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Segmentotopes

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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: top-base polytope atop bottom-base polytope. Symbolically one uses the parallelness of those bases: Ptop || Pbottom. (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 d-1 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 d-1 dimensional circumsphere used for a segmentotpal base of dimension d-k might well be larger than the radius r of its d-k dimensional circumsphere: for k > 1 the latter might well be placed within the former with some additional shift s:   R2 = r2 + s2. For a simple example just consider squippy, the 4-fold pyramid, which is just half of an oct, when being considered as line || triangle: the subdimensional base, the single edge, there is placed off-set.

In fact, let rk be the individual circumradii of the bases, let sk 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 R2 h2 = ((r22+s22)-(r12+s12))2 + 2 ((r12+s12)+(r22+s22)) h2 + h4.


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

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 (view-point at infinity) and central projections (finite view-point), 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:
fore color back color explanation

red

 -

Ptop

blue

 -

Pbottom

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 n-Prismatic or n-Antiprismatic Axis:
ortho n-gon n-prism gyrated n-prism 2n-prism n-antiprism  
  pt || n-p
(prism pyramids)		     pt || n-ap
(antiprism pyramids)		 point
line || ortho n-gon
(scalenes)		 line || n-p
(pyramid prisms)		       line segment
  n-g || n-p
(3,n-duoprisms)		 n-g || gyro n-p
(antifastegia)		 n-g || 2n-p
(magnabicupolic rings,
pucofastegia)		 n-g || n-ap
(= n-g || gyro n-p)		 n-gon
  2n-g || n-p
(orthobicupolic rings,
cupofastegia)		     2n-g || n-ap
(gyrobicupolic rings)		 2n-gon
  n-p || n-p
(4,n-duoprisms)		 n-p || gyro n-p
(antiprism prisms)		 n-p || 2n-p
(n-cupola prisms)		   n-prism
      2n-p || 2n-p
(4,2n-duoprisms)		   2n-prism
        n-ap || n-ap
(= n-p || gyro n-p)		 n-antiprism
Non-Lace-Prismatics:

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 1/4-lune 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 non-lace-prismatic 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:
CircumradiusSegmentochoron
0.632456 
K-4.1   pen = pt || tet        (regular)
K-4.1.1     = line || perp {3}
0.707107 
K-4.2   hex = tet || dual tet     (regular)

K-4.3   octpy = pt || oct         (segment of hex)
K-4.3.1       = {3} || gyro tet

K-4.4   squasc = pt || squippy    (luna of hex)
K-4.4.1        = line || tet
K-4.4.2        = {3} || incl {3}
K-4.4.3        = line || perp {4}
0.774597 
K-4.5   rap = tet || oct            (uniform)

K-4.6   traf = tet || squippy       (segment of rap)
K-4.6.1      = {3} || oct
K-4.6.2      = {3} || gyro trip

K-4.7   trippy = line || squippy    (segment of rap)
K-4.7.1        = {3} || tet
K-4.7.2        = pt || trip
K-4.7.3        = {3} || otho {4}
   (where 1 {3}-edge || 2 {4}-edges)

K-4.8   bidrap = {3} || squippy     (wedge of rap)
K-4.8.1        = {4} || tet
K-4.8.2        = line || ortho trip
0.790569 
K-4.9   tepe = tet || tet        (uniform)
K-4.9.1      = line || para trip
K-4.9.2      = {4} || ortho {4}
0.816497 
K-4.10   triddip = {3} || trip         (uniform)

K-4.11   ope = oct || oct              (uniform)
K-4.11.1     = trip || gyro trip

K-4.12   squippyp = squippy || squippy (segment of ope)
K-4.12.1          = {4} || trip
K-4.12.2          = line || cube

K-4.13   digytpuf = trip || refl ortho trip (gyrated ope)
0.879465 
K-4.14   squaf = {4} || squap     (wedge of oct || cube)
K-4.14.1       = {4} || gyro cube

K-4.15   octacube = oct || cube

K-4.16   squippy || gyro cube

K-4.17   squappy = {4} || gyro squippy
K-4.17.1         = pt || squap
0.866025 
K-4.18   tisdip = trip || trip (uniform)
K-4.18.1        = {4} || cube
0.962692 
K-4.19   squappip = squap || squap    (uniform)
K-4.19.1          = cube || gyro cube
1 
K-4.20   tes = cube || cube       (regular)

K-4.21   cubaike = cube || ike

K-4.22   paf = {5} || pap
K-4.22.1     = {5} || gyro pip

K-4.23   tetaco = tet || co       (segment of spid)

K-4.24   tet || tricu             (luna of spid)

K-4.25   tricuf = {3} || tricu    (luna of spid)
K-4.25.1        = {6} || trip

K-4.26   cubpy = {4} || squippy   (segment of ico)
K-4.26.1       = pt || cube

K-4.27   traw = {3} || gyro tricu (luna of ico)
K-4.27.1      = {6} || oct

K-4.28   bidoctaco = {4} || co    (wedge of ico)

K-4.29   octaco = oct || co       (segment of ico)

K-4.30   octatricu = oct || tricu (luna of ico)

K-4.31   squippiaco = squippy || co

K-4.32   squippiatricu = squippy || tricu

K-4.33   trateddi = {3} || teddi
1.028076 
K-4.34   trapedip = {5} || pip (uniform)
1.031784 
K-4.35   cubaco = cube || co
1.074481 
K-4.36   ipe = ike || ike             (uniform)

K-4.37   gyepippip = gyepip || gyepip

K-4.38   peppyp = peppy || peppy
K-4.38.1        = line || pip

K-4.39   pappip = pap || pap
K-4.39.1        = pip || gyro pip

K-4.40   mibdip = mibdi || mibdi

K-4.41   teddipe = teddi || teddi
1.106168 
K-4.42   squipdip = pip || pip (uniform)
1.118034 
K-4.43   cope = co || co          (uniform)

K-4.44   tobcupe = tobcu || tobcu

K-4.45   tricupe = tricu || tricu
K-4.45.1         = trip || hip
1.130454 
K-4.46   haf = {6} || hap
K-4.46.1     = {6} || gyro hip
1.154701 
K-4.47   thiddip = {6} || hip (uniform)
1.183216 
K-4.48   coatut = co || tut

K-4.49   tobcuatut = tobcu || tut

K-4.50   tricuatut = tricu || tut

K-4.51   tripuf = {6} || tricu
K-4.51.1        = {3} || hip

K-4.52   octatut = oct || tut
1.197085 
K-4.53   happip = hap || hap      (uniform)
K-4.53.1        = hip || gyro hip
1.224745 
K-4.54   shiddip = hip || hip  (uniform)

K-4.55   tuta = tut || inv tut (segment of rit)

