Guy Polyhedra General Theory
A ditelon is a 1-dimensional polytope, or closed line segment. It has two ends, or vertices, and (usually) a body, which is the bit of the line between the vertices. Several ditela may be chained end-to-end, and the chain bent round and closed in a loop to form a polygon.
This completes the pantheon of names for polytopes of up to 9 dimensions.
A defining characteristic of polytopes is that they are dyadic. A dyad is two objects treated as a single object. So-called complex "polytopes" are something else entirely.
This note grew out of attempts to systematise my ideas on polyhedra, and naturally expanded to cover some long-standing issues. It is essentially about naming things which, as Ursula K. Le Guin wrote in her Earthsea novels, is the key to gaining understanding and power over them.
I am indebted to Prof. Norman Johnson, "Dino" George Olshevsky, Wendy Krieger and others for their help and patience. No naming convention can please everybody, and I hope they will forgive me where I choose to differ from their ideas.
Modern theories of polygons and polyhedra understand these figures to be two- and three-dimensional examples of more general p-dimensional polytopes, or p-topes. In higher dimensions we have four-dimensional polycells or polychora. For five dimensions and up, three workers have jointly developed a naming system which is evident from the table and comments below. In lower dimensionalities Prof. Norman Johnson has named for us the 0-dimensional monon, or point, and for set theorists even the (-1)-dimensional nullitope, or null polytope.
But the poor old 1-tope still labours for the most part under the name "line segment", or more precisely "closed line segment". Though "dyad" has recently seen some use in this context, it is more of a description than a proper name (see below). I propose to call the general 1-tope a "polytelon", a many-ended thing (from the classical Greek τελος or telos, meaning an end). Many line segments are polytela. A real line segment has two ends, so is a ditelon. A complex 1-polytope having four vertices is a tetratelon, and so on.
A ditelon comprises two vertices, and (usually) a body. The vertices are just the two end points. The body is a line segment open at both ends, i.e. it does not include the end points: when we add the vertices to complete the ditelon, we add the end points and close the line segment. Some theories do not require the body to be present, but the geometry looks pretty odd without it.
Here is a table of the nouns and adjectives used for describing polytopes in the various numbers of dimensions. In some cases I have come across several variants in use: those which I regard as unsatisfactory are given in square brackets [ ], with comments below. My criteria for choice are mainly for clarity of meaning and expression in English, while also paying respect to the spirit of both English and classical Greek grammars.
|p (general)||p-Polytope or p-tope||p-Polytopes or p-topes||p-Polytopal or p-topal|
|0||Monon, [Monad]||Monons, [Monads]||Monal, [Monadic]|
|1||Ditelon, [Dion, Dyad]||Ditela, [Dions, Dyads]||Ditelic, [Dyadic]|
|4||Polycell or Polychoron||Polycells or Polychora||Polycellular or Polychoric|
"Monad" is a general descriptive term indicating something which exists in isolation. It is not appropriate to vertices, which are the ends and corners of higher polytopes.
"Dyad" is a general descriptive term indicating a pair of units treated as one. Other dyads may also be found in polytopes (see below). The natural extension to triads, tetrads and so on leads to a series of such terms which one might be tempted to apply to complex polytopes (having more than two vertices on an edge). Because of their wide usage (for example The fifty-nine icosahedra used "triad" and "tetrad" in a different context), these terms should not be given overly specialised meanings.
"Dion" is a later variation on ditelon, adopted by Norman W. Johnson in his Geometries and Transformations (CUP, 2018). It is notable as the first time that the 1-polytope has ever been specifically named in print. He originally suggested adapting my ditelon and polytelon respectively to "ditel" and "polytel" on the grounds that they are shorter and neater, but later changed his mind to dion. The complex equivalent would presumably now be "polyon". However I find his chosen contractions to be less informative - two or many of what? The standard names include such descriptive elements; "-gon" (knee or corner), "-hedron" (seat or face), etc., so I still think the name of the 1-tope should be similarly descriptive. "Polyon" would seem aprticularly uninformative. No doubt time will decide the accepted terms.
"Polyhedrons" is an attempt to anglicise the classical Greek ending of "Polyhedra", and seems to be becoming increasingly popular. The traditional spelling is widely used and understood, and the change adds as much confusion as it clears up.
"Polychoron" is the opposite - an attempt to Graec-ify the English root of "Polycell". While I find it needlessly obscurantist, it brings the benefit of a less tongue-twisting adjective. I haven't made up my mind on this one, though Johnson (2018) accepts it.
