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get replaced by some general n. Then the configuration space however would be n-1 dimensional (and those therefore no longer represent instances of lace cities, rather they then might be considered to represent some lace hyper-cities).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:

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.
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.
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

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    (up)
----

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 = ope 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 = ope 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

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 = ope
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 = ope
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    (up)
----

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 x3o3o3o3x = scad x3o3o3x3x = cappix x3o3x3o3x = card x3o3x3x3x = cograx x3x3o3x3x = captid x3x3x3x3x = gocad 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 oo3ox3xx3oo&#x = rapadeca oo3ox3xx3ox&#x ‡ oo3ox3xx3xo&#x = tipadeca oo3ox3xx3xx&#x = tipagrip oo3xx3oo3ox&#x = rapalsrip oo3xx3oo3xx&#x = srippip oo3xx3ox3ox&#x ‡ oo3xx3ox3xo&#x = sripadeca 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 ox3xo3xo3ox&#x † (spidadeca) 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 = thexarico 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


----
 6D    (up)
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(So far just some ...)

Wythoffians	Lace Prisms
x x x x x x = ax x x x x o3x = tratess x x x x x3x = higtess x x x o3o3x = tetcube x x x o3x3o = octcube ... x x o3x o3x = titstip x x x3x x3x = shihtip ... x x o3o3o3x = squapen x x o3o3x3o = squarap x x x3x3x3x = squagippid ... x x o3o3o d3x = squahex x x o3x3o d3o = squico ... x o3x o3o3x = tratepe x o3x o3x3o = trope x o3x x3o3x = tracope ... x o3o3o3o3x = hixip x o3o3o3x3o = rixip x o3o3o3x3x = tixip x o3o3x3o3o = dotip x o3o3x3o3x = sarxip x o3x3o3o3x = spixip x x3o3o3o3x = scadip x x3o3x3o3x = cardip x x3x3x3x3x = gocadip ... x o3o3o c3o3x = taccup x o3o3o c3x3o = ratip x o3o3x c3o3o = hinnip x o3o3x c3o3x = siphinnip x o3x3o c3o3o = nitip x x3x3x c3x3x = gippitip ... o3x o3x o3x = trittip o3x o3x x3x = tithtip x3x x3x x3x = hittip ... o3x o3o3o3x = trapen o3x o3o3x3o = trarap o3x o3x3o3x - trasrip o3x o3x3x3x = tragrip o3x x3o3o3x = traspid x3x x3x3x3x = hagippid ... o3x o3o3o d3x = trahex o3x o3x3o d3o = trico x3x x3x3x *d3x = hatico ... o3o3x o3o3x = tetdip o3o3x o3x3o = tetoct o3o3x o3x3x = tettut o3o3x x3o3x - tetco o3x3o o3x3o = octdip o3x3x o3x3x = tutdip x3x3x x3x3x = toedip ... o3o3o3o3o3x = hop o3o3o3o3x3o = ril o3o3o3o3x3x = til o3o3o3x3o3o = bril o3o3o3x3o3x = sril o3o3o3x3x3o = batal o3o3o3x3x3x = gril o3o3x3o3o3x = spil o3o3x3o3x3o = sabril o3o3x3o3x3x = patal o3o3x3x3o3o = fe o3o3x3x3o3x = pril o3o3x3x3x3o = gabril o3o3x3x3x3x = gapil o3x3o3o3o3x = scal o3x3o3o3x3o = sibpof o3x3o3o3x3x = catal o3x3o3x3o3x = cral