This page, and the one linked as "these
tables", are likely to change frequently throughout November
2024. In particular, the names of some hexads, and individual series, may change.
There are many infinite series of regular maps, with each member of a series being (generally) in a different genus of surface.
Each regular map is a member of a hexad of regular maps, as explained at hexads. Whenever we find a regular map, we can use duality and Petrie duality to find five (or sometimes two, rarely one or no) other members of its hexad. The same is true of series of regular maps. Whenever we find a series of regular maps, we have found a series of hexads of regular maps, which we may also regard as a hexad of series of regular maps.
These pages list 15 such hexads. Eight of them are in fact degenerate hexads, being triads.
Each hexad of series is denoted in these pages by an upper-case Greek letter, Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο . 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 ξ ο followed by some (possibly empty) list 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 number to identify individual hexads of series,
and individual members of one series. So:
Δ is a series of hexads, or equivalently a hexad of series;
Δ21 is a hexad, whose members all have 21 edges;
δ'°' is a series of regular maps; and
δ'°'21 is one of those maps, with 21 edges.
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.
Each Κ hexad includes a {4,4} (in orientable genus 0) with twice an
odd number of edges, the {4,4} being labelled <a,0>
with a odd.
Each λ hexad includes a {4,4} with a twice a square number of edges, the {4,4} being labelled <a,0>
with a even.
Each μ hexad includes a {4,4} with a square number of edges, the {4,4} being labelled <a,a>.
Some {4,4} regular maps are chiral; and so while they may be
regarded as regular maps, their Petrie duals are not regular
maps, and they don't belong in hexads of regular maps.
Each Ξ hexad includes a {6,3} (in orientable genus 1) with label (n,n), or equivalently Eisenstein numbers 0, n. Each Ο hexad includes a {6,3} with label (0,2n), or equivalently Eisenstein numbers n, n.
Some of the hexads listed here are more degenerate than their table indicates:
Α1 is a triad. Γ2 is a monad. Ε6 is a triad. Ζ4 is a triad. Θ8 is a monad. Λ8 is a monad. Μ16 is a monad. Ξ27 is a triad. Ο9 is a triad.
Some hexads (triads, monads) of regular maps are members of more than one of the series described in these pages.
Β2 = Κ2 (triad). Α3 = Δ3 = Ξ3. Γ4 = Ζ4 = Λ4 (triad). Γ6 = Ε6 (triad). Θ8 = Λ8 (monad). Ε12 = Ζ12.
The values of mV (vertex multiplicity) and mF (face multiplicity) for Α1, Β2 = Κ2, Γ2, Ε6, Ζ4, Ν6, Ξ27, Ο9 and Ο36 are exceptional, and not as given in their tables.
There is a θ° in every orientable genus of order 1 modulo 4. There is an ι° in every orientable genus of order 5 modulo 8. The θ°s in orientable genera of order 1 modulo 8 are listed in these pages. The θ°s and ι°s in orientable genera of order 5 modulo 8 are not listed in these pages, as I can't tell the θ°s from the ι°s.
More on Regular maps
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