﻿ Lace Simplices

### Lace Simplices

Besides of the extension of W. Krieger's notion of the axial lace prisms towards that of taller stacks, i.e. to lace towers, their multidimensional configuration, i.e. that of lace simplices proves most useful. Zero dimensional simplices are mere points. Accordingly the lace simplices with zero dimensional configuration space just reproduce the Wythoffian polytopes. One dimensional simplices are just a single dyad (edge). Accordingly the lace simplices with one dimensional configuration space are nothing but the lace prisms themselves (which as such are closely related, but not fully equivalent to the segmentotopes. Thus the simplest non-trivial lace simplices are the so called trigonics. As the triangle surely is an instance of a two dimensional configuration, the trigonics moreover are the simplest cases of lace cities too. That is, each trigonic can be represented by 3 variously marked Dynkin diagrams, all of them thus representing according Wythoffian polytopes, which all 3 belong to a common symmetry group representation. For general lace simplices this number 3 surely would get replaced by some general n. Then the configuration space however would be n-1 dimensional (and those therefore no longer represent instances of lace cities, rather they then might be considered to represent some lace hyper-cities).

Just as we had different edge types x (within the layers) and oo&#x (lacing ones across the layers) for lace prisms, we will get various face types for lace simplices too. There are the cases xNo and xNx (for various values of N) within the still "layers" being called configuration spots, there are the prismatic cases ox&#x and xx&#x, but in addition there is the truely two dimensional case ooo&#x as well, representing triangles having their corners in all 3 layers.

So far we have no individual names for the higher subsequent lace simplex cases. Instead one might just count the individual layers of the notation, i.e. the vertices of the simplex of configuration space. E.g. the uniforms then are 1-layered lace simplices, the lace prisms themselves become 2-layered lace simplices, and the trigonics then can be considered 3-layered lace simplices.

Within the following listings we want to enumerate the most often encountered, i.e. the simpler ones of the lace simplices. For obvious reasons the layer-wise Dynkin diagrams then would show up linkages between the nodes, which carry link markings 2 or 3 only (representing dihedral angles of the according orthoscheme which are π/2 = 90° and π/3 = 60° respectively). In order to be self-contained, we will list lace simplices of all lower dimensional configuration spaces, i.e. enclose the according Wythoffians and lace prisms too. (In contrast to the listings of the latter linked page, the non-fundamental cases and the reducible symmetry groups would be contained here too, as these are exactly the cases of most interest under the above chosen restriction.)

Within this context we provide 3 helpful tools on this site too. Those are:

1. Circumradius Calculator for Wythoffians: This excel spreadsheet allows to type in a Coxeter-Dynkin diagram by means of various pairwise node link mark numbers and various edge lengths for its nodes respectively, and returns the circumradius of the thus described Wythoffian polytope. Additionally it allows to calculate the height of a lace prism, when typing in the (signed) differences of the edge lengths of the top layer and bottom layer nodes and providing the lacing edge length too. Conversely it also allows to calculate the lacing edge length, when the height would be given instead.
2. Circumradius Calculator for Lace Prisms: This excel spreadsheet allows to type in the circumradii of the 2 base polytopes, possibly their additional off-center shifts (if subdimensional), as well as the height of the lace prism, and returns the circumradius of the to be considered lace prism. In fact this tiny tool just considers 2 parallel hyperspherical sections of an one plus dimensional hypersphere.
3. Circumradius Calculator for Trigonics: This excel spreadsheet allows to type in the circumradii of the 3 base polytopes, possibly their additional off-center shifts (if subdimensional), as well as the mutual pairwise perp-space displacements (i.e. heights of the according lateral lace prisms), and returns the circumradius of the to be considered trigonic. In fact this tiny tool just considers 3 parallel hyperspherical punctures of a two plus dimensional hypersphere, when considered in the 2+n dimensional geometry of lace cities.

Just as for segmentotopes not all lacings will be possible, when asking for unit edges throughout. This simply is a consequence of the triangle inequality and depends on the possibly too large difference of individual circumradii of the various layers. Within the listings below we use in those cases
† = the result would be sub-dimensional, as at least one of the contained lace prisms would become flat
‡ = the result would be impossible to be realised within an euclidean embedding space, as the required lacing edges would have to be larger than unity.

