## On tegums and joins.

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

### On tegums and joins.

URL pointed out in a different forum, that the various products have been studied under the names of join and incomplete join, citing, for example DMY Sommerville (1929). These were further studied in the seventies. However, these do not constitute a discovery of new products.

It is fairly easy to place points at the dual of prisms, and throw a convex hull over these. These points exist. But then to suppose that the result is a product is not discussed. The strong relation between the regular solids and the products is one of the points of my approach. Yet the science is not advanced enough to make the connection between the products and the regular solids in Coxeter (1946, 1971), the 1946 book considered the semenial reference in the subject.

The year 2002 was the first year that I had internet connection beyond downloading. This was when I joined Mangus Wenninger's polytope mailing list, to which the likes of John Conway, Norman Johnson, Richard Klitzing, Jonathon Bowers, Guy Inchbald, and other polytope enthusists (professional and ameteur), were members I introduced a number of new concepts and notations to the list, many of these being novel advancements.

The tegum product was introduced as a proclaimation that the dual of the prism shall be called a tegum, and its general discription followed. Antigegum was introduced at this time too. Various ill-recalled conversations took place, but essentially, it amounts to that until this time, the dual of a prism was a join, covered by a blanket. Norman Johnson said exactly that, including that the join was a sum, since it followed the p(XY)=p(X)+p(Y) where p() is the vertex-count

The meaning of product given was that some property p(x) exists where p(XY)=p(X).p(Y) is an algebraic calculation and that p(XY)=p(X)+p(Y) is a sum, resulting in an algebraic sum. Given that the join rarely consists of more than a blanket-show of points, and explicitly stated as a sum, that the product represents new thought.

So we come to that the various products consist of that over the surtope consist, pre-pended with content and/or appended with the nulloid. Appending a '1' at the right-hand side, will cause the next column to add, and this leads to what has been seen as a sum. This is why the face-count of a prism is the sum of face-counts, because the prism-function pre-pends a '1'.

The role of the nullitope and set theory was discussed as well. The previous model that the nulloid represents the empty set is clearly false, since it is the down-incidence of a polytope, and it is not shared with other polytopes. Specifically, the point of down-incidence represents the grouping of elements that are and are not part of the polytope. It is instead, the 'name' or 'identity' of the polytope, as much as a name represents an individual, but is no presence on the individual.

The set-theory model falls to bits when the notion of 'union' is discussed. This has no meaning in polytopes (it gives a multitope), and the up-incidences is a different polytope where the surtope represents the down incidence. Norman Johnson worked this idea into 'polytope clusters'.

The metric products of the various polytope products were never discussed outside of my hest. The various radiant products (tegum, prism, later crind) are coherent, in that measures against the powers of unit-lengths, represent a coherent scale of units against L^n, and that these units stand in the ratio of 1:n! for tegum:prism units. The crind-units represent units of sphere.

Given the request to review this by V Hernández, we come to the conclusion that that while the join name was used to describe arrangements of points of tegum and pyramid products, it never went past this and that as late as this century, mainstream science understood it to represent a sum, rather than a product. Further noted is that the pyramid product is an 'incomplete join', although this is the product of the full surtope consist including the units at both ends.
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wendy
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### Re: On tegums and joins.

I really don't think renaming something from "sum" to "product" is enough to claim it as your own discovery. You mention how people just did convex hulls for them, but from what i've seen they were defined as abstract polytopes too.
galoomba
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### Re: On tegums and joins.

I have little to go on to exactly what is implied by 'join' etc, except for the passing language of Conway and of Johnson.

Neither of these people advanced formulae for anything, except that Johnson said ver(x,y) = ver(x)+ver(y). The language of calling it a join, and Johnson specifically said that it was a 'sum of polytopes', suggests that even if they had worked more out, they were not venturing it in the discussion, either directly or through reference.

Violeta (of discord fame), gave a reference to Somerville (1929). It's not a standout item in the book, this spends a great deal of time discussing individual constructions like pyramid-pyramid figures, and a hedroid-wise construction of {3,3,5}. It mentions there were some 16 separate discoverers of the Schlafli symbol, and detailed the variations in construction. However, were the products as presented since 2002, of such importance to the regular solids, you would expect that Coxeter (Regular Polytopes, 1946), (Twelve Essays, 1868), and (Regular Complex Polytopes (1971)) would have devoted space to these products, since all but a small handful of regular polytopes and regular complex polytopes, are derived by application of these products, either iteratively or in mass. None of this passes.

