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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:

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 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 1/4-lune 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 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 pen = line || perp {3} (regular)
0.707107
K-4.2   hex = tet || dual tet     (regular)

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

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

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

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

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

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

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

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 squaf = {4} || gyro cube (wedge of oct || cube)

K-4.15   octacube = oct || cube

K-4.16   squippy || gyro cube

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

K-4.21   cubaike =  cube || ike

K-4.22   paf =  {5} || pap
K-4.22.1 paf =  {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 tricuf =  {6} || trip     (luna of spid)

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

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

K-4.28   {4} || co

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   {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 peppyp = line || pip

K-4.39   pappip = pap || pap
K-4.39.1 pappip = 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 tricupe = trip || hip
1.130454
K-4.46   haf = {6} || hap
K-4.46.1 haf = {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 tripuf = {3} || hip

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

K-4.55   tutcup = 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 oaf = {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   co || escu

K-4.63   coaop = co || op

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

K-4.67   gyesquibcupe = gyesquibcu || gyesquibcu

K-4.68   escupe = escu || escu

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

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

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

K-4.72   cube || gyesquibcu

K-4.73   squicuf = {4} || squacu   (wedge of sidpith)
K-4.73.1 squicuf = {8} || cube     (wedge of sidpith)
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 pappy = pt || pap

K-4.81   papadoe = pap || doe

K-4.82   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 peppypy = line || perp {5} (wedge of ex)

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 daf = {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 dappip = dip || gyro dip (uniform)
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 squipuf = {4} || op      (segment of srit)

K-4.106   op || tic

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 pecupe =  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   {5} || gyro pecu
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   peppy || pero

K-4.140   gyepipa pero = gyepip || pero

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

K-4.142   papaid = pap || id

K-4.143   mibdi || id

K-4.144   pap || pero

K-4.145   mibdi || pero

K-4.146   {5} || pero

K-4.147   teddi aid = teddi || id

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 pecuf = {10} || pip   (wedge of sidpixhi)

K-4.155   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 pepuf = {5} || dip     (segment of srahi)

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-af = {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-appip = n-p || gyro n-p (uniform)
...
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 so, 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. Quite immediate 4D cases could be e.g.
  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.

Further Reading



---- 5D ----

(Just some) Convex Segmentotera   (up)

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

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

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

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

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

triddaf =  trip || bidual trip (wedge of hin)
0.806226
penp =  pen || pen (uniform)
0.816497
rix =  pen || rap                    (uniform)

tepepy =  pt || tepe                 (segment of rix)

octatepe =  oct || tepe              (segment of rix)

trial triddip =  {3} || gyro triddip (segment of rix)

tedrix =  {4} || lacing-ortho ( {4} || lacing-ortho {4} )
tedrix = {4} || ortho tepe
0.841625
tratet = tet || tepe               (uniform)
tratet = {3} || triddip            (uniform)
tratet = trip || lacing-ortho trip (uniform)
0.866025
hexip = hex || hex        (uniform)

line || ope               (segment of hexip)

dot = rap || inv rap      (uniform)

triddippy = pt || triddip (segment of dot)

tet || ope                (segment of dot)

tridafup = triddip || bidual triddip (segment of dot)
0.895420
squiddaf = {4}-prism || bidual {4}-prism (scaliform)
0.912871
troct = oct || ope              (uniform)
troct = triddip || gyro triddip (uniform)
0.921954
rappip =  rap || rap  (uniform)
rappip =  ope || tepe (uniform)

trip || tisdip       (segment of rappip)
0.935414
squatet =  tepe || tepe              (uniform)
squatet =  {4} || tisdip             (uniform)
squatet =  cube || lacing-ortho cube (uniform)
0.951057
gadtaxhiap =  gadtaxhi || gadtaxhi (uniform)
0.957427
tratrip =  trip || tisdip     (uniform)
tratrip =  triddip || triddip (uniform)
1
squoct =  ope || ope                (uniform)
squoct =  tisdip || {3}-gyro tisdip (uniform)

traltisdip =  {3} || gyro tisdip

hexaico =  hex || ico        (segment of rat)

rapaspid =  rap || spid      (segment of rat)

opepy =  pt || ope           (segment of rat)

coahex =  co || hex          (wedge of rat)

penaspid =  pen || spid      (segment of scad)
1.040833
tracube =  cube || tes      (uniform)
tracube =  tisdip || tisdip (uniform)
1.048144
petet =  {5} || trapedip         (uniform)
petet =  pip || lacing-ortho pip (uniform)
1.050501
icoap =  ico || dual ico (scaliform)