```
----
0D
----
```
Wythoffians
```o = point
```

```
----
1D
----
```
Wythoffians Lace Prisms
```x = line
```
```oo&#x
```

```
----
2D
----
```
Wythoffians Lace Prisms Trigonics
```x x = {4}

x3o = {3}
x3x = {6}

...
```
```ox&#x = {3}
xx&#x = {4}
```
```ooo&#x = {3}
```

```
----
3D
----
```
Wythoffians Lace Prisms Trigonics 4-layered Lace Simplices
```x x x = cube

x o3x = trip
x x3x = hip

o3o3x = tet
o3x3o = oct
o3x3x = tut
x3o3x = co
x3x3x = toe

...
```
```ox ox&#x = squippy
ox xo&#x = tet
ox xx&#x = trip
xx xx&#x = cube

oo3ox&#x = tet
oo3xx&#x = trip
ox3ox&#x   † (hippy)
ox3xo&#x = oct
ox3xx&#x = tricu
xx3xx&#x = hip

...
```
```oox&#x = tet
oxx&#x = squippy
xxx&#x = trip
```
```oooo&#x = tet
```

```
----
4D
----
```
Wythoffians Lace Prisms Trigonics 4-layered Lace Simplices 5-layered Lace Simplices
```x x x x = tes

x x o3x = tisdip
x x x3x = shiddip

x o3o3x = tepe
x o3x3o = tope
x o3x3x = tuttip
x x3o3x = cope
x x3x3x = tope

o3x o3x = triddip
o3x x3x = thiddip
x3x x3x = hiddip

o3o3o3x = pen
o3o3x3o = rap
o3o3x3x = tip
o3x3o3x = srip
o3x3x3o = deca
o3x3x3x = grip
x3o3o3x = spid
x3o3x3x = prip
x3x3x3x = gippid

o3o3o *b3x = hex
o3x3o *b3o = ico
o3o3x *b3x = rit
o3x3o *b3x = thex
o3x3x *b3x = tah
x3o3x *b3x = rico
x3x3x *b3x = tico

...
```
```ox ox ox&#x = cubpy
ox ox xo&#x = squasc
ox ox xx&#x = squippyp
ox xo xx&#x = tepe
ox xx xx&#x = tisdip
xx xx xx&#x = tes

ox oo3ox&#x = trippy
ox oo3xo&#x = pen
ox oo3xx&#x = triddip
ox ox3ox&#x   ‡
ox xo3xo&#x   ‡
ox ox3xo&#x = traf
ox ox3xx&#x = tripuf
ox xo3xx&#x = tricuf
ox xx3xx&#x = thiddip
xx oo3ox&#x = tepe
xx oo3xx&#x = tisdip
xx ox3ox&#x   † (hippyp)
xx ox3xo&#x = tope
xx ox3xx&#x = tricupe
xx xx3xx&#x = shiddip

oo3oo3ox&#x = pen
oo3oo3xx&#x = tepe
oo3ox3oo&#x = octpy
oo3ox3ox&#x   ‡
oo3ox3xo&#x = rap
oo3ox3xx&#x = tetatut
oo3xx3oo&#x = ope
oo3xx3ox&#x = octatut
oo3xx3xx&#x = tuttip
ox3oo3ox&#x   † (copy)
ox3oo3xo&#x = hex
ox3oo3xx&#x = tetaco
ox3ox3ox&#x   ‡
ox3ox3xo&#x   † (tetaltut)
ox3ox3xx&#x   ‡
ox3xo3ox&#x = octaco
ox3xo3xx&#x = coatut
ox3xx3ox&#x   † (octatoe)
ox3xx3xo&#x = tuta
ox3xx3xx&#x = tutatoe
xx3oo3xx&#x = cope
xx3ox3xx&#x = coatoe
xx3xx3xx&#x = tope

...