o3x3o3x3x3o = bapril o3x3o3x3x3x = cagral o3x3x3o3o3x = copal o3x3x3o3x3x = captal o3x3x3x3o3x = copril o3x3x3x3x3o = gibpof o3x3x3x3x3x = gacal x3o3x3x3x3x = tagopal x3x3o3x3x3x = tacogral x3x3x3x3x3x = gotaf	o3o3o b3o3o3x = gee o3o3o b3o3x3o = rag o3o3o b3o3x3x = tag o3o3o b3x3o3o = brag o3o3o b3x3o3x = srog o3o3o b3x3x3o = botag o3o3o b3x3x3x = grog o3o3x b3o3o3o = hax o3o3x b3o3o3x = sochax o3o3x b3o3x3o = sophax o3o3x b3o3x3x = cophix o3o3x b3x3o3o = sirhax o3o3x b3x3o3x = crohax o3o3x b3x3x3o = prohax o3o3x b3x3x3x = caprohax o3x3o b3o3o3o = brox o3x3o b3o3o3x = spog o3x3o b3o3x3o = siborg o3x3o b3o3x3x = potag o3x3o b3x3o3o = xog o3x3o b3x3o3x = prog o3x3o b3x3x3o = gaborg o3x3o b3x3x3x = gopog o3x3x b3o3o3o = thax o3x3x b3o3o3x = cathix o3x3x b3o3x3o = pithax o3x3x b3o3x3x = capthix o3x3x b3x3o3o = girhax o3x3x b3x3o3x = cagrohax o3x3x b3x3x3o = gophax o3x3x b3x3x3x = gochax x3o3x b3o3o3o = rax x3o3x b3o3o3x = scag x3o3x b3o3x3o = sobpoxog x3o3x b3o3x3x = catog x3o3x b3x3o3o = saborx x3o3x b3x3o3x = crag x3o3x b3x3x3o = boprax x3o3x b3x3x3x = cagorg x3x3x b3o3o3o = botox x3x3x b3o3o3x = copog x3x3x b3o3x3o = boprag x3x3x b3o3x3x = captog x3x3x b3x3o3o = gaborx x3x3x b3x3o3x = coprag x3x3x b3x3x3o = gobpoxog x3x3x b3x3x3x = gocog o3o3o3o3o c3x = mo o3o3o3o3x c3o = jak o3o3o3o3x c3x = hejak o3o3o3x3o c3o = rojak o3o3o3x3o c3x = harjak o3o3o3x3x c3o = tojak o3o3o3x3x c3x = hotjak o3o3x3o3o c3o = ram o3o3x3o3o c3x = tim o3o3x3o3x c3o = sirjak o3o3x3o3x c3x = shorjak o3o3x3x3o c3o = botajik o3o3x3x3o c3x = titajak o3o3x3x3x c3o = girjak o3o3x3x3x c3x = ghorjak o3x3o3o3x c3o = shopjak o3x3o3o3x c3x = spojak o3x3o3x3o c3o = barm o3x3o3x3o c3x = sram o3x3o3x3x c3o = hopitjak o3x3o3x3x c3x = potjak o3x3x3o3x c3o = haprojak o3x3x3o3x c3x = projak o3x3x3x3o c3o = bitem o3x3x3x3o c3x = gram o3x3x3x3x c3o = ghopjak o3x3x3x3x c3x = gapjak x3o3o3o3x c3o = trim x3o3o3o3x c3x = spam x3o3o3x3x c3o = catjak x3o3o3x3x c3x = hictijik x3o3x3o3x c3o = sabrim x3o3x3o3x c3x = patom x3o3x3x3x c3o = cograjik x3o3x3x3x c3x = hocgarjik x3x3o3x3x c3o = titam x3x3o3x3x c3x = prom x3x3x3x3x c3o = gabrim x3x3x3x3x c3x = gopam	ox xo oo3oo3ox&#x = tepasc ox xo oo3oo3xx&#x = tetdip ox xx oo3oo3ox&#x = tepepyp ox xx ox3oo3xo&#x = tetafip ... ox oo3ox oo3xo&#x = trippete ox oo3xo oo3xo&#x = triddipasc ox oo3xo ox3xo&#x = tripal triddip ox oo3xx ox3xo&#x = titraf ox ox3xo ox3xo&#x = triddipabl tratrip xx oo3ox oo3xo&#x = hixip xx ox3xo ox3xo&#x = tridafupip ... ox oo3oo3oo3ox&#x = penppy ox oo3oo3oo3xo&#x = hop ox oo3oo3oo3xx&#x = trapen ox oo3oo3ox3xo&#x = penarappip ox oo3oo3xo3oo&#x = rapesc ox oo3oo3xo3ox&#x = rapa penp ox oo3oo3xx3oo&#x = trarap ox oo3xo3oo3ox&#x = rapalpenp ox ox3oo3oo3xo&#x = penaf ox xo3oo3oo3xx&#x = spidapenp ox xo3oo3ox3xo&#x = spida rappip xx oo3oo3oo3ox&#x = hixip xx oo3oo3oo3xx&#x = squapen xx oo3ox3xo3oo&#x = dotip xx xx3xx3xx3xx&#x = squagippid ... ox oo3oo3xo c3oo&#x = hexasc ox oo3oo3xo c3ox&#x = hexaf ...	