Segmentotopes
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:
 all vertices on the surface of 1 hypersphere
 all vertices on 2 parallel hyperplanes
 all edges of 1 length
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}.
Convex Segmentochora (up)
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
 regular triangle (point  line) and
 square (line  line).
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.
Color coding of the following pictures:
red 
 
P_{top} 
blue 
 
P_{bottom} 
gold 
 
lacing edges 
 
white 
spherical geometry 
 
light yellow 
euclidean geometry – cf. also: decompositions 
 
light green 
hyperbolic geometry 
Cases with Tetrahedral Axis:
Cases with Octahedral Axis:
Cases with Icosahedral Axis:
Cases with nPrismatic or nAntiprismatic Axis:
NonLacePrismatics:
Just as already often was used within the set of Johnson solids, several of the above segmentochora allow diminishings too.
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 though, 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,
although 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).
Full Table:
(For degenerate segmentochora, i.e. having zero heights, cf. page "finite flat complexes".)
Close Relatives:

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 though, 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:
 Lunae of hex: cells being squippy and tet
 1/4luna of hex = squasc = pt  squippy
(thus a segmentochoron) – hyperplanes intersect at 90°.
 Lunae of ico: cells being tricu, oct, and
squippy
 2/6luna of ico = oct  tricu
(thus a segmentochoron) – hyperplanes intersect at 120°.
 1/6luna of ico = {3}  gyro tricu
(thus a segmentochoron) – hyperplanes intersect at 60°.
 Lunae of ex: cells being pero, peppy, and
tet
 Lunae of spid resp. gyspid: cells being tricu,
tet, and trip
 0.290215luna of spid = tet  tricu
(thus a segmentochoron) – hyperplanes intersect at arccos(1/4) = 104.477512°.
 0.209785luna of spid = {3}  tricu
(thus a segmentochoron) – hyperplanes intersect at arccos(1/4) = 75.522488°.
 0.419569luna of gyspid = {3}  tricu  {3}
– hyperplanes intersect at 2 arccos(1/4) = arccos(7/8) = 151.044976°.
 Lunae of quawros: cells being squacu, cube, tet, and trip
 Lunae of stawros: cells being pecu, pip, tet, and trip

Surely the convexity restriction could be released.
Some 3D examples can be found on the page on axials.
Some 4D cases would be e.g.

0.618034  gikepy = pt  gike = ox3oo5/2oo&#x

0.618034  sissidpy = pt  sissid = ox5/2oo5oo&#x

0.618034  stasc = line  {5/2} = xo ox5/2oo&#x

0.658240  shasc = line  {7/2} = xo ox7/2oo&#x

0.662791  gissidagike = gissid  gike = ox3oo5/2xo&#x

0.692924  gissidagid = gissid  gid = oo3ox5/2xo&#x

0.726543  gikagid = gike  gid = xo3ox5/2oo&#x

0.707107  hossdap = pseudo sissid  pseudo sissid = reduced( β2β5o5/2o , by β5o5/2o )

0.774597  firp = tet  pseudo 2thah = reduced( xx3/2oo3ox&#x , by x3/2o3x )

0.866025  "coordaxes edge star"  cube

0.879465  "thahsquares star"  cube

0.912871  "coordplanes square star"  cube

0.951057  sidtidap = sidtid  gyro sidtid = xo5/2ox3oo3*a&#x

0.951057  gidtidap = gidtid  gyro gidtid = xo5/4ox3oo3*a&#x

0.951057  ditdidap = ditdid  gyro ditdid (the blend of the latter 2)

1  sissidagad = sissid  gad = xo5/2oo5ox&#x

1  co retrocuploid = co  pseudo 2oct+6{4} = reduced( ox3/2xx4oo&#x, by .x3/2.x4.o )

1.064815  gaddadid = gad  did = xo5ox5/2oo&#x

1.064815  sissidadid = sissid  did = xo5/2ox5oo&#x

1.328131  gaddaraded = gad  raded = ox5/2oo5xx&#x

1.336349  dida raded = raded  tigid = xx5ox5/2xo&#x

1.765796  siida = siid  gyro siid = xo5/2ox3xx3*a&#x

2.363565  radeda tigid = did  tigid = ox5xx5/2oo&#x

2.497212  sissidaraded = sissid  raded = ox5oo5/2xx&#x

4.352502  diddatigid = did  tigid = ox5xx5/2oo&#x

...  n/d scalene = line  perp {n/d} = xo oxn/doo&#x

...

