Naming Wythoffian polytopes

The Wythoffian polytopes are the uniform polytopes that can be constructed from standard Coxeter diagrams, "standard" meaning only unringed and ringed nodes and normal edges are allowed, excluding snub nodes, null edges or anything of the sort. Due to this, they form the more normal set of uniform polytopes, and in fact almost all convex uniform polytopes are Wythoffian, with only four (known) exceptions.^[a] Despite making up the most basic uniform polytopes of each dimension, there hasn't been all that much interest in making a consistent naming scheme for the more complicated ones, which can be seen with how dubiously the Gossetics E_n are usually named.

This page is then a full documentation, and creation if necessary, of a particular set of naming schemes for all convex spherical and Euclidean^[b] Wythoffian polytopes from rank 3 onwards, with dimension-specific morphemes for up to rank 9. Unlike in the rest of my site, as this page is tailored toward documenting naming schemes and making them compatible with the existing ones, I've expunged all of my personal naming preferences and instead used the standard names present in the Polytope Wiki... Mostly. My naming preferences are written in another article, Polytope naming scheme war.

This page is an expansion of a page posted about three weeks prior, which had about the same premise as this page but focused specifically on naming the Gossetics and more generally all branched Coxeter diagrams. An archive of the article can be found at Naming Gossetic Wythoffians.

Wythoffian operations

Before describing the individual naming schemes first, it is much more useful to describe Wythoffian operations, which give way to the Johnson–Ruen operations, off of which the naming schemes in this article are based.

Performing a Wythoffian operation on a base polytope P can be thought of as first replacing elements of a certain rank n with vertices and then repeatedly expanding elements corresponding to elements of certain ranks (i, j, k, ...) of the base polytopes. Both of these steps are optional and doing neither simply yields the base polytope. They can be notated by their truncation symbol, which is of the form t_n,i,j,k,...(P) for the notation given in that specific explanation. As an example, truncation, the operation that expands edges, can be given as t_0,1 as it replaces vertices (rank 0) with vertices (i.e., skips the first step) and then expands its edges (rank 1).

Let's take a more complicated example: t_1,3,4,7. It first replaces the edges (rank 1) of the polytope with vertices, then expands the elements corresponding to the zetta (rank 7) of the original polytope, then does the same thing for its tera (rank 4) and cells (rank 3). These expansions are commutative (can be done in any order).

This idea can be related to Coxeter diagrams in a neat way: for a truncation symbol t_i,j,k,... and selecting a base node from an unringed Coxeter diagram, the nodes i, j, k, ... away from the base node are ringed. This idea only fully applies to linear Coxeter diagrams, although in other diagrams this spread is limited to one node down a fork before becoming non-uniform. In general for all Coxeter diagrams, it can be said that ringing a node expands out the elements of the orbit corresponding to the diagram obtained from removing that node of the diagram.

Johnson–Ruen operations

The Johnson–Ruen operations are a method of naming Wythoffian operations based on their truncation symbol. They were named after mathematician Norman Johnson and Tom Ruen.^[c] Even though there are more Wythoffian operation naming schemes, the most prominent besides this being the Kepler–Bowers operations, this one is commonplace in places like Wikipedia for ≥4D and preferred in certain circumstances.^[d]

There are two distinct cases for these, depending on if the polytope has only one node ringed or multiple. For a truncation symbol t_i,j...z, the corresponding Johnson–Ruen operation is, depending on the case:

Case I: if there is only one node ringed, the operation will be the Latin numeral prefix for i plus -rectification, with the prefix not applied if i=1. If i=0, the operation is trivial (returns the polytope itself) and the its name is null. For example, t₁ is named rectification, t₂ is named birectification and t₃ is named trirectification. For completeness, below is a table for Latin numeral prefixes up to 4, sufficing up to 9D.

