Minersphere::Index::Erebean polytopes::Gossetoids

The Gossetoids, symmetry group ♯E_n, are the Erebean polytopes derived from the matrix form of the symbol [3,3,3³,3,...], that is, the matrix representing A_n with the furthest 1 moved to the third space leftwards/upwards. Their natural quotient order is 3, being the lowest prime to faithfully represent the three usual Gossetic symmetries.

Just like the usual Gossetics, they will be named, by increasing number of vertices, the ♯k₂₁ polytope, the ♯2_k1 polytope and the ♯1_k2 polytope, with k = n−4. They may alternatively be named in the elemental naming scheme as the siderotope, the argyrotope and the chrysotope, all coined by the Hi.gher.space community.

I have also tried the idea of modifying the double-node branch to begin with a 5. Due to irrelevance, I ended up dropping the idea after just having made the general table, available at Rhodogossetoids II.

Natural

Rank 9

The rank 9 cases are the ♯1₅₂ polyyotton (chrysoyotton, 6561-885735-yotton), the ♯2₅₁ polyyotton (argyroyotton, 6561-12597120-yotton) and the ♯5₂₁ polyyotton (sideroyotton, 885735-12597120-yotton). The first has 6561 1₄₂ polyzettal and 885735 demioctaractic yotta as a quotient of the 1₅₂ eightcomb, the second has 6561 2₄₁ polyzettal and 12597120 8-simplicial yotta as a quotient of the 2₅₁ eightcomb and the third has 885735 2₄₁ polyzettal and 12597120 8-orthoplicial yotta as a quotient of the 5₂₁ eightcomb.

Symmetry properties
Petrie polygon	Order
?	4571242905600
Symmetry axes
Symmetry	Count
A₈	12597120
D₈	885735
A₇×A₁	56687040
A₅×A₂×A₁	529079040
A₄×A₄	317447424
D₅×A₃	99202320
E₆×A₂	14696640
E₇×A₁	787320
E₈	6561
Discovery
MinersHavenM43	2 January 2026

Rank 10

Symmetry properties
Petrie polygon	Order
164?	2579025599882610278400
Symmetry axes
Symmetry	Count
A₉	710710317427968
D₉	27762121774530
A₈×A₁	3553551587139840
A₆×A₂×A₁	42642619045678080
A₅×A₄	29849833331974656
D₅×A₄	11193687499490496
E₆×A₃	2072905092498240
E₇×A₂	148064649464160
E₈×A₁	1850808118302
³♯E₉	564184764
Discovery
MinersHavenM43	2 January 2026

Rank 11

Symmetry properties
Petrie polygon	Order
?	152915585868239728626892800
Symmetry axes
Symmetry	Count
A₁₀	3830857830994461696
D₁₀	82303586212771638
A₉×A₁	21069718070469539328
A₇×A₂×A₁	316045771057043089920
A₆×A₄	252836616845634471936
D₅×A₅	110616019869965081472
E₆×A₄	24581337748881129216
E₇×A₃	2194762299007243680
E₈×A₂	36579371650120728
³♯E₉×A₁	16725821513544
³♯E₁₀	59292
Discovery
MinersHavenM43	2 January 2026

Rank 12

Symmetry properties
Petrie polygon	Order
363?	27088537289801063207068178841600
Symmetry axes
Symmetry	Count
A₁₁	56552081015597992171776
D₁₁	662719699401538970763
A₁₀×A₁	339312486093587953030656
A₈×A₂×A₁	6220728911715779138895360
A₇×A₄	5598656020544201225005824
D₅×A₆	2799328010272100612502912
E₆×A₅	725751706366840899537792
E₇×A₄	77759111396447239236192
E₈×A₃	1619981487425984150754
³♯E₉×A₂	987643034553259656
³♯E₁₀×A₁	5251699962
³♯E₁₁	177147
Discovery
MinersHavenM43	2 January 2026

Summary

Symmetry	³♯E_n	D_n	A_n
D₅	16	10	16
E₆	72	72	27
E₇	56	126	576
E₈	240	2160	17280
³♯E₉	6561	885735	12597120
³♯E₁₀	564184764	27762121774530	710710317427968
³♯E₁₁	59292	82303586212771638	3830857830994461696
³♯E₁₂	177147	662719699401538970763	56552081015597992171776

General

This table shows the flag counts for the symmetry ^p♯E_n, along with its Petrie polygon P. The Petrie polygons are dubious; when using the standard Petrie polygon function applied to the odd dimensional Gossetoids yield (likely) nonsense Petrie polygons.

n \ p	2	3	5
6	51840 P = 12	51840 P = 12	51840 P = 12
7	1451520	2903040	2903040
8	348364800 P = 15	696729600 P = 30	696729600 P = 30
9	89181388800	4571242905600	272160000000000
10	46998591897600 P = 31?	2579025599882610278400 P = 164?	P = 1638?
11	24815256521932800	152915585868239728626892800
12	103231467131240448000 P = 102?	27088537289801063207068178841600 P = 363?	P = 4686?