Site Map  Polytopes  Dynkin Diagrams  Vertex Figures, etc.  Incidence Matrices  Index 
The part about edgeexpanded polytopes was derived in a cooperation with J. McNeill (cf. e.g. his pages on EEBs or on n,n,3acrons).
Also lots of the EKFs provide interesting axial polytopes (even so these are not restricted to axial symmetries in general).
(E.g. in 3D cf. n/dpy for general {n/d} base.)
The pyramids are the outcome of the pyramid product. Take any polytope as base, a single point with a relative orthogonal offset. Then the projective scaling, centered at that point, will outline the derived pyramid within the interval from the point up to the given base. At this level nothing is said about dimensions, convexity of the base, nor about edge lengths.
As far as the base polytope is a nonsnub, has a Dynkin diagram description and is orbiform, the whole pyramid will have a Dynkin diagram too. At least in the sense of a lace prism with appropriately scaled lacings. Just prefix any node symbol (thus either being an x or a o) by an unringed node (o). And finally postfix the obtained symbol by "&#y", where the relative size of y is a function of the desired height of the pyramid, in fact it provides the lacing length. – Even so pyramids can be built on a snubbed base, the required symbol cannot be obtained in the just described way, because the processes of alternation and setting up the pyramid product do not commute. Orbiformity of the base was required, as else those lacings cannot all be of a single length.
The actual height of a pyramid can be calculated by means of theorem of Pythagoras as h = sqrt(y^{2}  r^{2}), where y will be the absolute length of the lacing edges and r is the circumradius of the base polytope. For 3D pyramids one further has r( xn/do ) = 1/(2 sin(π d/n)) and therefore, because of x being unit edges,
h( oxn/doo&#x ) = sqrt[1  1/(2 sin(π d/n))^{2}]
Generally this gives as restriction for possible base polytopes h > 0 or y > r. For those 3D pyramids therefore one derives
h( oxn/doo&#x ) > 0 or 1 > 1/(2 sin(π d/n)) or sin(π d/n) > 1/2 or π d/n > π/6 or n/d < 6
Because of the equivalence xn/do = xn/(nd)o we likewise get n/(nd) < 6, or in other words finally: 5/6 < n/d < 6.
Pyramids can be seen as segmentotopes, provided they fullfil their required axioms. In this context those boil down to the requirement that the base polytope has to be a polytope with unique circumradius r, which has to be strictly smaller than 1, and all edges are of unit length. Only then the lacings can be chosen to be of unit length as well. Furthermore the circumradius R of the whole pyramid can be provided then in terms of the circumradius r of the base as:
R^{2} = 1/[4 (1r^{2})]
The lacing facet polytopes all clearly are pyramids in turn, in fact they are pyramids based on the facets of the base polytope.
For convex pyramidal segmentotopes we have:
1D  oo&#x  pt  pt  line 
2D  ox&#x  pt  line  3g 
3D  ox3oo&#x  pt  3g  tet 
ox4oo&#x  pt  4g  squippy (J1)  
ox5oo&#x  pt  5g  peppy (J2)  
4D  ox3oo3oo&#x  pt  tet  pen 
ox3oo4oo&#x  pt  oct  octpy  
ox4oo3oo&#x  pt  cube  cubpy  
ox3oo5oo&#x  pt  ike  ikepy  
oox4ooo&#x  pt  squippy (J1)  squasc  
oox5ooo&#x  pt  peppy (J2)  peppypy  
ox ox3oo&#x  pt  trip  trippy  
ox ox5oo&#x  pt  pip  pippy  
  pt  squap  squappy  
  pt  pap  pappy  
  pt  gyepip (J11)  gyepippy  
  pt  mibdi (J62)  mibdipy  
  pt  teddi (J63)  teddipy 
A special subclass here are multipyramids. Again cf. to the pyramid product. Multipyramids are not to be missunderstood here in the sense of a bipyramid, i.e. not as "line  perp base", but rather are meant in the sense of iteratedly applying the pyramid operation instead, thereby adding a further dimension each time. Thus, these figures would be "pt  (pt  (...(pt  base)...))". J. Bowers here introduced a sequence of according names:
Qscalene = Qpypy, Qtettene = Qpypypy, Qpennene = Qpypypypy, Qhixene = Qpypypypypy, ...
Esp. the pointpyramidpyramid is nothing but a (generally) irregular triangle, which commonly is called a scalene (triangle).
This is, where this sequence derives from.
Some examples then would be:
dim.  scalenes  tettenes  pennenes  ... 
2D 
{3} = ptpypy 
 
 
 
3D 
tet = linepypy 
tet = ptpypypy 
 
 
4D 
pen = {3}pypy squasc = {4}pypy pesc = {5}pypy stasc = {5/2}pypy ... 
pen = linepypypy 
pen = ptpypypypy 
 
5D 
hix = tetpypy octasc = octpypy trippasc = trippypy squete = squippypypy ... 
hix = {3}pypypy squete = {4}pypypy ... 
hix = linepypypypy 
... 
6D 
hop = penpypy hexasc = hexpypy rapesc = rappypy tepasc = tepepypy squippypasc = squippyppypy octete = octpypypy trippete = trippypypy squepe = squascpypy ... 
hop = tetpypypy octete = octpypypy trippete = trippypypy squepe = squippypypypy ... 
hop = {3}pypypypy squepe = {4}pypypypy ... 
... 
7D 
oca = hixpypy rixasc = rixpypy hinsc = hinpypy hexete = hexpypypy ... 
oca = penpypypy hexete = hexpypypy ... 
oca = tetpypypypy ... 
... 
Applying here, the above note, that pyramids on orbiform bases with base radius lesser than 1 generally are segmentotopes, could be iterated. At the first level we have the alignment point  base. Scalenes with such an orbiform base generally do allow for a further such description as line  perp basebase. Tettenes with such an orbiform base generally do allow for a third such description as triangle  perp basebasebase. And the general item of that sequance then allows for the description as simplex  perp iterated base.