Number	Latin numeral
2	bi-
3	tri-
4	quadri-

Case II: if there is more than one node ringed, the operation name is given by the following process: for all ringed nodes except for the first, beginning from the last, append the operation indexed by its distance from the first ringed node by its prefixal form, until the last one in this step (which is the second ringed node), which gets appended with its nominal form. If the first ringed node is not the base node, the Latin numeral prefix for the index of the base node plus one (i+1) is attached to the beginning. If the truncation symbol includes every integer from i to z, the operation name may be preceded by the word full or great. In the case where the operation name is prefixed, full/great come before the prefix. Below is a list of the prefixal and nominal forms of each expansion name given its distance from first ringed node, sufficing up to 9D. The rank 8 expansions were named by me.^[e]

Rank	Prefixal form	Nominal form
1	—	truncation
2	canti-	cantellation
3	runci-	runcination
4	steri-	sterication
5	penti-	pentellation
6	hexi-	hexication
7	hepti-	heptellation
8	octi-	octication

As an example, t_1,3,4,7 becomes bihexiruncicantellation, with the individual morphemes being L(1+1)-O_p(7−1)-O_p(4−1)-O_m(3−1) = L(2)-O_p(6)-O_p(3)-O_n(2), with L(n) being the Latin numeral prefix for n and O(n) the expansion name, p and n indicating the prefixal and nominal forms respectively.

However, Johnson–Ruen operations usually leave two valid names for a polytope, which is solved with two criteria that take into account "which end of the diagram the ringed nodes are grouped towards" (calculated using bit indices) and the number of vertices of the base polytope, which are called the canonicalization criteria. Firstly, only the polytope with the smallest bit index, the sum of the exponents of 2 and each index of the truncation symbol, is considered, enough to solve conflicts such as truncated (t_0,1) cube = bitruncated (t_1,2) octahedron, wherein the former is canonicalized because 2⁰+2¹ < 2¹+2². If that condition still fails, in e.g., rectified (t₁) cube = rectified (t₁) octahedron, then only the polytope whose base polytope has the greater number of vertices is considered, with the former being canonicalized because the cube has more vertices than the octahedron (8 > 6). If the base polytope is self-dual or all the ends of its Coxeter diagram produce the same polytope, the latter case obviously does not matter.

Naming schemes

All of the naming schemes shown below except for the last are in the format of "operation base", where the Wythoffian polytope in question is the result of operation being applied to base. While operation is generally not a Johnson–Ruen operation, it is still usually a derivate that works for that specific symmetry with a special suffix attached.

The first three and last naming schemes in this section (reguliforms, demicubics, demicubic tesselations and prismatics) were already formalized long before the publication of this article. The naming for the cyclosimplicial tesselations was also likely conceived of before, but never written about to my knowledge.

Reguliforms

The reguliforms are the Wythoffian polytopes obtained from ringings of linear Coxeter diagrams, which can all be described as some Wythoffian operation applied to some regular polytope. For n>2, the relevant symmetries where this works for are, along with their Schläfli symbols:

• A_n {3,3,...,3,3}: 2ⁿ⁻¹+2^{⌈n/2−1⌉}−1 members,^[1] unless n=3 where there are 2, the rest gaining B₃ symmetry;
• B_n {4,3,...,3,3}: 2ⁿ−1 members, unless n=4 where there are 12, the rest gaining F₄ symmetry;
• ~C_n {4,3,...,3,4}: 2ⁿ+2^{⌈(n−1)/2⌉}−2 members, unless n=4 where there are 15, the rest gaining ~F₄ symmetry;
• ~G₂ {6,3}: 6 members;
• F₄ {3,4,3}: 9 members;
• ~F₄ {3,4,3,3}: 28 members;
• H₃ {5,3}: 7 members;
• H₄ {5,3,3}: 15 members.

Their Coxeter diagrams are related to their Schläfli symbol: for a Schläfli symbol of length n, the corresponding Coxeter diagram has n+1 nodes with all unringed except for the first while the edges correspond to every number in the symbol in order.^[2]

Of the ones described in this page, they have the simplest naming scheme: base is the name of the regular polytope where applying operation to it results in the Wythoffian polytope. For completeness, below is a table of the names for the relevant regular polytopes, following the long Greek format.^[3][4] When two polytopes are mentioned in one symmetry, the first has the greater amount of vertices. The number corresponding to the prefix can be viewed by hovering over it.