K-4.56   tetatut = tet || tut  (segment of rit)
1.274755 
K-4.57   tuttip = tut || tut (uniform)
1.409438 
K-4.58   oaf = {8} || oap
K-4.58.1     = {8} || gyro op
1.428440 
K-4.59   todip = {8} || op (uniform)
1.433724 
K-4.60   sniccup = snic || snic (uniform)
1.447009 
K-4.61   coasirco = co || sirco

K-4.62   coaescu = co || escu

K-4.63   coaop = co || op

K-4.64   squaw = {4} || gyro squacu
K-4.64.1       = {8} || squap
1.463603 
K-4.65   oappip = oap || oap    (uniform)
K-4.65.1        = op || gyro op (uniform)
1.48563 
K-4.66   sircope = sirco || sirco  (uniform)

K-4.67   esquigybcupe = esquigybcu || esquigybcu

K-4.68   escupe = escu || escu

K-4.69   squacupe = squacu || squacu
K-4.69.1          = cube || op

K-4.70   sodip = op || op          (uniform)

K-4.71   cubasirco = cube || sirco (segment of sidpith)

K-4.72   cube || esquibcu

K-4.73   squicuf = {4} || squacu   (wedge of sidpith)
K-4.73.1         = {8} || cube
1.487792 
K-4.74   dope = doe || doe (uniform)
1.582890 
K-4.75   sircoatoe = sirco || toe
1.612452 
K-4.76   tutatoe = tut || toe (segment of prip)
1.618034 
K-4.77   doaid = doe || id

K-4.78   ikadoe = ike || doe        (segment of ex)

K-4.79   gyepip || doe

K-4.80   pappy = {5} || gyro peppy
K-4.80.1       = pt || pap

K-4.81   papadoe = pap || doe

K-4.82   mibdiadoe = mibdi || doe

K-4.83   teddi adoe = teddi || doe

K-4.84   ikepy = pt || ike          (segment of ex)

K-4.85   gyepippy = pt || gyepip

K-4.86   peppypy = pt || peppy      (wedge of ex)
K-4.86.1         = line || perp {5}

K-4.87   mibdipy = pt || mibdi

K-4.88   teddipy = pt || teddi
CircumradiusSegmentochoron
1.658312 
K-4.89   tope = toe || toe (uniform)
1.693527 
K-4.90   iddip = id || id (uniform)

K-4.91   pobrope = pobro || pobro

K-4.92   perope = pero || pero
1.702385 
K-4.93   daf = {10} || dap
K-4.93.1     = {10} || gyro dip
1.717954 
K-4.94   tradedip = {10} || dip (uniform)
1.732051 
K-4.95   coatoe = co || toe (segment of rico)
1.747560 
K-4.96   dappip = dap || dap      (uniform)
K-4.96.1        = dip || gyro dip
1.765796 
K-4.97   squadedip = dip || dip (uniform)
1.785406 
K-4.98   toatic = toe || tic
1.847759 
K-4.99    ticcup = tic || tic      (uniform)

K-4.100   sircoatic = sirco || tic (segment of srit)

K-4.101   esquigybcu || tic

K-4.102   sircoagytic = sirco || gyro tic

K-4.103   escu || tic

K-4.104   escu || gyro tic

K-4.105   squipuf = {8} || squacu  (segment of srit)
K-4.105.1         = {4} || op

K-4.106   opatic = op || tic       (segment of srit)

K-4.107   octasirco = oct || sirco (segment of spic)

K-4.108   squippy || escu

K-4.109   squippy || squacu
2.213060 
K-4.110   sniddip = snid || snid (uniform)
2.288246 
K-4.111   sriddip = srid || srid    (uniform)

K-4.112   gyriddip = gyrid || gyrid

K-4.113   pabgyriddip = pabgyrid || pabgyrid

K-4.114   mabgyriddip = mabgyrid || mabgyrid

K-4.115   tagyriddip = tagyrid || tagyrid

K-4.116   diriddip = dirid || dirid

K-4.117   pecupe = pecu || pecu
K-4.117.1        = pip || dip

K-4.118   pagydriddip = pagydrid || pagydrid

K-4.119   magydriddip = magydrid || magydrid

K-4.120   bagydriddip = bagydrid || bagydrid

K-4.121   pabidriddip = pabidrid || pabidrid

K-4.122   mabidriddip = mabidrid || mabidrid

K-4.123   gybadriddip = gybadrid || gybadrid

K-4.124   tedriddip = tedrid || tedrid
2.370932 
K-4.125   gircope = girco || girco (uniform)
2.485450 
K-4.126   sridati = srid || ti
2.527959 
K-4.127   tipe = ti || ti (uniform)
2.613126 
K-4.128   ticagirco = tic || girco (segment of proh & srico)

K-4.129   coatic = co || tic       (segment of srico)
3.011250 
K-4.130   tiddip = tid || tid (uniform)
3.077684 
K-4.131   idasrid = id || srid    (segment of rox)

K-4.132   id || drid

K-4.133   paw = {5} || gyro pecu  (wedge of rox)
K-4.133.1     = {10} || pap

K-4.134   id || pabidrid

K-4.135   id || mabidrid

K-4.136   id || tedrid

K-4.137   ikaid = ike || id       (segment of rox)

K-4.138   gyepip || id

K-4.139   peppia pero = peppy || pero

K-4.140   gyepipa pero = gyepip || pero

K-4.141   pippy = {5} || peppy    (segment of rox)
K-4.141.1       = pt || pip

K-4.142   papaid = pap || id      (wedge of rox)

K-4.143   mibdi aid = mibdi || id (wedge of rox)

K-4.144   pap || pero

K-4.145   mibdi || pero

K-4.146   {5} || pero             (wedge of rox)

K-4.147   teddi aid = teddi || id (wedge of rox)

K-4.148   teddi || pero
3.498949 
K-4.149   toagirco = toe || girco (segment of prico)
3.835128 
K-4.150   griddip = grid || grid (uniform)

K-4.151   tiatid = ti || tid
5.236068 
K-4.152   doasrid = doe || srid (segment of sidpixhi)

K-4.153   doe || drid

K-4.154   pecuf = {5} || pecu   (wedge of sidpixhi)
K-4.154.1       = {10} || pip

K-4.155   doa pabidrid = doe || pabidrid

K-4.156   doe || mabidrid

K-4.157   doe || tedrid
6.073594 
K-4.158   idati = id || ti (segment of srix)
6.735034 
K-4.159   sridatid = srid || tid (segment of srahi)

K-4.160   gyrid || tid

K-4.161   pabgyrid || tid

K-4.162   mabgyrid || tid

K-4.163   tagyrid || tid

K-4.164   drid || tid

K-4.165   pepuf = {10} || pecu   (segment of srahi)
K-4.165.1       = {5} || dip

K-4.166   pagydrid || tid

K-4.167   magydrid || tid

K-4.168   bagydrid || tid

K-4.169   pabidrid || tid

K-4.170   mabidrid || tid

K-4.171   gybadrid || tid

K-4.172   tedrid || tid
9.744610 
K-4.173   tidagrid = tid || grid (segment of prix)
... 
K-4.174   n-af = {n} || n-ap
K-4.174.1      = {n} || gyro n-p
... 
K-4.175   3,n-dip = {n} || n-p (uniform)
... 
K-4.176   n-appip = n-ap || n-ap    (uniform)
K-4.176.1         = n-p || gyro n-p
... 
K-4.177   4,n-dip = n-p || n-p (uniform)

(For degenerate segmentochora, i.e. having zero heights, cf. page "finite flat complexes".)