oo3ox oo3oo3ox&#x = tratetpy oo3ox oo3oo3xo&#x = hop oo3ox oo3oo3xx&#x = tetdip oo3ox oo3ox3oo&#x = troctpy oo3ox oo3xo3oo&#x = octete oo3ox oo3xo3ox&#x = octa tratet oo3ox oo3xx3xx&#x = tettut oo3ox ox3oo3xo&#x = tetal tratet oo3xx oo3oo3ox&#x = trapen oo3xx oo3ox3xo&#x = trarap oo3xx ox3oo3xo&#x = trahex ox3xo oo3oo3ox&#x = trial tratet ox3xo oo3ox3xo&#x = tratet altroct ox3xo ox3oo3xo&#x = tratetdafup ... oo3oo3oo3oo3ox&#x = hop oo3oo3oo3oo3xx&#x = hixip oo3oo3oo3ox3oo&#x = rixpy oo3oo3oo3ox3xo&#x = ril oo3oo3oo3xx3oo&#x = rixip oo3oo3oo3xx3xx&#x = tixip oo3oo3ox3oo3oo&#x = dotpy oo3oo3ox3oo3xo&#x = hixadot oo3oo3ox3xo3oo&#x = bril oo3oo3ox3xo3ox&#x = rixasarx oo3oo3xx3oo3oo&#x = dotip oo3ox3oo3oo3xo&#x = hixalrix oo3ox3oo3xo3oo&#x = rixa oo3ox3oo3xo3ox&#x = rixaspix oo3ox3xo3oo3ox&#x = dottaspix oo3ox3xo3ox3oo&#x = dottasibrid oo3ox3xo3ox3xo&#x = sarxasibrid oo3ox3xx3xo3oo&#x = bittixa oo3xx3oo3oo3ox&#x = rixalspix oo3xx3oo3oo3xx&#x = spixip ox3oo3oo3oo3xo&#x = gee ox3oo3oo3xo3ox&#x = rixascad ox3xo3oo3ox3xo&#x = spixa ox3oo3xo3oo3ox&#x = dottascad ox3oo3xo3ox3xo&#x = spixalsarx ox3xo3oo3oo3xx&#x = scadaspix ox3xo3ox3xo3ox&#x = sibridacard xx3oo3oo3oo3xx&#x = scadip xx3oo3xx3oo3xx&#x = cardip xx3ox3oo3xo3xx&#x = cappixa xx3ox3xx3xo3xx&#x = cograxa xx3xx3xx3xx3xx&#x = gocadip ... oo3oo3oo b3oo3ox&#x = tacpy oo3oo3oo b3oo3xx&#x = taccup oo3oo3oo b3ox3oo&#x = ratpy oo3oo3oo b3ox3xo&#x = taccarat oo3oo3ox b3oo3oo&#x = hinpy oo3oo3ox b3oo3xo&#x = dijak oo3oo3ox b3xo3oo&#x = hinro oo3oo3ox b3xo3ox&#x = ratasiphin oo3oo3xx b3oo3oo&#x = hinnip oo3oo3xx b3oo3ox&#x = hinasiphin oo3oo3xx b3oo3xx&#x = siphinnip oo3ox3oo b3oo3xo&#x = taccanit oo3ox3oo b3xo3oo&#x = ratanit oo3ox3oo b3xo3ox&#x = ratasart oo3ox3xo b3oo3oo&#x = hinanit oo3ox3xo b3oo3xo&#x = nitasiphin oo3ox3xo b3xo3oo&#x = nitasirhin oo3ox3xo b3xo3ox&#x = sirhinasart ox3oo3ox b3xo3oo&#x = ratarin ox3oo3xo b3oo3oo&#x = hax ox3oo3xo b3oo3ox&#x = hinalsiphin ox3oo3xo b3oo3xx&#x = siphina ox3oo3xo b3xx3oo&#x = sirhina ox3oo3xx b3oo3oo&#x = hinarin ox3xo3ox b3oo3oo&#x = nitarin ox3xo3ox b3oo3ox&#x = nitaspat ox3xo3ox b3ox3xo&#x = sarta sibrant ox3xo3xx b3oo3oo&#x = rinathin ...
Trigonics	4-layered Lace Simplices	...
oox oxo ooo3oox&#x = hop oox oxo ooo3oxx&#x = tratetpy oox oxo ooo3xox&#x = triddipasc oox oxo xoo3oxx&#x = trial tratet oox oxx ooo3xoo&#x = trippete oox oxx oxo3xox&#x = tatraffip oxx xox ooo3xxo&#x = pedril oxx xox oox3xxo&#x = tisdip altratet ... oox ooo3ooo3oxo&#x = hop oox ooo3ooo3xxo&#x = tepasc oox ooo3oxo3ooo&#x = octete oox ooo3oxo3xoo&#x = rapesc oox ooo3oxo3xox&#x = octa tratet oox oxo3ooo3xoo&#x = hexasc oox oxo3ooo3xox&#x = tetal tratet oox oxx3ooo3xox&#x = hexcopedaw oox oxx3ooo3xxo&#x = tetaco altepetric oxx ooo3ooo3xoo&#x = squepe oxx oox3ooo3oxo&#x = hexippy oxx xox3ooo3xxo&#x = hexipaco ...	ooo3oox ooo3oxo&#x = hop ooo3oox ooo3xxo&#x = trippete ooo3oox oxo3xoo&#x = octete ooo3oox oxo3xox&#x = trial tratet oox3xxo oxo3xox&#x = triddippa tridafup ... ooo3ooo3ooo3oox&#x = hop ooo3ooo3ooo3oxx&#x = penppy ooo3ooo3ooo3xxx&#x = trapen ooo3ooo3oox3ooo&#x = rapesc ooo3oox3oxo3ooo&#x = dotpy ooo3oox3ooo3xxo&#x = rapalpenp ... oox3ooo3oxo b3xoo&#x = tedjak oox3xxx3oxo b3xoo&#x = thexgyt oxx3ooo3xox b3xxo&#x = ritgyt oxx3xxx3xox b3xxo&#x = tahgyt ...	ooxx oxox xoox&#x = hixacube ...