```
```oox oox&#x = squasc
oox oxo&#x = pen
oox oxx&#x = trippy
oox xxo&#x = squasc
oox xxx&#x = tepe
oxx oxx&#x = cubpy
oxx xox&#x = bidrap
oxx xxx&#x = squippyp
xxx xxx&#x = tisdip

ooo3oox&#x = pen
ooo3oxx&#x = trippy
ooo3xxx&#x = triddip
oox3oox&#x   ‡
oox3oxo&#x = octpy
oox3oxx&#x   † (decomposition of tricu)
oox3xxo&#x = traf
oox3xxx&#x = tricuf
oxx3oxx&#x   ‡
oxx3xox&#x = traw
oxx3xxx&#x = tripuf
xxx3xxx&#x = thiddip

...
```
```ooox&#x = pen
ooxx&#x = squasc
oxxx&#x = trippy
xxxx&#x = tepe
```
```oooo&#x = pen
```

```
----
5D
----
```
Wythoffians Lace Prisms
```x x x x x = pent

x x x o3x = tracube
x x x x3x = hacube

x x o3o3x = squatet
x x o3x3o = squoct
x x o3x3x = squatut
x x x3o3x = squaco
x x x3x3x = squatoe

x o3x o3x = tratrip
x o3x x3x = trahip
x x3x x3x = hahip

x o3o3o3x = penp
x o3o3x3o = rappip
x o3o3x3x = tippip
x o3x3o3x = srippip
x o3x3x3o = decap
x o3x3x3x = grippip
x x3o3o3x = spiddip
x x3o3x3x = prippip
x x3x3x3x = gippiddip

o3x o3o3x = tratet
o3x o3x3o = troct
o3x o3x3x = tratut
o3x x3o3x = traco
o3x x3x3x = tratoe
x3x o3o3x = hatet
x3x o3x3o = hoct
x3x o3x3x = hatut
x3x x3o3x = haco
x3x x3x3x = hatoe

o3o3o3o3x = hix
o3o3o3x3o = rix
o3o3o3x3x = tix
o3o3x3o3o = dot
o3o3x3o3x = sarx
o3o3x3x3o = bittix
o3o3x3x3x = garx
o3x3o3o3x = spix
o3x3o3x3o = sibrid
o3x3o3x3x = pattix
o3x3x3o3x = pirx
o3x3x3x3o = gibrid
o3x3x3x3x = gippix
x3o3o3x3x = cappix
x3o3x3o3x = card
x3o3x3x3x = cograx
x3x3o3x3x = captid

x o3o3o *c3x = hexip
x o3x3o *c3o = icope
x o3o3x *c3x = rittip
x o3x3o *c3x = thexip
x o3x3x *c3x = tahp
x x3o3x *c3x = ricope
x x3x3x *c3x = ticope

o3o3o *b3o3x = tac
o3o3o *b3x3o = rat
o3o3o *b3x3x = tot
o3o3x *b3o3o = hin
o3o3x *b3o3x = siphin
o3o3x *b3x3o = sirhin
o3o3x *b3x3x = pirhin
o3x3o *b3o3o = nit
o3x3o *b3o3x = sart
o3x3o *b3x3o = bittit
o3x3o *b3x3x = gart
o3x3x *b3o3o = thin
o3x3x *b3o3x = pithin
o3x3x *b3x3o = girhin
o3x3x *b3x3x = giphin
x3o3x *b3o3o = rin
x3o3x *b3o3x = spat
x3o3x *b3x3o = sibrant
x3o3x *b3x3x = pattit
x3x3x *b3o3o = bittin
x3x3x *b3o3x = pirt
x3x3x *b3x3o = gibrant
x3x3x *b3x3x = gippit

...