Close Relatives:

  1. 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 regular-faced 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 re-enter the range of valid segmentochora again: {n} || gyro n-py resp. {2n} || n-cu. In fact, here the necessary shifts of the bases could be assembled at the degenerate base alone. There a non-zero shift is known to be allowed.

    Without narrowing those findings, those 2 families could be de-mystified a bit by using different axes: the pyramidal case is nothing but the bipyramid of the n-antiprism, while the cupolaic case is nothing but the bistratic lace tower xxo-n-oxx 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 n-prismatic 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.

  2. 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.

  3. 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:

  4. 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.
    • 0.591923 - gissidpy = pt || gissid = oo3oo5/2ox&#x
    • 0.613502 - quithpy = pt || quith = oo3ox4/3ox&#x
    • 0.618034 - gikepy = pt || gike = ox3oo5/2oo&#x
    • 0.618034 - sissidpy = pt || sissid = ox5/2oo5oo&#x
    • 0.618034 - stasc = line || perp {5/2} = xo ox5/2oo&#x
    • 0.618034 - gissidragike = narrower gissid || gike = ox3oo5/3xo&#x
    • 0.618034 - shallower gissidasissid = gissid || sissid (with lacing starps)
    • 0.621876 - ogasc = line || perp {8/3} = xo ox4/3ox&#x
    • 0.658240 - shasc = line || perp {7/2} = xo ox7/2oo&#x
    • 0.662791 - gissidagike = taller gissid || gike = ox3oo5/2xo&#x
    • 0.662791 - taller gissidasissid = gissid || sissid (with lacing staps)
    • 0.662791 - stapepy = point || stap = oxo5/2oox&#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 )
    • 0.707107 - starpglassit = stappy || inv stappy = xooo5/2ooxo&#xr
    • 0.707107 - tho = tet || dual pseudo tet = hemi( x3o3/2o3o3*a )
    • 0.707107 - fitetaltet = faceted tet || dual tet = reduced( xx3/2ox3oo&#x , by x3/2x )
    • 0.739539 - quith || "thah-squares star" = reduced( xo4/3xx3/2ox&#x , by oct )
    • 0.774597 - tutrippapy = pt || tutrip = reduced( oox4/2oxx&#x , by ..x4/2..x )
    • 0.774597 - tubidrap = pseudo {4} || so = reduced( oxx4/2xox&#x , by ..x4/2..x )
    • 0.774597 - firp = tet || pseudo 2thah = reduced( xx3/2oo3ox&#x , by x3/2o3x )
    • 0.774597 - fitetaoct = faceted 3tet || oct = reduced( ox3/2xx3oo&#x , by x3/2x )
    • 0.774597 - hotetahoct = pseudo tet || pseudo oct = reduced( xx3/2ox3oo3*a&#x , by x3/2x )
    • 0.774597 - pafirp = tet || "thah-squares star" = reduced( xx3/2xo3ox&#x , by x3/2x )
    • 0.774597 - {3} || pseudo gyro trip = reduced( ox xx3/2ox&#x , by x3/2x )
    • 0.780850 - tistadip = {5/2} || stip = x3o x5/2o
    • 0.790569 - tutepe = pseudo {4} || {4/2}-prism = reduced( ox xo4/2xx&#x , by x.4/2x. )
    • 0.791345 - tistodip = {8/3} || stop = x3o x8/3o
    • 0.825168 - stappip = stap || stap = x s2s5/2s = narrower stip || stip = xx xo5/2ox&#x
    • 0.866025 - "coord-axes edge star" || cube
    • 0.874032 - stacupe = stip || 2pip = xx5/2ox&#x
    • 0.874032 - quit sissidagird = quit sissid || gird = reduced( ox5/4xx5/3xx&#x , by x5/4x )
    • 0.879465 - "thah-squares star" || cube
    • 0.881132 - sistadip = stip || stip = x4o x5/2o
    • 0.890446 - cubaquerco = cube || querco = ox3/2oo4xx&#x
    • 0.890446 - sistodip = stop || stop = x4o x8/3o
    • 0.912871 - "coord-planes 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              - sidtidpy = pt || sidtid = xo3oo3oo5/2*b&#x
    • 1              - gidtidpy = pt || gidtid = xo3oo3/2oo5*b&#x
    • 1              - ditdidpy = pt || ditdid = xo5oo3/2oo5/2*b&#x
    • 1              - tetacho = tet || cho = reduced( ox3/2ox3xx&#x , by x3/2x )
    • 1              - hotet aoho = pseudo 2tet || oho = reduced( xx3/2oo3ox3*a&#x , by x.3/2o.3o.3*a )
    • 1              - sissidagad = sissid || gad = xo5/2oo5ox&#x
    • 1              - co retro-cuploid = "thah-squares star" || co = reduced( ox3/2xx4oo&#x, by .x3/2.x4.o )
    • 1              - hoctaoho = pseudo 2oct || oho = reduced( xx3/2xo3ox3*a&#x, by x.3/2x.3o.3*a )
    • 1              - octacho = oct || cho = reduced( ox3/2xx3ox&#x, by x3/2x )
    • 1              - hothahacho = pseudo thah || cho = reduced( xx3/2ox3xx&&#x, by x3/2x )
    • 1              - gikaike = gike || ike
    • 1              - sirgikaike (edge-faceting of gikaike)
    • 1              - girgikaike (edge-faceting of gikaike)
    • 1              - cubagike = cube || gike
    • 1.064815 - gaddadid = gad || did = xo5ox5/2oo&#x
    • 1.064815 - sissidadid = sissid || did = oo5ox5/2xo&#x
    • 1.082392 - quithaquitco = quith || quitco = ox3xx4/3xx&#x
    • 1.120533 - tiggia gaquatid = tiggy || gaquatid = xx3xx5/3ox&#x
    • 1.136572 - gidtidagiid = gidtid || giid = oo3/2ox3xx5*a&#x
    • 1.183216 - ohoa hotut = oho || pseudo 2tut = reduced( xx3/2xo3xx3*a&#x, by x.3/2x.3o.3*a )
    • 1.183216 - choatut = cho || tut = reduced( ox3/2xx3xx&#x, by x3/2x )
    • 1.328131 - gaddaraded = gad || raded = xx5oo5/2ox&#x
    • 1.336349 - dida raded = raded || tigid = xx5ox5/2xo&#x
    • 1.414214 - didadoe = did || doe = reduced( oo5xx5/2ox&#x, by x5/2x )
    • 1.414214 - goccoa cotco = gocco || cotco = xx3ox4xx4/3*a&#x
    • 1.432173 - radedadoe = raded || doe = reduced( ox5xo5/2xx&#x , by x5/2x )
    • 1.618034 - gadadoe = gad || doe
    • 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
    • 3.835128 - sidtidasiid = sidtid || siid = xx5/2oo3ox3*a&#x
    • 4.352502 - diddatigid = did || tigid = ox5xx5/2oo&#x
    • 4.684471 - gidditdid aidtid = gidditdid || idtid = xx5/3xx3ox5*a&#x
    • ...             - n/d scalene = line || perp {n/d} = xo ox-n/d-oo&#x
    • ...
  5. 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