By means of external blends at the bases multistratic stacks can be derived, right in the
sense of lace towers. But in general, orbiformity
thereby is lost. Only few are known, which then still are. Among those clearly are the ones which can be deduced as
multistratic segments of uniform or even scaliform polytopes. But there are
also rare exceptional finds beyond those, e.g.

sidrebcu = sissid  (pseudo) did  gad = xoo5/2oxo5oox&#xt

...
Further Reading
 ↑
Convex Segmentochora – (PDF)
published as: "Convex Segmentochora", by Dr. R. Klitzing, Symmetry: Culture and Science, vol. 11, 139181, 2000

external link (hosted by J. McNeill)

5D

(Just some) Convex Segmentotera (up)

1.302772

pepip = pedip  pedip (uniform)

1.307032

tratut = tut  tuttip (uniform)

1.322876

hacube = shiddip  shiddip (uniform)
rittip = rit  rit (uniform)
tutcupip = tuttip  inv tuttip (segment of rittip)
tutcupip = tuta  tuta (segment of rittip)
tepeatuttip = tepe  tuttip (segment of rittip)
tepeatuttip = tetatut  tetatut (segment of rittip)

1.360147

tippip = tip  tip (uniform)

1.369306

squatut = tuttip  tuttip (uniform)

1.414214

pennatip = pen  tip (segment of rin)
tipadeca = tip  deca (segment of rin)
octatuttip = oct  tuttip (segment of sibrid)
sripadeca = srip  deca (half of sibrid)
hatricu = thiddip  hiddip (half of haco)

1.442951

otet = {8}  todip (uniform)
otet = op  lacingortho op (uniform)

1.462497

trasnic = snic  sniccup (uniform)

1.485633

owoct = todip  {3}gyro todip (uniform)

1.5

decap = deca  deca (uniform)

1.502958

rita sidpith = rit  sidpith

1.513420

trasirco = sirco  sircope (uniform)

1.515539

tradoe = doe  dope (uniform)

1.518409

squasnic = sniccup  sniccup (uniform)

1.567516

ocube = sodip  sodip (uniform)
squasirco = sircope  sircope (uniform)
sidpithip = sidpith  sidpith (uniform)
squicuffip = squicuf  squicuf (wedge of sidpithip)
squicuffip = tes  op (wedge of sidpithip)

1.569562

squadoe = dope  dope (uniform)

1.620185

ritag thex = rit  gyro thex (segment of sirhin)
thexa = thex  gyro thex (scaliform,
segment of sirhin)
teta tuttip = tet  tuttip (segment of sirhin)
rapadeca = rap  deca (segment of sirhin)
deca aprip = deca  prip (segment of sirhin)
tutcupa toe = tuta  toe (wedge of sirhin)

1.632993

tipalprip = tip  inv prip (segment of cappix)
tuttipa toe = tuttip  toe (wedge of cappix)

1.658312

triddippa hiddip = trddip  hiddip (segment of card)
pripa = prip  inv prip (segment of card)
sripaprip = srip  prip (segment of card)
thexip = thex  thex (uniform)

1.683251

tratoe = toe  tope (uniform)

1.688194

prippip = prip  prip (uniform)
tuttipa tope = tuttip  tope (segment of prippip)
tuttipa tope = tutatoe  tutatoe (segment of prippip)

