Acronym | ... |
Name | 4D corner hypercubera |
Circumradius | 1 |
Dual | combinatorically self-dual |
Face vector | 12, 31, 31, 12 |
Confer |
The name cubera is a portmanteau of the words 'cube' and 'chimera'. The 3D cubera achieves its self-duality as the result of being, essentially, half cube and half octahedron (the dual polyhedron of a cube). In fact it has three square facets joined at a vertex (the cube 'half') and four triangles, with three of them adjoining a central triangle (the octahedron 'half').
Dan Moore found a general dimensional analogy thereof in its sense of an hypercubical-corner-simplex atop full-hypercube. Moreover, as also proved by him here, those (corner) hypercubera happen to be combinatorically dual in general!
Incidence matrix according to Dynkin symbol
oq3oo&#x || oqoo3ooqo&#xt → height = 1 (where: x = 1, q = sqrt(2) = 1.414214, h = sqrt(3) = 1.732051) o.3o. .... .... | 1 * * * * * ♦ 3 1 0 0 0 0 0 0 0 | 3 3 0 0 0 0 0 0 0 | 1 3 0 0 0 0 verf: q-tet .o3.o .... .... | * 3 * * * * | 1 0 2 1 2 1 0 0 0 | 2 1 1 2 2 2 2 0 0 | 1 2 1 2 1 0 .. .. o...3o... | * * 1 * * * | 0 1 0 0 0 0 3 0 0 | 0 3 0 0 0 0 0 3 0 | 0 3 0 0 0 1 .. .. .o..3.o.. | * * * 3 * * | 0 0 0 1 0 0 1 2 0 | 0 1 0 0 0 2 0 2 1 | 0 2 0 0 1 1 .. .. ..o.3..o. | * * * * 3 * | 0 0 0 0 2 0 0 2 1 | 0 0 0 1 0 2 2 1 2 | 0 1 0 1 2 1 .. .. ...o3...o | * * * * * 1 ♦ 0 0 0 0 0 3 0 0 3 | 0 0 0 0 3 0 6 0 3 | 0 0 1 3 3 1 verf: q-oct -------------------------+-------------+-------------------+-------------------+------------ oo3oo&#x .... .... | 1 1 0 0 0 0 | 3 * * * * * * * * | 2 1 0 0 0 0 0 0 0 | 1 2 0 0 0 0 x o.3o. || o...3o... | 1 0 1 0 0 0 | * 1 * * * * * * * | 0 3 0 0 0 0 0 0 0 | 0 3 0 0 0 0 x .q .. .... .... | 0 2 0 0 0 0 | * * 3 * * * * * * | 1 0 1 1 1 0 0 0 0 | 1 1 1 1 0 0 q .o3.o || .o..3.o.. | 0 1 0 1 0 0 | * * * 3 * * * * * | 0 1 0 0 0 2 0 0 0 | 0 2 0 0 1 0 x .o3.o || ..o.3..o. | 0 1 0 0 1 0 | * * * * 6 * * * * | 0 0 0 1 0 1 1 0 0 | 0 1 0 1 1 0 q .o3.o || ...o3...o | 0 1 0 0 0 1 | * * * * * 3 * * * | 0 0 0 0 2 0 2 0 0 | 0 0 1 2 1 0 h .. .. oo..3oo..&#x | 0 0 1 1 0 0 | * * * * * * 3 * * | 0 1 0 0 0 0 0 2 0 | 0 2 0 0 0 1 x .. .. .oo.3.oo.&#x | 0 0 0 1 1 0 | * * * * * * * 6 * | 0 0 0 0 0 1 0 1 1 | 0 1 0 0 1 1 x .. .. ..oo3..oo&#x | 0 0 0 0 1 1 | * * * * * * * * 3 | 0 0 0 0 0 0 2 0 2 | 0 0 0 1 2 1 x -------------------------+-------------+-------------------+-------------------+------------ oq ..&#x .... .... | 1 2 0 0 0 0 | 2 0 1 0 0 0 0 0 0 | 3 * * * * * * * * | 1 1 0 0 0 0 oo3oo&#x || oo..3oo..&#x | 1 1 1 1 0 0 | 1 1 0 1 0 0 1 0 0 | * 3 * * * * * * * | 0 2 0 0 0 0 .q3.o .... .... | 0 3 0 0 0 0 | 0 0 3 0 0 0 0 0 0 | * * 1 * * * * * * | 1 0 1 0 0 0 .q .. || ..o.3..o. | 0 2 0 0 1 0 | 0 0 1 0 2 0 0 0 0 | * * * 3 * * * * * | 0 1 0 1 0 0 .q .. || ...o3...o | 0 2 0 0 0 1 | 0 0 1 0 0 2 0 0 0 | * * * * 3 * * * * | 0 0 1 1 0 0 .o3.o || .oo.3.oo.&#x | 0 1 0 1 1 0 | 0 0 0 1 1 0 0 1 0 | * * * * * 6 * * * | 0 1 0 0 1 0 .o3.o || ..oo3..oo&#x | 0 1 0 0 1 1 | 0 0 0 0 1 1 0 0 1 | * * * * * * 6 * * | 0 0 0 1 1 0 .. .. oqo. ....&#xt | 0 0 1 2 1 0 | 0 0 0 0 0 0 2 2 0 | * * * * * * * 3 * | 0 1 0 0 0 1 .. .. .... .oqo&#xt | 0 0 0 1 2 1 | 0 0 0 0 0 0 0 2 2 | * * * * * * * * 3 | 0 0 0 0 1 1 -------------------------+-------------+-------------------+-------------------+------------ oq3oo&#x .... .... ♦ 1 3 0 0 0 0 | 3 0 3 0 0 0 0 0 0 | 3 0 1 0 0 0 0 0 0 | 1 * * * * * top corner pyramid of cube oq .. || oqo. ....&#xt ♦ 1 2 1 2 1 0 | 2 1 1 2 2 0 2 2 0 | 1 2 0 1 0 2 0 1 0 | * 3 * * * * cubera = half-square || square .q3.o || ...o3...o | 0 3 0 0 0 1 | 0 0 3 0 0 3 0 0 0 | 0 0 1 0 3 0 0 0 0 | * * 1 * * * straight q-{3} || pt .q .. || ..oo3..oo&#x | 0 2 0 0 1 1 | 0 0 1 0 2 2 0 0 1 | 0 0 0 1 1 0 2 0 0 | * * * 3 * * oblique q-line || x-line .o3.o || .... .oqo&#xt | 0 1 0 1 2 1 | 0 0 0 1 2 1 0 2 2 | 0 0 0 0 0 2 2 0 1 | * * * * 3 * oblique pt || square .. .. oqoo3ooqo&#xt ♦ 0 0 1 3 3 1 | 0 0 0 0 0 0 3 6 3 | 0 0 0 0 0 0 0 3 3 | * * * * * 1 bottom full cube
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