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)

spiddip =  spid || spid (uniform)
1.143215
trapip =  pip || squipdip      (uniform)
trapip =  trapedip || trapedip (uniform)
1.154701
traco =  co || cope (uniform)
1.172604
hatet =  {6} || thiddip          (uniform)
hatet =  hip || lacing-ortho hip (uniform)

pexhix = {6} || triddip
1.185120
squike =  ipe || ipe (uniform)
1.190238
rap || inv srip (segment of spix)

spid || srip    (segment of spix)

tet || cope     (segment of spix)

pabex hix = thiddip || antipara thiddip
1.213922
pecube =  squipdip || squipdip (uniform)
1.224745
hoct =  thiddip || {3}-gyro thiddip (uniform)

squaco =  cope || cope              (uniform)

icarit =  ico || rit         (segment of nit)

rapasrip =  rap || srip      (segment of nit)

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

oct || cope                 (segment of nit)

pt || tisdip                (segment of nit)
                             
ica tutcup =  ico || tutcup  (segment of nit)
                             
coa tutcup =  co || tutcup   (wedge of nit)
1.227160
tessarit  = tes || rit
1.274755
ritag rit = rit || gyro rit (scaliform)

octa tutcup = oct || tutcup

spidatip = spid || tip
1.284523
srippip =  srip || srip (uniform)
1.290994
rap || tip     (segment of sarx)

srip || tip    (segment of sarx)

co || tuttip   (segment of sarx)

{3} || thiddip (segment of sarx)
1.307032
tratut =  tut || tuttip (uniform)
1.322876
hacube =  shiddip || shiddip (uniform)

rittip =  rit || rit         (uniform)

tepeatuttip =  tepe || tuttip
tepeatuttip =  tetatut || tetatut
1.360147
tippip =  tip || tip (uniform)
1.369306
squatut =  tuttip || tuttip (uniform)
1.414214
pen || tip              (segment of rin)

tipadeca =  tip || deca  (segment of rin)

srip || deca            (segment of sibrid)
1.442951
otet =  {8} || todip          (uniform)
otet =  op || lacing-ortho 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)
1.569562
squadoe =  dope || dope (uniform)
CircumradiusSegmentoteron
1.620185
ritathex = rit || thex          (segment of sirhin)

thexag thex = thex || gyro thex (scaliform,
                                 segment of sirhin)

toa tutcup = toe || tutcup
1.632993
prip || tip (segment of cappix)
1.658312
hiddip || triddip     (segment of card)

prip || inv prip      (segment of card)

prip || srip          (segment of card)

thexip =  thex || thex (uniform)
1.683251
tratoe =  toe || tope (uniform)
1.688194
prippip =  prip || prip (uniform)
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)

ico || thex            (segment of sart)

rico || thex           (segment of sart)

prip || inv srip       (segment of sart)

{3} || shiddip         (segment of sart)
1.738546
rico || sadi
1.765796
squid =  iddip || iddip (uniform)
1.778824
ricoalrico =  rico || gyro rico (scaliform)
1.802776
ricope =  rico || rico (uniform)
1.847759
ica sidpith =  ico || sidpith

ricasrit =  rico || srit

ricoaspic =  rico || spic
1.848423
pripalgrip =  prip || gyro grip (segment of pattix)

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 =  grip || prip
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)

deca || grip               (segment of pirx)
1.994779
ofx3xoo4ooo&#xt || o3o4x
2.150581
tahagtah =  tah || gyro tah (scaliform)
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
rico || tah  (segment of sibrant)

grip || srip (segment of sibrant)

co || tope   (segment of sibrant)
2.268840
squasnid =  sniddip || sniddip (uniform)
2.291288
gippiddip =  gippid || gippid (uniform)
2.306383
trasrid =  srid || sriddip (uniform)
2.321762
sidtaxhiap =  sidtaxhi || sidtaxhi (uniform)
2.327373
gippid || grip (segment of cograx)
2.342236
squasrid =  sriddip || sriddip (uniform)
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

sricoalsrico =  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
gippid || prip (segment of pattit)
2.878460
proh || tat (segment of capt)
3
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
grit || proh  (segment of pattin)

tut || tuttip (segment of pattin)
3.450631
contip =  cont || cont (uniform)
3.522336
pricoalprico =  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
gidpith || grit (segment of cogrin)
4.311477
gricoalgrico =  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)
11.263210
grahip =  grahi || grahi (uniform)
12.796423
gidpixhip =  gidpixhi || gidpixhi (uniform)
...
3,n-dippip =  n-p || 4,n-dip     (uniform)
3,n-dippip =  3,n-dip || 3,n-dip (uniform)
...
n,cube-dip =  4,n-dip || 4,n-dip (uniform)
...
n,m-dippip =  n,m-dip || n,m-dip (uniform)
...
n,m-dafup =  n,m-dip || bidual n,m-dip (scaliform)

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



---- 6D ----

(Just some) Convex Segmentopeta   (up)

CircumradiusSegmentopeton
0.654654
hop =  pt || hix        (regular)
hop =  line || perp pen (regular)
hop =  {3} || perp tet  (regular)
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)
hixip =  trip || part. ortho trip (uniform)


dijak = hin || tac        (segment of jak)