----
 5D
----

(Just some) Convex Segmentotera   (up)

CircumradiusSegmentoteron
0.645497 
hix = pt || pen        (regular)
    = line || perp tet
    = {3} || perp {3}
0.707107 
tac = pen || dual pen     (regular)

hexpy = pt || hex         (segment of tac)

octasc = line || perp oct (lune of tac)

squete = {3} || perp {4}  (wedge of tac)
0.790569 
hin = hex || gyro hex         (uniform)

rappy = pt || rap             (segment of hin)

trafpy = pt || traf           (wedge of hin
                               segment of rappy)

bidrappy = pt || bidrap       (segment of hin)

trippasc = line || perp trip  (segment of hin)

editetaf = line || tepe       (segment of hin)

tetaf = tet || inv tepe       (segment of hin)
      = tet || hex

ditetaf = {3} || gyro tepe    (wedge of hin)
        = pen || gyro trip

squasquasc = {4} || squasc    (wedge of hin)

tedhin = {4} || hex           (wedge of hin)

bidhin = oct || hex           (wedge of hin)
       = tet || inv rap

dihin = pen || inv rap        (segment of hin)

nibdihin = pen || inv traf    (wedge of hin)

triddaf = trip || bidual trip (wedge of hin)
0.806226 
penp = pen || pen        (uniform)
     = line || tepe
     = {4} || ortho trip
0.816497 
rix = pen || rap                    (uniform)

tepepy = pt || tepe                 (segment of rix)

octatepe = oct || tepe              (segment of rix)

bidrix = pen || bidrap              (segment of rix)
       = squippy || ortho tepe

tedrix = {4} || lacing-ortho ( {4} || lacing-ortho {4} )
       = {4} || ortho tepe          (wedge of rix)

trial triddip = {3} || gyro triddip (segment of rix)
0.841625 
tratet = tet || tepe               (uniform)
       = {3} || triddip
       = trip || lacing-ortho trip
0.866025 
hexip = hex || hex                      (uniform)
      = tepe || inv. tepe

octpyp = line || ope                    (segment of hexip)
       = octpy || octpy

squascop = {4} || part. ortho cube      (segment of hexip)
         = squasc || squasc

dot = rap || inv rap                    (uniform)

triddippy = pt || triddip               (segment of dot)

tetcubedaw = {2}-ap || axis-ortho {4}-p (lune of dot)

teta ope = tet || ope                   (segment of dot)
         = oct || rap

taope = {3} || ope                      (wedge of dot)

tridafup = triddip || bidual triddip    (segment of dot)

teddot = bidrap || gyro-inv bidrap      (diminishing of dot)
0.895420 
squiddaf = {4}-prism || bidual {4}-prism (scaliform)

cubaope = cube || ope
0.912871 
troct = oct || ope              (uniform)
      = triddip || gyro triddip

tisquippy = {3} || tisdip       (segment of troct)
          = trip || triddip
0.921954 
rappip = rap || rap          (uniform)
       = ope || tepe

traffip = trip || inv tisdip (segment of rappip)

tepacube = tepe || cube      (wedge of rappip)
0.935414 
squatet = tepe || tepe              (uniform)
        = {4} || tisdip
        = cube || lacing-ortho cube
0.957427 
tratrip = trip || tisdip     (uniform)
        = triddip || triddip
1 
squoct = ope || ope                         (uniform)
       = tisdip || {3}-gyro tisdip

squasquippy = {4} || tes                    (segment of squoct)

cubasquasc = cube || squasc                 (segment of traltisdip)

traltisdip = {3} || gyro tisdip

tisdipah = tisdip || {6}

hexaico = hex || ico                        (segment of rat)

rapaspid = rap || spid                      (segment of rat)

opepy = pt || ope                           (segment of rat)

squippyippy = pt || squippyp                (segment of rat)
            = line || cubpy

hexaco = hex || co                          (lune of rat)

rapaco = rap || co                          (lune of rat)

octpy || octaco                             (lune of rat)

opeaco = ope || co                          (segment of rat)

opeah = ope || {6}                          (wedge of rat)

tripgytricudaw = trip || ortho-gyro tricu   (wedge of rat)

oct-first lune of rat = rap || tetaco       (lune of rat)

tet-first lune of rat = hex || octaco       (lune of rat)
                      = rap || alt. tetaco

penatrip = pen || trip                      (wedge of scad)

penaspid = pen || spid                      (segment of scad)

tepaco = tepe || co                         (lune of scad)

pexhix = trip || axis-ortho base-para tricu (wedge of scad)
1.014647 
piddaf = pip || bidual pip
1.040833 
tracube = cube || tes      (uniform)
        = tisdip || tisdip
1.048144 
petet = {5} || trapedip         (uniform)
      = pip || lacing-ortho pip
1.050501 
icoap = ico || dual ico (scaliform)

coates = co || tes

icates = tes || ico

tessap = hex || tes

{4} || gyro tes         (wedge of tessap)
1.106168 
poct = trapedip || {3}-gyro trapedip (uniform)
1.112583 
trike = ike || ipe (uniform)
1.118034 
pent = tes || tes        (regular)

icope = ico || ico       (uniform)

opeacope = ope || cope   (segment of icope)

cubpyp = line || tes     (segment of icope)

trawp = ope || hip       (wedge of icope)
      = traw || traw

spiddip = spid || spid   (uniform)

tepeacope = tepe || cope (segment of spiddip)

tricufip = hip || tisdip
         = tricuf || tricuf
1.120019 
cubacope = cube || cope
1.143215 
trapip = pip || squipdip      (uniform)
       = trapedip || trapedip
1.154701 
traco = co || cope            (uniform)

tritricu = triddip || thiddip (segment of traco)
1.172604 
hatet = {6} || thiddip          (uniform)
      = hip || lacing-ortho hip

pexhix = {6} || triddip
1.185120 
squike = ipe || ipe (uniform)
1.190238 
opeatut = ope || tut            (wedge of spix)

copatut = cope || tut           (wedge of spix)

