The digonal-octagonal tetraprismantiprismoid is a convex isogonal polychoron that consists of 8 rectangular gyroprisms , 16 digonal-rectangular gyrowedges , 8 rhombic disphenoids , and 32 phyllic disphenoids of two kinds. 1 rhombic disphenoid, 4 phyllic disphenoids, 2 rectangular gyroprisms, and 4 digonal-rectangular gyrowedges join at each vertex. It can be obtained as a subsymmetrical faceting of the octagonal-dioctagonal duoprism . However, it cannot be made scaliform.
Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:
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{\displaystyle {\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{2}}}
≈ 1:2.60607.
The vertices of a digonal-octagonal tetraprismantiprismoid, assuming that the edge length differences are minimized, using the ratio method, are given by all even permutations of the first two coordinates of:
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{\displaystyle \pm \left({\frac {1}{2}},\,{\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{4}},\,0,\,{\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{4}}\right),}
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{\displaystyle \pm \left({\frac {1}{2}},\,-{\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{4}},\,0,\,{\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{4}}\right),}
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{\displaystyle \pm \left({\frac {2-{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,{\frac {2+3{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,{\frac {2+{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,{\frac {2+{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}}\right),}
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{\displaystyle \pm \left({\frac {2+3{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,{\frac {2-{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,{\frac {2+{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,{\frac {2+{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}}\right),}
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{\displaystyle \pm \left({\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{4}},\,{\frac {1}{2}},\,{\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{4}},\,0\right),}
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{\displaystyle \pm \left({\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{4}},\,-{\frac {1}{2}},\,{\frac {1+{\sqrt {2}}+{\sqrt {5+2{\sqrt {2}}}}}{4}},\,0\right),}
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{\displaystyle \pm \left({\frac {2+3{\sqrt {2}}+{\sqrt {10+4{\sqrt {5}}}}}{8}},\,-{\frac {2-{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,{\frac {2+{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,-{\frac {2+{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}}\right),}
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{\displaystyle \pm \left({\frac {2-{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,-{\frac {2+3{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,{\frac {2+{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}},\,-{\frac {2+{\sqrt {2}}+{\sqrt {10+4{\sqrt {2}}}}}{8}}\right).}