(E.g. in 3D cf. n/dp for general {n/d} bases.)
Similar to the pyramids, prisms are the outcome of the prism product of any base polytope with a lacing edge. Again nothing is said in general about dimension, convexity nor edge lengths.
If the base polytope has any Dynkin diagram description, this product will have one too. Just add a further ringed node (inline: x) to the diagram but no further links. Note, this would work for snubbed base polytopes alike. For nonsnubbed ones however, the Dynkin diagram can be rewritten as a lace prism just by doubling any node symbol of the base polytope, and by postfixing a "&#y", where y provides the relative edge length of the lacings. – Again snubbing does not commute with the product. In fact, for 3D prisms, commutation would lead to the antiprisms.
Prisms can be seen as segmentotopes, provided they fullfil the required axioms. In this context those boil down to the requirement, that the base polytope just has to be orbiform. Further, the lacings will have to be of unit length as well. I.e. for the height of prismatic segmentotopes one generally has h = 1.
The lacing facet polytopes all clearly are prisms in turn, in fact they are prisms based on the facets of the base polytope.
For convex prismatic segmentotopes we have:
1D 
x o = oo&#x  pt  pt  line 
2D 
x x = xx&#x  line  line  4g 
3D 
x xno = xxnoo&#x x x4o = xx4oo&#x 
ng  ng
4g  4g 
np
cube 
x xnx = xxnxx&#x  2ng  2ng  2np  
4D 
x x3o3o = xx3oo3oo&#x  tet  tet  tepe 
x x3x3o = xx3xx3oo&#x  tut  tut  tuttip  
x x3o4o = xx3oo4oo&#x x o3x3o = oo3xx3oo&#x  oct  oct  ope  
x o3x4o = oo3xx4oo&#x  co  co  cope  
x o3o4x = oo3oo4xx&#x  cube  cube  tes  
x x3x4o = xx3xx4oo&#x x x3x3x = xx3xx3xx&#x  toe  toe  tope  
x x3o4x = xx3oo4xx&#x  sirco  sirco  sircope  
x o3x4x = oo3xx4xx&#x  tic  tic  ticcup  
x x3x4x = xx3xx4xx&#x  girco  girco  sircope  
x x3o5o = xx3oo5oo&#x x s3s3s  ike  ike  ipe  
x o3x5o = oo3xx5oo&#x  id  id  iddip  
x o3o5x = oo3oo5xx&#x  doe  doe  dope  
x x3x5o = xx3xx5oo&#x  ti  ti  tipe  
x x3o5x = xx3oo5xx&#x  srid  srid  sriddip  
x o3x5x = oo3xx5xx&#x  tid  tid  tiddip  
x x3x5x = xx3xx5xx&#x  grid  grid  griddip  
x s3s4s  snic  snic  sniccup  
x s3s5s  snid  snid  sniddip  
x x xno = xx xxnoo&#x  np  np  4,ndip  
x s2sns  nap  nap  nappip  
all orbiform Johnson solids  J##  J##  J##p 
Of course, multiprisms do exist as well. And this not only with respect to several perpendicular axes (as in: prism of (prism of (prism of ...)) ), but also in the sense of larger perpendicular objects than a mere product with a line. Cf. here again to the prism product, then within its full generality.
(E.g. in 3D cf. n/dap for general {n/d} bases.)
As such an antiprism is defined only for 3D. There it can be derived as the snub (i.e. alternated faceting) of the prisms with even numbered base polygons. Sure, this concept could be extended to higher dimensions as well, but because of the decreasing relative amount of degrees of freedom when trying to come back to uniform figures (i.e. equal edge lengths) after the (generally applicable) alternated faceting, this ansatz becomes not too effective. (Rare examples in that sense would be sidtidap and gidtidap.)
An alternate idea would be to consider the base polygons of 3D antiprisms as a dual pair of regular polytopes. This ansatz, via lace prisms, clearly extends to any dimension, for 1D it just is point  point, and else just take any linear reflection group graph, assign for the top layer (left symbol at each node position of the graph) the ringed node "x" at the leftmost position, all others will be marked "o", while for the bottom layer (right symbol at each node position) the ringed node "x" then will be placed at the rightmost position, and again all others will be marked "o". Finally postfix at this Dynkin diagram "&#y", where y gives the relative length of the lacing edges. – As an aside, extending beyond the topic of axials, this ansatz furthermore could be extended onto ndental reflection group diagrams as well, replacing the lace prisms by (n layered) lace simplices, with a single ringed node at a different end for each layer.
It should be emphasized here, by taking dual pairs of regular polytopes, the bases generally will not be the same polytopes, i.e. the topbottom symmetry generally is lost. It is retained only whenever those are a selfdual pair (as this was the case for any regular polygon).
Sometimes antiprisms with lateral facet pyramids, which do cross the axis of global symmetry, are also called retroprisms. E.g. for the 5/2ap the lateral triangles don't, whereas for 5/3ap they do.
Whenever those lacing edges can be chosen to be of the same length as the ones of the base polytopes, we will have a valid segmentotope. Just as for any segmentotope, the lacing facet polytopes all will be segmentotopes in turn. In fact their bases always will be codimensional: vertex atop facet (i.e. bottomup pyramids), edge atop ridge, etc. ..., ridge atop edge, facet atop vertex (i.e. topdown pyramids).