rapatut = rap || tut            (wedge of spix)

rapalsrip = rap || inv srip     (segment of spix)

spidasrip = spid || srip        (segment of spix)

tetacope = tet || cope          (segment of spix)

triddipa hip = {3} || tricupe   (wedge of spix)
             = trip || tripuf
             = triddip || hip

tripa thiddip = {6} || tricupe  (wedge of spix)
              = trip || thiddip

pabex hix = thiddip || antipara thiddip (scaliform)

triddip althiddip = triddip || gyro thiddip
1.213922 
pecube = squipdip || squipdip (uniform)
1.224745 
octhipdaw = {3}-ap || axis-ortho {6}-p

hoct = thiddip || {3}-gyro thiddip (uniform)

hisquippy = {6} || shiddip         (segment of hoct)

squaco = cope || cope              (uniform)

squatricu = tisdip || shiddip      (segment of squaco)
          = tricupe || tricupe

icarit = ico || rit            (segment of nit)

octco tuttric = octaco || tut  (wedge of nit)
              = oct || coatut
              = co || octatut

rapasrip = rap || srip         (segment of nit)

pabdinit = srip || inv srip    (scaliform,
                                segment of nit)

octacope = oct || cope         (segment of nit)

trial tricupe = {3} || tricupe (wedge of nit)

tisdippy = pt || tisdip        (segment of nit)
                             
ica tutcup = ico || tuta       (segment of nit)
                             
coa tutcup = co || tuta        (wedge of nit)
                             
coarit = co || rit             (wedge of nit)
1.227160 
tessarit = tes || rit
1.258306 
trahip = thiddip || thiddip (uniform)
       = hip || shiddip
1.274755 
tepatut = tepe || tut         (wedge of siphin)

tutaf = tut || inv tuttip     (wedge of siphin)

tutas = tuta || alt. tuta     (scaliform,
                               segment of siphin)

rita = rit || gyro rit        (scaliform,
                               segment of siphin)

hexalrit = hex || gyro rit    (segment of siphin)

octa tutcup = oct || tuta

tripalhip = trip || lacing-ortho hip

tetco tuttric = tet || coatut (segment of penasrip)
              = co || tetatut
              = tut || tetaco

penasrip = pen || srip

spidatip = spid || tip

sripaltip = srip || inv tip
1.284523 
srippip = srip || srip        (uniform)

opeatuttip = ope || tuttip    (segment of srippip)

copea tuttip = cope || tuttip (segment of srippip)

tripufip = trip || shiddip    (wedge of srippip)
         = tripuf || tripuf
1.290994 
rapatip = rap || tip        (segment of sarx)

sripatip = srip || tip      (segment of sarx)

coatuttip = co || tuttip    (segment of sarx)

trathiddip = {3} || thiddip (segment of sarx)
1.302772 
pepip = pedip || pedip (uniform)
1.307032 
tratut = tut || tuttip (uniform)
CircumradiusSegmentoteron
1.322876 
hacube = shiddip || shiddip      (uniform)

rittip = rit || rit              (uniform)

tutcupip = tuttip || inv tuttip  (segment of rittip)
         = tuta || tuta

tepeatuttip = tepe || tuttip     (segment of rittip)
            = tetatut || tetatut
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)
     = op || lacing-ortho op
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

sirco acope = sirco || cope
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,
                                 segment of scant)

tessasidpith = tes || sidpith   (segment of scant)

squicuffip = squicuf || squicuf (wedge of sidpithip)
           = tes || op
1.569562 
squadoe = dope || dope (uniform)
1.583946 
toa sircope = toe || sircope
1.618035 
idadope = id || dope
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)
1.645329 
otricu = todip || hodip (half of oco)
1.683251 
tratoe = toe || tope (uniform)
1.688194 
prippip = prip || prip            (uniform)

tuttipa tope = tuttip || tope     (segment of prippip)
             = tutatoe || tutatoe
1.693527 
exip = ex || ex      (uniform)

gappip = gap || gap  (uniform)

sadip = sadi || sadi (uniform)
1.695040 
cuba sircope = cube || sircope
1.717954 
trid = id || iddip (uniform)
1.719624 
haop = hodip || hodip (uniform)
1.732051 
squatoe = tope || tope        (uniform)

icathex = ico || thex         (segment of sart)

thexarico = thex || rico      (segment of sart)

sripalprip = srip || inv prip (segment of sart)

trashiddip = {3} || shiddip   (segment of sart)

cotut totric = co || tutatoe  (wedge of sart)
             = coatut || toe
             = tut || coatoe
1.738546 
ricoasadi = rico || sadi

doa iddip = doe || iddip
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)
         = coatoe || coatoe
1.847759 
coasircope = co || sircope

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 
ticca sircope = tic || sircope

squatic = ticcup || ticcup (uniform)

spiccup = spic || spic     (uniform)

srittip = srit || srit     (uniform,
                            segment of span)

sidpith || srit            (segment of span)
1.914854 
deca agrip = deca || grip (segment of pirx)
1.994779 
cuba octu = cube || octu
2.117085 
twacube = sitwadip || sitwadip (uniform)
2.150581 
taha = tah || gyro tah (scaliform,
                        segment of pirhin)

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)

sircoa ticcup = 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)

prohalsrico = proh || gyro srico (diminishing of sricoa)
2.647378 
tico || prissi
2.660531 
prohp = proh || proh     (uniform,
                          segment of carnit)

prit || proh             (segment of carnit)

sritapritsritaprit = srit || prit (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 
tataproh = tat || proh (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)

ticca gircope = tic || gircope
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,n-dippip = n-p || 4,n-dip     (uniform)
           = 3,n-dip || 3,n-dip
... 
n,tet-dip = n-gon || 3,n-dip        (uniform)
          = n-p || lacing-ortho n-p
... 
n,cube-dip = 4,n-dip || 4,n-dip (uniform)
... 
n,tricu-dip = 3,n-dip || 6,n-dip (half of n,co-dip)
... 
n,m-dippip = n,m-dip || n,m-dip (uniform)
... 
(n,m-ap)-dip = n,m-dip || para-gyro n,m-dip (uniform)
... 
n,m-dafup = n,m-dip || bidual n,m-dip (scaliform)
... 
n-daf = n-p || bidual n-p (scaliform)
... 
(n, dual n, n-p)-tric = n-g || gyro 3,n-dip
... 
(n, n-p, gyro n-p)-tric = n-g || n-appip

(For degenerate segmentotera, i.e. having zero heights, cf. page "finite flat complexes".)

