Here is what i know of ursulatopes.
Ursulatopes are a kind of partially truncated fostrums, literally the prototype is xu&#ft. If the truncation happens at 1/F of the height from the large base, and the large base is rectified, the style of the figure is xfo&xt, a pentagon. The base figure is rectified.
The ursulation of a polygon gives xfoPoop&#t, where p is the shortchord of P (eg x=3, q=4, f=5, h=6, u=infinity. The largest value of P which leads to mostly unit-edge ursulate is P=10. A polytope with this size is x3o3o5o, so xfo3oox3ooo5ooo&#xt is in fact flat.
All of the parameters of the ursulates are given by the a single property: since i use the square of the diameter as the size-property, the discussion shall be with this.
The ursulates have four kinds of face: the top, the base, the lacing faces (eg the pentagons), and the valences (a valance here is something that covers the legs of furniture, or the wheels where these are open.
The base is a rectate specifically of the top face of edge 2. This produces in general, a figure for which the D2 is (in L2 form:) b = 4t-4 So, eg 3.618*4-4 = 10.944. ie an ursulated x3o5o gives a base o3x5o, which is a decagon-round.
The valance faces slope inwards on the teddi xfo3oox&#t, but on xfo10oop&#xt, they are clearly facing upwards. One can find out the condition where the valence faces become vertical, by equating fPo with oPp. This means that the apex of the valence-faces are directly over the base (they're in essence op&#t pyramids. The value gives a base diameter of t=5.236067=2F. For a top base larger than this, the figure is "vertically convex": you can stand it on a face, and all the falls from any point perpendicular to the base is inside the figure. This means we can use this as a capping for a pyramid extension, or for back to back constructions.
One can place a peaking pyramid on the top of the ursulate, in the form of oxfo3ooox&#xt. This is allowed as long as the diameter2 of the base is less than 4.
So we get
- 1.3333 < t< 4 top face can be raised to a peak
- 4.000 <= t < 5.236 can be neither top-peaked or used as a pyramid cap.
- 5.236< t <10.472 The figure is vertically convex, and might be used as pyramid-capping.
While this might be applied to conventional simple-figures + truncates (eg t = x3oXo.... b=o3xXo....), it also applies to the rectates, t=..o3x3o.., b=..x3o3x..
One can derive a formula for the diameter of a truncate, directly from the dynkin symbol, where D is the diameter square of the 'regular figure' (including the likes of 2_21 &c).
D' = (n+1)² D - 4(n)², where n is the order of rectate, When one sets D to the diameter of a simplex D=2n/(n+1), then the n'th rectate corresponds to a point.
One puts D'=10.472, and finds D for various values of N, ie D = (D'+4n²)/(n+1). For the cross polytope, and simplexes in high dimensions, n is limited to 4, but for lower dimensions, eg up to 22D, all values of N work with the simplex.
So, apart from 2_21 = /4B with d=2.66666, one has 1/3B with d=6.666666, both work.
Stott-expansionsIt is permissible to make any node, not being used in either the top or base figures, to cause all nodes in the tower to be marked. This creates a new kind of face along the edges running from the vertices of the sections xPo to fPo.
One can comprise the stott-expanded figure of an ursulate by eg T = t+S, B=b+S. For example, one might have oxfo3ooox5xxxx&#xt, can be constructed by way of the stott-sum u = oxfo3ooox5oooo&xt and the stott-expansion consist S=oooo3oooo5xxxx&xt. The element s is S with the the nodes involved in the ursulation, removed, ie s = .3.5x = x.
The top and bottom bases are T=t+S, B=b+S. The lacing-faces which have pentagons and teddies, are expansions of the face-elements as well (ie the lacing-faces are ursulates of the faces of t). So these expand to T'=t'+S and B'=b'+S. The valence faces start as pyramids, from point || vf of t, becomes expanded by s to s || s+vf. New faces follow the former edges from the vertices of t to the valences apex, gives now a {5}*s pyramid.
So if ü represents the ursulated bottom face node, we get then o3ü5x has 30 pentagonal prisms following the edges, and a further 12 prisms following the pyramids from the pentagons of x3o5x to f3o5x.
The parameters evaluated above do not affect the stott expansions, so the total diameter of t can be much larger than 10.944.
[b]Multiple ursulates[b]
Not only can one have ursulates top-and-tailing a pyramid (especially if the top diameter2 is > 5.236), but when expansion is taken to account, one can have different ursulates at each end. A simple case might be
o3x3o3o3oBo is vertical-convex, so it might be used as an an apex, eg
ox3xo3ox3oo3ooBoo (omitting medial layers)
We see that the bottom is given by x3o3xo3oBo. This can be stott-expanded by x3o3xo3xBo (or the last node by 'x'). But this same base can arise from a different figure, eg
oox3xfo3oox3ooo3xxxBooo&xt and xxx3ooo3oox3xfo3oox&Booo both have the same base, but are the ursulates of 1/3B and 3/1B.
So, we can produce an assymetric (like the gyrated and gyrated-extended bicupolae), like
ooxxx 3 xfooo 3 ooxoo 3 ooofx 3 xxxoo B ooooo.
What is happening here is the top is an ursulate of node 2 into 1 and 3, the total expanded along 5 (and probably 6). The vertex-layers 3,4,5 are another ursulate from node 4 into 3 and 5, the lot expanded by 1. So the middle layer is to have nodes 1,3,5 marked, which can be expanded into a prism.
One can use a similar process on the simplex-as-simplex faces. For example, the larger simplex-rectates give rise to vertical-convex ursulates, which might be used as face adornments on the simplex and cross polytopes having these faces.