Acronym | ... |
Name |
axially pyritohedral octa-diminishing of telex, hexa-diminishing of oofoo3oxoxo5xooox&#xt |
Lace city in approx. ASCII-art |
o2x x2f f2o x2f o2x -- J91 (bilbiro) a b c f2f F2x o2F F2x f2f d e f F2o f2F f2F F2o g h x2F F2f x2F i j F2o f2F f2F F2o f2f F2x o2F F2x f2f o2x x2f f2o x2f o2x -- J91 (bilbiro) | +-- o3o5x (doe) +---------------------------------- o3o5x (doe) |
Face vector | 100, 348, 340, 92 |
Confer |
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This axially pyritohedral polychoron could be augmented with 6 bilbiro wedges, thus restoring the full axially icosahedral symmetry, then resulting in oofoo3oxoxo5xooox&#xt. That one in turn would be the medial parabidiminished telex, i.e. the tele-elongated ex.
Consider a dodecahedron as a base. Then attach 6 bilunabirotundae (J91) edgewise in an axially pyritohedral manner. Bend those in 90 degrees to the bottom dodecahedron. Thus you could attach at the top a second dodecahedron, again adjoining edgewise. Now note that the pentagons of the dodecahedra and those of the bilunabirotundae come always in pairs. In fact you could connect them dyadically by inserting a metabidiminished icosahedron (J62) each. Further, when speaking of axially pyritohedral subsymmetry and dodecahedral bases, it becomes clear that the cubical subset of dodecahedral vertices becomes somehow special. And indeed, there you can apply 3D counterparts of Norman's "luna"s: linear stacks of a tetrahedron, a triangular prism, and another tetrahedron. - By now nearly all faces are matched dyadically. Only all squares remain open and some of the triangles too. And this is the remaining part: apply 24 square pyramids (J1) onto the squares of the triangular prisms, and further 12 square pyramids onto the sides of the bilunabirotundae.
Incidence matrix according to Dynkin symbol
oFx|xFf|fFo|f-2-xoF|fxF|ofF|f-2-Fxo|Ffx|Fof|f-2-FFF|xxx|ooo|F-&#zx → all existing heights = 0 agi beh cjf d o.. ... ... . 2 o.. ... ... . 2 o.. ... ... . 2 o.. ... ... . & | 24 * * * | 1 2 2 0 0 0 0 0 0 | 2 3 1 2 0 0 0 0 0 0 0 0 | 1 3 1 0 0 0 0 a,g,i ... o.. ... . 2 ... o.. ... . 2 ... o.. ... . 2 ... o.. ... . & | * 48 * * | 0 1 0 1 1 2 1 1 1 | 1 0 1 1 1 2 1 1 1 1 2 2 | 0 2 1 1 2 1 1 b,e,h ... ... o.. . 2 ... ... o.. . 2 ... ... o.. . 2 ... ... o.. . & | * * 12 * | 0 0 0 0 0 0 4 4 0 | 2 0 0 0 0 0 0 2 2 2 4 0 | 0 2 1 0 2 1 0 c,f,j ... ... ... o 2 ... ... ... o 2 ... ... ... o 2 ... ... ... o | * * * 16 | 0 0 3 0 0 0 0 0 3 | 0 3 0 3 0 0 0 0 0 0 0 3 | 1 3 0 0 0 0 1 d ---------------------------------------------------------------------+-------------+----------------------------+-------------------------------------+------------------ ... ... ... . x.. ... ... . ... ... ... . ... ... ... . & | 2 0 0 0 | 12 * * * * * * * * | 2 2 0 0 0 0 0 0 0 0 0 0 | 1 2 1 0 0 0 0 aa,gg,ii o.. o.. ... . 2 o.. o.. ... . 2 o.. o.. ... . 