```
```ox ox ox ox&#x   † (tespy)
ox ox ox xo&#x   † (cubasc)
ox ox ox xx&#x = cubpyp
ox ox xo xo&#x   † ({4} || perp {4})
ox ox xo xx&#x = squascop
ox ox xx xx&#x = squasquippy
ox xo xx xx&#x = squatet
ox xx xx xx&#x = tracube
xx xx xx xx&#x = pent

ox ox oo3ox&#x = tisdippy
ox ox oo3xo&#x = squete
ox ox oo3xx&#x = tisquippy
ox ox ox3ox&#x   ‡
ox ox ox3xo&#x = traltisdip
ox ox ox3xx&#x = trashiddip
ox ox xo3xo&#x   ‡
ox ox xo3xx&#x = tisdipah
ox ox xx3xx&#x = hisquippy
ox xo oo3ox&#x = trippasc
ox xo oo3xx&#x = tratet
ox xo ox3ox&#x   ‡
ox xo ox3xo&#x = triddaf
ox xo ox3xx&#x = triphipdaw
ox xo xx3xx&#x = hatet
ox xx oo3ox&#x = trippyp
ox xx oo3xo&#x = penp
ox xx oo3xx&#x = tratrip
ox xx ox3ox&#x   ‡
ox xx ox3xo&#x = traffip
ox xx ox3xx&#x = tripufip
ox xx xo3xo&#x   ‡
ox xx xo3xx&#x = tricufip
ox xx xx3xx&#x = trahip
xx xx oo3ox&#x = squatet
xx xx oo3xx&#x = tracube
xx xx ox3ox&#x   †
xx xx ox3xo&#x = squoct
xx xx ox3xx&#x = squatricu
xx xx xx3xx&#x = hacube

ox oo3oo3ox&#x = tepepy
ox oo3oo3xo&#x = hix
ox oo3oo3xx&#x = tratet
ox oo3ox3oo&#x = opepy
ox oo3ox3ox&#x   ‡
ox oo3ox3xo&#x = teta ope
ox oo3ox3xx&#x = teta tuttip
ox oo3xo3oo&#x = octasc
ox oo3xo3ox&#x = octatepe
ox oo3xo3xo&#x   ‡
ox oo3xo3xx&#x = tepatut
ox oo3xx3oo&#x = troct
ox oo3xx3ox&#x = octatuttip
ox oo3xx3xo&#x = opeatut
ox oo3xx3xx&#x = tratut
ox ox3oo3ox&#x   ‡
ox ox3oo3xo&#x = tetaf
ox ox3oo3xx&#x = tetacope
ox ox3ox3ox&#x   ‡
ox ox3ox3xo&#x   ‡
ox ox3ox3xx&#x   ‡
ox ox3xo3ox&#x = octacope
ox ox3xo3xo&#x   ‡
ox ox3xo3xx&#x = copatut
ox ox3xx3ox&#x   ‡
ox ox3xx3xo&#x = tutaf
ox ox3xx3xx&#x = tutatope
ox xo3oo3xo&#x   ‡
ox xo3oo3xx&#x = tepaco
ox xo3ox3xo&#x = opeaco
ox xo3ox3xx&#x = coatuttip
ox xo3xo3xo&#x   ‡
ox xo3xo3xx&#x   ‡
ox xo3xx3xo&#x   ‡
ox xo3xx3xx&#x = tuttipa toe
ox xx3oo3xx&#x = traco
ox xx3ox3xx&#x = coatope
ox xx3xo3xx&#x = copatoe
ox xx3xx3xx&#x = tratoe
xx oo3oo3ox&#x = penp
xx oo3oo3xx&#x = squatet
xx oo3ox3oo&#x = octpyp
xx oo3ox3ox&#x   ‡
xx oo3ox3xo&#x = rappip