The thing you should watch with terms, is though while they can give deep insight into what is going on, they can also profoundly block the view. Calling the products a 'sum' would prevent one seeing it as a product, unless someone else had already done so. When I was searching for methods for multiplying numbers in base 120, the standard notation is to refer to all of the column-holders as 'digits', as in decimal. If you look at say, the sexagesimal number system on the wikipedia, you might see that they list the 'digits' all the way to 59. This is a fault one also finds in the leading experts in ancient numbers (eg O Neugebauer, 1959), that one needs to use tables and reckoners to handle the 3600 times tables. I am still asked to this day what are the 120 digits that I use for the system.

In reality, the word 'digit' represents three different things, which are all distinct in number-systems like 60 and 120. One can appreciate the issue a little more, if one supposes that instead of just columns, one also uses rows. The calculations ard done at cell level, and the order of carry is handled as to whether one moves 'up' the column (ie 10), or to the next column left (6, 12). You can then pretty much handle everything in the units row, with a fast conversion between dickers (groups of 10) and dozens (12). No one asks you to multiply 83 by 76 directly in decimal or in twelfty, although in the mathematical illusion, these are 'digits' for which the product and sum must be learnt.

The time I spend in the discord list, arguing about words like 'face', 'plane', etc, is because I have well learnt over many years, that one can freely drift from one meaning of the word to another, and suppose that 'plane' = dividing space, or 'plane' = two-dimensions, is true only in three dimensions. When one starts to look into the higher dimensions, one must be more careful as to preserve exactly one meaning for the word, and not let the word drift to other parts.

There are discussions on the discord, to which there is a measure of clouded thinking under way. For example, the nulloid is not the empty set, and knowing this, every surtope of a polytope is a nulloid to the incident-upwards parts of a polytope. Likewise, while polytopes have symmetry groups, it is not back-reflected. That is, you can demonstrate things like 'colour-groups' which are symmetry maps which do not descend to mathematical groups. The same alternating group A5 is the rotational symmetry of the icosahedron and of the pentachoron, and both of these are direct subgroups of the [3,3,5]+. But there is no transformation between the two. Instead, one has to head to a subgroup of the group AA6 to see the connection, and even this is faulty.

It is a hard row to hoe, to imagine an entirely new idiom for something long established. The mind holds tight to the words it has learnt, and it is not easy to see these things without seeing what has been given before. I derived the products in terms of the radiant function, and the products of draught by looking at the progression from end to end. The joins etc do not as far as i can see, cover non-polytopes figures. The products do. There is a bi-circular tegum, and amazingly, its surtope consist is derived by the tegum-product of two circles (ie 1e -> 1,1,0,1 total = #,1,0,1 tegum form; bi-circle = #,1,0,2,0,1 = 1 choron, two edges). Solids have surfaces, divided into surtopes, but they don't follow polytope rules [eg they don't have the diamond property). They are a subset of solids, and the products work on solids (including polytopes).
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wendy
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### Re: On tegums and joins.

wendy wrote:The role of the nullitope and set theory was discussed as well. The previous model that the nulloid represents the empty set is clearly false, since it is the down-incidence of a polytope, and it is not shared with other polytopes. Specifically, the point of down-incidence represents the grouping of elements that are and are not part of the polytope. It is instead, the 'name' or 'identity' of the polytope, as much as a name represents an individual, but is no presence on the individual.

Every element of the polytope is incident with the nulloid, but not everything incident with the nulloid is an element of the polytope.

Things that are not part of the polytope are irrelevant; it doesn't matter whether they're incident with the nulloid. Thus there is nothing wrong with the nulloid being the empty set, and with several different polytopes having the same nulloid.

For example, the nulloid is not the empty set, and knowing this, every surtope of a polytope is a nulloid to the incident-upwards parts of a polytope.

True, the nulloid is not necessarily the empty set. A vertex is a nulloid to the vertex figure. A polytope's body is a nulloid to the dual polytope.
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