- ambification pre-image:
- scad
- general polytopal classes:
- isogonal
links
| Acronym | rescad |
| Name | rectified scad |
| Circumradius | sqrt(3) = 1.732051 |
| Face vector | 120, 720, 1170, 660, 92 |
| Confer |
|
|
External links |
|
Rectification wrt. a non-regular polytope is meant to be the singular instance of truncations on all vertices at such a depth that the hyperplane intersections on the former edges will coincide (provided such a choice exists). Within the specific case of scad all edges belong to a single orbit of symmetry, i.e. rectification clearly is applicable, without any recourse to Conway's ambification (chosing the former edge centers generally). None the less, wrt. the individual facets of the pre-image, in some cases there an ambification is possible only, in fact, tepe cannot be rectified (within this stronger sense).
Still, because the pre-image uses different polygonal faces, this would result in 2 different edge sizes in the outcome polychoron. That one here is scaled such so that the shorter one becomes unity. Then the larger edge will have size q=sqrt(2).
Incidence matrix according to Dynkin symbol
uo3ox3oo3xo3ou&#zq → height = 0 o.3o.3o.3o.3o. | 60 * | 6 6 0 | 6 3 3 6 12 0 0 | 2 3 3 6 2 6 6 0 0 | 1 1 3 3 2 0 .o3.o3.o3.o3.o | * 60 | 0 6 6 | 0 0 3 12 6 3 6 | 0 0 6 3 6 6 2 3 2 | 0 3 3 1 2 1 -------------------+-------+-------------+--------------------------+-------------------------------+---------------- .. .. .. x. .. | 2 0 | 180 * * | 2 1 0 0 2 0 0 | 1 2 0 2 0 1 2 0 0 | 1 0 1 2 1 0 oo3oo3oo3oo3oo&#q | 1 1 | * 360 * | 0 0 1 2 2 0 0 | 0 0 2 2 1 2 1 0 0 | 0 1 2 1 1 0 .. .x .. .. .. | 0 2 | * * 180 | 0 0 0 2 0 1 2 | 0 0 2 0 2 1 0 2 1 | 0 2 1 0 1 1 -------------------+-------+-------------+--------------------------+-------------------------------+---------------- .. .. o.3x. .. | 3 0 | 3 0 0 | 120 * * * * * * | 1 1 0 0 0 0 1 0 0 | 1 0 0 1 1 0 .. .. .. x.3o. | 3 0 | 3 0 0 | * 60 * * * * * | 0 2 0 2 0 0 0 0 0 | 1 0 1 2 0 0 uo .. .. .. ou&#zq | 2 2 | 0 4 0 | * * 90 * * * * | 0 0 2 2 0 0 0 0 0 | 0 1 2 1 0 0 .. ox .. .. ..&#q | 1 2 | 0 2 1 | * * * 360 * * * | 0 0 1 0 1 1 0 0 0 | 0 1 1 0 1 0 .. .. .. xo ..&#q | 2 1 | 1 2 0 | * * * * 360 * * | 0 0 0 1 0 1 1 0 0 | 0 0 1 1 1 0 .o3.x .. .. .. | 0 3 | 0 0 3 | * * * * * 60 * | 0 0 2 0 0 0 0 2 0 | 0 2 1 0 0 1 .. .x3.o .. .. | 0 3 | 0 0 3 | * * * * * * 120 | 0 0 0 0 1 0 0 1 1 | 0 1 0 0 1 1 -------------------+-------+-------------+--------------------------+-------------------------------+---------------- .. o.3o.3x. .. ♦ 4 0 | 6 0 0 | 4 0 0 0 0 0 0 | 30 * * * * * * * * | 1 0 0 0 1 0 .. .. o.3x.3o. ♦ 6 0 | 12 0 0 | 4 4 0 0 0 0 0 | * 30 * * * * * * * | 1 0 0 1 0 0 uo3ox .. .. ou&#zq ♦ 3 6 | 0 12 6 | 0 0 3 6 0 2 0 | * * 60 * * * * * * | 0 1 1 0 0 0 uo .. .. xo3ou&#zq ♦ 6 3 | 6 12 0 | 0 2 3 0 6 0 0 | * * * 60 * * * * * | 0 0 1 1 0 0 .. ox3oo .. ..