# Snub bitetrahedral tetracontoctachoron

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Snub bitetrahedral tetracontoctachoron
File:Snub bitetrahedral tetracontoctachoron.png
Rank4
TypeIsogonal
Notation
Bowers style acronymSebtic
Elements
Cells288 phyllic disphenoids, 192 chiral triangular antipodiums, 48 snub tetrahedra
Faces576+576 scalene triangles, 288 isosceles triangles, 192+192 triangles
Edges144+288+576+576
Vertices288
Vertex figure11-vertex polyhedron with 2 pentagons, 4 tetragons, and 4 triangles
Measures (edge length 1)
Central density1
Related polytopes
ArmySebtic
RegimentSebtic
DualHendecahedral diacosioctacontoctachoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA3●B3, order 576
ConvexYes
NatureTame

The snub bitetrahedral tetracontoctachoron or sebtic is a convex isogonal polychoron that consists of 48 snub tetrahedra, 192 chiral triangular antipodiums and 288 phyllic disphenoids. 2 snub tetrahedra, 4 chiral triangular antipodiums, and 4 phyllic disphenoids join at each vertex. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle {\sqrt {\frac {3+{\sqrt {2}}}{2}}}}$ ≈ 1:1.48563.

## Vertex coordinates

Vertex coordinates for a snub bitetrahedral tetracontoctachoron, assuming that the edge length differences are minimized, are given by all even permutations with an even number of sign changes of:

• ${\displaystyle \left({\frac {\sqrt {2-{\sqrt {2}}}}{4}},\,{\frac {\sqrt {2+{\sqrt {2}}}}{4}},\,{\frac {\sqrt {4+{\sqrt {10+4{\sqrt {2}}}}}}{4}},\,{\frac {\sqrt {12+8{\sqrt {2}}+{\sqrt {266+188{\sqrt {2}}}}}}{4}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {8-2{\sqrt {2}}-2{\sqrt {14-8{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {16+6{\sqrt {2}}+2{\sqrt {46+32{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {24+10{\sqrt {2}}+2{\sqrt {158+104{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {32+18{\sqrt {2}}+6{\sqrt {46+32{\sqrt {2}}}}}}{8}}\right),}$

as well as all even permutations with an odd number of sign changes of:

• ${\displaystyle \left({\frac {\sqrt {8-2{\sqrt {2}}+2{\sqrt {14-8{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {16-2{\sqrt {2}}+2{\sqrt {62-16{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {16+10{\sqrt {2}}+2{\sqrt {46+32{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {40+26{\sqrt {2}}+2{\sqrt {670+472{\sqrt {2}}}}}}{8}}\right).}$

Another set of coordinates for a snub bitetrahedral tetracontoctachoron, assuming that the ratio method is used, are given by all even permutations with an even number of sign changes of:

• ${\displaystyle \left({\frac {\sqrt {2-{\sqrt {2}}}}{4}},\,{\frac {\sqrt {2+{\sqrt {2}}}}{4}},\,{\frac {\sqrt {6+2{\sqrt {2}}+{\sqrt {26+16{\sqrt {2}}}}}}{4}},\,{\frac {\sqrt {26+18{\sqrt {2}}+{\sqrt {826+584{\sqrt {2}}}}}}{4}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {12+2{\sqrt {2}}-2{\sqrt {14-4{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {20+10{\sqrt {2}}+2{\sqrt {142+100{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {52+30{\sqrt {2}}+2{\sqrt {478+332{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {60+38{\sqrt {2}}+6{\sqrt {142+100{\sqrt {2}}}}}}{8}}\right),}$

as well as all even permutations with an odd number of sign changes of:

• ${\displaystyle \left({\frac {\sqrt {12+2{\sqrt {2}}+2{\sqrt {14-4{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {20+2{\sqrt {2}}+2{\sqrt {94+20{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {44+30{\sqrt {2}}+2{\sqrt {142+100{\sqrt {2}}}}}}{8}},\,{\frac {\sqrt {68+46{\sqrt {2}}+2{\sqrt {2078+1468{\sqrt {2}}}}}}{8}}\right).}$