New naming scheme experiment

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

New naming scheme experiment

Postby polychoronlover » Sun Nov 22, 2020 8:44 pm

Throughout the years, I have found that a good system of nomenclature helps me understand and categorize objects better in my mind. Starting around 5 years ago, I got the idea of creating a new system of nomenclature for polytopes, that would be consistent and avoid the quirks in existing naming systems.

When performing a uniform polytope search by regiment enumeration, two useful things to know are (1) a polytope’s verf structure and (2) the possible Wythoffian families. In addition, convex polytopes are often described by (3) a particular Wythoffian operation applied to a base shape, distinguished by a pattern of nodes ringed in the Coxeter diagram. All of these features are therefore reasonable choices to be the basis of a naming system. In particular, when I was studying families and regiments of uniform honeycombs, I found many surprising cases where members of the same family found their way into many varied regiments. This led me to consider the Wythoffian family to be an important aspect for a naming system designed to promote this cool way of classifying polytopes.

For ease of name memorization, I also decided that the following features would be important for a nomenclature system:
• The name should be almost always uniquely derivable from the shape itself given a short set of rules. (Like the Bowers names of convex Wythoffian polytopes, but very much unlike the Bowers names of nonconvex uniform polytopes.)
• The name as generated by the rules should be unique (at least up to symmetry); even if the construction method behind the name is not unique, there should at least be a canonical one preferred.

I have been working on a new nomenclature for uniform polytopes. The basis of my system is the elemental naming scheme, a nomenclature used on the hi.gher.space wiki[1][2] and referenced on Richard Klitzing’s website[3][4][5], but extended to nonconvex shapes as well as Euclidean tesselations. I am a fan of this naming scheme for several reasons:
• Each name can be decomposed into 3 pieces for analysis: a prefix, an infix, and a suffix.
• The prefix always refers to the symmetry or to a regular polytope in the same family.
• The infix always refers to the identity of the Wythoffian operation on said regular polytope; thus, the index already indirectly indicated the verf shape.
• The suffix refers to the number of dimensions. There was even talk of using suffixes like terid, petid, ectid, etc. in place of choron, teron, peton[6] because the prefix and infix referred to the shape itself, not its facets. (It should be noted that this choice is more in line with Anton Sherwood’s original intention for this naming system, which used -choron for polyhedra and -tetron for polychora[7], but this was considered to be confusing as the suffix -choron already gave the impression of a 4D shape to almost everyone.)
• There was no historical baggage or motivation to keep the names consistent with traditional names; this allowed the scheme to be almost a blank slate for experimentation.

In the original elemental naming scheme, the names were mostly limited to shapes that resulted from Wythoffian operations on regular polytopes. The prefix represents a regular polytope (eg cosmo) or a family if the shape could be derived identically from two regular polytopes (eg rhodo). The infix represents a particular operation on the polytope (eg recti, canti, cantitomo, etc.). The suffix indicates the dimension.

I realized that, when extended to nonconvex polytopes, the infix could take two roles at once: identifying the verf shape and, assuming the shape can be constructed with a linear Dynkin diagram, identifying the operation on the base. All in all, it should be easy to interpret the infix both ways: as an identifier of the Wythoffian operation when the Coxeter diagram is linear and as an identifier of the verf shape in general. Non-Wythoffians in a Wythoffian regiment could be identified by modifying the name of one of the Wythoffians in the regiment. As this was already proving to be a very ambitious endeavor, I decided to restrict my names to the subclass of uniform polytopes that I call locally uniform.

The unfinished product is looking to have several other interesting features:
• The names are usually shorter than the common name of the same shape[8].
• Every polytope has a canonical name for each Wythoffian symmetry in which it is locally uniform.
• A polytope’s conjugate is always named by either adding or removing “quasi” from its name.

Scope of this project
This naming scheme is planned to be able to name all polytopes that are:
• Locally uniform (a sub-category of uniform; see [9] for definition)
• Either embeddable in a space of the same dimension, with a finite number of facets (spherical) or embeddable in a space of one less dimension, with an infinite number of facets (Euclidean). I decided not to make names for hyperbolic tessellations, because the number of possible symmetries (each symmetry gets an elementary name), verf shapes, and Wythoffian families would be too large.
• True polytopes under Bowers’ definition (not fissaries, coincidics, or compounds (yet)).
There should be at most one name for each symmetry, and one name for the highest Wythoffian symmetry.
Note: Naming non-Wythoffians in Wythoffian regiments seems like the most ambitious part of my project right now. They will be named based on the facets they share with Wythoffians, and I don’t expect to ever have rules for covering every Wythoffian regiment. I plan to create a partial list of rules sometime, but right now I’m mostly focusing on naming Wythoffians and hemi-Wythoffians.

