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In 2000 Klitzing published^{[1]} on segmentotopes in general resp. his research on convex segmentochora in special. Segmentotopes are polytopes which bow to the following conditions:
The first condition shows that the circumradius is well defined. Moreover, in union with condition 3 this implies that all 2D faces have to be regular. The sandwich type 2nd condition implies that all edges, which aren't contained completely within one of the hyperplanes, would join both, i.e. having one vertex each in either plane. Thence segmentotopes have to be monostratic. In fact, segmentotopes are the monostratic orbiforms. (Orbiforms, a terminus which had been created several years later only, just use the first and third condition of those.) Finally it follows, that the bases have to be orbiforms, while all the lacing elements have to be (lower dimensional) segmentotpes.
The naming of individual segmentotopes usually is based on the 2 base polytopes: topbase polytope atop bottombase polytope. Symbolically one uses the parallelness of those bases: P_{top}  P_{bottom}. (For the smaller segmentotopes the choice of bases need not be unique.)
Segmentotopes are closely related to lace prisms. In fact those concepts have a large common intersection, but they are not identical. While lace prisms need an axis of symmetry (in fact one which is describable as Dynkin symbol), one uses the terminus "axis" in the context of segmentotopes only in the sense of the line defined by the centers of the circumspheres of d1 dimensional bases. (Thus the topic of axial polytopes does more relate to lace towers (as multistratic generalizations of lace prisms) than to segmetotopes. None the less there is a large overlap.) – Note that the radius R of the d1 dimensional circumsphere used for a segmentotpal base of dimension dk might well be larger than the radius r of its dk dimensional circumsphere: for k > 1 the latter might well be placed within the former with some additional shift s: R^{2} = r^{2} + s^{2}. For a simple example just consider squippy, the 4fold pyramid, which is just half of an oct, when being considered as line  triangle: the subdimensional base, the single edge, there is placed offset.
In fact, let r_{k} be the individual circumradii of the bases, let s_{k} be the respective shifts away from the axis (if subdimensional), let h be the axial height and R the global circumradius, then we have the following interrelation formula between all these sizes 4 R^{2} h^{2} = ((r_{2}^{2}+s_{2}^{2})(r_{1}^{2}+s_{1}^{2}))^{2} + 2 ((r_{1}^{2}+s_{1}^{2})+(r_{2}^{2}+s_{2}^{2})) h^{2} + h^{4}.
Just as polytopes are distinguished dimensionally as polygons, polyhedra, polychora, etc.
so too are segmentotopes.
The only segmentogons (without any further adjectivic restriction) clearly are
Segmentohedra already include infinite series like
(*) Grünbaumian bottom base understood to be withdrawn.
For convex ones d clearly has to be 1 and several of the above series then become finite.
Polychora generally have the disadvantage not being visually accessible. Segmentochora are not so hard for that: Just consider the 2 base polyhedra displayed concentrically, perhaps slightly scaled, while the lacings then get slightly deformed. Mathematically speaking, axial projections (viewpoint at infinity) and central projections (finite viewpoint), either one being taken on the axis for sure, from 4D onto 3D generally are well "viewable" for such monostratic figures. Especially when using real 3D graphics (like the here used VRMLs) instead of further projections onto 2D as in usual pictures: Note that all pictures at "segmentochoron display" within individual polychoron files will be linked to such VRML files.
fore color  back color  explanation 
red 
  P_{top} 
blue 
  P_{bottom} 
gold 
  lacing edges 
  white 
spherical geometry 
  light yellow 
euclidean geometry – cf. also: decompositions 
  light green 
hyperbolic geometry 
ortho ngon  nprism  gyrated nprism  2nprism  nantiprism  