Wythoffians

Lace Prisms

x x x x x x = ax

x x x x o3x = tratess
x x x x x3x = higtess

x x x o3o3x = tetcube
x x x o3x3o = octcube
...

x x o3x o3x = titstip
x x x3x x3x = shihtip
...

x x o3o3o3x = squapen
x x o3o3x3o = squarap
x x x3x3x3x = squagippid
...

x x o3o3o *d3x = squahex
x x o3x3o *d3o = squico
...

x o3x o3o3x = tratepe
x o3x o3x3o = trope
x o3x x3o3x = tracope
...

x o3o3o3o3x = hixip
x o3o3o3x3o = rixip
x o3o3o3x3x = tixip
x o3o3x3o3o = dotip
x o3o3x3o3x = sarxip
x o3x3o3o3x = spixip
x x3o3o3o3x = scadip
x x3o3x3o3x = cardip
x x3x3x3x3x = gocadip
...

x o3o3o *c3o3x = taccup
x o3o3o *c3x3o = ratip
x o3o3x *c3o3o = hinnip
x o3o3x *c3o3x = siphinnip
x o3x3o *c3o3o = nitip
x x3x3x *c3x3x = gippitip
...

o3x o3x o3x = trittip
o3x o3x x3x = tithtip
x3x x3x x3x = hittip
...

o3x o3o3o3x = trapen
o3x o3o3x3o = trarap
o3x o3x3o3x - trasrip
o3x o3x3x3x = tragrip
o3x x3o3o3x = traspid
x3x x3x3x3x = hagippid
...

o3x o3o3o *d3x = trahex
o3x o3x3o *d3o = trico
x3x x3x3x *d3x = hatico
...

o3o3x o3o3x = tetdip
o3o3x o3x3o = tetoct
o3o3x o3x3x = tettut
o3o3x x3o3x - tetco
o3x3o o3x3o = octdip
o3x3x o3x3x = tutdip
x3x3x x3x3x = toedip
...