xx oo3ox3xx&#x = tepeatuttip
xx oo3xx3oo&#x = squoct
xx oo3xx3ox&#x = opeatuttip
xx oo3xx3xx&#x = squatut
xx ox3oo3ox&#x   † (copy-prism)
xx ox3oo3xo&#x = hexip
xx ox3oo3xx&#x = tepeacope
xx ox3ox3ox&#x   ‡
xx ox3ox3xo&#x   † (tetaltut-prism)
xx ox3ox3xx&#x   ‡
xx ox3xo3ox&#x = opeacope
xx ox3xo3xx&#x = copea tuttip
xx ox3xx3ox&#x   † (octatoe-prism)
xx ox3xx3xo&#x = tutcupip
xx ox3xx3xx&#x = tuttipa tope
xx xx3oo3xx&#x = squaco
xx xx3ox3xx&#x = copatope
xx xx3xx3xx&#x = squatoe
```
```oo3ox oo3ox&#x = triddippy
oo3ox oo3xo&#x = hix
oo3ox oo3xx&#x = tratet
oo3ox ox3ox&#x   ‡
oo3ox ox3xo&#x = trial triddip
oo3ox ox3xx&#x = trathiddip
oo3ox xo3xo&#x   ‡
oo3ox xo3xx&#x = pexhix
oo3ox xx3xx&#x = hatet
oo3xx oo3xx&#x = tratrip
oo3xx ox3xo&#x = troct
oo3xx ox3xx&#x = tritricu
oo3xx xx3xx&#x = trahip
ox3ox ox3ox&#x   ‡
ox3ox ox3xo&#x   ‡
ox3ox ox3xx&#x   ‡
ox3ox xo3ox&#x   ‡
ox3ox xo3xo&#x   ‡
ox3ox xo3xx&#x   ‡
ox3ox xx3xx&#x   †
ox3xo ox3xo&#x = tridafup
ox3xo ox3xx&#x = triddip althiddip
ox3xo xx3xx&#x = hoct
ox3xx ox3xx&#x = triddippa hiddip
ox3xx xo3xx&#x = pabex hix
ox3xx xx3xx&#x = hatricu
xx3xx xx3xx&#x = hahip

oo3oo3oo3ox&#x = hix
oo3oo3oo3xx&#x = penp
oo3oo3ox3oo&#x = rappy
oo3oo3ox3ox&#x   ‡
oo3oo3ox3xo&#x = rix
oo3oo3ox3xx&#x = pennatip
oo3oo3xx3oo&#x = rappip
oo3oo3xx3ox&#x = rapatip
oo3oo3xx3xx&#x = tippip
oo3ox3oo3ox&#x   ‡
oo3ox3oo3xo&#x = dihin
oo3ox3oo3xx&#x = penasrip
oo3ox3ox3oo&#x   ‡
oo3ox3ox3ox&#x   ‡
oo3ox3ox3xo&#x   ‡
oo3ox3ox3xx&#x   ‡
oo3ox3xo3oo&#x = dot
oo3ox3xo3ox&#x = rapasrip
oo3ox3xo3xo&#x   † (rapaltip)
oo3ox3xo3xx&#x = sripatip
oo3ox3xx3ox&#x   ‡
oo3ox3xx3xx&#x = tipagrip
oo3xx3oo3ox&#x = rapalsrip
oo3xx3oo3xx&#x = srippip
oo3xx3ox3ox&#x   ‡
oo3xx3ox3xx&#x = sripagrip
oo3xx3xx3oo&#x = decap
oo3xx3xx3ox&#x = deca agrip
oo3xx3xx3xx&#x = grippip
ox3oo3oo3ox&#x   † (spidpy)
ox3oo3oo3xo&#x = tac
ox3oo3oo3xx&#x = penaspid
ox3oo3ox3ox&#x   ‡
ox3oo3ox3xo&#x   † (penalsrip)
ox3oo3ox3xx&#x   ‡
ox3oo3xo3ox&#x = rapaspid
ox3oo3xo3xo&#x   ‡
ox3oo3xo3xx&#x = spidatip