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

hinpy =  pt || hin         (segment of jak)

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

hixalrix =  hix || inv rix (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)

octatratet =  oct || tratet        (segment of ril)
0.856349
trapen =  pen || penp (uniform)
0.866025
hax =  hin || gyro hin     (uniform)

taccup =  tac || tac       (uniform)

tetdip =  tet || tratet    (uniform)

tetal tratet =  tet || inv tratet

tepasc =  line || perp tepe

rixpy =  pt || rix         (segment of hax)

hixadot =  hix || dot      (segment of hax)

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

hexaf =  hex || gyro hexip (segment of hax)
CircumradiusSegmentopeton
0.925820
bril =  rix || dot       (uniform)

tratetpy =  pt || tratet (segment of bril)

tratet || gyro troct    (segment of bril)

tet || troct            (segment of bril)
0.935414
hinnip =  hin || hin (uniform)
0.948683
squapen =  penp || penp (uniform)
0.957427
rixip =  rix || rix (uniform)
1
dotip =  dot || dot           (uniform)
dotip =  rappip || inv rappip (uniform)

hixascad =  hix || scad       (segment of staf)

taccarat =  tac || 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)
1.224745
ax =  pent || pent    (regular)

ratanit =  rat || nit (segment of brag)
1.228783
pentanit = pent || nit
1.290994
ritgyt = rit || gyro ritag rit (scaliform)

ratasiphin =  rat || siphin     (segment of rojak)
1.414214
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
bittixalbittix =  bittix || inv bittix (scaliform)
1.632993
thexgyt =  thex || gyro thexag thex (scaliform)
2.160247
cappixalcappix =  cappix || alt cappix (scaliform)
2.160247
tahgyt =  tah || gyro tahagtah (scaliform)


---- 7D ----

(Just some) Convex Segmentoexa   (up)

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

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

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

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

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

jakpy =  pt || jak          (segment of naq)

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

gahax =  gee || hax         (segment of naq)

rapete =  {3} || perp rap
0.935414
hesa =  hax || gyro hax       (uniform)

rilpy =  pt || ril            (segment of hesa)

hopabril =  hop || bril       (segment of hesa)

rilalbril =  ril || alt. bril (segment of hesa)
0.957427
jakip =  jak || jak (uniform)
0.968246
broc =  ril || bril (uniform)
0.981981
rillip =  ril || ril (uniform)
0.991632
trippen =  trapen || trapen (uniform)
CircumradiusSegmentoexon
1
he =  bril || inv bril       (uniform)

tettepe =  tetdip || tetdip  (uniform)

haxpy =  pt || hax           (segment of laq)

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

garag =  gee || rag          (segment of rez)

rilastaf =  ril || staf      (segment of rez)

hopastaf =  hop || staf     (segment of suph)
1.052209
brillip =  bril || bril (uniform)
1.060660
haxip =  hax || hax (uniform)
1.118034
tratratrip =  trittip || trittip (uniform)
1.224745
ragabrag =  rag || brag (segment of barz)
1.322876
hept =  ax || ax                    (regular)

haxabrag =  hax || brag             (segment of lin)

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

moarojak =  mo || rojak             (segment of lin)

bragasochax =  brag || sochax       (segment of lin)

brilpy =  pt || bril                (segment of lin)
1.414214
bragabrox =  brag || brox (segment of sez)
1.581139
broxarax =  brox || rax (segment of bersa)
1.732051
rojaka hejak =  rojak || hejak (segment of rolaq)
...
n,n,n-tippip =  n,n,n-tip || n,n,n-tip (uniform)


---- 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)
0.829156
ocpe  = oca || oca (uniform)
0.866025
zeep  = zee || zee (uniform)
0.872872
trihop  = hop || hopip (uniform)
0.881917
rene  = oca || roc (uniform)
0.963624
squahop  = hopip || hopip (uniform)
CircumradiusSegmentozetton
1
brene  = roc || broc           (uniform,
                               segment of fy)

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

triphix  = trahix || trahix    (uniform)

rocpy  = pt || roc             (segment of hocto)

rocahe  = roc || he           (segment of hocto)

ocabroc  = oca || broc        (segment of hocto)

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

ocasuph  = oca || suph        (segment of soxeb)

naqpy  = pt || naq            (segment of fy)

naqalaq  = naq || laq         (segment of fy)

naqpe  = naq || naq            (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)
1.224745
rezabarz =  rez || barz (segment of bark)
1.414214
octo  = hept || hept     (regular)

barzasez  = barz || sez (segment of tark)

hesapy  = pt || hesa     (segment of bay)


---- 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)
CircumradiusSegmentoyotton
0.894427
reday  = ene || rene (uniform)
0.968246
squoc  = ocpe || ocpe (uniform)
1.024695
breday  = rene || brene (uniform)
1.060660
henne  = hocto || gyro hocto (uniform)
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)


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