2 o.. o.. ... . &#x & | 1 1 0 0 | * 48 * * * * * * * | 1 0 1 1 0 0 0 0 0 0 0 0 | 0 2 1 0 0 0 0 ab,eg,hi o.. ... ... o 2 o.. ... ... o 2 o.. ... ... o 2 o.. ... ... o &#x & | 1 0 0 1 | * * 48 * * * * * * | 0 2 0 1 0 0 0 0 0 0 0 0 | 1 2 0 0 0 0 0 ad,dg,di ... x.. ... . ... ... ... . ... ... ... . ... ... ... . & | 0 2 0 0 | * * * 24 * * * * * | 0 0 1 0 1 0 0 0 1 0 0 0 | 0 1 1 0 0 1 0 bb,ee,hh (ortho) ... ... ... . ... ... ... . ... ... ... . ... x.. ... . & | 0 2 0 0 | * * * * 24 * * * * | 0 0 0 0 1 2 0 1 0 1 0 0 | 0 0 1 1 2 1 0 bb,ee,hh (para) ... oo. ... . 2 ... oo. ... . 2 ... oo. ... . 2 ... oo. ... . &#x & | 0 2 0 0 | * * * * * 48 * * * | 0 0 0 0 0 1 1 0 0 0 1 1 | 0 1 0 1 1 0 1 be,bh,eh ... o.. o.. . 2 ... o.. o.. . 2 ... o.. o.. . 2 ... o.. o.. . &#x & | 0 1 1 0 | * * * * * * 48 * * | 1 0 0 0 0 0 0 1 0 0 1 0 | 0 1 1 0 1 0 0 bc,ej,fh ... o.. ..o . 2 ... o.. ..o . 2 ... o.. ..o . 2 ... o.. ..o . &#x & | 0 1 1 0 | * * * * * * * 48 * | 0 0 0 0 0 0 0 0 1 1 1 0 | 0 1 0 0 1 1 0 bf,ce,hj ... o.. ... o 2 ... o.. ... o 2 ... o.. ... o 2 ... o.. ... o &#x & | 0 1 0 1 | * * * * * * * * 48 | 0 0 0 1 0 0 0 0 0 0 0 2 | 0 2 0 0 0 0 1 bd,de,dh ---------------------------------------------------------------------+-------------+----------------------------+-------------------------------------+------------------ ... ... ... . x.. f.. o.. . ... ... ... . ... ... ... . &#xt & | 2 2 1 0 | 1 2 0 0 0 0 2 0 0 | 24 * * * * * * * * * * * | 0 1 1 0 0 0 0 aabcb,ggeje,iihfh ... ... ... . x.o ... ... f ... ... ... . ... ... ... . &#xt & | 3 0 0 2 | 1 0 4 0 0 0 0 0 0 | * 24 * * * * * * * * * * | 1 1 0 0 0 0 0 aadgd,ggdid,iidad o.. x.. ... . ... ... ... . ... ... ... . ... ... ... . &#x & | 1 2 0 0 | 0 2 0 1 0 0 0 0 0 | * * 24 * * * * * * * * * | 0 1 1 0 0 0 0 abb,gee,ihh o.. o.. ... o 2 o.. o.. ... o 2 o.. o.. ... o 2 o.. o.. ... o &#x & | 1 1 0 1 | 0 1 1 0 0 0 0 0 1 | * * * 48 * * * * * * * * | 0 2 0 0 0 0 0 abd,deg,dhi ... x.. ... . ... ... ... . ... ... ... . ... x.. ... . & | 0 4 0 0 | 0 0 0 2 2 0 0 0 0 | * * * * 12 * * * * * * * | 0 0 1 0 0 1 0 bbbb,eeee,hhhh ... ... ... . ... ... ... . ... ... ... . ... xx. ... . &#x & | 0 4 0 0 | 0 0 0 0 2 2 0 0 0 | * * * * * 24 * * * * * * | 0 0 0 1 1 0 0 bbee,bbhh,eehh (para) ... ooo ... . 2 ... ooo ... . 2 ... ooo ... . 2 ... ooo ... . &#x | 0 3 0 0 | 0 0 0 0 0 3 0 0 0 | * * * * * * 16 * * * * * | 0 0 0 1 0 0 1 beh ... ... ... . ... ... ... . ... ... ... . ... x.. o.. . &#x & | 0 2 1 0 | 0 0 0 0 1 0 2 0 0 | * * * * * * * 24 * * * * | 0 0 1 0 1 0 0 bbc,eej,hhf (para) ... x.. ..o . ... ... ... . ... ... ... . ... ... ... . &#x & | 0 2 1 0 | 0 0 0 1 0 0 0 2 0 | * * * * * * * * 24 * * * | 0 1 0 0 0 1 0 bbf,eec,hhj (ortho) ... ... ... . ... ... ... . ... ... ... . ... x.. ..o . &#x & | 0 2 1 0 | 0 0 0 0 1 0 0 2 0 | * * * * * * * * * 24 * * | 0 0 0 0 1 1 0 bbf,eec,hhj (para) ... oo. o.. . 2 ... oo. o.. . 2 ... oo. o.. . 