The rules for naming polytopes will be posted soon.

References
1. http://hi.gher.space/wiki/Elemental_naming_scheme
2. http://hi.gher.space/wiki/List_of_uniform_polychora – try mousing over the third column of the table
3. https://bendwavy.org/klitzing/explain/product.htm
4. https://bendwavy.org/klitzing/incmats/ico.htm – look at the third name given
5. https://bendwavy.org/klitzing/incmats/pent.htm – look at the fourth name given
6. viewtopic.php?f=25&t=1510&p=18105#p18102
7. https://bendwavy.org/wp/?p=1049
8. viewtopic.php?f=25&t=1510#p15284
9. viewtopic.php?f=25&t=2440&p=27003#p27003
Last edited by polychoronlover on Tue Nov 24, 2020 5:15 am, edited 1 time in total.
Climbing method and elemental naming scheme are good.
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Re: New naming scheme experiment

Postby polychoronlover » Tue Nov 24, 2020 5:05 am

Rules for non-prismatic shapes:
  • The name is divided into 3 parts: a prefix, an infix, and a suffix.
  • The prefix is chosen based on the prefix table: https://docs.google.com/spreadsheets/d/ ... sp=sharing.
    • If the shape is Wythoffian, hemi-Wythoffian, or some other multiple covering is Wythoffian (where elements and element figures of dimension 3 or higher are allowed to be multiple coverings themselves), the prefix is usually the distinct name given to the family (the graph of nodes and labeled edges on which nodes are ringed in the Coxeter-Dynkin diagram). This name is unique up to dimension.
      • Example: o3x3o3x3/2*a (2ratho) counts as a valid double covering despite containing 2thahs as cells, but x3o3x5/2*a (2id) does not because it contains faces (10/2-gons) that are multiple coverings.
    • Most shapes can be constructed from multiple families because translations of 3 ↔ 3/2, 4 ↔ 4/3, 5 ↔ 5/4, etc. for all edges surrounding a node of the CD diagram are equivalent to making an edge retrograde if the node is ringed, and have no effect if the node is not ringed.
      • Thus, any n-dimensional family belongs to a super-family of up to 2^(n – 1) distinct families producing the same polytopes except with retrograde edges.
      • If all edges in the graph are of the form 2, 3/n, 5/n, or ∞, the entire super-family has a single name. (Exception: o3o3...o3o3/2o is called repyro while o3o3...o3o3o is called pyro. Repyro is only used for the special case x3o3...o3o3/2x/2. As far as I know, this is the only case where a (hemi-) Wythoffian polytope becomes another (hemi-) Wythoffian when edges are made retrograde without the two shapes being conjugates; namely, the expanded and facetorectified simplexes.).
      • If some edges in the graph are of the form 4/n or 6/n, two families are chosen from the super-family which are the same except that all 4’s and 4/3’s or 6’s and 6/5’s are swapped, so that all valid polytopes in the super-family are in at least one of the two categories (hopefully this should always be possible). Unless the two families are the same, the name of one is formed by appending quasi- to the beginning of the other’s name. Any members shared between both families are named as part of the non-quasi-family. The canonical families chosen are described in the table.
        • Families with hypercubic honeycomb symmetry sometimes have 4’s and 4/3’s that belong to two separate zones of symmetry. In this case, the 4’s and 4/3’s from each zone can be swapped independently of each other to produce four families from the same super-family. These four families have separate names formed by adding quasi- in various places. When a shape belongs to multiple families, the family with the least amount of “quasi”s is used.
      • If all the ringed nodes on the Coxeter-Dynkin diagram are along a linear subgraph (of at least two nodes) that trails off to a dead end in one direction, then the polytope is the result of a linear Wythoffian operation applied to the shape formed by ringing the node at the end of the line. In this case, the prefix is a special “endpoint” name denoting the particular figure by ringing the end node. If the CD diagram consists of more than two trails with a single ringed node at their junction (e.g. o3o3o3x3o3o *c3o), the name of the shortest line is used.
      • Example: sidero is the special prefix for x3o3o3o3o *b3o (the diagono family), so x3x3o3o3o *b3o is called sidero-tomo-ectid, not diagono-tomo-ectid.
      • If the entire Coxeter-Dynkin diagram is linear, both endpoints are tested by converting the pattern of on and off nodes to a binary string, starting from the other endpoint and ending at the one being tested. That binary string is converted to a numeric value and the endpoint with the smaller value is used in the name. If the values of both endpoints are the same, the Wythoffian operation has the same result when applied to the base and the dual. In this case, the family prefix itself is used.
      • Example: x3x3o3o4x could be geo-(11001)-petid or aero-(10011)-petid; the latter name is chosen. However, x3o3x3o4x is stauro-(10101)-petid. (Strings of numbers are not actually used as infixes but the infix is derived from the Wythoffian operation as explained below.)
Soon I will explain the rules for generating infixes and suffixes.
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Re: New naming scheme experiment