1.693527

exip = ex  ex (uniform)
gappip = gap  gap (uniform)
sadip = sadi  sadi (uniform)

1.717954

trid = id  iddip (uniform)

1.732051

squatoe = tope  tope (uniform)
icathex = ico  thex (segment of sart)
rico  thex (segment of sart)
sripalprip = srip  inv prip (segment of sart)
trashiddip = {3}  shiddip (segment of sart)
cotut totric = co  tutatoe (wedge of sart)
cotut totric = coatut  toe (wedge of sart)
cotut totric = tut  coatoe (wedge of sart)

1.738546

rico  sadi

1.765796

squid = iddip  iddip (uniform)

1.778824

ricoa = rico  gyro rico (scaliform)

1.802776

ricope = rico  rico (uniform)
copatope = cope  tope (segment of ricope)
copatope = coatoe  coatoe (segment of ricope)

1.847759

ica sidpith = ico  sidpith
ricasrit = rico  srit
ricoaspic = rico  spic

1.848423

pripalgrip = prip  gyro grip (segment of pattix)
tutatope = tut  tope (segment of pattix)

1.870173

tratic = tic  ticcup (uniform)

1.870829

ritarico = rit  rico (segment of spat)
spidaprip = spid  prip (segment of spat)
pripagrip = prip  grip (segment of spat)
gripa = grip  inv grip (segment of spat)
copatoe = cope  toe (wedge of spat)

1.910497

grippip = grip  grip (uniform)

1.914214

squatic = ticcup  ticcup (uniform)
spiccup = spic  spic (uniform)
srittip = srit  srit (segment of span)
sidpith  srit (segment of span)

1.914854

deca agrip = deca  grip (segment of pirx)

1.994779

ofx3xoo4ooo&#xt  o3o4x

2.117085

twacube = sitwadip  sitwadip (uniform)

2.150581

taha = tah  gyro tah (scaliform,
wedge of hejak)
hinarin = hin  rin (segment of hejak)
tipagrip = tip  grip

2.179449

tahp = tah  tah (uniform)

2.207107

tattip = tat  tat (uniform)
srit  tat (segment of sirn)
rap  tip (segment of sirn)
sirco  ticcup (segment of sirn)
{4}  todip (segment of sirn)

2.231808

trasnid = snid  sniddip (uniform)

2.236068

ricatah = rico  tah (segment of sibrant)
sripagrip = srip  grip (segment of sibrant)
coatope = co  tope (wedge of sibrant)

2.268840

squasnid = sniddip  sniddip (uniform)

2.291288

gippiddip = gippid  gippid (uniform)

2.306383

trasrid = srid  sriddip (uniform)

2.327373

gripagippid = grip  gippid (segment of cograx)

2.342236

squasrid = sriddip  sriddip (uniform)

2.371708

thexagtah = thex  gyro tah

2.388442

tragirco = girco  gircope (uniform)

2.423081

squagirco = gircope  gircope (uniform)
prittip = prit  prit (uniform)

2.544388

trati = ti  tipe (uniform)

2.576932

squati = tipe  tipe (uniform)

2.632865

ritasrit = rita  srit
sricoa = srico  gyro srico (scaliform)

2.647378

tico  prissi

2.660531

prohp = proh  proh (uniform,
segment of carnit)
prit  proh (segment of carnit)
prit  srit (segment of carnit)
hodip  tisdip (segment of carnit)
sricope = srico  srico (uniform)

2.692582

ticope = tico  tico (uniform)

2.738613

pripa gippid = prip  gippid (segment of pattit)

2.878460

proh  tat (segment of capt)

3

tahatico = tah  tico (segment of pirt)

3.025056

tratid = tid  tiddip (uniform)

3.047217

grittip = grit  grit (segment of prin)

3.052479

squatid = tiddip  tiddip (uniform)

3.118034

roxip = rox  rox (uniform)

3.239235

prohagrit = proh  grit (segment of pattin)

3.450631

contip = cont  cont (uniform)