2 ... oo. o.. . &#x & | 0 2 1 0 | 0 0 0 0 0 1 1 1 0 | * * * * * * * * * * 48 * | 0 1 0 0 1 0 0 bce,bfh,ehj ... oo. ... o 2 ... oo. ... o 2 ... oo. ... o 2 ... oo. ... o &#x & | 0 2 0 1 | 0 0 0 0 0 1 0 0 2 | * * * * * * * * * * * 48 | 0 1 0 0 0 0 1 bde,bdh,deh ---------------------------------------------------------------------+-------------+----------------------------+-------------------------------------+------------------ oFx ... ... f 2 xoF ... ... f 2 Fxo ... ... f ... ... ... . &#zx ♦ 12 0 0 8 | 6 0 24 0 0 0 0 0 0 | 0 12 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * o.x x.f ..o f ... ... ... . ... ... ... . ... ... ... . &#xr & ♦ 3 4 1 2 | 1 4 4 1 0 2 2 2 4 | 1 1 1 4 0 0 0 0 1 0 2 2 | * 24 * * * * * cycle (abfhid) o.. x.. f.. . 2 x.. f.. o.. . ... ... ... . F.. x.. o.. . &#zx & ♦ 4 8 2 0 | 2 8 0 4 4 0 8 0 0 | 4 0 4 0 2 0 0 4 0 0 0 0 | * * 6 * * * * ... ... ... . ... ... ... . ... ... ... . ... xxx ... . &#x ♦ 0 6 0 0 | 0 0 0 0 3 6 0 0 0 | 0 0 0 0 0 3 2 0 0 0 0 0 | * * * 8 * * * ... ... ... . ... ... ... . ... ... ... . ... xx. o.. . &#x & ♦ 0 4 1 0 | 0 0 0 0 2 2 2 2 0 | 0 0 0 0 0 1 0 1 0 1 2 0 | * * * * 24 * * ... x.. ..o . ... ... ... . ... ... ... . ... x.. ..o . &#x & ♦ 0 4 1 0 | 0 0 0 2 2 0 0 4 0 | 0 0 0 0 1 0 0 0 2 2 0 0 | * * * * * 12 * ... ooo ... o 2 ... ooo ... o 2 ... ooo ... o 2 ... ooo ... o &#x ♦ 0 3 0 1 | 0 0 0 0 0 3 0 0 3 | 0 0 0 0 0 0 1 0 0 0 0 3 | * * * * * * 16
8 * * * * * * * * * | 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 2 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 a (o2x2F F) * 16 * * * * * * * * | 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 b (x2f2F x) * * 4 * * * * * * * | 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 0 0 0 0 0 2 0 0 0 0 4 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 2 0 0 2 0 0 0 1 0 0 0 0 0 c (f2o2F o) * * * 16 * * * * * * | 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 d (f2f2f F) * * * * 16 * * * * * | 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 | 0 0 1 0 0 1 1 0 1 1 1 1 1 0 0 e (F2x2f x) * * * * * 4 * * * * | 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 | 0 0 0 2 1 0 0 2 0 0 0 0 0 1 0 f (o2F2f o) * * * * * * 8 * * * | 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 1 0 0 0 0 0 | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 2 1 0 0 0 0 0 0 0 0 0 | 0 1 1 0 0 0 0 0 0 0 2 1 0 0 0 g (F2o2x F) * * * * * * * 16 * * | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 | 0 0 0 1 0 0 1 1 1 0 1 0 1 1 1 h (f2F2x x) * * * * * * * * 8 * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 1 | 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 2 0 0 1 0 0 | 0 1 0 2 0 0 0 0 0 0 1 0 0 1 0 i (x2F2o F) * * * * * * * * * 4 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 4 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 2 4 0 0 0 0 2 2 | 0 0 0 0 0 0 0 0 0 0 2 1 2 0 1 j (F2f2o o) 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