o3o3o3o3o3x = hop
o3o3o3o3x3o = ril
o3o3o3o3x3x = til
o3o3o3x3o3o = bril
o3o3o3x3o3x = sril
o3o3o3x3x3o = batal
o3o3o3x3x3x = gril
o3o3x3o3o3x = spil
o3o3x3o3x3o = sabril
o3o3x3o3x3x = patal
o3o3x3x3o3o = fe
o3o3x3x3o3x = pril
o3o3x3x3x3o = gabril
o3o3x3x3x3x = gapil
o3x3o3o3o3x = scal
o3x3o3o3x3o = sibpof
o3x3o3o3x3x = catal
o3x3o3x3o3x = cral
o3x3o3x3x3o = bapril
o3x3o3x3x3x = cagral
o3x3x3o3o3x = copal
o3x3x3o3x3x = captal
o3x3x3x3o3x = copril
o3x3x3x3x3o = gibpof
o3x3x3x3x3x = gacal
x3o3x3x3x3x = tagopal
x3x3o3x3x3x = tacogral
x3x3x3x3x3x = gotaf

o3o3o *b3o3o3x = gee
o3o3o *b3o3x3o = rag
o3o3o *b3o3x3x = tag
o3o3o *b3x3o3o = brag
o3o3o *b3x3o3x = srog
o3o3o *b3x3x3o = botag
o3o3o *b3x3x3x = grog
o3o3x *b3o3o3o = hax
o3o3x *b3o3o3x = sochax
o3o3x *b3o3x3o = sophax
o3o3x *b3o3x3x = cophix
o3o3x *b3x3o3o = sirhax
o3o3x *b3x3o3x = crohax
o3o3x *b3x3x3o = prohax
o3o3x *b3x3x3x = caprohax
o3x3o *b3o3o3o = brox
o3x3o *b3o3o3x = spog
o3x3o *b3o3x3o = siborg
o3x3o *b3o3x3x = potag
o3x3o *b3x3o3o = xog
o3x3o *b3x3o3x = prog
o3x3o *b3x3x3o = gaborg
o3x3o *b3x3x3x = gopog
o3x3x *b3o3o3o = thax
o3x3x *b3o3o3x = cathix
o3x3x *b3o3x3o = pithax
o3x3x *b3o3x3x = capthix
o3x3x *b3x3o3o = girhax
o3x3x *b3x3o3x = cagrohax
o3x3x *b3x3x3o = gophax
o3x3x *b3x3x3x = gochax
x3o3x *b3o3o3o = rax
x3o3x *b3o3o3x = scag
x3o3x *b3o3x3o = sobpoxog
x3o3x *b3o3x3x = catog
x3o3x *b3x3o3o = saborx
x3o3x *b3x3o3x = crag
x3o3x *b3x3x3o = boprax
x3o3x *b3x3x3x = cagorg
x3x3x *b3o3o3o = botox
x3x3x *b3o3o3x = copog
x3x3x *b3o3x3o = boprag
x3x3x *b3o3x3x = captog
x3x3x *b3x3o3o = gaborx
x3x3x *b3x3o3x = coprag
x3x3x *b3x3x3o = gobpoxog
x3x3x *b3x3x3x = gocog

o3o3o3o3o *c3x = mo
o3o3o3o3x *c3o = jak
o3o3o3o3x *c3x = hejak
o3o3o3x3o *c3o = rojak
o3o3o3x3o *c3x = harjak
o3o3o3x3x *c3o = tojak
o3o3o3x3x *c3x = hotjak
o3o3x3o3o *c3o = ram
o3o3x3o3o *c3x = tim
o3o3x3o3x *c3o = sirjak
o3o3x3o3x *c3x = shorjak
o3o3x3x3o *c3o = botajik
o3o3x3x3o *c3x = titajak
o3o3x3x3x *c3o = girjak
o3o3x3x3x *c3x = ghorjak
o3x3o3o3x *c3o = shopjak
o3x3o3o3x *c3x = spojak
o3x3o3x3o *c3o = barm
o3x3o3x3o *c3x = sram
o3x3o3x3x *c3o = hopitjak
o3x3o3x3x *c3x = potjak
o3x3x3o3x *c3o = haprojak
o3x3x3o3x *c3x = projak
o3x3x3x3o *c3o = bitem
o3x3x3x3o *c3x = gram
o3x3x3x3x *c3o = ghopjak
o3x3x3x3x *c3x = gapjak
x3o3o3o3x *c3o = trim
x3o3o3o3x *c3x = spam
x3o3o3x3x *c3o = catjak
x3o3o3x3x *c3x = hictijik
x3o3x3o3x *c3o = sabrim
x3o3x3o3x *c3x = patom
x3o3x3x3x *c3o = cograjik
x3o3x3x3x *c3x = hocgarjik
x3x3o3x3x *c3o = titam
x3x3o3x3x *c3x = prom
x3x3x3x3x *c3o = gabrim
x3x3x3x3x *c3x = gopam

ox xo oo3oo3ox&#x = tepasc
ox xo oo3oo3xx&#x = tetdip
ox xx oo3oo3ox&#x = tepepyp
ox xx ox3oo3xo&#x = tetafip
...