Rank	An	Bn		~Cn	F4	~G2/~F4		Hn
3	Tetrahedron	Cube	Octahedron	Square tiling		Hexagonal tiling	Triangular tiling	Dodecahedron	Icosahedron
4	Pentachoron	Tesseract	Hexa­deca­choron	Cubic honeycomb	Icosi­tetra­choron			Hecaton­icosa­choron	Hexacosi­choron
5	Hexateron	Penteract	Triaconta­diteron	Tesseractic tetracomb		Icosi­tetra­choric tetracomb	Hexa­deca­choric tetracomb
6	Heptapeton	Hexeract	Hexeconta­tetra­peton	Penteractic pentacomb
7	Octaexon	Hepteract	Hecaton­icos­octa­exon	Hexeractic hexacomb
8	Enneazetton	Octeract	Diacosi­penteconta­hexa­zetton	Hepteratic heptacomb
9	Decayotton	Enneract	Pentacosi­dodeca­yotton	Octeractic octacomb

Demicubics

The demicubics are the Wythoffian polytopes with D_n symmetry, which has 2ⁿ⁻² members for n≥5, becoming unique only in 5D and above. The D_n Coxeter diagram is like that of A_n−1, but with a single-node branch on the second node. For a demicubic polytope to be distinct, one of the two single-node branches must be ringed and the other unringed, with the node of the ringed branch considered the base node for the purposes of naming. Otherwise, it could be formed as a Wythoffian operation on the n-orthoplex.

There are two main cases in the demicubic naming scheme. For the simpler case which occurs when it can be described as a Wythoffian operation on the n-demicube without spreading to the branches, base becomes the name for the n-cube prefixed by demi- and operation is the Johnson–Ruen operation that when applied to the n-demicube yields the Wythoffian polytope. Even though it is possible to name the birectification, cantellation, bitruncation and cantitruncation of the n-demicube as operations on it, they instead use the more complex naming scheme.

The more complex naming scheme is based off of the fact that most demicubics cannot be described by Wythoffian operations on some member, unlike the reguliforms. They are instead best described by an alternated faceting of a Wythoffian operation on the n-cube.^[5] The allowed Wythoffian operations done before the alternated faceting include any which have a truncation symbol with a 0 rank but without a 1 rank.

Due to this, the complex naming scheme is instead based on these alternated facetings. Similarly to the reguliforms, operation is the Johnson–Ruen operation that, when applied to the n-cube and taken its alternated faceting, produces the polytope, plus the suffix -ic appended to the prefixal form of the operation. Base is then the n-cube.

Alternatively, it is possible to name the demicubic polytope directly off of its Coxeter diagram. Operation is the Johnson–Ruen operation of the truncation symbol of the diagram with the unringed single-node branch removed (which yields a linear diagram), with every rank after 0 increased by one and with the suffix -ic added to the prefixal form of the operation name. Again, base is then the n-cube.

As an example, the demicubic polytope that is the alternated faceting of the steriruncinated hexeract has Coxeter diagram   o
x o x x o, whose underlying linear diagram has truncation symbol t_0,2,3, which is then named the steriruncic hexeract.

As stated before in the Wythoffian operation explanation section, for a more physical intuition, it can be said that the demicubic operations, save for cantic, expand the rank n elements with n-demicubic symmetry, e.g., steric expands the demitesseractic (hexadecachoric) elements of the polytope, as sterication expands the tera (rank 4) elements of a polytope.

Demicubic tesselations

The demicubic tesselations are the Wythoffian polytopes with ~B_n symmetry, which has 2ⁿ⁻¹ members for n≥3, except for n=4 where it has four members, becoming unique only in 3D (rank 4) and above. The ~B_n Coxeter diagram is like that of D_n+1, but the farthest edge of the longest branch is a 4-edge. These work similarly to the demicubics, with two single-node branches, one ringed and the other unringed, with the base node being the ringed branch.

Their naming scheme is identical to the demicubics; a demicubic tesselation is named as if it were a demicubic D_n+1 by replacing the 4-edge with a 3-edge and keeping the node ringings. If the polytope can be made as a Wythoffian operation on the alternated n-cubic tesselation, base is the name of the n-cubic tesselation prefixed with demi-, unless n=3 where it is tetrahedral-octahedral honeycomb. Otherwise, base is the name of the n-cubic tesselation. As an example,   o
x o x x o4x is named the hexisteriruncic hexeractic hexacomb, from   o
x o x x o x being named the hexisteriruncic hepteract.

Quarter cubic tesselations

The quarter cubic tesselations are the Wythoffian polytopes with ~D_n symmetry, which has 2ⁿ⁻⁴+2^{⌈(n−5)/2⌉} members for n≥5, becoming unique only in 5D (rank 6) and above. Their Coxeter diagram is like that of A_n−1, but with two single-node branches, one on the second node and another on the second last node. Like with the demicubics, one of the nodes between the first node and the node of the first branch must be ringed and the other unringed, and same goes for the last node and the node of the second branch. They are also best named as alternated facetings of demicubic tesselations, or double alternated facetings of cubic tesselations. The allowed Wythoffian operations done before the double alternated faceting include any which have a truncation symbol with a 0 rank and a n rank but without a 1 rank nor a n−1 rank.