ox3oo3xx3ox&#x   † (rapalprip)
ox3oo3xx3xo&#x = sripaltip
ox3oo3xx3xx&#x = tipalprip
ox3ox3oo3xx&#x   ‡
ox3ox3ox3ox&#x   ‡
ox3ox3ox3xo&#x   ‡
ox3ox3ox3xx&#x   ‡
ox3ox3xo3ox&#x   ‡
ox3ox3xo3xo&#x   ‡
ox3ox3xo3xx&#x   † (tipaprip)
ox3ox3xx3ox&#x   ‡
ox3ox3xx3xo&#x   ‡
ox3ox3xx3xx&#x   ‡
ox3xo3oo3xx&#x = spidasrip
ox3xo3ox3xo&#x = sripa
ox3xo3ox3xx&#x = sripalprip
ox3xo3xo3xx&#x   ‡
ox3xo3xx3ox&#x = deca aprip
ox3xo3xx3xo&#x   † (sripalgrip)
ox3xo3xx3xx&#x = pripalgrip
ox3xx3oo3xx&#x = sripaprip
ox3xx3ox3xx&#x   ‡
ox3xx3xo3xx&#x = pripagrip
ox3xx3xx3ox&#x   † (deca || gippid)
ox3xx3xx3xo&#x = gripa
ox3xx3xx3xx&#x = gripagippid
xx3oo3oo3xx&#x = spiddip
xx3oo3ox3xx&#x = spidaprip
xx3oo3xx3xx&#x = prippip
xx3ox3ox3xx&#x   ‡
xx3ox3xo3xx&#x = pripa
xx3ox3xx3xx&#x = pripa gippid
xx3xx3xx3xx&#x = gippiddip
```
```oo3oo3oo *b3ox&#x = hexpy
oo3oo3oo *b3xx&#x = hexip
oo3oo3ox *b3ox&#x   ‡
oo3oo3ox *b3xo&#x = hin
oo3oo3ox *b3xx&#x = hexalrit
oo3oo3xx *b3xx&#x = rittip
oo3ox3oo *b3oo&#x   † (icopy)
oo3ox3oo *b3ox&#x   ‡
oo3ox3oo *b3xo&#x = hexaico
oo3ox3oo *b3xx&#x   † (hexathex)
oo3ox3ox *b3ox&#x   ‡
oo3ox3ox *b3xo&#x   ‡
oo3ox3ox *b3xx&#x   ‡
oo3ox3xo *b3xo&#x   ‡
oo3ox3xo *b3xx&#x = ritag thex
oo3ox3xx *b3xx&#x   † (ritatah)
oo3xx3oo *b3oo&#x = icope
oo3xx3oo *b3ox&#x = icathex
oo3xx3oo *b3xx&#x = thexip
oo3xx3ox *b3ox&#x   ‡
oo3xx3ox *b3xo&#x = thexa
oo3xx3ox *b3xx&#x = thexagtah
oo3xx3xx *b3xx&#x = tahp
ox3oo3ox *b3ox&#x   ‡
ox3oo3ox *b3xo&#x   † (hexarit)
ox3oo3ox *b3xx&#x   ‡
ox3oo3xo *b3xx&#x = rita
ox3oo3xx *b3xx&#x = ritarico
ox3ox3ox *b3ox&#x   ‡
ox3ox3ox *b3xo&#x   ‡
ox3ox3ox *b3xx&#x   ‡
ox3ox3xo *b3xo&#x   † (ritathex)
ox3ox3xo *b3xx&#x   ‡
ox3ox3xx *b3xx&#x   ‡
ox3xo3ox *b3ox&#x   † (icarico)
ox3xo3ox *b3xx&#x   ‡
ox3xo3xx *b3xx&#x = ricatah
ox3xx3ox *b3ox&#x   ‡
ox3xx3ox *b3xo&#x   † (thexatah)
ox3xx3ox *b3xx&#x   ‡
ox3xx3xo *b3xx&#x = taha
ox3xx3xx *b3xx&#x = tahatico
xx3oo3xx *b3xx&#x = ricope
xx3ox3xx *b3xx&#x   † (ricoatico)
xx3xx3xx *b3xx&#x = ticope

...