ox oo3ox oo3xo&#x = trippete
ox oo3xo oo3xo&#x = triddipasc
ox oo3xo ox3xo&#x = tripal triddip
ox oo3xx ox3xo&#x = titraf
ox ox3xo ox3xo&#x = triddipabl tratrip
xx oo3ox oo3xo&#x = hixip
xx ox3xo ox3xo&#x = tridafupip
...

ox oo3oo3oo3ox&#x = penppy
ox oo3oo3oo3xo&#x = hop
ox oo3oo3oo3xx&#x = trapen
ox oo3oo3ox3xo&#x = penarappip
ox oo3oo3xo3oo&#x = rapesc
ox oo3oo3xo3ox&#x = rapa penp
ox oo3oo3xx3oo&#x = trarap
ox oo3xo3oo3ox&#x = rapalpenp
ox ox3oo3oo3xo&#x = penaf
ox xo3oo3oo3xx&#x = spidapenp
ox xo3oo3ox3xo&#x = spida rappip
xx oo3oo3oo3ox&#x = hixip
xx oo3oo3oo3xx&#x = squapen
xx oo3ox3xo3oo&#x = dotip
xx xx3xx3xx3xx&#x = squagippid
...

ox oo3oo3xo *c3oo&#x = hexasc
ox oo3oo3xo *c3ox&#x = hexaf
...

oo3ox oo3oo3ox&#x = tratetpy
oo3ox oo3oo3xo&#x = hop
oo3ox oo3oo3xx&#x = tetdip
oo3ox oo3ox3oo&#x = troctpy
oo3ox oo3xo3oo&#x = octete
oo3ox oo3xo3ox&#x = octa tratet
oo3ox oo3xx3xx&#x = tettut
oo3ox ox3oo3xo&#x = tetal tratet
oo3xx oo3oo3ox&#x = trapen
oo3xx oo3ox3xo&#x = trarap
oo3xx ox3oo3xo&#x = trahex
ox3xo oo3oo3ox&#x = trial tratet
ox3xo oo3ox3xo&#x = tratet altroct
ox3xo ox3oo3xo&#x = tratetdafup
...

oo3oo3oo3oo3ox&#x = hop
oo3oo3oo3oo3xx&#x = hixip
oo3oo3oo3ox3oo&#x = rixpy
oo3oo3oo3ox3xo&#x = ril
oo3oo3oo3xx3oo&#x = rixip
oo3oo3oo3xx3xx&#x = tixip
oo3oo3ox3oo3oo&#x = dotpy
oo3oo3ox3oo3xo&#x = hixadot
oo3oo3ox3xo3oo&#x = bril
oo3oo3ox3xo3ox&#x = rixasarx
oo3oo3xx3oo3oo&#x = dotip
oo3ox3oo3oo3xo&#x = hixalrix
oo3ox3oo3xo3oo&#x = rixa
oo3ox3oo3xo3ox&#x = rixaspix
oo3ox3xo3oo3ox&#x = dottaspix
oo3ox3xo3ox3oo&#x = dottasibrid
oo3ox3xo3ox3xo&#x = sarxasibrid
oo3ox3xx3xo3oo&#x = bittixa
oo3xx3oo3oo3ox&#x = rixalspix
oo3xx3oo3oo3xx&#x = spixip
ox3oo3oo3oo3xo&#x = gee
ox3oo3oo3xo3ox&#x = rixascad
ox3xo3oo3ox3xo&#x = spixa
ox3oo3xo3oo3ox&#x = dottascad
ox3oo3xo3ox3xo&#x = spixalsarx
ox3xo3oo3oo3xx&#x = scadaspix
ox3xo3ox3xo3ox&#x = sibridacard
xx3oo3oo3oo3xx&#x = scadip
xx3oo3xx3oo3xx&#x = cardip
xx3ox3oo3xo3xx&#x = cappixa
xx3ox3xx3xo3xx&#x = cograxa
xx3xx3xx3xx3xx&#x = gocadip
...

