{\sqrt {2}}}}}{4}},\,{\frac {2+{\sqrt {2}}+{\sqrt {16+8{\sqrt {2}}}}}{4}}\right),}$
• ${\displaystyle \left({\frac {{\sqrt {2}}+{\sqrt {4-2{\sqrt {2}}}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}},\,{\frac {2+{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{4}},\,{\frac {2+{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}}\right),}$

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

• ${\displaystyle \left({\frac {\sqrt {2{\sqrt {2}}-2}}{4}},\,{\frac {\sqrt {2+2{\sqrt {2}}}}{4}},\,{\frac {2+{\sqrt {2+2{\sqrt {2}}}}}{4}},\,{\frac {2+2{\sqrt {2}}+{\sqrt {14+10{\sqrt {2}}}}}{4}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {2}}{4}},\,{\frac {{\sqrt {2}}+2{\sqrt {1+{\sqrt {2}}}}}{4}},\,{\frac {2+{\sqrt {2}}+2{\sqrt {1+{\sqrt {2}}}}}{4}},\,{\frac {2+{\sqrt {2}}+2{\sqrt {2+2{\sqrt {2}}}}}{4}}\right),}$
• ${\displaystyle \left({\frac {{\sqrt {2}}+{\sqrt {2{\sqrt {2}}-2}}}{4}},\,{\frac {{\sqrt {2}}+{\sqrt {2+2{\sqrt {2}}}}}{4}},\,{\frac {2+{\sqrt {2}}+{\sqrt {14+10{\sqrt {2}}}}}{4}},\,{\frac {2+{\sqrt {2}}+{\sqrt {2+2{\sqrt {2}}}}}{4}}\right),}$

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