```
Trigonics
```oox oox oox&#x   † (cubasc)
oox oox oxo&#x = squete
oox oox oxx&#x = squippyippy
oox oox xxo&#x   † ({4} || perp {4})
oox oox xxx&#x = squascop
oox oxo oxx&#x = tepepy
oox oxo xoo&#x = hix
oox oxo xox&#x = trippasc
oox oxo xxx&#x = penp
oox oxx oxx&#x = tisdippy
oox oxx xox&#x = tetcubedaw
oox oxx xxo&#x = editetaf
oox oxx xxx&#x = trippyp
oox xxo xxo&#x   † (cubasc)
oox xxo xxx&#x = squascop
oox xxx xxx&#x = squatet
oxx oxx oxx&#x   ‡
oxx oxx xox&#x = cubasquasc
oxx oxx xxx&#x = cubpyp
oxx xox xxo&#x = tedrix
oxx xox xxx&#x = tepacube
oxx xxx xxx&#x = squasquippy
xxx xxx xxx&#x = tracube
```
```oox ooo3oox&#x = trippasc
oox ooo3oxo&#x = hix
oox ooo3oxx&#x = triddippy
oox ooo3xxo&#x = trippasc
oox ooo3xxx&#x = tratet
oox oox3oox&#x   ‡
oox oox3oxo&#x = trafpy
oox oox3oxx&#x   ‡
oox oox3xxo&#x = triddaf
oox oox3xxx&#x = triphipdaw
oox oxo3oxo&#x   ‡
oox oxo3oxx&#x   † (trip || hippy)
oox oxo3xoo&#x = octasc
oox oxo3xox&#x = trial triddip
oox oxo3xxo&#x   ‡
oox oxo3xxx&#x = pexhix
oox oxx3oxx&#x   ‡
oox oxx3xox&#x = octhipdaw
oox oxx3xxo&#x = tripgytricudaw
oox oxx3xxx&#x = trathiddip
oox xxo3xxo&#x   ‡
oox xxo3xxx&#x = triphipdaw
oox xxx3xxx&#x = hatet
oxx ooo3oox&#x = tepepy
oxx ooo3oxx&#x = tisdippy
oxx ooo3xoo&#x = squete
oxx ooo3xox&#x = penatrip
oxx ooo3xxx&#x = tisquippy
oxx oox3oox&#x   ‡
oxx oox3oxo&#x = opepy
oxx oox3oxx&#x   ‡
oxx oox3xoo&#x = ditetaf
oxx oox3xox&#x   † ({3} || hippyp)
oxx oox3xxo&#x = taope
oxx oox3xxx&#x = triddipa hip
oxx oxx3oxx&#x   ‡
oxx oxx3xoo&#x = traltisdip
oxx oxx3xox&#x = trial tricupe
oxx oxx3xxx&#x = trashiddip
oxx xoo3xoo&#x   ‡
oxx xoo3xox&#x   ‡
oxx xoo3xxx&#x = tisdipah
oxx xox3xox&#x   ‡
oxx xox3xxo&#x = opeah
oxx xox3xxx&#x = tripa thiddip
oxx xxx3xxx&#x = hisquippy
xxx ooo3oox&#x = penp
xxx ooo3oxx&#x = trippyp
xxx ooo3xxx&#x = tratrip
xxx oox3oox&#x   ‡
xxx oox3oxo&#x = octpyp
xxx oox3oxx&#x   † (trip || hippyp)
xxx oox3xxo&#x = traffip
xxx oox3xxx&#x = tricufip
xxx oxx3oxx&#x   ‡
xxx oxx3xox&#x = trawp
xxx oxx3xxx&#x = tripufip
xxx xxx3xxx&#x = trahip
```
```ooo3ooo3oox&#x = hix
ooo3ooo3oxx&#x = tepepy
ooo3ooo3xxx&#x = tratet
ooo3oox3ooo&#x = octasc
ooo3oox3oox&#x   ‡
ooo3oox3oxo&#x = rappy
ooo3oox3oxx&#x   ‡
ooo3oox3xxo&#x = octatepe
ooo3oox3xxx&#x = tepatut
ooo3oxx3ooo&#x = opepy
ooo3oxx3oox&#x   ‡
ooo3oxx3oxx&#x   ‡
ooo3oxx3xoo&#x = teta ope
ooo3oxx3xox&#x = rapatut
ooo3oxx3xxx&#x = teta tuttip
ooo3xxx3ooo&#x = troct
ooo3xxx3oox&#x = opeatut
ooo3xxx3oxx&#x = octatuttip
ooo3xxx3xxx&#x = tratut
