Erebean polytopes

The Erebean polytopes (/əˈrɛbiːən/) are abstract regular polytopes derived from groups made using a specific construction method, which takes in a Cartan matrix and a prime quotient order p. A simple explanation of how Erebean polytopes are built is that instead of building the polytope based on the Cartan matrix within the real numbers, it is built within the finite field of the quotient order or an extension of it. This series of pages was mainly made to place all the research done on them in the #hyperbolic-tilings channel in the Polytope Discord somewhere else than Discord.

The first known instance of this method employed to generate a non-fictitious higher dimensional "analog" of a series of shapes was on 9 December 2024 by Milo Jacquet, generating a quotient of {5,3,3,3} composed of 650 hecatonicosachoral tera, functioning as a 5D "analog" of the dodecahedron and the hecatonicosachoron (or H_n in general). The method was based on the field quotient technique used in David Madore's Hyperbolic maze. Later in June next year, PlanetN9ne used this method to build such analogs of H_n up to rank 8, as well as rank 5 and 6 analogs of the Klein quartic and the 4900-cell. The Erebean polytopes were named after the latter's preferred name for the theoretical ninth planet of the Solar System, Erebus.

In these pages, Erebean polytopes are arranged in series, denoting Cartan matrices that follow some structure between ranks. The best known example of this is the rhodoid series, which extends the H_n symmetries. In this series, the Cartan matrices follow the structure:

That is, the diagonal is 2 and its adjacents are 1 except for the two closest to the top-left corner that are φ.

Erebean symmetries sometimes have a "natural quotient order". This is usually either the non-3 number within its diagram if it is prime or the smallest prime that successfully constructs the base polytopes of a series. Iff the quotient order is natural, the polytope is not built within an extension of the finite field of order p. In some (at least for Rhodoids and Tychoids) cases, symmetries with an odd rank and p = 2 are non-orientable.

Objects generated through this procedure can be notated by prepending its notation with a sharp "♯" symbol, with the quotient order in superscript preceding the sharp. As an example, the procedure done on the symmetry E₉ with p = 5 is noted ⁵♯E₉.

The code used to research all of this can be found at erebean.gap, utilizing the GAP system. Of course, it is also a precise description of the construction method.

Series

These are structured by first describing the series' properties, the polytopes generated by the natural quotient order and then a general table showing values for polytopes generated by non-natural quotient orders.

• Rhodoids — notated ♯H_n, made from the diagram [5,3,3,...]. Forms analogs of the dodecahedron and the hecatonicosachoron.
• Tychoids — made from the diagram [7,3,3,...]. Its natural quotients forms analogs of the Klein map and the 4900-cell.
• Gossetoids — notated ♯E_n, made from the diagram [3,3,3³,3,3,...]. Forms analogs of the Gossetic polytopes, E₆, E₇ and E₈.
• Iridoids — notated ♯Ir_n(k), made from the diagram [k,3,3,...]. Generalizes the rhodoids and tychoids to face types higher than pentagons and heptagons.