Some non-convex segmentotera would be: 



----
 6D
----

(Just some) Convex Segmentopeta   (up)

CircumradiusSegmentopeton
0.654654 
hop = pt || hix        (regular)
    = line || perp pen
    = {3} || perp tet
0.707107 
gee = hix || inv hix      (regular)

tacpy = pt || tac         (segment of gee)

hexasc = line || perp hex (lune of gee)

octete = {3} || perp oct  (wedge of gee)

squepe = {4} || perp tet  (wedge of gee)
0.816497 
hixip = hix || hix                (uniform)
      = line || penp
      = trip || part. ortho trip

dijak = hin || tac                (segment of jak)

tedjak = hex || hin               (scaliform, 
                                   tridiminished jak)

hedjak = {4} || hin               (wedge of jak)

hinpy = pt || hin                 (segment of jak)

dihinpy = pt || dihin             (segment of jak)

rapesc = line || perp rap         (segment of jak)

tripal triddip = trip || dual-perp triddip
                                  (wedge of jak)

hixalrix = hix || inv rix         (segment of jak)

hixaltriddip = hix || alt triddip (wedge of jak)

penaf = pen || inv penp           (segment of jak)

rapalpenp = rap || inv penp       (segment of jak)

trippete = {3} || perp trip       (segment of jak)
0.845154 
ril = hix || rix                  (uniform)

penppy = pt || penp               (segment of ril)

rapa penp = rap || penp           (segment of ril)

trial tratet = {3} || dual tratet (segment of ril)

octa tratet = oct || tratet       (segment of ril)

tidril = hix || tedrix            (wedge of ril)

pedril = {4} || tratet            (segment of ril)
0.856349 
trapen = pen || penp   (uniform)
       = {3} || tratet
0.866025 
hax = hin || gyro hin          (uniform)

taccup = tac || tac            (uniform)

tetdip = tet || tratet         (uniform)

tratetdafup = tratet || bidual tratet (scaliform)

tetal tratet = tet || inv tratet

tepasc = line || perp tepe

rixpy = pt || rix              (segment of hax)

hixadot = hix || dot           (segment of hax)

hixacube = hix || cube         (wedge of hax)

rixa = rix || inv rix          (scaliform,
                                segment of hax)

hexaf = hex || gyro hexip      (segment of hax)

octatepepy = pt || octatepe    (wedge of hax)
           = tet || rappy
           = oct || tepepy

hixa tetaope = hix || teta ope (half of hax)

hixaope = hix || ope           (quarter of hax)

tritra ope = (trig, dual trig, para trip, gyro trip) lace simplex
0.912871 
trahex = hex || hexip (uniform)
0.918559 
trial troct = {3} || gyro troct
0.925820 
bril = rix || dot                        (uniform)

tratetpy = pt || tratet                  (segment of bril)

tratet altroct = tratet || gyro troct    (segment of bril)

teta troct = tet || troct                (segment of bril)

pena rappip = pen || rappip              (segment of bril)

rappaf = rap || inv. rappip              (segment of bril)

triddippa tridafup = triddip || tridafup (segment of bril)

tisdip altratet = tisdip || gyro tratet

tatraffip = {3} || traffip
          = trip || gyro tisquippy
          = traf || tisdip
0.935414 
hinnip = hin || hin                (uniform)
       = hexip || alt hexip

tetafip = tepe || inv squatet      (segment of hinnip)
        = tepe || hexip

penpal rappip = penp || alt rappip (segment of hinnip)
              = dihin || dihin

rappyp = line || rappip            (segment of hinnip)
0.948683 
squapen = penp || penp   (uniform)
        = {4} || squatet
0.957427 
rixip = rix || rix         (uniform)

tepepyp = line || squatet  (segment of rixip)
        = tepepy || tepepy
0.966092 
trarap = rap || rappip                 (uniform)
       = tratet || troct

titraf = triddip || gyro-para tratrip  (segment of trarap)
CircumradiusSegmentopeton
1 
dotip = dot || dot                (uniform)
      = rappip || inv rappip

tridafupip = tridafup || tridafup (segment of dotip)
           = tratrip || bidual tratrip

squahex = hexip || hexip          (uniform)

hixascad = hix || scad            (segment of staf)

spidapenp = spid || penp          (segment of staf)

hexippy = pt || hexip             (segment of rag)

hexipaico = hexip || ico          (wedge of rag)

hexipaco = hexip || co            (wedge of rag)

taccarat = tac || rat             (segment of rag)

hexarat = hex || rat              (segment of rag)

rixascad = rix || scad            (segment of rag)

dotpy = pt || dot                 (segment of mo)

dottascad = dot || scad           (segment of mo)

hinro = hin || rat                (segment of mo)

hinatedrat = hin || tedrat        (segment of mo)

icoahin = ico || hin              (segment of mo)

hinaco = hin || co                (segment of mo)

penal rappip = pen || alt rappip  (segment of mo)

spidarappip = spid || rappip      (wedge of mo)

tetaopepy = pen || ope            (wedge of mo)
          = tet || opepy
          = pt || teta ope

hexaoctaco tettic = co || bidhin  (wedge of dottascad)
                  = oct || hexaco
                  = tet || alt. rapaco

tetaco altepetric = co || tetaf
                  = tet || hexaco
                  = tet || inv tepaco

triddipasc = line || perp triddip
1.118034 
ratip = rat || rat    (uniform)

scadip = scad || scad (uniform)
1.123033 
icaf = ico || dual icope
1.195229 
rixalspix = rix || inv spix (segment of scal)

scadaspix = scad || spix    (segment of scal)
1.224745 
ax = pent || pent                   (regular)

troctpy = pt || troct               (segment of brag)

dottaspix = dot || spix             (segment of brag)

spixa = spix || inv spix            (segment of brag)

ratanit = rat || nit                (segment of brag)

hexcopedaw = hex || axis-ortho cope (segment of brag)

hexaicope = hex || icope            (segment of brag)

troctal traco = troct || inv. traco
1.228783 
pentanit = pent || nit
1.290994 
spixip = spix || spix         (uniform)

ritgyt = rit || gyro rita     (scaliform,
                               wedge of rojak)

ritahin = rit || hin          (wedge of rojak)

hinanit = hin || nit          (segment of rojak)

nitasiphin = nit || siphin    (segment of rojak)

ratasiphin = rat || siphin    (segment of rojak)

rixaspix = rix || spix        (segment of rojak)

spixalsarx = spix || inv sarx (segment of rojak)

icarita = ico || rita         (wedge of rojak)

hexaico alrittric = hex || alt. icarit (wedge of rojak)
                  = ico || hexalrit
                  = rit || alt. hexaico

rappippy = pt || rappip       (segment of rojak)

sripaf = srip || srippip
1.316561 
trasrip = srip || srippip (uniform)
1.322876 
tettut = tut || tratut          (uniform)

hinasiphin = hin || siphin      (segment of sochax)

siphina = siphin || alt. siphin (segment of sochax)

ritas = rita || alt. rita       (segment of siphina)
1.362770 
rixatix = rix || tix (segment of sril)
1.369306 
siphinnip = siphin || siphin (uniform)
1.384437 
sarxip = sarx || sarx (uniform)
1.414214 
tixip = tix || tix           (uniform)