oo3oo3oo *b3oo3ox&#x = tacpy
oo3oo3oo *b3oo3xx&#x = taccup
oo3oo3oo *b3ox3oo&#x = ratpy
oo3oo3oo *b3ox3xo&#x = taccarat
oo3oo3ox *b3oo3oo&#x = hinpy
oo3oo3ox *b3oo3xo&#x = dijak
oo3oo3ox *b3xo3oo&#x = hinro
oo3oo3ox *b3xo3ox&#x = ratasiphin
oo3oo3xx *b3oo3oo&#x = hinnip
oo3oo3xx *b3oo3ox&#x = hinasiphin
oo3oo3xx *b3oo3xx&#x = siphinnip
oo3ox3oo *b3oo3xo&#x = taccanit
oo3ox3oo *b3xo3oo&#x = ratanit
oo3ox3oo *b3xo3ox&#x = ratasart
oo3ox3xo *b3oo3oo&#x = hinanit
oo3ox3xo *b3oo3xo&#x = nitasiphin
oo3ox3xo *b3xo3oo&#x = nitasirhin
oo3ox3xo *b3xo3ox&#x = sirhinasart
ox3oo3ox *b3xo3oo&#x = ratarin
ox3oo3xo *b3oo3oo&#x = hax
ox3oo3xo *b3oo3ox&#x = hinalsiphin
ox3oo3xo *b3oo3xx&#x = siphina
ox3oo3xo *b3xx3oo&#x = sirhina
ox3oo3xx *b3oo3oo&#x = hinarin
ox3xo3ox *b3oo3oo&#x = nitarin
ox3xo3ox *b3oo3ox&#x = nitaspat
ox3xo3ox *b3ox3xo&#x = sarta sibrant
ox3xo3xx *b3oo3oo&#x = rinathin
...

Trigonics

4-layered Lace Simplices

...

oox oxo ooo3oox&#x = hop
oox oxo ooo3oxx&#x = tratetpy
oox oxo ooo3xox&#x = triddipasc
oox oxo xoo3oxx&#x = trial tratet
oox oxx ooo3xoo&#x = trippete
oox oxx oxo3xox&#x = tatraffip
oxx xox ooo3xxo&#x = pedril
oxx xox oox3xxo&#x = tisdip altratet
...

oox ooo3ooo3oxo&#x = hop
oox ooo3ooo3xxo&#x = tepasc
oox ooo3oxo3ooo&#x = octete
oox ooo3oxo3xoo&#x = rapesc
oox ooo3oxo3xox&#x = octa tratet
oox oxo3ooo3xoo&#x = hexasc
oox oxo3ooo3xox&#x = tetal tratet
oox oxx3ooo3xox&#x = hexcopedaw
oox oxx3ooo3xxo&#x = tetaco altepetric
oxx ooo3ooo3xoo&#x = squepe
oxx oox3ooo3oxo&#x = hexippy
oxx xox3ooo3xxo&#x = hexipaco
...

ooo3oox ooo3oxo&#x = hop
ooo3oox ooo3xxo&#x = trippete
ooo3oox oxo3xoo&#x = octete
ooo3oox oxo3xox&#x = trial tratet
oox3xxo oxo3xox&#x = triddippa tridafup
...

ooo3ooo3ooo3oox&#x = hop
ooo3ooo3ooo3oxx&#x = penppy
ooo3ooo3ooo3xxx&#x = trapen
ooo3ooo3oox3ooo&#x = rapesc
ooo3oox3oxo3ooo&#x = dotpy
ooo3oox3ooo3xxo&#x = rapalpenp
...

oox3ooo3oxo *b3xoo&#x = tedjak
oox3xxx3oxo *b3xoo&#x = thexgyt
oxx3ooo3xox *b3xxo&#x = ritgyt
oxx3xxx3xox *b3xxo&#x = tahgyt
...

ooxx oxox xoox&#x = hixacube
...

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Lace Simplices

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---- 1D ----

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---- 3D ----

---- 4D (up) ----

---- 5D (up) ----

---- 6D (up) ----