Just like with the demicubics and the demicubic tesselations, their naming scheme is based on alternated facetings. Operation is the Johnson–Ruen operation that, when applied to the n-cubic tesselation and taken its alternated faceting twice for both 4-edges, produces the polytope, plus the suffix -ical appended to the prefixal form of the operation. The leading rank n expansion is omitted for being redundant, and if the polytope is simply the double alternated faceted n-cubic tesselation its name consists of the name of the n-cubic tesselation after the word quarter. Base is then the name of the n-cubic tesselation.

Likewise, it is possible to name the Wythoffian polytope directly off of its Coxeter diagram. Operation is the Johnson–Ruen operation of the truncation symbol of the diagram with the two unringed single-node branches removed (which yields a linear diagram), with every rank after 0 increased by one except for rank n which is ignored entirely, and with the suffix -ical added to the prefixal form of the operation name. Again, base is then the name of the n-cubic tesselation.

As an example, the polytope that is the double alternated faceting of the heptistericantellated hepteractic heptacomb has Coxeter diagram   o     o
x x o x o x, whose underlying linear diagram has truncation symbol t_0,1,3,5, which is then named the stericantical hepteractic heptacomb.

Cyclosimplicial tesselations

The cyclosimplicial tesselations are the Wythoffian polytopes with ~A_n symmetry, which has A000029(n+1)−1 members for n≥4, except for n=3 where it has one member (the cyclotruncated tetrahedral-octahedral honeycomb), becoming unique only in 3D (rank 4) and above. The ~A_n Coxeter diagram is a cycle of n+1 nodes connected by 3-edges.

The naming scheme has two cases depending on whether it's possible to construct the polytope with Wythoffian operations on its only quasiregular member. If so, operation is simply the Johnson–Ruen operation that yields the polytope when applied to the quasiregular member. Otherwise, operation is based on cyclic operations, being the Johnson–Ruen operation that when applied to a (n+1)-simplex and wrapped (two opposite nodes of the diagram are connected by a 3-edge) results in the Wythoffian polytope, prefixed by cyclo-. As for base, in n=3 it is simply tetrahedral-octahedral honeycomb, otherwise, base consists of the name of the n-simplex prefixed by cyclo-, with the name for the n-tesselation added at the end as a word. For clarity, below is a table for the relevant tesselations.

Dimension (= Rank−1)	Name
3	Tetrahedral-octahedral honeycomb
4	Cyclopentachoric tetracomb
5	Cyclohexateric pentacomb
6	Cycloheptapetic hexacomb
7	Cyclooctaexic heptacomb
8	Cycloenneazettic octacomb

As an example, the diagram  x o
o   o
 x x is named the cycloruncitruncated cyclohexateric pentacomb, as it can be made by wrapping a runcitruncated (n+1)-simplex (x x o x o o). The canonicalization feature is also much more notable in this symmetry, as the aforementioned polytope can be named in eleven other ways, ordered by bit index:

• cyclosteritruncated [...] (ct_0,1,4)
• cycloruncicantellated [...] (ct_0,2,3)
• cyclopenticantellated [...] (ct_0,2,5)
• cyclosteriruncinated [...] (ct_0,3,4)
• cyclopentiruncinated [...] (ct_0,3,5)
• cyclobiruncitruncated [...] (ct_1,2,4)
• cyclobisteritruncated [...] (ct_1,2,5)
• cyclobiruncicantellated [...] (ct_1,3,4)
• cyclobisteriruncinated [...] (ct_1,4,5)
• cyclotriruncitruncated [...] (ct_2,3,5)
• cyclotriruncicantellated [...] (ct_2,4,5)

Gossetics

The Gossetics are the Wythoffian polytopes with E_n symmetry, although this subsection will also include the tesselations ~E₇ and ~E₈. More generally, it applies to any polytope whose Coxeter diagram is like that of A_n with a single node branch between (exclusive) the second and second last nodes. The relevant symmetries are:

• E₆: diagram like that of A₅ with a single-node branch on the third node, has 39 members;
• E₇: diagram like that of A₆ with a single-node branch on the third node, has 127 members;
• E₈: diagram like that of A₇ with a single-node branch on the third node, has 255 members;
• ~E₇: diagram like that of A₇ with a single-node branch on the fourth node, has 143 members;
• ~E₈: diagram like that of A₈ with a single-node branch on the third node, has 511 members.