The height of a (uniform) 3D antiprism can be calculated using the polygonal circumradius r( xn/do ) = 1/(2 sin(π d/n)), the inradius ρ( xn/do ) = sqrt[r^{2}  (1/2)^{2}] = 1/(2 tan(π d/n)) and the height of the lacing triangle h( x3o ) = r( x3o ) + ρ( x3o ) = sqrt(3)/2
h( xon/dox&#x ) = sqrt[(r( xn/do )  ρ( xn/do ))^{2}  (h( x3o ))^{2}] = sqrt[(1 + 2 cos(π d/n))/(2 + 2 cos(π d/n))]
For convex antiprismatic segmentotopes we have:
1D  oo&#x  pt  pt  4g 
2D  xx&#x  line  line  4g 
3D 
xonox&#x xo ox&#x xo3ox&#x 
ng  dual ng line  perp line 3g  dual 3g 
nap tet oct 
4D  xo3oo3ox&#x  tet  dual tet  hex 
xo3oo4ox&#x  oct  cube  octacube (alt.: octap)  
xo3oo5ox&#x  ike  doe  ikadoe (alt.: ikap) 
Finally a further extension should be mentioned within this context, even so no longer belonging truely to axial polytopes. The so far emphasized extension here was to consider antiprisms as using linear Dynkin diagrams displaced by some offset in a further dimensional direction, which have inverted decorations. Whenever the (individual undecorated) Dynkin diagram would have inversional symmetry, this also could be read, as if the total (now decorated) Dynkin diagram would have been inverted as a whole. And, using the around symmetry, the point inversion of a linear diagram happens to be nothing but a rotation by 180°. – And right this observations now opens for a further extension. Consider the 4D tridental Dynkin diagrams o3o3o *b3o. Those also allow for some gyrational symmetry, this time by 120°. Accordingly we no longer shall consider the mere stack of 2 gyrations, but we well can consider the simplectical (here: trigonal) arrangement of all gyrated versions at the same time. Here we consider a 4D "base" space and a further 2D position space. Accordingly those here described polytopes then would live within 6D. – As antiprisms with symmetrical (undecorated) base diagrams in this view well could be considered gyroprisms, that considered extended class of polypetons thus generally is called the one of gyrotrigonisms. – Right as is true for the all the gyroprisms, these gyrotrigonisms too all happen to be scaliform, because obviously all vertices do fall into a single orbit of global symmetry. Moreover, because a trigon clearly is a 2D segmentotope, these gyrotrigonisms could be seen as a monostratic stack of one such 4D base atop of a 5D gyrostack of the 2 other such 4D bases. Accordingly these gyrotrigonisms all qualify as convex segmentopeta. – Despite of this involved description, there happen to be just 4 gyrotrigonisms in total. These are named according to their respective base polychora as hexgyt, thexgyt, ritgyt, and tahgyt.
(E.g. in 3D cf. n/dcu for general {n/d} and {2n/d} bases.)
As such a cupola is defined only for 3D. Even so it is ment as monostratic faceparallel topsection of larger (uniform) polyhedra, it is best defined directly as lace prism xxn/dox&#y. As such, the cupola is nothing but a Stott expansion of the pyramid (as the first node position changes from "oo" to "xx"), accordingly for y = x we get the same heights, i.e. h( xxn/dox&#x ) = h( oon/dox&#x ), and therefrom the same restriction: 6/5 < n/d < 6. Further the base polygon, in order to not become a Grünbaumian double cover, requires d to be odd.
In order to extrapolate cupolas into spaces of higher dimensions, there are different valid possibilities, even within the realm of segmentotopes:
This extrapolation is based on the observation, that the bottom polytope is the kernel of intersection of a dual pair of the top polytope. (Here the base xn/dx of a 3D cupola is read as being the kernel of the compound of xn/do with on/dx.) Speaking of dual, the top figures here will be restricted to regular polytopes. Dealing with their Dynkin diagrams those kernels of intersection, (as is described in the truncation series) in case of odd dimensional top facets, just have the single middle node ringed, resp., for even dimensional top facets, just the two central nodes ringed.
For according segmentochora xPoQo  oPxQo, i.e. the lace prisms xoPoxQoo&#x, the lacings thus would be antiprisms (as subdiagrams: xoPox ..&#x) and pyramids (as: .. oxQoo&#x) only.
This different extrapolation sticks to the idea of being a cap of a larger uniform polytope. It also starts with regular polytopes for top facets, but asking the bottom facet being the corresponding Stott expanded version, i.e. its Dynkin diagram has both end nodes ringed. The Dynkin diagram of that larger uniform polytope (of which the cupola would be a cap of) furthermore could be derived by adding "...3x" to the diagram of the top facet.
The accordingly extrapolated segmentochora xPoQo  xPoQx, i.e. the lace prisms xxPooQox&#x have for lacing facets prisms (as subdiagrams: xxPoo ..&#x), trips (as: xx .. ox&#x, i.e. used as digonal 3D cupola in here), and pyramids (as: .. ooQox&#x). Those then would be the xPoQocap of the polychoron xPoQo3x.
It should be noted, that in a much looser sense, sometimes any possible monostratic stacking of Dynkin symbols, i.e. any lace prism, with nondegenerate bases (esp. neither pyramid nor wedge), which not qualifies as prism or other more specific terms (e.g. not an antiprism), might be termed "cupola".
Case A) is the reading of the term "cupola", which the author prefers. For case of B) the author rather prefers the term cap. Finally C) is mentioned here for awareness only, and not too much endorsed by the author.
