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[Bihlbo]

I bet since some of you are mathematicians and/or smarter than me, you'd know the answer to this.

What are the terms for symmetry that centers on a unit, with an odd number of units, versus symmetry that centers between units, with an even number of total units?

Example: The Parthenon's face has _____ symmetry since it has 8 columns, but the sides have _____ symmetry since there's a column in the center of the 17 that run it's length.

Example: These two stairways have different symmetry: https://imgur.com/qm2v5f0
When talking about this, I could say "I made this stair well with an odd number of stairs in each direction so it would line up with a 3-width hallway, but since the other stairs have an even number, to keep the stair well in line I can only do 2 or 4-wide halls."
But surely there's a term for this type of symmetry. I want these terms so I can instead say, "I made this stair well _____ so it would line up with a 3-width hallway, but since the other stairs are ____, to keep the stair well in line I can only do 2 or 4-wide halls."

We have a term for a line that goes from one vertex to another vertex that doesn't share a side with the first vertex. Diagonal. Nice and simple. I'm looking for similar terms.

So far I can't find any terms for the two types of symmetry in architecture, tile-setting, or what limited stuff I could find in geometry (I mean, I could be at this for years, there's just too much to learn out there).

[Maximum Spin]

You know, I don't think there are actual specific terms for this. There might be in some obscure subject, but I usually only hear those described as "odd parity" and "even parity".

[Bihlbo]

I also don't know what the terms are. Maybe someone who does will see this.

[Maximum Spin]

I think you've misunderstood. I'm not saying that I don't know what the terms are, I'm saying that there are no such terms in common use in any of the fields you've mentioned or any other one I'm familiar with where symmetry is important (crystallography, for example). Considering that the concept only even applies to discrete objects with only one specific kind of symmetry (reflectional), it's probably not something that would need to have a term anyway.

[Urist mcbayblade]

I don't think there a are any actual terms for this because in really life 2 tile symmetry can't really exist.  I mostly just use one block/tile center or two bock/tile center when referring to this.

[Maximum Spin]

I don't think there a are any actual terms for this because in really life 2 tile symmetry can't really exist.  I mostly just use one block/tile center or two bock/tile center when referring to this.

Symmetry lines that either go through a vertex or between them can exist in math, though. We just don't call it anything special.

[Starver]

(I first thought the question wasvabout chirality, until I fought through imgur's overlaid "agree to terms"/"get the App" mess of browser-killing stuff and realised I'd completely managed to misread the premise.)

I'm not entirely sure what is used mathematically, but in metallurgy (and crystallography in general, I presume) there's "face-centred" and "body-centred" used for intersticial atoms in a (usually cubic) 3d matrix being either centred to the 'frame' of the host structure or centred upon each frame's face.

For 2d (and, in this case, looking for the sole 'geographic centre' rathed than merely a homogenously repeatingly pattern) I'd go for terms something like "tile-centric" for the focus of an odd-odd array, "corner(s)-centric" for an even-even one and "edge-centric" for an odd-even (or verse vica) array.  I'm sure someone like Roger Penrose has used (if not coined) the actually accepted terms you might want, though. Perhaps look up some papers from him?