## Elements of Crinds and Tegums

Higher-dimensional geometry (previously "Polyshapes").

### Elements of Crinds and Tegums

So you may have seen a flurry of activity from me on the wiki, adding counts and descriptions of elements to all the 4D and 5D tapertopes.

I quickly realised I'd need a convention for naming these, as there are many different ways to curve n-surfaces.

I think I have a way of enumerating and naming the curved elements of tapertopes now, which I'll elaborate on later, but suffice to say a side effect of this is to create a set of not only all tapertopes and their curved elements, but all toratopes too, since all toratopes are elements of higher dimensional tapertopes (I think). So I figured, why stop there - I've been looking for a way to reunite the tapertopes, toratopes and bracketopes ever since they were first split up.

This brought me to the question of curved elements of bracketopes - specifically those of crinds and tegums since those are the ones we don't yet have.

The standard 3D crind, or circle{square,line}, has four curved faces which I call lunes. circle{triangle,line} has 3 lunes, and in general circle{n-gon,line} has n lunes. I'd like to refer to all these lunes as the same object, regardless of how many sides the crind has, much like how the iscosceles triangles of n-spindles are all called triangles even though they have different angles.

With that in mind, my question is - how does one identify different types of lune in higher-dimensional crinds? And do unique elements exist in higher-dimensional tegums? The 3D bicone, for example, only repeats curved surfaces already known - specifically, it has two curved cone-surfaces, whereas the cone only has one, but they are the same object. Does this continue into higher dimensions, or are there higher-dimensional tegums with objects not seen in their cone/pyramid equivalents?

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### Elements of Tegums

The tegum product is a drawing of surface. Its surface elements are then both the surface of the original elements, and pyramid products of them.

For example, the drawing of a hexagon onto a line creates in four dimensions, a hexagon-pyramid-pyramid, or hexagon-line pyramid. Starting at the hexagon, the subsequent sections become hexagonal prisms, such that the percentages of the hexagon and the line add to 1 (ie 30% hexagon * 70% line).

The number of elements can be realised from the tegum product. The 'tegum property' of a polytope is the surface elements + an element of dimension -1. The 'tegum property' of a tegum product is the algebraic property of the tegum product of the elements.

Here is the surtope consists of the four products, applied to a pentagon and a hexagon. N is the nulloid, of -1 D. V=vertices, E=edges, H=2D hedra (hedrons), C=3D chora (chorons), T = 4D tera (terons), and P = 5d peta (petons). The multiplication is algebraic. You can verify it by multiplying 1005005001 * 1006006001 etc. That's how i got the numbers.

A polytope usually consists of a 'volume' and a 'recriprocal volume'. The four products either include these in the product (1), or suppress them (*).

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Pentagon-hexagon tegum                           Pentagon-hexagon-prism                   T   C   H   E   V   N        T   C   H   E   V   N                   4   3   2   1   0  -1    pentagon               *   5   5  (1)              (1)  5   5  *    hexagon                *   6   6  (1)              (1)  6   6  *                -------------------------    --------------------------                   *  30  60  41  11  (1)     (1)  11  41  60  30  *                -------------------------    --------------------------Pentagon-hexagon pyramid                       Pentagon-hexagon-comb              P   T   C   H   E   V   N            C   H   E   V   N              5   4   3   2   1   0  -1            3   2   1   0  -1                         (1)  5   5  (1)               *   5   5   *                         (1)  6   6  (1)               *   6   6   *             ---------------------------     -----------------------            (1)  11  41  62  41  11  (1)          *  30   60  30   *            ----------------------------    -------------------------

The derived elements in the products of draught (the ones that end in (1), are pyramid products. Pyramid products lead to regular simplexes.

The derived elements in the products of repetition (the ones with an * at the end) give prisms (squares, etc).

The products of content (the ones with a (1) at the front), include the content as a surtope. The interior of a pentagon occurs as part of the base of a prism and of a pyramid, but not in the tegum.

The products of surface (the ones with a leading *), do not include the content.

This is a pentagonal prism and tegum, disected into an array of the different surtopes and their consists.

Code: Select all
                        Prism                                    Tegum                    (line)      2 vertex                       2 vertices      (1 null)  pentagon        (pent prism )  2 pentagons          5 edg   2 vert^5 edg     null^ 5 edg                                                               = 10 triang     =  5  edg   5 edges        5 line*line    5 line*2 point      5 vert   2 vert^5vert    nul ^ 5 vert                   = 5 squares    = 10 lines                    = 10 edges      = 5 vert  5 vertex       5 vertex*line   5 point*2 point    (1 null)  2 vert^ null    null ^ null                   = 5 lines      = 10 vertices                 = 2 vert       = 1 (null)
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### Re: Elements of Crinds

A crind is in essence a round tegum, and this should be enough to derive its surface consist.

A crind of the type (line * N-1-tope), has gores or lunes that go pole to pole through the faces of the N-1 tope. You multiply as per a tegum, but suppress the face-count of the second element.

A crind of the product of higher order elements, then follows that of the tegum, because there is nowhere that the gores can go through.

