Updated 30 Dec 2020

### The short version

A ditelon is a 1-dimensional polytope, or closed line segment. It has two ends, or vertices, and (usually) a body, which is the segment 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.

Together with the nullon and monon, 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.

### The long version

#### Preface

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 stories, 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 theirs.

#### Ditela

Leonhard Euler started it all when in 1750 he first named the edges of a polyhedron and came up with his formula VE + F = 2. Until then, two thousand years of theory had dealt only with vertices and faces.

Around 150 years later analogous figures in higher dimensions were discovered and polygons and polyhedra came to be undersottd as two- and three-dimensional examples of more general p-dimensional polytopes, or p-topes. In four dimensions we have polycells or polychora. For five dimensions and up, three investigators have jointly developed a naming system which is evident from what follows.

In lower dimensionalities Prof. Norman Johnson named for us the 0-dimensional monon, or point, and for set theorists even the (-1)-dimensional nullitope, or null polytope (a name which I suggest be simplified to the nullon).

But the same loose end that Euler fixed back in 1750 has reared its ugly head again; 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. The classical Greek τελος or telos means an end, as for example the end of a rope. Such a rope or line segment always has two ends, so I propose to call the 1-polytope a "ditelon", a two-ended thing.

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.

Complex polytopes are not in general dyadic and a complex "edge" may have many vertices (see also below). Such complex line segments may be called polytela. A complex 1-polytope having four vertices is a tetratelon, and so on.

#### Polytope names

The table gives the nouns and adjectives which describe 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 lexicons and grammars.

Only the dimensions up to four are important, from five and up there are few interesting polytopes and naming them has relatively little value. Above 8 dimensions the value tails off altogether, but I include the 9th for completeness.

p (general) p-Polytope or p-tope p-Polytopes or p-topes p-Polytopal or p-topal
-1 Nullon [Nullitope] Nullons [Nullitopes] Nullonal
1 (complex) Polytelon Polytela Polytelic
2 Polygon Polygons Polygonal
3 Polyhedron Polyhedra, [Polyhedrons] Poyhedral
4 Polychoron [Polycell] Polychora [Polychorons, Polycells] Polychoric [Polycellular]
5 Polyteron Polytera Polyteric
6 Polypeton Polypeta Polypetal
7 Polyexon Polyexa Polyexal
8 Polyzetton Polyzetta Polyzettal
9 Polyyotton Polyyetta Polyyettal

"Nullitope" is expressive and has gained some use, but it is not how we conventionally name polytopes. They are all "poly-something-ons". The most appropriate equivalent here is the "nothing-on" or nullon.

"Monad" is a more general descriptive term indicating something indivisible. The idea formed the basis of the Monadology, a philosophical work by Gottfried Leibniz.

"Dyad" is a more 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 particularly 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. The same applies to "Polychorons".

"Polycell" saw early usage but, with the wider usage of cells in even higher dimensions, appears to the modern eye to be more a synonym for "polytope". "Polychoron" is rapidly becoming the accepted modern term (e.g. Johnson 2018).

For higher dimensions, George Olshevsky developed a system 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 spherical surface embedded 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.

#### The dyadic nature of polytopes

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. In abstract polytope theory it is known as the diamond condition, after the characteristic connectivity or incidences of the various elements.

#### Polytopes and configurations

A real polytope (a polytope in some real space, such as Euclidean) has a closed surface, typically finite, which bounds an identifiable interior. This surface is of such importance that it is often treated as the polytope itself.

By contrast, 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. It is anything but a finite surface.

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. The habit stuck in three dimensions too. It was only later, as the number of dimensions studied crept steadily higher and the use of topological decomposition to characterise and even define polytopes came to the fore, that more or less finite polytopes became a definitively distinct species and dyadic polygons were reclassified accordingly. Nowadays, such 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.

Before going further, some preliminary explanation of the principles of complex geometry may be helpful. A complex polytope is a polytope in a complex space, where coordinate vaues are expressed as complex numbers. This contrasts with real spaces, such as the familiar Euclidean, where coordinates are usually given as real numbers. One can also give a real space a complex metric, in which one of the coordinates is expressed as imaginary, i.e. a multiple of the square root of –1. This leads to the awkward situation that a complex line has one complex dimension but, when treated as a metric on a real space, that space has two dimensions as in the Argand diagram. Real polytopes can be constructed in such spaces provided thay have even dimension. But they cannot be constructed in true complex spaces. A complex polytope may be constructed in either, but say a complex polygon will require either two complex dimensions of four real ones (imbued with a complex metric).

Because of this the points on a complex line can not be ordered in the way they are on a real line. Given any two points on the line, there is no way to define any idea of "between" them. A complex edge may therefore have many vertices on a given line, with none in between any of the others; it need not be dyadic. This fundamentally affects the nature of the geometric conception.

Because complex polytopes are not in general dyadic, they are not really polytopes either.

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 oddity for its originators to pass comment on it, and sufficient of an anachronism to confuse the modern reader. Less regular complex "polytopes" are a logical corollary of the regular theory but, not being (regular) configurations either, have no other analogous name. They might perhaps be described as properly-connected arrangements of lines, but do not seem to have been studied.

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.