This page, and the one linked as "these tables", are likely to change frequently throughout November 2024. In particular, the names of some hexads will change.
There are many infinite series of regular maps, with each member of a series being in a different genus of surface.
Just as regular maps can be grouped in hexads, so can these infinite series; so we can think of hexads of infinite series, and equivalently, of infinite series of hexads. These pages list 13 such hexads – eight of them are in fact degenerate hexads, being triads.
See these tables showing the 13 hexads.
Each hexad is denoted by an upper-case Greek letter, Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν. The members of each hexad are denoted by the lower-case version of the Greek letter for the hexad α β γ δ ε ζ η θ ι κ λ μ ν followed by some of the symbols ' °. These symbols mean "dual" and "Petrie dual" respectively, and correspond to the δ and π used at the hexad page. So the Α hexad of infinite series comprises the individual series α, α', α'°, α'°', α°', α°.
Each member of a hexad of regular maps has the same number of edges. We use this to identify individual hexads of series, and individual members of one series. So, for example, Δ is a series of hexads, or equivalently a series of hexads; Δ21 is a hexad, δ'°' is a series of regular maps, and δ'°'21 is one of those maps.
Each Α hexad includes a digon (in orientable genus 0) and
a hemi-digon (in non-orientable genus 1) with an odd number
of edges.
Each Β hexad includes a hemidigon with an even number of edges.
Each Γ hexad includes a digon with an even number of edges.
The series member indicators α'°, α'°, α°', α°, β°', β°, γ°' and γ°
are not shown at the pages for genera orientable-0 and non-orienable 1.
Some of the hexads listed here are more degenerate than their table indicates:
Α1 is a triad. Γ2 is a monad. Ζ4 is a triad. Ε6 is a triad. Θ8 is a monad. Λ8 is a monad. Μ16 is a monad.
Some regular maps are members of more than one of the series described on these pages.
β2 is the same as κ2'°. Α3 is the same as Δ3. Γ4 is the same as Ζ4. Ε6 is the same as Γ6. Θ8 is the same as Λ8. Ε12 is the same as Ζ12. The last four are doubtful.
The values of mV (vertex multiplicity) and mF (face multiplicity) for Γ2, Δ3, Ε6, Ζ4, Κ2, and Μ2 are exceptional, as noted alongside their tables.
There is a θ° in every orientable genus of order 1%4. There is an ι° in every orientable genus of order 5%8. The θ°s in orientable genera of order 1%8 are listed in these pages. The θ°s and ι°s in orientable genera of order 5%8 are not listed in these pages, as I can't tell the θ°s from the ι°s.