For convex cupolaic segmentotopes we have:
A  B  
1D  oo&#x  pt  pt  line  (same: line = ptcap of line itself)  
2D  xx&#x  line  line  4g  (same: 4g = linecap of 4g itself)  
3D  xx ox&#x  line  4g  trip  (same: trip = linecap of trip itself)  
xx3ox&#x  3g  6g  tricu  (same: tricu = 3gcap of co)  
xx4ox&#x  4g  8g  squacu  (same: squacu = 4gcap of sirco)  
xx5ox&#x  5g  10g  pecu  (same: pecu = 5gcap of srid)  
4D  xo ox3oo&#x  line  perp {3}  pen  xx oo3ox&#x  line  trip  tepe (linecap of tepe itself)  
xo ox4oo&#x  line  perp {4}  squasc  xx oo4ox&#x  line  cube  squippyp (linecap of squippyp itself)  
xo ox5oo&#x  line  perp {5}  pesc  xx oo5ox&#x  line  pip  peppyp (linecap of peppyp itself)  
xo3ox oo&#x  3g  dual 3g  (subdimensional: oct)  xx3oo ox&#x  3g  trip  triddip (3gcap of triddip itself)  
xo3ox3oo&#x  tet  oct  rap  xx3oo3ox&#x  tet  co  (tetcap of spid)  
xo3ox4oo&#x  oct  co  (octcap of ico)  xx3oo4ox&#x  oct  sirco  (octcap of spic)  
xo3ox5oo&#x  ike  id  (ikecap of rox) 
(unit lacing impossible in spherical space: xx3oo5ox&#x would be the hyperbolic ikecap of x3o5o3o)  
xo4ox oo&#x  4g  dual 4g  (subdimensional: squap)  xx4oo ox&#x  4g  cube  tisdip (4gcap of tisdip itself)  
xo4ox3oo&#x  cube  co  xx4oo3ox&#x  cube  sirco  (cubecap of sidpith)  
xo5ox oo&#x  5g  dual 5g  (subdimensional: pap)  xx5oo ox&#x  5g  pip  trapedip (5gcap of trapedip itself)  
xo5ox3oo&#x  doe  id  xx5oo3ox&#x  doe  srid  (cubecap of sidpixhi) 
Cuploids are a completely 3D specific concept. They are somehow related to several of the uniform polyhedra, which do not emanate directly by Wythoff's construction, but in fact are reduced forms of Grünbaumian polyhedra. The same holds true here: as pointed out above, the denominator d of the top polygon xn/do has to be odd, else the cupola gets a Grünbaumian doublecover polygon for bottom base. Exactly in those prohibited cases, i.e. for d being even, that offending face will just be withdrawn, and the open but pairwise coincident edges will be reconnected in the obvious way.
This picture, showing a {7/4}cuploid, reflected in a mirror, was rendered by C. Tuveson in 2001 in reply to a post of mine: "But your structure reminds me to a true polyhedron with a {7/2}heptagrammic edge circuit at the bottom and a {7/3}face at the top side, joined to one another by squares plus trigons (the latter pointing towards the vertices of the topface). It is the retrograd {7/3}cuploid, or might also be called the {7/4}cuploid. As it is well known, the bottomface of a {n/d}cupola is a {(2n)/d}; but for 7/4 this becomes the reducible number 14/4, which is nothing but the (reduced) {7/2} with a double circuit. Thereby the latteral sides (squares and trigons) join at the bottom edges, and the bottom face is obsolete (or would have to be counted twice, giving rise to pairwise coincident edges)." 
It should be noted additionally that, as long as the top face is not retrograde, i.e. as long as n/d > 2, central parts of the top base offers both sides to the outside because of the bottom base reduction. Faces having this property generally are called membranes.
Within the bounds, also provided above, cuploids exist as segmentohedra. Their bottom face just will have to be marked "pseudo". As this adjective is not transferable into Dynkin diagrams, a lace prism description does not exist. Further, as d has to be even and thus n/d can no longer be integral, clearly there is no convex segmentotope. – As (nonconvex) examples {3/2}  pseudo {6/2} and {5/2}  pseudo {10/2} might serve.
Also being ment for 3D in the first run, those are miming the cuploids in the complemental cases, i.e. for d being odd again. In those cases nomal cupolas do exist. Sure, in a locally similar manner, 2 copies each can be blended in an axially gyrated way. This blending operation thereby withdraws the doubled up bottom face, while the top face becomes a regular compound. (Even so those clearly are segmentotopes, those top face compounds will not be convex.) The easiest ones here are:
Nonetheless, this type of operation surely does apply also to pairs of (either way) higher dimensionally extrapolated cupolas with dual top bases. But because then those dual top bases generally no longer have the same circumradius, the height of the to be blended cupolas too must no longer be the same. Thus the top bases then generally would not result in a compound but are arranged in parallel layers, i.e. the blend would not be monostratic anymore. Therefore the research for cupolaic blend segmentotopes (i.e. being monostratic) has to restrict to top bases which are selfdual only (or other wise would assure to have the same circumradius). Examples here are
Fastegium here derives from latin fastigium, the pediment. Accordingly, as arbitrary dimensional analogue just
Thus, written as a lace prism, it is just oy ...xx...&#z, where x, y, z all define edges of possibly different lengths. The lacing facets accordingly are 2 prisms, similar the bottom one, but now with lacing edges z (while the bottom one has ylacings), plus, as far as the starting figure had any facets itself, the subdimensional fastegiums derived by those. (Sometimes even the top layer is allowed to differ in size, yielding oy ...wx...&#z.) So, fastegia are special cases of wedges.
If z = y (and w = x) this figure can be rewritten as lace simplex or even as duoprism. oy ...xx...&#y = ...xxx...&#y = y3o ...x.... If further all edges have equal size and the starting polytope was some orbiform, say Q, then the fastegium will be a valid segmentotope (in fact Q  Qp). Accordingly the segmentotopal height then generally will be h = sqrt(3)/2.
Moreover, whenever Q would be additionally uniform, then the fastegium clearly will be uniform itself, being nothing but the 3,Qdip.
For convex fastegmal segmentotopes we have:
2D 
ox oo&#x = ooo&#x = x3o o  pt  line  {3} 
3D 
ox xx&#x = xxx&#x = x3o x  line  {4}  trip 
4D 
ox xxnoo&#x = xxxnooo&#x = x3o xno  {n}  np  3,ndip 
This small table already shows, provided P would be any convex, unitedged, and orbiform polytope of any dimension, that the set of convex fastegmal segmentotopes could equivalently be described as the set of according 3,Pduoprisms.
The antifastegium is essentially built the same way as a (normal) fastegium, just that its subdimensional top layer polytope is replaced by its dual. Speaking of duals, this already asks the starting figure itself to be a regular polytope. Those generally are Wythoffian and therefore moreover orbiform.
Any antifastegium can be written as lace prism, which is oy xo...ox&#z in general, where again x, y, z all define edges of possibly different lengths. The facets of an antifastegium are its bottom prism (.y) (.o)...(.x), the 2 antiprism connecting either bottom base to the top base .. xo...ox&#z, and finally for any other subelement of the bottom prism there will by an according codimensional subelement of the top base, which has to be adjoined.
