Square double prismantiprismoid
Rank 4 Type Isogonal Elements Cells 32 tetragonal disphenoids , 64 digonal disphenoids , 128 isosceles trapezoidal pyramids , 16 square prisms , 32 rectangular trapezoprisms , 16 square antiprisms Faces 128+128 isosceles triangles , 256 scalene triangles , 32 squares , 64 rectangles , 128 isosceles trapezoids Edges 64+128+128+256 Vertices 128 Vertex figure Laterobietrakis digonal-isosceles trapezoidal notch Measures (based on variant with uniform square prisms and antiprisms of edge length 1) Circumradius
5
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2
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14
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10
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2
≈
2.46306
{\displaystyle {\sqrt {\frac {5+2{\sqrt {2}}+{\sqrt {14+10{\sqrt {2}}}}}{2}}}\approx 2.46306}
Central density 1 Related polytopes Dual Square double tegmantitegmoid Abstract & topological properties Euler characteristic 0 Orientable Yes Properties Symmetry I2 (8)≀(S2 /2) , order 256Convex Yes Nature Tame
The square double prismantiprismoid is a convex isogonal polychoron and the third member of the double prismantiprismoid family. It consists of 16 square antiprisms , 16 square prisms , 32 rectangular trapezoprisms , 128 isosceles trapezoidal pyramids , 32 tetragonal disphenoids , and 64 digonal disphenoids . 1 square antiprism, 1 square prism, 2 rectangular trapezoprisms, 5 isosceles trapezoidal pyramids, 1 tetragonal disphenoid, and 2 didgonal disphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal square-octagonal prismantiprismoids . However, it cannot be made scaliform.
A variant with uniform square antiprisms and regular cubes can be vertex-inscribed into a bitruncatotetracontoctachoron . Another variant can be vertex-inscribed into a biambotetracontoctachoron .
The vertices of a square double prismantiprismoid, assuming that the square antiprisms and square prisms are uniform of edge length 1, centered at the origin, are given by:
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {2+2{\sqrt {2}}}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {2+2{\sqrt {2}}}}}{2}},\,\pm {\frac {1}{2}}\right),}
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{\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {1+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {1+{\sqrt {2}}}}}{2}}\right),}
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{\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {1+{\sqrt {1+{\sqrt {2}}}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {1+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {1+{\sqrt {2}}}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {1+{\sqrt {1+{\sqrt {2}}}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {2+2{\sqrt {2}}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {2}}+{\sqrt {2+2{\sqrt {2}}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {1+{\sqrt {2}}}}}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {2}}+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {1+{\sqrt {1+{\sqrt {2}}}}}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {2}}+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {1+{\sqrt {1+{\sqrt {2}}}}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right).}