Postby Klitzing » Wed Nov 25, 2020 6:00 pm

Most shapes can be constructed from multiple families because translations of 3 ↔ 3/2, 4 ↔ 4/3, 5 ↔ 5/4, etc. for all edges surrounding a node of the CD diagram are equivalent to making an edge retrograde if the node is ringed, and have no effect if the node is not ringed.

  • First of all it is more that conjugation interchanges 5 <--> 5/3 and 5/2 <--> 5/4 instead.
  • Next it is that for cases, which are open ended and onesided ringed only, you have that links n/d result in the same figure as do (n/d)' = n/(n-d).
  • But within a triangular circuit like oPoQoR*a this same has to be applied at 2 links simultaneously, i.e. resulting in according quadruples: oPoQoR*a, oPoQ'oR'*a, oP'oQoR'*a, and oP'oQ'oR*a (as long as applied decorations wouldn't break down these replacements).
  • Accordingly in a cut open triangular circuit, i.e. like oPoQo, a decoration of type xPoQx produces the same shape as would xP'oQ'x.
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Re: New naming scheme experiment

Postby polychoronlover » Wed Nov 25, 2020 10:20 pm

Klitzing wrote:First of all it is more that conjugation interchanges 5 <--> 5/3 and 5/2 <--> 5/4 instead.


Yes, which is why my retrograde-edge operation is not the same as conjugation (usually).

Klitzing wrote:Next it is that for cases, which are open ended and onesided ringed only, you have that links n/d result in the same figure as do (n/d)' = n/(n-d).
But within a triangular circuit like oPoQoR*a this same has to be applied at 2 links simultaneously, i.e. resulting in according quadruples: oPoQoR*a, oPoQ'oR'*a, oP'oQoR'*a, and oP'oQ'oR*a (as long as applied decorations wouldn't break down these replacements).


Yes, the transformation needs to be applied to every branch connected to a particular node at once. Furthermore, applying the transformation twice to the same node gives the same result as not applying it at all, and I'm pretty sure that any sequence of transformations around some nodes is commutative. But transforming a set of nodes has the same effect as transforming its complement set. That's where I got at most 2^(n - 1) distinct families in a super-family.
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Re: New naming scheme experiment

Postby polychoronlover » Sun Dec 13, 2020 8:10 pm

Here are the rules for generating infixes. The infix represents the pattern of ringed nodes in the Coxeter-Dynkin diagram, and is constructed in such a way that it usually provides some information about the vertex figure.
  • Rules for Wythoffian objects:
    Processing the Coxeter-Dynkin diagram
    • Restructure the Coxeter-Dynkin diagram as shown by replacing each connected subgraph of unringed nodes with a single “clump,” which will be called an A-node. The remaining ringed nodes will be called x-nodes.
      • Each A-node connects only to x-nodes, while each x-node can connect to x-nodes and A-nodes.
      • There is at most one link between any given A-node and x-node, even if different off-nodes in the A-node connect to the same x-node.
      ENS_infix_fig1.jpg
      (480.33 KiB) Not downloaded yet