The naming scheme has two cases depending on whether it's possible to construct the polytope as a Wythoffian operation on one of its three fininodal (quasiregular with its ringed node at one of its ends) members without going down a branch. If so, operation is the operation that when applied to base, a fininodal, produces the polytope. Otherwise, it resorts to a more complex naming scheme.

Beginning from a base node, operation is the Johnson–Ruen operation of the truncation symbol of the diagram with the unringed single-node branch removed, with every rank after the single-node branch increased by one and with the suffix -ial added to the prefixal form of the operation name, unless the last operation is of rank 1 (truncation) in which case it becomes truncal. If the single-node branch is ringed, demi- is prefixed to operation. If there is no operation to prefix demi- to, operation will then be demified. The second canonicalization criterion is reversed for these symmetries: instead of picking the one with the greatest amount of vertices, it instead picks the one with the least amount of vertices. Base is then the name of the fininodal corresponding to that base node.

As an example, the incorrectly named "trirectified 1₂₂ polytope"^[f], diagram     o
x o o o x, is instead named the pential icosiheptahebdomecontadipeton, as its underlying linear diagram's truncation symbols is t_0,4, which with the ranks after the branched node (i>2) increased by one is named pential. Below is a table of relevant base polytopes. The Coxeter symbol relates to the Coxeter diagram by having the first number be the number of nodes in the base branch and the subsequent subscript numbers being the lengths of the other branches after a single fork.

Symmetry	Coxeter symbolic	Long numeric	Long Greek
E6	122 polytope/polypeton	54-peton	Pentecontatetrapeton
	221 polytope/polypeton	27-72-peton	Icosihepta­hebdomecontadipeton
E7	132 polytope/polyexon	56-126-exon	Pentecontahexa­hecatonicosihexaexon
	231 polytope/polyexon	56-576-exon	Pentecontahexa­pentacosihebdomecontahexaexon
	321 polytope/polyexon	126-576-exon	Hecatonicosihexa­pentacosihebdomecontahexaexon
E8	142 polytope/polyzetton	240-2160-zetton	Diacositessaraconta­dischiliahecatonhexecontazetton
	241 polytope/polyzetton	240-17280-zetton	Diacositessaraconta­myriaheptachiliadiacosogdoëcontazetton
	421 polytope/polyzetton	2160-17280-zetton	Dischiliahecatonhexeconta­myriaheptachiliadiacosogdoëcontazetton
~E7	133 polytope/heptacomb	—	Pentecontahexa­hecatonicosihexaexic heptacomb
	331 polytope/heptacomb	—	Octaexic-hecatonicosihexa­pentacosihebdomecontahexaexic heptacomb
~E8	152 polytope/octacomb	—	Demiocteractic-diacositessaraconta­dischiliahecatonhexecontazettic octacomb
	251 polytope/octacomb	—	Enneazettic-diacositessaraconta­myriaheptachiliadiacosogdoëcontazettic octacomb
	521 polytope/octacomb	—	Enneazettic-diacosipentecontahexazettic octacomb

The default in the naming scheme presented in this article is the long Greek format, but due to being extremely unwieldy (even moreso than the regulars) the Coxeter symbolic form is recommended for these.

Hexagossetic hexacombs

The hexagossetic hexacombs are the Wythoffian polytopes with ~E₆ symmetry, which has 39 members. Its Coxeter diagram has three branches of two nodes each, meeting at a central node. Alternatively, it can be thought of as like that of A₅ but with a double-node branch on the third node. More generally, it applies to any polytope whose Coxeter group consists of three branches that each have at least two nodes.

The naming scheme is very similar to the Gossetics. The main member, which base always is, is named the It has two cases depending on whether it's possible to construct the polytope as a Wythoffian operation on one of its three fininodal members without going down a branch. The simpler case is identical.