pt  np (prism pyramids) 
pt  nap (antiprism pyramids) 
point  
line  ortho ngon (scalenes) 
line  np (pyramid prisms) 
line segment  
ng  np (3,nduoprisms) 
ng  gyro np (antifastegia) 
ng  2np (magnabicupolic rings, pucofastegia) 
ng  nap (= ng  gyro np) 
ngon  
2ng  np (orthobicupolic rings, cupofastegia) 
2ng  nap (gyrobicupolic rings) 
2ngon  
np  np (4,nduoprisms) 
np  gyro np (antiprism prisms) 
np  2np (ncupola prisms) 
nprism  
2np  2np (4,2nduoprisms) 
2nprism  
nap  nap (= np  gyro np) 
nantiprism 
Just as used within Johnson solids several of the above segmentochora allow diminishings. Those generally break some symmetries of one or both bases, and therefore too of the axial symmetry. Thus there is no longer a unifying symmetry according to which a Dynkin symbol could be designed, being the premise for lace prisms. Even so, such diminished ones are fully valid segmentotopes (following the above 3 conditions). An easy 3D example could be again line  triangle, the alternate description of the above mentioned squippy. The easiest 4D one clearly is squasc (i.e. the 1/4lune of the hex). – Gyrations in some spare cases might apply as well, even so those in general would conflict to the requirement of exactly 2 vertex layers.
But there is also a generic nonlaceprismatic convex segmentochoron with octahedral symmetry at one base, with icosahedral symmetry at the other. (The common subsymmetry clearly is the pyrital. But that one does not bow to Dynkin symbol description.) That one uses as faces 1 cube (as bottom base), 6 trip (as line  square), 12 squippy (as triangle  line), 8 tet (as triangle  point), and 1 ike (as top base).

(For degenerate segmentochora, i.e. having zero heights, cf. page "finite flat complexes".)
In 2012 two sets of closely related monostratic polytopes where found:
In fact the constraint, of the lacing edges all having unit lengths too, results in some specific shift values for the base polyhedra. As those base polyhedra on the other hand are not degenerate, this conflicts to having a common circumsphere in general: within the case of pyramids only the special value n=3 results in a true convex segmentochoron (being the hex), within the case of cupolae only the special value n=2 results in a true convex segmentochoron (being trip  refl ortho trip). Even so, the values n=3,4,5 resp. n=2,3,4,5 would well generate monostratic convex regularfaced polychora (CRFs).
The closeness even could be pushed on a bit. If one of the (equivalent) bases each, which are segmentohedra in turn, would be diminished by omitting the smaller top face, i.e. maintaining therefrom only the larger bottom face, those would reenter the range of valid segmentochora again: {n}  gyro npy resp. {2n}  ncu. In fact, here the necessary shifts of the bases could be assembled at the degenerate base alone. There a nonzero shift is known to be allowed.
Without narrowing those findings, those 2 families could be demystified a bit by using different axes: the pyramidal case is nothing but the bipyramid of the nantiprism, while the cupolaic case is nothing but the bistratic lace tower xxonoxx oxo&#xt.
Later the author realised also the existance of their ortho counterparts:
Again those can be splitted at their now always prismatic pseudofacial equator For the pyramidal ones, the case n=2 becomes subdimensional, there representing nothing but the oct. The general case here is nothing but the nprismatic bipyramid. For the cupolaic case N=2 clearly is full dimensional, but pairs of contained squippies become corealmic and thus unite into octs. Therefore that case happens to become nothing but ope.
In 2014 three close relatives to pyramids have been found, which are not orbiform just because their base is not. Those are line  bilbiro, {3}  thawro, resp. {5}  pocuro. A further such case would be line  esquidpy.
Besides of considering 2 parallel hyperplanes cutting out monostratic layers of larger polychora, in 2012 (and more deeply in 2013) considerations about 2 intersecting equatorial hyperplanes were done. Those clearly cut out wedges. The specific choice of those hyperplanes made up the terminus lunae. Only the CRF ones were considered. The following lunae are known so far:
 5D 
(For degenerate segmentotera, i.e. having zero heights, cf. page "finite flat complexes".)
 6D 


 7D 


 8D 


 9D 


 10D 

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