George Olshevsky developed a system for higher dimensions working from the observation that any n-tope is made by assembling (n-1)-topes. It is based on use of the Greek numerical prefix for the dimension of the facet with an -on suffix, giving "polytetron," "polypenton," "polyhexon," and so on. This parallels the nomenclature for spherical surfaces, for example a sphere in 5 dimensions is a 4-sphere. However names like "icositetron" would be ambiguous - is it a 24-dimensional polytope, or a polytetron with 20 facets? Meanwhile Wendy Krieger came up with the idea of using established powers-of-ten prefixes on a "thousands rule" best explained by example. The prefix for 1012 is Tera-. We note that 1012 = 1,0004, so a poly-4-tope, or 5-tope, is a polyteron. Similarly a 6-tope is a polypeton and a 7-tope a polyexon. It looks as if the strict Greek numerical prefix of Dinogeorge has been modified by subtracting or changing a letter here and there: I do not know if this is coincidence or if the standard power-of-ten prefixes were arrived at on this basis. My choice of -ic or -al endings for the associated adjectives are made according to the rules I set out earlier, with -ic being closer to the Greek. Jonathan Bowers has come up with names up to far higher dimensions. Alternatively one may stick to plain old "5-tope," "6-tope", etc. despite their being, slightly uncomfortably, of equivalent dimensionality to the lower-numbered 4-sphere, 5-sphere, etc.
One of the defining characteristics of real polytopes is that they are dyadic. If line segments are no longer "dyads" but "ditela", how can polytopes remain dyadic? There are two ways to answer this.
Firstly, theories formalising the dyadic aspect have to date been based on the observation that a real line segment joins just two vertices (ends), so for theoretical purposes these vertices may be treated as a dyad. This remains true - the dyadic property has not disappeared because we renamed the thing.
Applying the duality theorems of projective geometry to this observation, in successively higher dimensions we obtain a series of dyadic properties. A vertex of a polygon joins just two sides. An edge of a polyhedron joins just two faces. A wall of a polychoron joins just two cells. And so on: in general, the (p-2)-elements of a p-tope are dyadic. Thus, the dyadic property is not unique to ditela, but is characteristic of all polytopes.
Alternatively we can find a slightly different dyadic property. Just for fun, let's look at the dual properties the other way round. The vertices of a polygon are dyadic in the sense that the ends of two adjacent edges are treated as a single vertex, the edges of a polyhedron are dyadic in the sense that the sides of two adjacent faces are treated as a single edge, in a polychoron the faces of two adjacent cells are treated as a single wall, and so on in higher dimensions: again, in general the (p-2)-elements of a p-tope are dyadic.
Applying projective duality to these higher-dimensional objects, in all cases we obtain the same reciprocal rule, viz. that "the edges of (i.e. incident on) two adjacent vertices are treated as a single edge". This is just confirmation that the two ends are joined to the same edge. This dyadic property applies to the body of the ditelon, rather than to its ends as before.
Johnson (2018) defines the dyadic property on p.224.
Complex polytopes are not in general dyadic. However they are not really polytopes either. A real polytope has a closed surface which bounds an identifiable interior. This surface is of such importance that it is often treated as the polytope itself.
A configuration is a regular arrangement of points and lines (and, by extension planes, etc.) having the same connectivity for all lines and the same for all meeting points on those lines.
Historically, geometers made little distinction between polygons and configurations. To them a dyadic polygon was just a trivial, even degenerate, configuration. Configurations in general, even the so-called "complete" polygons, have a richer connectivity. It was only later, as the number of dimensions studied crept steadily higher and the use of surface decomposition to characterise and even define polytopes came to the fore, that polytopes became a definitively distinct species and dyadic polygons were reclassified accordingly. Nowadays, geometric polytopes are understood as "realizations" of underlying dyadic and properly-connected set theoretic structures known as abstract polytopes. There can be no going back on a century of such theoretical advances.
The first complex polytopes to be described in any detail were regular but non-dyadic "polygons". A complex polygon extends indefinitley and has no closed surface. Because it is non-dyadic it cannot be a realization of any abstract polytope. It is more akin to a complete polygon than to a dyadic polygon. A regular complex "polytope" is best understood as a complex configuration and its naming as a polytope was, even at the time, sufficient of an anachronism to confuse the modern reader. Less regular complex polytopes have no other analogous name, but might be described as properly-connected arrangements of lines.
The polytope terminology has been applied unchallenged to these complex constructions for a long time now, so is probably too entrenched to be undone. But the distinction is an important one which needs to be borne in mind.
29 May 2011: Up to 9 dimensions and other minor revisions/corrections.
10 June 2014: Two corruptions to table fixed.
30 Oct 2018: Johnson's Geometries and Transformations. Configurations.
Guy Polyhedra General Theory