Fiddled it out, finally!
This is, how
quidex could be designed directly bottom-up as tegum sum, rather than the top-down description so far known only (eg. as dual of sadi, as quatri-ico-diminished ex, as ico-stellated hi):
- Code: Select all
ooV|xoF|o-3-ooo|ooo|F-3-Voo|Fxo|o *b3-oVo|oFx|o-&#z(v,f,F) → existing heights=0
V = 2f = 3.236067977
F = ff = 2.618033989
v = 1/f = 0.618033989
f = 2 cos(36°) = 1.618033989
o.. ... .-3-o.. ... .-3-o.. ... . *b3-o.. ... . | 8 * * * * * * | 4 0 0 0 0 0 0 | 6 0 0 0 0 0 | 4 0 0
.o. ... .-3-.o. ... .-3-.o. ... . *b3-.o. ... . | * 8 * * * * * | 0 4 0 0 0 0 0 | 0 6 0 0 0 0 | 0 4 0
..o ... .-3-..o ... .-3-..o ... . *b3-..o ... . | * * 8 * * * * | 0 0 4 0 0 0 0 | 0 0 6 0 0 0 | 0 0 4
... o.. .-3-... o.. .-3-... o.. . *b3-... o.. . | * * * 32 * * * | 1 0 0 3 0 0 0 | 3 0 0 3 0 0 | 3 0 1
... .o. .-3-... .o. .-3-... .o. . *b3-... .o. . | * * * * 32 * * | 0 1 0 0 3 0 0 | 0 3 0 0 3 0 | 1 3 0
... ..o .-3-... ..o .-3-... ..o . *b3-... ..o . | * * * * * 32 * | 0 0 1 0 0 3 0 | 0 0 3 0 0 3 | 0 1 3
... ... o-3-... ... o-3-... ... o *b3-... ... o | * * * * * * 24 | 0 0 0 4 4 4 8 | 2 2 2 8 8 8 | 4 4 4
-----------------------------------------------------------+-------------------+----------------------+-------------------+---------
o.. o.. .-3-o.. o.. .-3-o.. o.. . *b3-o.. o.. .-&#v | 1 0 0 1 0 0 0 | 32 * * * * * * | 3 0 0 0 0 0 | 3 0 0 v-edges
.o. .o. .-3-.o. .o. .-3-.o. .o. . *b3-.o. .o. .-&#v | 0 1 0 0 1 0 0 | * 32 * * * * * | 0 3 0 0 0 0 | 0 3 0 v-edges
..o ..o .-3-..o ..o .-3-..o ..o . *b3-..o ..o .-&#v | 0 0 1 0 0 1 0 | * * 32 * * * * | 0 0 3 0 0 0 | 0 0 3 v-edges
... o.. o-3-... o.. o-3-... o.. o *b3-... o.. o-&#f | 0 0 0 1 0 0 1 | * * * 96 * * * | 1 0 0 2 0 0 | 2 0 1 f-edges
... .o. o-3-... .o. o-3-... .o. o *b3-... .o. o-&#f | 0 0 0 0 1 0 1 | * * * * 96 * * | 0 1 0 0 2 0 | 1 2 0 f-edges
... ..o o-3-... ..o o-3-... ..o o *b3-... ..o o-&#f | 0 0 0 0 0 1 1 | * * * * * 96 * | 0 0 1 0 0 2 | 0 1 2 f-edges
... ... . ... ... F ... ... . ... ... . | 0 0 0 0 0 0 2 | * * * * * * 96 | 0 0 0 1 1 1 | 1 1 1 F-edges
-----------------------------------------------------------+-------------------+----------------------+-------------------+---------
o.. x.. o ... ... . ... ... . ... ... .-&#(v,f)t | 1 0 0 2 0 0 1 | 2 0 0 2 0 0 0 | 48 * * * * * | 2 0 0 kite
... ... . ... ... . .o. .x. o ... ... .-&#(v,f)t | 0 1 0 0 2 0 1 | 0 2 0 0 2 0 0 | * 48 * * * * | 0 2 0 kite
... ... . ... ... . ... ... . ..o ..x o-&#(v,f)t | 0 0 1 0 0 2 1 | 0 0 2 0 0 2 0 | * * 48 * * * | 0 0 2 kite
... ... . ... o.. F ... ... . ... ... .-&#f | 0 0 0 1 0 0 2 | 0 0 0 2 0 0 1 | * * * 96 * * | 1 0 1 golden triangle
... ... . ... .o. F ... ... . ... ... .-&#f | 0 0 0 0 1 0 2 | 0 0 0 0 2 0 1 | * * * * 96 * | 1 1 0 golden triangle
... ... . ... ..o F ... ... . ... ... .-&#f | 0 0 0 0 0 1 2 | 0 0 0 0 0 2 1 | * * * * * 96 | 0 1 1 golden triangle
-----------------------------------------------------------+-------------------+----------------------+-------------------+---------
o.. xo. o-3-o.. oo. F ... ... . ... ... .-&#(v,f,f)t | 1 0 0 3 1 0 3 | 3 0 0 6 3 0 3 | 3 0 0 3 3 0 | 32 * * tower a-d-g-e
... ... . .o. .oo F-3-.o. .xo o ... ... .-&#(v,f,f)t | 0 1 0 0 3 1 3 | 0 3 0 0 6 3 3 | 0 3 0 0 3 3 | * 32 * tower b-e-g-f
... ... . ..o o.o F ... ... . *b3-..o o.x o-&#(v,f,f)t | 0 0 1 1 0 3 3 | 0 0 3 3 0 6 3 | 0 0 3 3 0 3 | * * 32 tower c-f-g-d
Obviously this incidence matrix shows up the same cyclical symmetry of the legs of this symmetry representation as in the according representation sadi does. That is, both are [3,4,3]
+ symmetrical figures, which is the common subgroup of [3,3,5] and [3,4,3].
The first 6 vertex types display tetrahedral vertex figures, the 7th then is a dodecehardal one. The first 3 face types are a kite, which happens to be nothing but a mono-stellated regular pentagon. The other 3 then are a large obtuse golden triangle, which happens to be a bistellated regular pentagon. The cells then are all tri-trigonally diminished icosahedra, or, conversely described, a mono-trigonally stellated dodecahedron. A picture of that cell looks like this:
Quidex is the one but last member of that multi-ico-diminishing of ex:
0-idex = ex,
1-idex = sadi,
2-idex = bidex (noble, cells being teddies),
3-idex = tridex,
4-idex = quidex,
5-idex = hi.
Moreover we have: k-idex is dual to (5-k)-idex ! (More details on that can be found
here (scroll a bit down).
--- rk