Finally, antifastegia, having unit edges only, clearly are segmentotopes.
For convex antifastegmal segmentotopes we have:
2D 
ox oo&#x = ooo&#x  pt  line  {3} 
3D 
ox xx&#x = xxx&#x  line  {4}  trip 
4D 
ox xonox&#x = xxonoox&#x  {n}  gyro np  naf 
(The more general, not necessarily convex case of 4D then is the n/daf.)
A duoantifastegiaprism always can be written as a lace prism in the form xo...ox yo...oy&#z, where again x, y, z all define edges of possibly different lengths. Here the subelements xo...ox&#z and yo...oy&#z would be required to describe antiprisms each. Accordingly the bases here would be required to be (bidually aligned copies of) duoprisms of two regulars each. (In fact, the term duoantifastegiaprism was chosen as a contraction from duoprism duoantifastegium.)
The general convex duoantifastegiaprismal segmentoteron here would be n,mdafup = xonox xomox&#x. (In other dimensions so, the 2 subelemental antiprisms no longer would be bound to equating dimensions.)
A duoantifastegium then can be given as the specific cases, where one of the required subelemental antiprisms becomes a (stretched) tet (xo ox&#z). This clearly makes the duoprisms of the 2 bases degenerate, i.e. the bases would become subdimensional. (This is what reduces one prismatic part from the defining duoprism, and therefore too from the contracted name.)
The general duoantifastegial segmentoteron here would be n/ddaf = xo ox xon/dox&#x (which moreover is convex for d=1).
As an aside it should be pointed out, that the sequence of used operations (i.e. prism product and stacking) here does not commute: e.g. the number of vertices of (x3o x3o)  (o3x o3x) clearly is (3·3) + (3·3) = 18, whereas (x3o  o3x) (x3o  o3x) would result in (3+3) · (3+3) = 36.
The bipyramids are the outcome of the tegum product. Take any polytope as a base and a single line segment in orthogonal space, either one being centered at the origin. Then the projective scaling, centered at the ends of the segment, will outline the derived pyramid within the interval from the point up to the given base. At this level nothing is said about dimensions, convexity of the base, nor about edge lengths.
Clearly, bipyramids are closely related to pyramids. In fact those are just external blends of 2 pyramids, being adjoined at their base polytopes. Therefore that defining polytope itself will not be contribute as a true facet of the outcome, as it thereby would be blended out. Yet it can be considered a pseudo face. Likewise it might be possible to dissect the bipyramid into 2 pyramids while squeezing inbetween an equatorial prism. This then is what is meant by elongated bipyramids.
As far as the base polytope is a nonsnub, has a Dynkin diagram description and is orbiform, the whole pyramid, bipyramid and even the elongated bipyramids will have a Dynkin diagram too. At least in the sense of a lace tower with appropriately scaled lacings. For bipyramids just pre and postfix any node symbol (thus either being an x or a o) by an unringed node (o). And finally add to the obtained symbol a final "&#y", where the relative size of y is again a function of the desired height of the pyramid on either side, because literally it provides the lacing length. – Even so pyramids can be built on a snubbed base, the required symbol cannot be obtained in the just described way, because the processes of alternation and setting up the tegum product do not commute. Orbiformity of the base was required, as else those lacings cannot all be of a single length.
For convex bipyramids, which generally are external blends of segmentotopes, we have:
2D  oqo&#xt  pt  qline  pt  4g 
3D  oxo3ooo&#xt  pt  3g  pt  tridpy (J12) 
oxo4ooo&#xt  pt  4g  pt  oct  
oxo5ooo&#xt  pt  5g  pt  pedpy (J13)  
4D  oxo3ooo3ooo&#xt  pt  tet  pt  tete 
oxo3ooo4ooo&#xt = ooo3oxo3ooo&#xt  pt  oct  pt  hex  
oxo4ooo3ooo&#xt  pt  cube  pt  cute  
oxo3ooo5ooo&#xt  pt  ike  pt  it  
ooxo4oooo&#xr  pt  squippy (J1)  pt  octpy  
oxo oxo3ooo&#xt  pt  trip  pt  tript  
oxo oxo5ooo&#xt  pt  pip  pt  pipt  
  pt  squap  pt  squapdpy (?)  
  pt  pap  pt  papdpy (?)  
  pt  gyepip (J11)  pt  gyepipdpy (?)  
  pt  mibdi (J62)  pt  mibdidpy (?)  
  pt  teddi (J63)  pt  teddidpy (?) 
(Lately Bowers prefers the acronym suffix Xt, here representing X(,line)tegum, rather then the older Xdpy.)
As the base angle of a pyramid always is smaller than 90 degrees, an according external blend with a matching prism thus is convex whenever the pyramid itself was.
For convex elongated pyramids, which generally are external blends of segmentotopes, we have e.g.:
3D  oxx3ooo&#xt  pt  3g  3g  etripy (J7) 
oxx4ooo&#xt  pt  4g  4g  esquipy (J8)  
oxx5ooo&#xt  pt  5g  5g  epeppy (J9)  
4D  oxx3ooo3ooo&#xt  pt  tet  tet  etepy 
oxx3ooo4ooo&#xt = ooo3oxx3ooo&#xt  pt  oct  oct  eoctpy  
oxx4ooo3ooo&#xt  pt  cube  cube  ecubpy  
oxx3ooo5ooo&#xt  pt  ike  ike  eikepy  
oxx oxx3ooo&#xt  pt  trip  trip  etrippy  
oxx oxx5ooo&#xt  pt  pip  pip  epippy 
Obviously the same construction as for elongated pyramids applies for their Stott expansions wrt. their across symmetry. This is what is understood within 3D as the wellknown elongated cupolas in the set of Johnson solids. But, as has been outlined above, the extension of the meaning of the term cupola is quite ambiguous beyond 3D. In fact, those expansions only would build a subset of the type C cupolas, or would be a superset of the type B cupolas. The type A cupolas on the other hand have to be investigated individually whether the 90 degrees condition is being met for all bottom base angles.