3.522336

pricoa = prico  gyro prico (scaliform)

3.534493

pricope = prico  prico (uniform)
gidpithip = gidpith  gidpith (uniform,
segment of cogart)

3.736068

hipe = hi  hi (uniform)

3.845977

tragrid = grid  griddip (uniform)

3.867584

squagrid = griddip  griddip (uniform)

3.988340

gritta gidpith = grit  gidpith (segment of cogrin)

4.311477

gricoa = grico  gyro grico (scaliform)

4.328427

gricope = grico  grico (uniform)

4.562051

rahipe = rahi  rahi (uniform)

4.670365

thipe = thi  thi (uniform)

4.749980

texip = tex  tex (uniform)

4.776223

grixip = grix  grix (uniform)

5.194028

gippiccup = gippic  gippic (uniform)

5.259887

sidpixhip = sidpixhi  sidpixhi (uniform)

6.094140

srixip = srix  srix (uniform)

6.753568

srahip = srahi  srahi (uniform)

7.596108

xhip = xhi  xhi (uniform)

8.294035

prahip = prahi  prahi (uniform)

9.757429

prixip = prix  prix (uniform)

11.263210

grahip = grahi  grahi (uniform)

12.796423

gidpixhip = gidpixhi  gidpixhi (uniform)

...

3,ndippip = np  4,ndip (uniform)
3,ndippip = 3,ndip  3,ndip (uniform)

...

n,cubedip = 4,ndip  4,ndip (uniform)

...

n,mdippip = n,mdip  n,mdip (uniform)

...

n,mdafup = n,mdip  bidual n,mdip (scaliform)


(For degenerate segmentotera, i.e. having zero heights, cf. page "finite flat complexes".)
Some nonconvex segmentotera would be:

0.623054  stadow = pentagram  fully perp. pentagram (aka: star disphenoid)

0.636010  sissidisc = line  fully perp. sissid (aka: sissid scalene)

0.674163  shadow = small heptagram  fully perp. small heptagram (aka: small heptagrammic disphenoid)

0.680827  gashia = gashi  dual gashi (aka: gashi antiprism)

0.951057  gadtaxhiap = gadtaxhi  alt. gadtaxhi (aka: gadtaxhi alterprism, uniform)

0.951057  gadtaxadiap = gadtaxady  alt. gadtaxady (aka: gadtaxady alterprism)

1.074481  sishiap = narrower sishi  sishi

1.248606  ragashia = ragashi  inv ragashi (aka: ragashi alterprism)

1.618034  sirgashia = sirgashi  inv sirgashi (aka: sirgashi alterprism)

1.693527  gohip = gohi  gohi

1.917564  righia = righi  inv righi (aka: righi alterprism)

2.321762  sidtaxhiap = sidtaxhi  alt. sidtaxhi (aka: sidtaxhi alterprism, uniform)

2.321762  sirdtaxadiap = sirdtaxady  alt. sirdtaxady (aka: sirdtaxady alterprism)

3.404434  sirghia = sirghi  inv sirghi (aka: sirghi alterprism)

3.855219  stut phiddixa = stut phiddix  alt. stut phiddix (aka: stut phiddix alterprism)

4.923348  pirghia = pirghi  inv pirghi (aka: pirghi alterprism)

5.345177  wavhiddixa = wavhiddix  alt. wavhiddix (aka: wavhiddix alterprism)

6.893126  sphiddixa = sphiddix  alt. sphiddix (aka: sphiddix alterprism)

6D

(Just some) Convex Segmentopeta (up)
Some nonconvex segmentopeta would be:

7D

(Just some) Convex Segmentoexa (up)
Some nonconvex segmentoexa would be:

0.831254  sissiddow = sissid  fully perp. sissid (aka: sissid disphenoid)

8D

(Just some) Convex Segmentozetta (up)

9D

(Just some) Convex Segmentoyotta (up)

10D

(Just some) Convex Segmentoxenna (up)