The complex case differs, though. First, from a base node a main branch and an over branch are chosen, where if the Coxeter diagram has these two of different lengths, then the longest is considered the main branch. If the over branch has at least one node ringed, there will be two distinct operation lines, the main line and the over line. Beginning from a base node and following the main branch after the fork, the main line is constructed simililarly to the normal Gossetics, but the increase after the "single-node branch" (over branch) is determined by the over branch's length. The over line is constructed by stringing together the prefixal forms of the expansion names indexed by the distance between that node and the first ringed node, in order of decreasing rank, for every ringed node in the over branch. If the over branch is not ringed, operation simply consists of the main line. Otherwise, it consists of joining both lines with the infix -super-. If there is no main line for whatever reason, operation will consist of supered prefixed by the over line. If the main line has a Latin numeral prefix (i.e., first ringed node is not the base node), it will be appended before the over line. Finally, base is then the name of the fininodal corresponding to that base node. The name of the only fininodal in the relevant symmetry is icosiheptahebdomecontadipetic hexacomb, or 2₂₂ hexacomb in the Coxeter symbolic form.

In the case of polytopes with ~E₆ symmetry, where there is no longest branch, this causes various conflicting names. These are solved by choosing the combination of base node and branch with the smallest bit index with the branch with the greatest bit index being considered the main branch.

As an example, the polytope represented by the diagram       x o
x x x
      o x is named the pentisuperhexicantitruncal icosiheptahebdomecontadipetic hexacomb. It, however, does have five other possible names from choosing different branches:

• hexisuperpenticantitruncal [...] (Et_0,1,2,3|4)
• pentisuperhexipenticantial [...] (Et_0,2,5,6|3)
• hexipentisuperpenticantial [...] (Et_0,2,3|3,4)
• bipentisuperpentisteritruncal [...] (Et_1,2,3,4|4)
• bipentisterisuperpentitruncal [...] (Et_1,2,4|3,4)

Prismatics

Finally, the prismatics are the Wythoffian polytopes that can be made as a Cartesian product of two or more Wythoffian polytopes. Their Coxeter diagram includes the Coxeter diagrams of the polytopes of which it is a Cartesian product, disconnected (or, well, connected by 2-edges).

These don't follow the "operation base" pattern mentioned at the start of this section and the name is instead comprised of all the operands. The naming scheme consists of stringing together the adjectival forms of the operands, separated by dashes, finished by the word prism prefixed by a modified Latin numeral prefix of the number of operands, which are generally a standard prefix with -o attached. The order operands appear in is given by a hierarchy of 1. growing rank, 2. growing base symmetry order and 3. decreasing bit index. If there are multiple (n) dyads as operands, they will be combined into a single n-cube operand. If there is only one dyad present though, the name will be given by the adjectival form of the polytope with the dyad removed from the operands plus the word prism, which also applies if the polytope is a prism of a multiprism. Below is a table of these modified Latin numeral prefixes.

Number	Modified Latin numeral prefix
2	duo-
3	trio-
4	quadro-

As an example, the Cartesian product of a dyad, a tetrahedron, a truncated tetrahedron and a hexadecachoron is named the tetrahedral-truncated tetrahedral-hexadecachoric trioprismatic prism.

References

1. ↑ Calculated using: Wikipedia contributors. Burnside's lemma.
2. ↑ Klitzing, Richard. Dynkin Diagrams§Schläfli symbols.
3. ↑ MinersHaven. Numeral prefix generator.
4. ↑ Klitzing, Richard. 3D, 4D, 5D, 6D, 7D, 8D, 9D.
5. ↑ Klitzing, Richard. Dynkin Diagrams§Snub polytopes.

Notes

1. ↑ The snub cube, the snub dodecahedron, the snub disicositetrachoron and the grand antiprism.
2. ↑ For Euclidean and hyperbolic tilings, convexity implies that the embedding the polytope in either a paraboloid and a hyperboloid respectively is convex. For a non-convex Euclidean honeycomb that is made out of convex polytopes and does not have self-intersection, see the stack of truncated square tiling alterprisms.
3. ↑ According to R. Klitzing on the Polytope Discord server, Johnson coined the lower dimensional cases while Ruen continued it to higher dimensions, elaborating it in detail especially within Wikipedia.
4. ↑ In a poll I hosted on the Polytope Discord on 19 September 2025, 8 of 13 people preferred the Johnson–Ruen runcitruncated over the Kepler–Bowers prismatorhombated/prismatotruncated.
5. ↑ While not officially on the table for irrelevancy, I've also thought of ennellation, decimation, undecillation, duodecimation etc.
6. ↑ In reality, the true trirectified 1₂₂ polytope is not even uniform, having two types of vertices corresponding to the two types of tetrahedral cells.