For convex elongated cupolas (in the sense of across symmetrically Stott expanded elongated pyramids), which then generally are external blends of segmentotopes, we have e.g.:
3D  oxx3xxx&#xt  3g  6g  6g  etcu (J18) 
oxx4xxx&#xt  4g  8g  8g  escu (J19)  
oxx5xxx&#xt  5g  10g  10g  epcu (J20)  
4D  xoo3oxx4ooo&#xt  oct  co  co  eoctaco 
xoo3oxx4xxx&#xt  sirco  tic  tic  esircoatic  
xoo3oxx5ooo&#xt  ike  id  id  eikaid  
xoo3oxx5xxx&#xt  srid  tid  tid  esridatid  
oxx3ooo3xxx&#xt  tet  co  co  etetaco  
oxx3ooo4xxx&#xt  cube  sirco  sirco  ecuba sirco  
oxx4ooo3xxx&#xt  oct  sirco  sirco  eocta sirco  
oxx3ooo5xxx&#xt  doe  srid  srid  edoasrid  
oxx3xxx3ooo&#xt  oct  tut  tut  eoctatut  
oxx3xxx4ooo&#xt  co  toe  toe  ecoatoe  
oxx4xxx3ooo&#xt  co  tic  tic  ecoatic  
oxx3xxx5ooo&#xt  id  ti  ti  eidati  
oxx3xxx3xxx&#xt  tut  toe  toe  etutatoe  
oxx3xxx4xxx&#xt  tic  girco  girco  etica girco  
oxx4xxx3xxx&#xt  toe  girco  girco  etoa girco  
oxx3xxx5xxx&#xt  tid  grid  grid  etidagrid  
oxx oxx3xxx&#xt  3g  hip  hip  etripuf  
oxx oxx4xxx&#xt  4g  op  op  esquipuf  
oxx oxx5xxx&#xt  5g  dip  dip  epepuf 
This concept is meant for 3D and was invented as an infinite series of polyhedra in summer 1999 by the author. It starts with the (exterior) blend of 2 prisms, i.e. the lace tower xxxn/dooo&#xt, with n ≥ 2d (i.e. progrades). This one would be the {n/d}gonal (0,0)EEB.
Now unconnect the lacing edges, and insert triangles inbetween the lacing squares. Here generally there are 2 possibilities: either by bending the squares inward, thus looking like an exterior blend at the bottom face of 2 retrograde cupolas (or cuploids), the {n/d}gonal (exo) (1,1)EEB; or by bending the squares outward, thus looking like the corresponding blend of 2 prograde cupolas (or cuploids), {n/d}gonal (endo) (1,0)EEB.
Instead of inserting a single triangle into that lacing gap at either segment, one equally could insert k triangles each. I.e. the basevertices are [n/d,4,3^{k},4] (up to windings, see below). While at the central layer there are vertex types [3^{2},4^{2}] (for k>0) and [3^{4}] (for k>1). Again there are exo and endotypes, relating to emanating triangles to the relative outside resp. to the inside (best seen in the equatorial section); equivalently exo means that the lacing squares will bend inward, endo describes the cases where those squares bend outward. Here one also speaks of kextended (exo/endo) EEBs.
For k>1 there even are several possibilities when the sequence of the equatorial edges of a square pair, of those inserted triangle pairs, and of the next square pair winds less or more than once around their orthogonal, vertical axis. This winding number w will be the second parameter of the general {n/d}gonal (k,w)EEB. Those are restricted to 0 ≤ w ≤ k1. (In order to get a planar equatorial layer, the bending of the squares will have to be adapted accordingly.) – Here some examples are in place. The following pictures show parts of the base polygons in red, the equatorial edges between the squares in blue, and those between triangles in black.
exo (3,0)EEB, the winding KTSL around O is < 2π 
endo (3,0)EEB, the winding KTSL around O is < 2π 
endo (3,1)EEB, the winding KSTL around O is > 2π, but < 2 · 2π 
In order to calculate the height consider the internal vertex angle of the base polygon xn/do. That one is
∠AOB = π(12d/n)
For endoEEBs (with k>0) one adds 2 right angles (∠AOK, ∠LOB) plus 2πw. That total angle then will be devided into k equal parts, each being the angle sustained by any equatorial triangle edge:
∠TOS = 2π(1+wd/n)/k
On the other hand, for exoEEBs (with k>0) one has to subtract from ∠AOB those 2 right angles. But one will have to add 2π(w+1) here. Thus the total angle a posteriori will be the same, and so too each angle sustained by any equatorial triangle edge gets the same number as for the endo case.
Using unit edges and the radius of those equatorial vertex circles r = OK = OL = OT = OS within the right triangle, which is the half of TOS, one gets r sin(∠TOS/2) = 1/2. On the other hand the total height of the EEB is h = 2 sqrt(1  r^{2}). Thus (independing of exo or endo)
h( {n/d}gonal (k,w)EEB ) = sqrt[4  1/sin^{2}(π(1+wd/n)/k)]
This height formula also shows the range of n/d such that for any given value of (k,w) both the exo or endoforms of an {n/d}gonal (k,w)EEB do exist, i.e. provide a height with h>0.
Some special cases clearly are the {n/d}gonal endo (1,0)EEBs. Those are also known as (prograde) orthobicupola. The special cases n/d = 3/1, 4/1, 5/1 belong to the Johnson solids, in fact those are tobcu (J27), squobcu (J28), resp. pobcu (J30). – The {n/d}gonal exo (1,0)EEBs then are the corresponding retrograde orthobicupola, i.e. orthobicupola with top base {n/(nd)} (with n ≥ 2d).
Other special cases are the {n/d}gonal endo (2,1)EEBs. Those are also known as sphenoprisms, i.e. the connection of the bases by (pairs of) trianglesquaretriangle sphenoids. The special case n/d=2 here again is a Johnson solid, in fact the esquidpy (J15). But even within the range 2≤n/d<3 all those {n/d}gonal endo (2,1)EEBs are at least locally convex. (The limiting case n/d = 3 then would become flat.)
