Klitzing wrote:Or, when the empty space between the "4/3" and the "*b" was placed by will (i.e. representing an independent graph, only re-connected by the astersks), then you aim for a second connection between the b-th and the e-th node by that "3" (besides the already mentioned one ...o3o3o3x...), but then there remains that other link "4/3" unconnected, having no other node symbol to which it has to be connected to.

That hypothesis is right. In fact, I meant x4o3o3o3x4/3*a *b3*e (and by analogy, x4x3o3o3o4/3*b3e is supposed to be x4x3o3o3o4/3*a *b3e).

Lesson learned: label the nodes before linearizing the symbol.

username5243 wrote:Neat. This regiment should be pretty large because it contains the octet regiment as a facet. The octet regiment here should act like it has 53 members because it is appearing in cyclotetrahedral symmetry, with the cyclotetrahedral members of the octet regiment appearing twice each. (a similar thing happens in stut cadoca - ditatha appears in two different orientations, so if you were to make a list of its members, you'd have to distinguish between "a-ditathas" and "b-ditathas".) In fact, the stut cadoca regiment may well appear as a facet of this one, making it even larger. (I'm not sure what the largest regiment of tetracombs will actually be.)

This could well be the largest. Other incredibly large regiments will probably include those of cypit (probably not quite as big), stadit (has a cube || co verf), skivtadia (x4o || x4o || o4x verf), scyropot, aftadia, and its conjugate (which I don't see on your spreadsheet), and icot (due to the large range of symmetries it can take on).

username5243 wrote:Anyway, I'd really like to fill in some of these counts, but I have no clue how Hedrondude does it for polytera. (And I suspect some of these regiments will be larger than any polyteron regiment.)

Don't worry, I have a plan for when it comes time to search these regiments

Anyway, on to the subject I was planning to talk about.

Prismatic honeycombs and regimentsBesides the one I listed in my last list of uniform honeycombs, there are 3 more copycats of the square hemiapeirogonal prismatic honeycomb. One has only half of the azips blended into squats, one has rows of azips going perpendicular to columns of azips (the columns being perpendicular to the prismatic layers), and one has half of the azips blended into squats and the other half in columns instead of rows.

There are also 5 more copycats of the ditrigonal triangular hemiapeirogonal (ditatha) prismatic honeycomb, formed by starting at the ditatha pseudoprismatic honeycomb (containing trips and squats) and selectively un-blending the squats into rows or columns of azips.

There is one new copycat each of the prismatic honeycombs of sossa, gossa, satsa, hatha, snassa, rassersa, rarsisresa, rosassa, and rorisassa, besides the one where the azips blend into squats. In each case, it has columns instead of rows of azips.

The alternate triangular prismatic honeycomb (which is like the ditatha prismatic honeycomb except each layer gets the opposite set of trips) has 13 copycats, if my count is correct, formed by replacing rows of azips by squats or columns of azips. In 7 cases, the trats decompose into sets of azaps as well.

The alternate trihexagonal prismatic honeycomb has one copycat, where the square tilings decompose into rows of azips.

Finally, there are an uncounted number of other chon regiment members, by letting it act like x~x * x~x * x~x (the least density of symmetry that keeps it uniform.) One of these has the same verf as the koho, the skew-octahemioctachoron.

Climbing method and elemental naming scheme are good.