Uncovesseltuxe, the character, is named for the Universal cover of SL(2,R) x E1, his home geometry, and I have occasionally
thought up similar names like "Uncovesseltuxaichtu" or "Uncovesseltuxsolthrite" (now Uncovesseltuxsoldri).
So why not complete this system of Englishy names for non-Euclidean geometries?
This system will of course, not count every geometry (there are uncountably many mathematically important ones in 5D, and also for 3D if you allow geometries with no finite/wrapped versions)
Euclidean geometry is simply "e". In most contexts it is pronounced [ɛ] (like the e in bed), but at the end of a word it converts to a silent e.
This will only happen in a product with E1, where the x will disambiguate them (this is what happens with Uncovesseltuxe).
When it has a dimension other than 1, you must qualify it with a number for the dimension. If the number starts with a vowel, put a z in between.
In the special case of just E1 alone, the dimension is known as "Emono" [ɛmanou].
Examples: Emono, Etu, Edri, Ededra, Epenta, Ehexa, Ehepta, Ezocta, Ezennea, Edeca, Ezundeca, Edodeca, Edrideca, Ededradeca, Epentadeca, Ehexadeca, Eheptadeca, Ezoctadeca, Ezenneadeca, Ezicosa.
Spherical geometry is "ess", pronounced [ɛs] (like the letter S). You must qualify it with a number for the dimension.
Examples: Esstu, Essdri, Essdedra, Esspenta, Esshexa, Esshepta, Essocta, Essennea, Essdeca, Essundeca, Essdodeca, Essdrideca, Essdedradeca, Esspentadeca, Esshexadeca, Essheptadeca, Essoctadeca, Essenneadeca, Essicosa.
Hyperbolic geometry is "aich", pronounced [e.ɪk] (two syllables: ay-ick). You must qualify it with a number for the dimension.
Examples: Aichtu, Aichdri, Aichdedra, Aichpenta, Aichhexa, Aichhepta, Aichocta, Aichennea, Aichdeca, Aichundeca, Aichdodeca, Aichdrideca, Aichdedradeca, Aichpentadeca, Aichhexadeca, Aichheptadeca, Aichoctadeca, Aichenneadeca, Aichicosa.
Nil geometry is simply "nil", pronounced [nɪl] (how you would pronounced nil as an English speaker). You must qualify it with a number for the dimension.
Examples: Nildri, Nildedra, Nilpenta, Nilhexa.
Sol3 geometry is "soldri", pronounced [soldɹʷai] ("soul dry"). Higher-dimensional Sol geometries with subscripts follow the formula: Sol{dimension}sub{subscript numbers, separated by "sub"}ven.
For example Sol4_1 is Soldedrasubmono, Sol4_0 is Soldedrasuboudena, Sol4_5_7 is Soldedrasubpentasubhepta.
In higher dimensions like 5, where multiple infinite families can be subjectively combined into one Sol family with multiple subscripts, do that.
Geometries that are the same curvature as SL(n, R) are referred to as Uncovessel{n}, regardless of whether it is the universal cover. This is Uncovesseltu for SL(2,R), Uncovesseldri for SL(3,R), Uncovesselocta for SL(8,R), Uncovesselicosa for SL(20,R), etc. "Uncovessel" is pronounced [ʌnkʌvɛsəl] (like "uncover" joined at the v with "vessel").
Geometries that are titled Fn are called Eff{n}, pronounced [ɛf] (like the letter F). For example, F4 is Effdedra.
The Cartesian product is simply an "x" between the two geometries' names under this system. They can be ordered either way, though it's preferable to put the ones with more dimensions first. So H2 x E2 is Aichtuxetu, Nil4 x H3 is
Nildedraxaichdri, Nil7 x Nil5 x H4 x S3 x E3 is Nilheptaxnilpentaxaichdedraxessdrixedri (pronounced [nɪl.hɛp.tæks.nɪl.pɛn.tæks.eɪk.dɛ.dɹʷɪks.ɛs.dɹʷɪks.ɛ.dɹʷai]).
An A converts its pronunciation to [æ] (the a in bat) when before an x, while an I converts to [ɪ] (the i in bit).
The "twisted product" of two geometries, whatever that actually means, is notated as Twiss{Cartesian products}ven. For example, ~SL(2,R)/Uncovesseltu can also be called Twissaichtuxeven ([twɪ.se.ɪk.tuks.ɛ.vɛn]).
Multiple Cartesian products within Twiss...ven indicate some sort of multi-way twisting that may exist (there's some paper about multiply-twisted products). To take a twisted product involving a twisted product as its arguments,
nest the Twiss...ven.
Numbers are basically Greek prefixes, but they use "outena" for just 0 and "tu" for just 2. "Tri" and "tetra" in any part of another are replaced with "dri" and "dedra" because the former have unfortunately become dimensional slurs in the fictional universe this naming scheme came from.