nitarin = nit || rin         (segment of brox)

rixasarx = rix || sarx       (segment of brox)

sarxasibrid = sarx || sibrid (segment of brox)
1.418705 
pentarin = pent || rin
1.581139 
bittixa = bittix || inv bittix (scaliform)
1.614654 
open = {8} || otet (uniform)
1.632993 
thexgyt = thex || gyro thexa (scaliform,
                              wedge of hejak)

hinarin = hin || rin         (segment of hejak)
1.658312 
cappixa = cappix || alt cappix (scaliform)

sirhina = sirhin || alt sirhin (scaliform,
                                segment of sophax)
1.732051 
cardip = card || card        (uniform,
                              segment of ram)

dottasibrid = dot || sibrid  (segment of ram)

sibridacard = sibrid || card (segment of ram)

nitasirhin = nit || sirhin   (segment of ram)

sirhinasart = sirhin || sart (segment of ram)
1.936492 
thina = thin || alt thin     (scaliform)
                              segment of sirhax)

sirhinathin = sirhin || thin (segment of sirhax)
2.160247 
tahgyt = tah || gyro taha (scaliform,
                           wedge of harjak)

rinathin = rin || thin    (segment of harjak)
2.236068 
sartabittit = sart || bittit        (segment of siborg)

bittita sibrant = bittit || sibrant (segment of siborg)
2.345208 
squagippid = gippiddip || gippiddip (uniform)
3 
gocadip = gocad || gocad (uniform)
4.855004 
gacnetip = gacnet || gacnet (uniform)
... 
(n,pen)-dip = {n} || (n,tet)-dip (uniform)
... 
(n-g, dual n-g, para n-p, gyro n-p) lace simplex
... 
(n,m)-dip || (n,m)-dafup

Some non-convex segmentopeta would be: 



----
 7D
----

(Just some) Convex Segmentoexa   (up)

CircumradiusSegmentoexon
0.661438 
oca = pt || hop        (regular)
    = line || perp hix
    = {3} || perp pen
    = tet || perp tet
0.707107 
zee = hop || inv hop       (regular)

geepy = pt || gee          (segment of zee)

taccasc = line || perp tac (lune of zee)

hexete = {3} || perp hex   (wedge of zee)

squix = {4} || perp pen    (wedge of zee)

octepe = tet || perp oct   (wedge of zee)
0.823754 
hopip = hop || hop (uniform)
0.866025 
roc = hop || ril                (uniform)

hixippy = pt || hixip           (segment of roc)

trialtrapen = {3} || inv trapen (segment of roc)

octatetdip = oct || tetdip      (segment of roc)

rapatrapen = rap || trapen      (segment of roc)

rixahixip = rix || hixip        (segment of roc)

tedroc = tedrix || hixip        (segment of roc)

geep = gee || gee               (uniform)
     = hixip || inv hixip

tacpyp = line || taccup         (segment of geep)

trahix = hix || hixip           (uniform,
                                 wedge of naq)

rila = ril || alt. ril          (scaliform,
                                 segment of naq)

jaka = jak || alt. jak          (scaliform,
                                 segment of naq)

jakpy = pt || jak               (segment of naq)

dijakpy = pt || dijak           (segment of naq)

hixaf = hix || inv hixip        (segment of naq)

hopalril = hop || inv ril       (segment of naq)

gahax = gee || hax              (segment of naq)

hinsc = line || perp hin        (wedge of naq)

hexalhax = hex || alt. hax      (wedge of naq)

trapendafup = trapen || bidual trapen (scaliform)

rapete = {3} || perp rap

trippepe = trip || perp tet
0.912871 
tratac = tac || taccup (uniform)
0.935414 
hesa = hax || gyro hax                    (uniform)

rilpy = pt || ril                         (segment of hesa)

hexal trahex = hex || gyro trahex         (segment of hesa)

hopabril = hop || bril                    (segment of hesa)

rilalbril = ril || alt. bril              (segment of hesa)

hinaf = hin || alt. hinnip                (segment of hesa)

trahex dafup = trahex || dual-gyro trahex (segment of hesa)

tetal tetdip = tet || dual tetdip

tethex = hex || trahex                    (uniform)
       = tetdip || para-dual tetdip
       = hexip || lacing-ortho hexip

penpasc = line || perp penp

tridafupap = tridafup || lacing-ortho tridafup (scaliform)
0.957427 
squahix = hixip || hixip (uniform)

jakip = jak || jak       (uniform)
0.968246 
broc = ril || bril               (uniform)

rapaltrarap = rap || alt. trarap (segment of broc)

tetocta = tetoct || alt. tetoct  (scaliform,
                                  segment of broc)
0.978945 
trahin = hin || hinnip (uniform)
0.981981 
rillip = ril || ril (uniform)
0.991632 
trippen = trapen || trapen (uniform)
0.992157 
trial trarap = {3} || alt. trarap
CircumradiusSegmentoexon
1 
he = bril || inv bril       (uniform)

tetdippy = pt || tetdip     (segment of he)

tetdippa octdip = tetdip || octdip     (segment of he)

trarapdafup = trarap || bi-alt. trarap (segment of he)

trapen altralrap = trapen || bi-alt. trarap

squarat = ratip || ratip    (uniform)
                             diminishing of barz)

tettepe = tetdip || tetdip  (uniform)

trarix = rix || rixip       (uniform,
                             wedge of laq)

haxpy = pt || hax           (segment of laq)

ratahax = rat || hax        (wedge of laq)

haxarag = hax || rag        (segment of laq)

jakamo = jak || mo          (segment of laq)

hopalbril = hop || inv bril (segment of laq)

brilastaf = bril || staf    (segment of laq)

rixapy = pt || rixa         (wedge of laq)

taccuppy = pt || taccup     (segment of rez)

garag = gee || rag          (segment of rez)

rilastaf = ril || staf      (segment of rez)

hopastaf = hop || staf      (segment of suph
                             diminishing of laq)

rixasc = line || perp rix

rixaf = rix || inv rixip

trial trahex = {3} || inv trahex

trahex aico = trahex || ico

ratahinnip = rat || hinnip
1.040833 
titridafup = trittip || bidual trittip (scaliform)

rixap = rixip || inv rixip             (scaliform)
1.052209 
brillip = bril || bril (uniform)
1.060660 
haxip = hax || hax (uniform)
1.118034 
tratratrip = trittip || trittip (uniform)

stafip = staf || staf           (uniform)

hinnipa ratip = hinnip || ratip
1.154701 
trarat = rat || ratip (uniform)
1.224745 
trahexpy = pt || trahex  (segment of barz)

ragabrag = rag || brag   (segment of barz)

scala = scal || inv scal (scaliform,
                          segment of barz)
1.322876 
hept = ax || ax                   (regular)

brilalspil = bril || inv spil     (segment of sco)

