Tychoids

The tychoids are the Erebean polytopes derived from the matrix form of the symbol [7,3,3,...]. They are usually used to make analogs of the Klein map and dual Klein map in the sense that they have heptagonal faces/peak figures and simplicial vertex figures/facets, although it can also generate some other [7,3,3,...] quotients. Their natural quotient order is 7, generating Q_n symmetry, named after the Klein quartic.

Each tychoidal symmetry has two regular polytopes, the one with the least amount of vertices being known as the thalassoplex, analogous to the dual Klein map, and the one with the greatest amount of vertices being known as the tychoplex, analogous to the Klein map.

They are named after the elemental name for Q_n symmetry, coined by me. Yes, this does mean that, unlike the rhodoids, the symmetry name and its regular polytopes' names are not distinct...

Information that couldn't fit nicely into here (incidence matrices and vertex layers) was put in tychoids.txt.

Natural

Rank 5

The rank 5 cases are the 5-tychoplex (2352-teron) and the 5-thalassoplex (2304960-teron), being duals of each other with ⁷♯Q₅ symmetry. The former has 2352 350-cell tera as a quotient of {7,3,3,3} while the latter has 2304960 pentachoral tera as a quotient of {3,3,3,7}. It is known that the 5-thalassoplex has octagonal cycles, that is the polygon obtained by going from edge to the opposing edge on each vertex. For the 4900-cell (4-thalassoplex), this yields a cyclic polyhedron, the Heawood map, as its dual Klein maps vertex figures have three antipodal vertices as opposed to the normal two.

Symmetry properties
Petrie polygon	Order
24	276595200
Symmetry axes
Symmetry	Count
Q4	2352
Q3×A1	411600
I2(7)×A2	3292800
A1×A3	5762400
A4	2304960
Discovery
PlanetN9ne	2 June 2025

Rank 6

Symmetry properties
Petrie polygon	Order
200	4635182361600
Symmetry axes
Symmetry	Count
7♯Q5	16758
Q4×A1	19707408
Q3×A2	2299197600
I2(7)×A3	13795185600
A1×A4	19313259840
A5	6437753280
Discovery
PlanetN9ne	3 June 2025

Rank 7

Symmetry properties
Petrie polygon	Order
342	546914437209907200
Symmetry axes
Symmetry	Count
7♯Q6	117992
7♯Q5×A1	988654968
Q4×A2	775105494912
Q3×A3	67821730804800
I2(7)×A4	325544307863040
A1×A5	379801692506880
A6	108514769287680
Discovery
MinersHavenM43	18 June 2025

Rank 8

Symmetry properties
Petrie polygon	Order
600	450219964711195607040000
Symmetry axes
Symmetry	Count
7♯Q7	823200
7♯Q6×A1	48565507200
7♯Q5×A2	271286923219200
Q4×A3	159516710852889600
Q3×A4	11166169759702272000
I2(7)×A5	44664679038809088000
A1×A6	44664679038809088000
A7	11166169759702272000
Discovery
MinersHavenM43	18 June 2025

Summary

Symmetry	7♯Qn	An
I2(7)	7	7
Q3	24	56
Q4	350	4900
7♯Q5	2352	2304960
7♯Q6	16758	6437753280
7♯Q7	117992	108514769287680
7♯Q8	823200	11166169759702272000

General

This table shows the flag counts for the symmetry ^p♯[7,3,3,...], along with its Petrie polygon P.

n \ p	2	3	5	7
3	504 P = 9	19656 P = 26	1953000 P = 126	336 P = 8
4	524160 P = 63	387419760 P = 365	P = 601	117600 P = 25
5	1056706560 P = 63	P = 364	P = 124	276595200 P = 24
6	69387579555840 P = 171	P = 9490	P = 1116	4635182361600 P = 200
7	9077005607176765440 P = 585	P = 728	P = 3906	546914437209907200 P = 342
8	P = 63			450219964711195607040000 P = 600