One even could extrapolate EEBs to retrograde {n/d}, i.e. to {n/(nd)} within the so far assumed prograde bound n ≥ 2d. This extension would switch endo and exo EEBs. For sure, the parameter k is unaffected. But with w' = kw1 one gets the identity
{n/d}gonal exo (k,w)EEB = {n/(nd)}gonal endo (k,kw1)EEB {n/d}gonal endo (k,w)EEB = {n/(nd)}gonal exo (k,kw1)EEB
which shows, that such retrograde bases not truely produce anything new.
Just as the EEBs start with the exterior blend of 2 {n/d}prisms, the EEAs, i.e. edgeexpanded antiprisms, are meant to start with the appropriate blend of 2 {n/d}antiprisms. But, in fact, this would become rather the k=1 cases. We even could start instead by a pair of {n/d}pyramids, which are mirrored at their tips. This then will become the general {n/d}gonal (0,0)EEA. Iterated insertion of triangle pairs at the lacing edges of that bipyramid produces – similar to the EEBs – a new set of {n/d}gonal exo/endo (k,w)EEAs. Here the former prismsquares (of the EEBs) clearly are replaced by the lacing antiprismtriangles. – The EEAs where found by J. McNeill.
k = 2  k = 3  k = 4  k = 5  
w = 0 
endo {7} (2,0)EEA 
endo {7} (3,0)EEA 
endo {7} (4,0)EEA  
w = 1 
endo {7} (4,1)EEA 
endo {7} (5,1)EEA 
k = 2  k = 3  k = 4  k = 5  
w = 0 
exo {7} (2,0)EEA 
exo {7} (3,0)EEA 
exo {7} (4,0)EEA 
exo {7} (5,0)EEA 
w = 1 
exo {7} (3,1)EEA 
exo {7} (4,1)EEA 
exo {7} (5,1)EEA  
w = 2 
exo {7} (5,2)EEA 
The height can be calculated along the same lines as were shown for the EEBs. The height of the {n/d}gonal endo (k,w)EEA reads as follows. (Those for the exo versions could be deduced by substituting w → w* = kw1.)
h( {n/d}gonal endo (k,w)EEA ) = sqrt[4  1/sin^{2}(π(1+wd/n)/(k+1))]
In 2004 A. Weimholt came up with an interesting set of polytopes which later became known as ursatopes (or ursulates). The 3D member of this large family, which since got lots of extensions, is teddi. This is why the whole family (which then will be forced to contain it as subdimensional boundary) was named ursatopes.
The general setup here later was mainly elaborated by W. Krieger. In fact, these polytopes derive from a cone where the edges of the bottom base become acute golden triangular sides (i.e. ox&#f) of a pyramid. From that cone one consideres first a section of lacing edge length 2f. The remainder then is a truncation of that pyramid: Chopping off the tip at half of its height, revealing thus the ursatopal bottom base as this crosssection, resp. chopping off the other vertices in such a way, that the remaining lacings get reduced from their flength down to xsized edges only, while the pyramidal bottom edges get reduced to zero, i.e. this ursatopal top face will be the rectification of the bottom one. – The new additional lacing edges introduced by the latter choppings by construction will have unit size. But the existence of the to be produced rectification as well as the to be obtained unit edge sizes all over raise some restrictions on the possible ursatopal bottom bases. And also the height of that starting pyramid would be required to be positive.
All this can be cast into the following necessary and sufficient conditions:
These then could be subject to Stott expansions within the across subsymmetry.
2D  ofx&#xt = {5}  vertex figure of s3s4o 
3D  ofx3xoo&#xt = teddi  vertex figure of s3s4o3o 
4D  ofx3xoo3ooo&#xt = tetu  vertex figure of s3s4o3o3o 
ofx3xoo4ooo&#xt = xoo3ofx3xoo&#xt = octu  vertex figure of s3s4o3o4o  
ofx3xoo5ooo&#xt = iku  diminishing of x3o3o5o  
ofx3xoo3xxx&#xt = coatutu  
ofx3xoo4xxx&#xt = sirco aticu  
ofx3xoo5xxx&#xt = sridatidu  
5D  ofx3xoo3ooo3ooo&#xt = penu  vertex figure of s3s4o3o3o3o 
xoo3ofx3xoo3ooo&#xt = rapu  
ofx3xoo3ooo4ooo&#xt = ofx3xoo3ooo *b3ooo&#xt = hexu  vertex figure of s3s4o3o3o4o  
ofx3xoo4ooo3ooo&#xt = xoo3ofx3xoo4ooo&#xt = xoo3ofx3xoo *b3xoo&#xt = icou  
ofx3xoo3xxx3ooo&#xt = sripadecu  
ofx3xoo3ooo3xxx&#xt = spidasripu  
ofx3xoo3xxx3xxx&#xt = pripal gridu  
xoo3ofx3xoo3xxx&#xt = sripal pripu  
ofx3xoo3xxx4ooo&#xt = ofx3xoo3xxx *b3xxx&#xt = ricoatahu  
ofx3xoo3ooo4xxx&#xt = sidpitha sritu  
ofx3xoo3xxx4xxx&#xt = prohagritu  
ofx3xoo4xxx3ooo&#xt = xoo3ofx3xoo4xxx&#xt = sricoacontu  
ofx3xoo4ooo3xxx&#xt = spica sricou  
ofx3xoo4xxx3xxx&#xt = pricoal gricou 
It happens that the ursatopes are orbiform in general. E.g. for the nonexpanded ursatopes one even can calculate their fulldimensional circumradius R from the subdimensional circumradius r of the quasiregular base according to: R^{2} = f^{2} (f r^{2} + 1/4) / (f^{2}  r^{2}) , where f = (1+sqrt(5))/2 = 1.618034 is the golden ratio.