scala spil = scal || spil         (segment of sco)

cubico = squico || squico         (uniform,
                                   segment of lin)

rojaka = rojak || alt. rojak      (scaliform,
                                   segment of lin)

haxabrag = hax || brag            (segment of lin)

haxa cytedbrag = hax || cytedbrag (segment of lin)

moarojak = mo || rojak            (segment of lin)

bragasochax = brag || sochax      (segment of lin)

brilpy = pt || bril               (segment of lin)

hinaratip = hin || ratip          (segment of lin)

spila = spil || inv spil          (scaliform)
1.384437 
rojakip = rojak || rojak (uniform)
1.414214 
trasarx = sarx || sarxip   (uniform)

bragabrox = brag || brox   (segment of sez)

ragasochax = rag || sochax (segment of ranq)

jakarojak = jak || rojak   (segment of ranq)

rojakatrim = rojak || trim (segment of ranq)
1.451600 
srillip = sril || sril (uniform)
1.457317 
ohix = {8} || open (uniform)
1.581139 
broxarax = brox || rax          (segment of bersa)

sabrila = sabril || inv. sabril (scaliform)
1.658312 
jakatrim = jak || trim       (segment of stanq)

hejaka = hejak || alt. hejak (segment of stanq)
1.732051 
haxabrox = hax || brox        (segment of rolaq)

broxa sophax = brox || sophax (segment of rolaq)

rojaka hejak = rojak || hejak (segment of rolaq)
1.936492 
crala = cral || inv. cral          (scaliform)

shopjaka = shopjak || alt. shopjak (scaliform,
                                    segment of branq)
2.207940 
thaxa = thax || alt. thax (scaliform,
                           segment of sirhesa)
2.5 
hagippiddip = hagippid || hagippid (uniform)
3.774917 
gotafip = gotaf || gotaf (uniform)
6.264982 
gopamp = gopam || gopam (uniform)
... 
n,n,n-tippip = n,n,n-tip || n,n,n-tip (uniform)
... 
(n,m)-dafupap = (n,m)-dafup || lacing-ortho (n,m)-dafup (scaliform)

Some non-convex segmentoexa would be: 



----
 8D
----

(Just some) Convex Segmentozetta   (up)

CircumradiusSegmentozetton
0.666667 
ene  = pt || oca (regular)
0.707107 
ek  = oca || dual oca      (regular)

zeepy  = pt || zee         (segment of ek)

geeasc = line || perp gee (wedge of ek)
       = pt || geepy

taccete = {3} || perp tac (wedge of ek)
        = line || perp tacpy
        = pt || taccasc

hexepe = tet || perp hex (lune of ek)
       = {3} || perp hexpy
       = line || perp hexasc
       = pt || hexete
0.829156 
ocpe  = oca || oca (uniform)
0.866025 
zeep  = zee || zee (uniform)
0.872872 
trihop  = hop || hopip (uniform)
0.881917 
rene  = oca || roc (uniform)
0.894427 
pendip  = pen || tetpen (uniform)
0.948683 
penhex = hex || tethex (uniform)
       = hexip || trahex
       = tetpen || first-dual tetpen
0.957427 
octhix = oct || octpen (uniform)
       = ope || tetoct
       = trahix || first-dual trahix
0.963624 
squahop  = hopip || hopip (uniform)
CircumradiusSegmentozetton
1 
trajak = jak || jakip        (uniform,
                              wedge of fy)

brene  = roc || broc          (uniform,
                              segment of fy)

hocto  = hesa || gyro hesa    (uniform,
                              segment of fy)

penrap = rap || tetrap       (uniform,
       = rappip || trarap     segment of fy)
       = tetpen || octpen

triphix  = trahix || trahix   (uniform)

tethin  = hin || trahin       (uniform)

rocip  = roc || roc           (uniform)

rocpy  = pt || roc            (segment of hocto)

rocahe = roc || he           (segment of hocto)

ocabroc = oca || broc        (segment of hocto)

broca = broc || inv broc     (scaliform,
                              segment of hocto)

ocasuph = oca || suph        (segment of soxeb)

naqpy = pt || naq            (segment of fy)

jakapy = pt || jaka          (wedge of fy)

jakaf = jak || alt. jakip    (wedge of fy)

jakap = jaka || jaka         (scaliform,
                              segment of fy)

naqalaq = naq || laq         (segment of fy)

naqpe  = naq || naq           (uniform,
                              segment of fy)

zahesa = zee || hesa         (segment of fy)

hesa arez = hesa || rez      (segment of fy)

ocalroc  = oca || inv roc     (segment of fy)

rocalbroc  = roc || inv broc  (segment of fy)

rocasuph  = roc || suph       (segment of fy)

hexal hesa  = hex || alt hesa (wedge of codify)

zarez = zee || rez           (segment of rek)

octa tethex  = oct || tethex

tethex aico  = tethex || ico

rapepe = tet || perp rap
1.054093 
tetrappal octrap = tetrap || alt. octrap
1.118034 
laqpe  = laq || laq (uniform)
1.224745 
rezabarz = rez || barz (segment of bark)
1.414214 
octo  = hept || hept              (regular)

trarojak = rojak || rojakip      (uniform)

barzasez = barz || sez           (segment of tark)

hesapy  = pt || hesa              (segment of bay)

laqalin  = laq || lin             (segment of bay)

linaranq  = lin || ranq           (segment of bay)

rojakaf = rojak || alt rojakip   (segment of bay)

rojakap = rojakip || alt rojakip (segment of bay)
        = rojaka || rojaka

moarojka = mo || rojaka          (wedge of bay)
         = rojak || moarojak



----
 9D
----

(Just some) Convex Segmentoyotta   (up)

CircumradiusSegmentoyotton
0.670820 
day = pt || ene (regular)
0.707107 
vee = ene || dual ene (regular)
0.833333 
enep = ene || ene (uniform)
0.866025 
ekip = ek || ek (uniform)
0.877971 
trioc = oca || ocpe (uniform)
0.894427 
reday = ene || rene (uniform)
CircumradiusSegmentoyotton
0.903696 
penhix = hix || tethix (uniform)
       = pen || pendip
0.968246 
squoc = ocpe || ocpe (uniform)
1.024695 
breday = rene || brene (uniform)
1.060660 
henne = hocto || gyro hocto (uniform)
1.095445 
treday = brene || trene (uniform)
1.118034 
rapdafup = rapdip || bi-alt. rapdip (scaliform)
1.5 
enne = octo || octo (regular)



-----
 10D
-----

(Just some) Convex Segmentoxenna   (up)

CircumradiusSegmentoxennon
0.674200 
ux = pt || day (regular)
0.707107 
ka = day || dual day (regular)
0.904534 
ru = day || reday (uniform)
CircumradiusSegmentoxenna
1.044466 
bru = reday || breday (uniform)
1.128152 
tru = breday || treday (uniform)