The square double antiprismoid or squidiap is a convex isogonal polychoron and the third member of the double antiprismoid family. It consists of 16 square antiprisms , 64 tetragonal disphenoids , and 128 sphenoids . 2 square antiprisms, 4 tetragonal disphenoids, and 8 sphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal square-square duoantiprisms or by alternating the octagonal ditetragoltriate . However, it cannot be made uniform. It is the first in an infinite family of isogonal square antiprismatic swirlchora .
Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:
6
−
2
2
{\displaystyle {\frac {\sqrt {6-{\sqrt {2}}}}{2}}}
≈ 1:1.07072. For this variant the edges of the squares of the inscribed duoantiprisms have ratio 1:
2
+
2
2
{\displaystyle {\frac {2+{\sqrt {2}}}{2}}}
≈ 1:1.70711. A variant with uniform square antiprisms also exists; this variant is based on a duoantiprism based on squares with edge length ratio 1:
1
+
2
{\displaystyle {\sqrt {1+{\sqrt {2}}}}}
≈ 1:1.55377.
The vertices of a square double antiprismoid, assuming that the square antiprisms are uniform of edge length 1, centered at the origin, are given by:
(
0
,
±
2
2
,
0
,
±
1
+
2
2
)
,
{\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\sqrt {\frac {1+{\sqrt {2}}}{2}}}\right),}
(
0
,
±
2
2
,
±
1
+
2
2
,
0
)
,
{\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\sqrt {\frac {1+{\sqrt {2}}}{2}}},\,0\right),}
(
±
2
2
,
0
,
0
,
±
1
+
2
2
)
,
{\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,\pm {\sqrt {\frac {1+{\sqrt {2}}}{2}}}\right),}
(
±
2
2
,
0
,
±
1
+
2
2
,
0
)
,
{\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\sqrt {\frac {1+{\sqrt {2}}}{2}}},\,0\right),}
(
±
1
2
,
±
1
2
,
±
1
+
2
2
,
±
1
+
2
2
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {2}}}}{2}}\right),}
(
0
,
±
1
+
2
2
,
±
1
2
,
±
1
2
)
,
{\displaystyle \left(0,\,\pm {\sqrt {\frac {1+{\sqrt {2}}}{2}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}
(
±
1
+
2
2
,
0
,
±
1
2
,
±
1
2
)
,
{\displaystyle \left(\pm {\sqrt {\frac {1+{\sqrt {2}}}{2}}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}
(
±
1
+
2
2
,
±
1
+
2
2
,
0
,
±
2
2
)
,
{\displaystyle \left(\pm {\frac {\sqrt {1+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {2}}}}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}}\right),}
(
±
1
+
2
2
,
±
1
+
2
2
,
±
2
2
,
0
)
.
{\displaystyle \left(\pm {\frac {\sqrt {1+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right).}
An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:
(
0
,
±
2
2
,
0
,
±
1
+
2
2
)
,
{\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {1+{\sqrt {2}}}{2}}\right),}
(
0
,
±
2
2
,
±
1
+
2
2
,
0
)
,
{\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,0\right),}
(
±
2
2
,
0
,
0
,
±
1
+
2
2
)
,
{\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,\pm {\frac {1+{\sqrt {2}}}{2}}\right),}
(
±
2
2
,
0
,
±
1
+
2
2
,
0
)
,
{\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {1+{\sqrt {2}}}{2}},\,0\right),}
(
±
1
2
,
±
1
2
,
±
2
+
2
4
,
±
2
+
2
4
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {2}}}{4}},\,\pm {\frac {2+{\sqrt {2}}}{4}}\right),}
(
0
,
±
1
+
2
2
,
±
1
2
,
±
1
2
)
,
{\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}
(
±
1
+
2
2
,
0
,
±
1
2
,
±
1
2
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}
(
±
2
+
2
4
,
±
2
+
2
4
,
0
,
±
2
2
)
,
{\displaystyle \left(\pm {\frac {2+{\sqrt {2}}}{4}},\,\pm {\frac {2+{\sqrt {2}}}{4}},\,0,\,\pm {\frac {\sqrt {2}}{2}}\right),}
(
±
2
+
2
4
,
±
2
+
2
4
,
±
2
2
,
0
)
.
{\displaystyle \left(\pm {\frac {2+{\sqrt {2}}}{4}},\,\pm {\frac {2+{\sqrt {2}}}{4}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right).}