&#q ♦ 1 3 | 0 3 3 | 0 0 0 3 0 0 1 | * * * * 120 * * * * | 0 1 0 0 1 0 .. ox .. xo ..&#q ♦ 2 2 | 1 4 1 | 0 0 0 2 2 0 0 | * * * * * 180 * * * | 0 0 1 0 1 0 .. .. oo3xo ..&#q ♦ 3 1 | 3 3 0 | 1 0 0 0 3 0 0 | * * * * * * 120 * * | 0 0 0 1 1 0 .o3.x3.o .. .. ♦ 0 6 | 0 0 12 | 0 0 0 0 0 4 4 | * * * * * * * 30 * | 0 1 0 0 0 1 .. .x3.o3.o .. ♦ 0 4 | 0 0 6 | 0 0 0 0 0 0 4 | * * * * * * * * 30 | 0 0 0 0 1 1 -------------------+-------+-------------+--------------------------+-------------------------------+---------------- .. o.3o.3x.3o. ♦ 10 0 | 30 0 0 | 20 10 0 0 0 0 0 | 5 5 0 0 0 0 0 0 0 | 6 * * * * * uo3ox3oo .. ou&#zq ♦ 4 12 | 0 24 24 | 0 0 6 24 0 8 8 | 0 0 4 0 8 0 0 2 0 | * 15 * * * * uo3ox .. xo3ou&#zq ♦ 9 9 | 9 36 9 | 0 3 9 18 18 3 0 | 0 0 3 3 0 9 0 0 0 | * * 20 * * * uo .. oo3xo3ou&#zq ♦ 12 4 | 24 24 0 | 8 8 6 0 24 0 0 | 0 2 0 4 0 0 8 0 0 | * * * 15 * * .. ox3oo3xo ..&#q ♦ 4 4 | 6 12 6 | 4 0 0 12 12 0 4 | 1 0 0 0 4 6 4 0 1 | * * * * 30 * .o3.x3.o3.o .. ♦ 0 10 | 0 0 30 | 0 0 0 0 0 10 20 | 0 0 0 0 0 0 0 5 5 | * * * * * 6
or o.3o.3o.3o.3o. & | 120 | 6 6 | 6 3 3 18 | 2 3 9 8 6 | 1 4 3 2 ---------------------+-----+---------+----------------+-------------------+------------ .. .. .. x. .. & | 2 | 360 * | 2 1 0 2 | 1 2 2 2 1 | 1 2 1 1 oo3oo3oo3oo3oo&#q | 2 | * 360 | 0 0 1 4 | 0 0 4 2 2 | 0 2 2 1 ---------------------+-----+---------+----------------+-------------------+------------ .. .. o.3x. .. & | 3 | 3 0 | 240 * * * | 1 1 0 1 0 | 1 1 0 1 .. .. .. x.3o. & | 3 | 3 0 | * 120 * * | 0 2 2 0 0 | 1 2 1 0 uo .. .. .. ou&#zq | 4 | 0 4 | * * 90 * | 0 0 4 0 0 | 0 2 2 0 .. ox .. .. ..&#q & | 3 | 1 2 | * * * 720 | 0 0 1 1 1 | 0 1 1 1 ---------------------+-----+---------+----------------+-------------------+------------ .. o.3o.3x. .. & ♦ 4 | 6 0 | 4 0 0 0 | 60 * * * * | 1 0 0 1 .. .. o.3x.3o. & ♦ 6 | 12 0 | 4 4 0 0 | * 60 * * * | 1 1 0 0 uo3ox .. .. ou&#zq & ♦ 9 | 6 12 | 0 2 3 6 | * * 120 * * | 0 1 1 0 .. ox3oo .. ..&#q & ♦ 4 | 3 3 | 1 0 0 3 | * * * 240 * | 0 1 0 1 .. ox .. xo ..&#q ♦ 4 | 2 4 | 0 0 0 4 | * * * * 180 | 0 0 1 1 ---------------------+-----+---------+----------------+-------------------+------------ .. o.3o.3x.3o. & ♦ 10 | 30 0 | 20 10 0 0 | 5 5 0 0 0 | 12 * * * uo3ox3oo .. ou&#zq & ♦ 16 | 24 24 | 8 8 6 24 | 0 2 4 8 0 | * 30 * * uo3ox .. xo3ou&#zq ♦ 18 | 18 36 | 0 6 9 36 | 0 0 6 0 9 | * * 20 * .. ox3oo3xo ..&#q ♦ 8 | 12 12 | 8 0 0 24 | 2 0 0 8 6 | * * * 30
© 2004-2024 | top of page |