quickfur wrote:Hahaha I've been making a fool of myself.
Wendy was right all along; the square pyramid antiprism is the same as the square antiprism bipyramid. The pentagonal pyramid antiprism is also the same thing as the pentagonal antiprism bipyramid. Both are CRF.
This CRF is a nice find! The equivalence is clearly understood in terms of lace cities: the n-gonal pyramid can be given as a lace prism ox-n-oo&#x, i.e. as o-n-o (point) || x-n-o (n-gon). The according pyramid in dual positioning clearly would be o-n-x (dual n-gon) || o-n-o (point at the opposite side). So you would get for lace city (= stack of towers):
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o-n-o o-n-x
x-n-o o-n-o
And this very city, rotated by 45 degrees, shows: you would have 3 towers: o-n-o (point), o-n-x || x-n-o (n-antiprism), o-n-o (point). Thus indeed, it is the n-antiprism-dipyramid.
Next. You don't find that fellow within my list of segmentochorons. In the (unrotated) representation it would well be a lace prism (monostratic), not a true tower. So, why it isn't a segmentochoron?
In fact, it is the restriction having the edge length, connecting the (nearer) base vertices of either pyramid the same as those of the pyramid edges them selves, and further the edge length connecting the base vertices of one pyramid to the tip of the other as the same length as well. This is what defines the values of G and H. H just is the height of that lace prism (in fact twice the height, as your chosen edge length is 2). But G represents an shift of the pyramids out of their (vertex) circumcenter. Therefore the very alignment (shift) of the base polytopes disallows a single vertex distance, and so too the overall arangement of the 2 antiparallel, gyrated pyramids does not allow for one. This is why it is not a segmentochoron for n=4 and n=5. - For n=3 it would be, in fact, it just is a different description of the hexadecachoron!
But there is a small spin-off! If you would diminish one of the pyramids down to its base polygon, you would be allowed to shift that degenerate (flat) polytope out of its circumcenter without leaving the restriction of having a unique 3d circumcenter (of that base). So you migh, as in your coordinates) attach that total shift G to the diminished pyramid, and the other one would get shift-free. Accordingly you re-enter the realm of segmentochorons:
3g || gyro tet,
4g || gyro 4pyr,
5g || gyro 5pyr
all are listed in my enumeration!
By virtue of that lace city you even could understand the there being found equivalences:
3g || gyro tet = point || oct,
4g || gyro 4pyr = point || 4ap,
5g || gyro 5pyr = point || 5ap.
--- rk