      ENS_infix_fig2.jpg
      (563.53 KiB) Not downloaded yet
    • If the new graph has cycles, remove as many as you can by cutting connections between pairs of x-nodes. (To do: There may be multiple ways to do this, so elaborate where necessary.) Whether or not a pair of ringed nodes is connected is irrelevant to the shape of the vertex figure.
      ENS_infix_fig3.jpg
      (509.89 KiB) Not downloaded yet
    • If multiple A-nodes are connected to the exact same set of x-nodes, merge them into a single A-node.
      ENS_infix_node_combining.jpg
      (386.45 KiB) Not downloaded yet
    • A connected subgraph of x-nodes where no x-node is connected to an x-node outside the subgraph, with no two x-nodes connected to the same A-node, is replaced by a super-X-node.
      ENS_infix_x-node_combining.jpg
      (210.89 KiB) Not downloaded yet
    • The A-nodes, x-nodes, and super-X-nodes in the processed graph can be called “macro-nodes”.
    Obtaining the infix name from the processed graph
    • Assuming that the resulting graph has no cycles:
    • The entire graph is “traversed” starting at one endpoint (macro-node with only one neighbor) and ending at another. The graph is now imagined as the linear path of traversal with possible branches on the sides. The list of nodes along the path alternates between A-nodes and either x-nodes or super-X-nodes.
      • Here is a table of the preferred traversals for each family: https://docs.google.com/spreadsheets/d/ ... sp=sharing. Look for all traversals where the branches off of the path have the shortest combined length (by number of macro-nodes). Of all those traversals, if there are any that end in an x-node or a super-X-node, choose them. Otherwise, choose the ones where the ending A-node consists of a cluster of the fewest unringed nodes. Of those, choose the one that is highest on the list of preferred traversals. (In most cases, this process is easier than it sounds, because the processed graph rarely has branches of more than one macro-node and hardly ever has more than 6 traversals.)
      • Exception: If all the ringed nodes are along a unique linear path in the original CD diagram with no branches and where only one side is connected to the rest of the graph, the direction of traversal is along the linear path towards the unconnected end. (In this case, the polytope is a direct Wythoffian operation on the shape where the ending node is ringed, so the infix must reflect this operation as well as the verf shape.)
      ENS_infix_traversal.jpg
      (447.03 KiB) Not downloaded yet
    • The identity of each A-node is given by the subgraph it was generated from, and which nodes of the subgraph were connected to each of the neighboring ringed nodes.
    • Each A-node can be associated with a lace simplex (call it C) between the figures formed when ringing the nodes connected to each of its ringed neighbors. This C is an element of the full vertex figure.
      ENS_infix_lace_simplices.jpg
      (261.49 KiB) Not downloaded yet
    • Each type of lace simplex C has a name, which is associated with the A-node. These names can be looked up in the infix table: https://docs.google.com/spreadsheets/d/ ... sp=sharing
      • The name is different depending on which component of the lace simplex is in the direction of traversal. Example: x3o3o3x3o *b3o and x3o3o3x3o *c3o get different infixes.
    • To obtain the infix, normally the names of all A-nodes in order of traversal are strung together starting with the first A along the traversal after the first x or super-X (if the traversal starts with a A, just ignore it). Any super-X-nodes have their names inserted between A’s or at the beginning or end, as appropriate, but normal x-nodes have no names.
      • If the traversal ends in an A-node made of n unringed nodes, then (n + 1) is appended as a prefix to the last of the names.
      • Example: o3x3o3x3x3x3o *c3o (hibcagorfy) gets the infix fastegi-biscali.
        ENS_infix_example.jpg
        (117.41 KiB) Not downloaded yet
      • To do: what if the processed graph has branches not on traversal? (e.g. o3x3o3x3o *c3x)
    • If only one node is ringed in the original CD diagram, special naming rules apply.
      • In this case, the processed diagram consists of one x-node connected tpo one A-node. The lace simplex C associated with the A-node is Wythoffian itself; it is the vertex figure of the polytope.
      • If C has only one ringed node in its own CD diagram, no infix is used. (This case corresponds to the regular polytopes, and any other shape with a single ringed node at the end of a linear chain. If the family has multiple shapes formed by ringing different endpoint nodes, their names are still distinct because different endpoint prefixes are used.)
      • Otherwise, a special infix is used depending on the type of vertex figure. These infixes can also be looked up in the infix table.
    • To do: what if the processed graph still has cycles? (e.g. o3x3o3x3o3x*a)
  • If the shape is hemi-Wythoffian, the above rules are used to give an infix for the double covering, which is then prefixed by “hemi”.
  • To do: what if the shape is non-Wythoffian?
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Re: New naming scheme experiment

Postby polychoronlover » Fri Dec 18, 2020 8:29 pm

Here are the rules for generating suffixes:
  • The suffix indicates the dimension number.
    • In one through four dimensions, the suffix is -telon, -gon, -hedron, or -choron respectively because these suffixes are so recognizable already. (I might change them to latrid, hedrid, chorid, and terid if this system gets more popular.)
    • In 5 dimensions and up, the suffix for spherical polytopes follows a pattern suggested by Wendy Krieger [1]. The suffix ends with -id (as in solid) and refers to the dimension of the shape itself, not the dimension of the facets: petid, ectid, zettid, yottid, etc.
      The suffixes for tessellations of 4D space and higher end in -on and refer to the dimension of the facets: teron, peton, ecton, zetton, etc. This is because the prefixes for Euclidean families do tend to refer to the identities of possible facets.
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