| Site Map | Polytopes | Dynkin Diagrams | Vertex Figures, etc. | Incidence Matrices | Index |
---- 5D ----
This page is available sorted by point-group symmetry (below)
or by complexity (older version).
(For obvious reasons only the existing 4D graph types, which exist as subgroups in 5D as well, have to be extended here.)
| Linear | Tridental | Loop & Tail | |
|---|---|---|---|
o--P--o--Q--o--R--o--S--o
|
o-P-o-Q-o *b-R-o-S-o =
o_
-P_
>o--R--o--S--o
_Q-
o-
|
o-P-o-Q-o-R-o-S-o-T-*c =
_o
_R- |
o--P--o--Q--o< | S
-T_ |
-o
|
o-P-o-Q-o-R-o-S-o-T-*b =
o---P---o---Q---o
| |
T R
| |
o---S---o
|
| Two Armed | Two Legged | Prolate Crossed Rhomb & Tail | Oblate Crossed Rhomb & Tail |
o-P-o-Q-o-R-o *b-S-o-T-*c =
o--P--o--Q--o--R--o
\ /
S \ / T
o
|
o-P-o-Q-o-R-o-S-*b-T-o =
o_ _o
-P_ _Q- |
>o< | R
_T- -S_ |
o- -o
|
o-P-o-Q-o-R-o-S-o-T-*b *c-U-*e =
o---P---o---Q---o
\ / \
T U R
\ / \
o---S---o
|
o-P-o-Q-o-R-o-S-o-T-*b-U-*d =
o---Q---o---P---o
\ / \
R U T
\ / \
o---S---o
|
| Bowtie | Loop | House | Three Looped |
o-P-o-Q-o-R-*a-S-o-T-o-U-*a =
o_ _o
| -P_ _U- |
Q | >o< | T
| _R- -S_ |
o- -o
|
o-P-o-Q-o-R-o-S-o-T-*a =
_o_
_T- -P_
o- -o
\ /
S Q
\ /
o---R---o
|
o-P-o-Q-o-R-o-S-o-T-*a-U-*c =
o---T---o_
| | -P_
S U >o
| | _Q-
o---R---o
|
o-P-o-Q-o-R-*a-S-o-T-o-U-*a *c-V-*e =
o---P---o---U---o
\ / \ /
Q R S T
\ / \ /
o---V---o
|
| Tetrahedron & Tail | Others | ||
o-P-o-Q-o-R-o-S-o-T-*b-U-*d *c-V-*e =
_o
_- /|
_Q- R |
_- / |
o---P---o<---U---o V
-_ \ |
-T_ S |
-_ \|
-o
| |||
In the following symmetry listings "etc." means replacments according to 3 ↔ 3/2, to 4 ↔ 4/3, to 5 ↔ 5/4, or to 5/2 ↔ 5/3.
Polytera with Grünbaumian elements so far are not investigated any further. Those are Grünbaumian a priori, usually because of some subgraph -x-n/d-x-, where d is even. Others, which come out as being Grünbaumian a posteriori will be given none the less.
| tetrahedron & tail ones |
|---|
o-P-o-Q-o-R-o-S-o-T-*b-U-*d *c-V-*e =
_o
_- /|
_Q- R |
_- / |
o---P---o<---U---o V
-_ \ |
-T_ S |
-_ \|
-o
|
By close inspection of the possible subgraphs on the tetrahedral part within 4D (tetrahedral graph ones), if done with respect to their according pentachoral or tesseractic symmetries, not only does restrict the there possible link marks Q,R,S,T,U,V drastically, rather in fact there is just a single non-Grünbaumian polychoral facet at all, kavahto, i.e. even restricting its vertex decorations as well. That having said, by further inspectation of the 4D (loop & tail graph ones), it occurs moreover, that P has to be 3 or 3/2, while only one out of Q,T,U is either 4 or 4/3.
As an aside, because 4D (two-loop graph ones), as add-ons to a tetrahedral subgraph, has for according compatible (sub-)symmetries only the demi-tesseractic symmetries where all exiting link marks being 3 or 3/2, therefore it becomes obvious directly from the fact that the graph of kavahto requires some link marks 4 and 4/3 too, that similar hypothetical graph types like "tetrahedron & handle" or "trigonal bipyramid" cannot exist.
As a further aside we note, that the tetrahedral graph of kavahto has a singel trigonal subgraph as base, all of its link marks are 3 or 3/2, while all other lacing link marks are of type 4 or 4/3. Thus, subsuming that a hypothetical "full pentachoral graph" might exist, the restriction to kavahto only facets clearly becomes impossible: one might attach two tetrahedral graphs at their 3-3/2 bases, but then, when trying to squeeze in further tetrahedral graphs, these would need to have a lacing zig-zag of 4-4/3 edges between two non-connected 3-3/2 edges. But that however, cf. 4D (tetrahedral graph ones), then would be icosatetrachoric symmetrical, i.e. does not occur as a possible subsymmetry within spherical 5D. Thence such a hypothetical "full pentachoral graph" cannot exist either.
Thus, in summary, these "tetrahedron & tail" graphs are the most complex graphs within spherical 5D geometry.
o3x3x3o4x4*b3/2*d *c4/3*e - skevatant x3x3x3o4x4*b3/2*d *c4/3*e - scatnit |
o3o3x3x4x4*b3/2*d *c4/3*e - raktatant x3o3x3x4x4*b3/2*d *c4/3*e - (contains "2thah") |
o3x3o4x4x3*b4/3*d *c3/2*e - gakevatant x3x3o4x4x3*b4/3*d *c3/2*e - gactanet |
o3/2x3x3o4x4*b3/2*d *c4/3*e - skevatant x3/2x3x3o4x4*b3/2*d *c4/3*e - [Grünbaumian] |
o3/2o3x3x4x4*b3/2*d *c4/3*e - raktatant x3/2o3x3x4x4*b3/2*d *c4/3*e - (contains "2thah") |
o3/2x3o4x4x3*b4/3*d *c3/2*e - gakevatant x3/2x3o4x4x3*b4/3*d *c3/2*e - [Grünbaumian] |
o3x3x3/2o4/3x4*b3*d *c4/3*e - skevatant x3x3x3/2o4/3x4*b3*d *c4/3*e - scatnit |
o3o3/2x3x4x4/3*b3*d *c4/3*e - raktatant x3o3/2x3x4x4/3*b3*d *c4/3*e - (contains "2thah") |
o3x3/2o4/3x4x3*b4/3*d *c3*e - gakevatant x3x3/2o4/3x4x3*b4/3*d *c3*e - gactanet |
o3/2x3x3/2o4/3x4*b3*d *c4/3*e - skevatant x3/2x3x3/2o4/3x4*b3*d *c4/3*e - [Grünbaumian] |
o3/2o3/2x3x4x4/3*b3*d *c4/3*e - raktatant x3/2o3/2x3x4x4/3*b3*d *c4/3*e - (contains "2thah") |
o3/2x3/2o4/3x4x3*b4/3*d *c3*e - gakevatant x3/2x3/2o4/3x4x3*b4/3*d *c3*e - [Grünbaumian] |
© 2004-2025 | top of page |