The tutsatopes (or tutisms) are a quite similar genuinely bistratic polytopic family as the ursatopes. They just substitute within their role the teddies by tuts. The accordingly changed necessary and sufficient conditions then reads like this:
But this class of polytopes happens to be describable in a closed form. In fact those are nothing but the Pfirst bistratic caps of Q, where P clearly is just that quasiregular base of the tutsatope, and Q can be obtained in Dynkin symbol description from that of P by adding to that single ringed node a further leg, marked 3, the other end of which has to be ringed as well: E.g. the dotbased tutsatope ooo3oox3xux3oox3ooo&#xt is just the dotfirst bistratic cap of tim.
Again those tutsatopes could be subject to Stott expansions within the across subsymmetry. Here the implied additional ringings on the Dynkin diagram of the base layer (P) can be correspondingly transfered onto that of Q.
Therefore both, all the tutsatopes and all their expansions, trivially are orbiform in general.
2D  xux&#xt = {6}  (full) x3x = {6} 
3D  xux3xoo&#xt = tut  (full) x3x3o = tut 
4D  xux3xoo3ooo&#xt = tip  (full) x3x3o3o = tip 
xux3xoo4ooo&#xt = xoo3xux3xoo&#xt = octum  rotunda of x3x3o4o  
xux3xoo5ooo&#xt = iktum  bistratic cap of x3x3o5o  
xux3xoo3xxx&#xt = coatotum  bistratic cap of x3x3o3x  
xux3xoo4xxx&#xt = sircoa gircotum  bistratic cap of x3x3o4x  
xux3xoo5xxx&#xt = srida gridtum  bistratic cap of x3x3o5x  
5D  xux3xoo3ooo3ooo&#xt = tix  (full) x3x3o3o3o = tix 
xoo3xux3xoo3ooo&#xt = raptum  bistratic cap of x3x3o *b3o3o  
xux3xoo3ooo4ooo&#xt = xux3xoo3ooo *b3ooo&#xt = hextum  rotunda of x3x3o3o4o  
xux3xoo3xxx3ooo&#xt = sripa griptum  bistratic cap of x3x3o3x3o  
xux3xoo3ooo3xxx&#xt = spida priptum  bistratic cap of x3x3o3o3x  
xux3xoo3xxx3xxx&#xt = pripa gippidtum  bistratic cap of x3x3o3x3x  
xoo3xux3xoo3xxx&#xt = sripa gippidtum  bistratic cap of x3x3o *b3o3x  
xux3xoo3xxx4ooo&#xt = xux3xoo3xxx *b3xxx&#xt = ricoa ticotum  bistratic cap of x3x3o3x4o  
xux3xoo3ooo4xxx&#xt = sidpitha prittum  bistratic cap of x3x3o3o4x  
xux3xoo3xxx4xxx&#xt = proha gidpithtum  bistratic cap of x3x3o3x4x 
These can be considered as dissected bipyramids with a squeezed in equatorial prism. Therefore, whenever the bipyramid was Dynkin describable, the elongated one will be too, just double up the medial vertex layer symbol at each node position. I.e. whenever the (sectional) base had an x, then the pyramid would have an ox, the bipyramid an oxo, and the elongated bipyramid would require an oxxo node. (Similar for o nodes.)
For convex elongated bipyramids, which generally are external blends of segmentotopes, we have:
2D  ohho&#xt  pt  hline  hline  pt  6g 
3D  oxxo3oooo&#xt  pt  3g  3g  pt  etidpy (J14) 
oxxo4oooo&#xt  pt  4g  4g  pt  esquidpy (J15)  
oxxo5oooo&#xt  pt  5g  5g  pt  epedpy (J16)  
4D  oxxo3oooo3oooo&#xt  pt  tet  tet  pt  etedpy (?) 
oxxo3oooo4oooo&#xt  pt  oct  oct  pt  pex hex  
oxxo4oooo3oooo&#xt  pt  cube  cube  pt  ecubedpy (?)  
oxxo3oooo5oooo&#xt  pt  ike  ike  pt  eikedpy (?)  
  pt  squippy  squippy  pt  esquippidpy (?)  
  pt  pap  pap  pt  epapdapy (?)  
  pt  gyepip (J11)  gyepip (J11)  pt  egyepipdapy (?)  
  pt  mibdi (J62)  mibdi (J62)  pt  emibdidpy (?)  
  pt  teddi (J63)  teddi (J63)  pt  eteddidpy (?)  
oxxo oxxo3oooo&#xt  pt  trip  trip  pt  etripdapy (?)  
oxxo oxxo5oooo&#xt  pt  pip  pip  pt  epipdapy (?) 
In the research for n/d,n/d,3acrohedra – an acrohedron is a polyhedron containing acrons (or vertices), where acron stems from Greek ακροσ (acros, i.e. summit), as in Acropolis – M. Green found in October 2005 an 7,7,3acrohedron, which he called a supersemicupola. Based on that finding a small family of n/d,n/d,3acrohedra was set up according the generalized building rules thereof:
Clearly n/d only ranges according to 12/5 < n/d < 12, as in those extremal values the acrons would become flat.
For cases with n being even there generaly would be an easier acrohedron too. In fact, those according to Green's rule then just describes a gyrated blend of 2 such easier ones. (This is quite similar as for the cupolaic blends.)
The following list provides the known n/d,n/d,3acrohedra which follow that Green's rule.
{n/d}  Name  (related easier acrohedron)  

3/1  (degenerate): just the Grünbaumian doublecovered skinsurface of the 3fold pyramid  tetrahedron  
4/1  tutrip, "Phillips head"  trigonal prism (as digonal cupola) 

5/1  ikefaceting ike55  (none)  
5/2  sissidfaceting sissid55  (none)  
6/1  tutut (has a membrane)  truncated tetrahedron  
7/1  (small) supersemicupola (has a membrane)  (none)  
7/2  great supersemicupola  (none)  
8/1  tutic (is a tube)  truncated cube  
8/3  tuquith  quasitruncated cube  
10/1  tutid (is a tube)  truncated dodecahedron  
10/3  tuquit gissid  quasitruncated great stellated dodecahedron 
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