oox3ooo3oox&#x   ‡
oox3ooo3oxo&#x = hexpy
oox3ooo3oxx&#x   † (tet || copy)
oox3ooo3xxo&#x = tetaf
oox3ooo3xxx&#x = tepaco
oox3oox3ooo&#x   ‡
oox3oox3oox&#x   ‡
oox3oox3oxo&#x   ‡
oox3oox3oxx&#x   ‡
oox3oox3xxo&#x   ‡
oox3oox3xxx&#x   ‡
oox3oxo3oox&#x   † (oct || copy)
oox3oxo3oxx&#x   † (tut || copy)
oox3oxo3xoo&#x = bidhin
oox3oxo3xox&#x = rapaco
oox3oxo3xxo&#x   † (tet || inv tetaltut)
oox3oxo3xxx&#x = tetco tuttric
oox3oxx3oox&#x   ‡
oox3oxx3oxo&#x   ‡
oox3oxx3oxx&#x   ‡
oox3oxx3xoo&#x   † (oct || tetaltut)
oox3oxx3xox&#x   ‡
oox3oxx3xxo&#x   † (tut || tetaltut)
oox3oxx3xxx&#x   ‡
oox3xxo3oox&#x = opeaco
oox3xxo3oxx&#x = octco tuttric
oox3xxo3xxo&#x   ‡
oox3xxo3xxx&#x = coatuttip
oox3xxx3oox&#x   ‡
oox3xxx3oxo&#x = octa tutcup
oox3xxx3oxx&#x   † (tut || octatoe)
oox3xxx3xxo&#x = tutaf
oox3xxx3xxx&#x = tuttipa toe
oxx3ooo3oxx&#x   ‡
oxx3ooo3xox&#x = hexaco
oxx3ooo3xxx&#x = tetacope
oxx3oox3oxx&#x   ‡
oxx3oox3xox&#x   ‡
oxx3oox3xxo&#x   † (co || tetaltut)
oxx3oox3xxx&#x   ‡
oxx3oxx3oxx&#x   ‡
oxx3oxx3xox&#x   ‡
oxx3oxx3xxx&#x   ‡
oxx3xoo3oxx&#x = octacope
oxx3xoo3xxx&#x = copatut
oxx3xox3oxx&#x   † (co || octatoe)
oxx3xox3xxo&#x = coa tutcup
oxx3xox3xxx&#x = cotut totric
oxx3xxx3oxx&#x   ‡
oxx3xxx3xox&#x = tutcupa toe
oxx3xxx3xxx&#x = tutatope
xxx3ooo3xxx&#x = traco
xxx3oox3xxx&#x = copatoe
xxx3oxx3xxx&#x = coatope
xxx3xxx3xxx&#x = tratoe
```
```...
```
4-layered Lace Simplices 5-layered Lace Simplices
```ooox ooox&#x = squete
ooox ooxo&#x = hix
ooox ooxx&#x = trippasc
ooox oxxo&#x = squete
ooox oxxx&#x = tepepy
ooox xxxo&#x = trippasc
ooox xxxx&#x = penp
ooxx ooxx&#x   † (cubasc)
ooxx oxox&#x = bidrappy
ooxx oxxx&#x = squippyippy
ooxx xxoo&#x   † ({4} || perp {4})
ooxx xxox&#x = squasquasc
ooxx xxxx&#x = squascop
oxxx oxxx&#x = tisdippy
oxxx xoxx&#x = tetcubedaw
oxxx xxxx&#x = trippyp
xxxx xxxx&#x = squatet
```
```oooo3ooox&#x = hix
oooo3ooxx&#x = trippasc
oooo3oxxx&#x = triddippy
oooo3xxxx&#x = tratet
ooox3ooox&#x   ‡
ooox3ooxo&#x = octasc
ooox3ooxx&#x   ‡
ooox3oxxo&#x = trafpy
ooox3oxxx&#x   † (tricuf-py)
ooox3xxxo&#x = trial triddip
ooox3xxxx&#x = pexhix
ooxx3ooxx&#x   ‡
ooxx3oxox&#x   † (traw-py)
ooxx3oxxx&#x   ‡
ooxx3xxoo&#x = triddaf
ooxx3xxox&#x = tripgytricudaw
ooxx3xxxx&#x = triphipdaw
oxxx3oxxx&#x   ‡
oxxx3xoxx&#x = octhipdaw
oxxx3xxxx&#x = trathiddip
xxxx3xxxx&#x = hatet
```
```...
```
```oooox&#x = hix
oooxx&#x = squete
ooxxx&#x = trippasc
oxxxx&#x = tepepy
xxxxx&#x = penp

```
6-layered Lace Simplices
```oooooo&#x = hix

```