The actual elements are either flat for the various factors (element) of the product, or curved in a generalised ellipse for the derived element. In essence, you draw an ellipse quadrant between every pair of points one of each element. The axies of the ellipses are the radii from the centre of product to where the points lie on the surface.
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### Re: Content of Prisms, Crinds and Tegums

The 'brick' products are "choerent". This means that when a particular product is equated with the algberaic multiply, the powers of a line L^n, represent a series of units, where the volume in L^(a+b) = (L^a)(L^b).

The prism powers of a linear inch are inch = linear inch, inch^2 = square inch, inch^3 = cubic inch, inch^4 = tesseractic inch, etc. Generally PL^n. The volume of two shapes, of content x square inch (ie PL2), and y cubic inches (ie PL3) is a penteractic inch PL5.

Likewise, the tegum powers of a inch = diagonal inch, gives a inch^2 = rhombic inch, an inch^3 = octahedral inch, a inch^4 16ch inch, etc, Generally TL^n. The volume of a tegum-product, of two volumes, of x TL3 and y TL2 is xy TL5.

Likewise, the crind powers of an inch gives a CL1 diametric inch, a CL2 circular inch, a CL3 spheric inch, a CL4 glomic inch, with the same property.

The ratios vary according to dimension. But the PLn = n! TLn (ie the prism is n factorial of the inscribed tegum), and the CLn lies inbetween,

PL1 = CL1 = TL1, PL2 = (4/p) CL2 = 2 TL2, PL3 = (6/p) CL3 = 6 TL3, PL4 = (32/p^2) CL4 = 24 TL4, and so on.

The problem of finding the volume of a crind, as ( [ xx ] x ), can be thus solved.

The inner product is [], a prism. So we find 1 PL1 * 1 PL1 = 1 PL2.

The outer product is the crind product, so we have to convert these to CL units. 1 PL2 = (4/p) CL2, 1 PL1 = 1 CL1, product gives 4/p CL3

The actual product is required in cubic measure PL3, so we multiply CL3 = (p/6) PL3 to get 4p/6p = 2/3 PL3.

The volume of a pyramid, (which represents powers of its vertices), is the same as that of tegum product of the verious apexes and the altitude. So if you are multipying the volume of a pyramid, whose base consist of a square and a point, and the altitude is 1, then we have

{ [ | | ] ^ } where {} represents a pyramid product, and ^ is the manditorary altitude between bases,

gives [ || ] is a prism product of lines, is 1 PL1 * 1 PL1 = 1 PL2 = 2 TL2.

The tegum product of a point, the square and the altitude, is 1 TL0 * 2 TL2 * 1 TL1 = 2 TL3 = 2/6 PL3 = 1/3 PL3. The pyramid on the base and height of a cube is 1/3 of its volume.
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### Re: Names fot Elements of Crinds and Tegums

The word used to describe the slices that make up a globe of the earth, is 'gore'.

There are no new elements that appear in the products that are not themselves parts of the base(s), or pyramid or prism products. For example, in the dodecagonal-pentagon prism, evidently the dodecagon and the pentagons are not products (they're in the bases), but the derived elements are things like pentagon-line prism or pentagon-pentagon prism, are lesser-dimension products.

The tegum and comb products are surface products, so these products have surfaces that consist of pyramid and prism products of the surface elements. For example, the comb product of a dodecagon and a dodecahedron, contains neither of these shapes, but the product of pentagonal prisms and their consists.

In general, i do not make special use of words like cone etc for the higher dimension circle products. The circle, sphere etc are solids, and the derived products are named in the same way that regular polytope products are, eg 'bi-circular prism' vs 'bi-octagon prism'. There is still the matter of how to deal with nested products, the current thinking is to use duo-, tri- etc to represent the level of brackets demanded.

The surtope consist of a round thing, under one of the four surtope products (all except crind), follow exactly the same rules as do polytopes, in that there is an extra 1 available at each end, except that all circles/spheres/etc have just 1 face (N-1 surtope).

So, eg the bi-circular tegum, comes as -h,1e.0v.(1n) * -h,1e,0v,(1n) = 1c,0h,2e,0v,(1n). This means it has one face, 2 edges. It is possible to find its volume, because it's <(xx)(xx)> The product (xx) is 1 CL2 or circular inch. We need this in rhombic inches, so it's p/2 TL2. The tegum product of volumes p/2 TL2 and p/2 TL2 is p^2/4 TL4. If we want this in cubic measure, we note that 1 CL4 = 24 PL2, so the volume of this figure is PL/96 PL4 (tesseracts), or (32/p^2) = 1/3 CL4 = 1/3 of the enclosing glome.
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### Re: Notations for round things, etc

Because the circles participate in various products, they are given 'fake' CD diagrams, to match their symmetry. One can then represent various ellipsoids etc in terms of this scheme.

A circle is as a polygon, xOo. A sphere is like a polyhedron, ie xOoOo and so forth. You have N-1 x [ Oo ] with the bit inbrackets repeated N-1 times.

The addition of extra 'x' into the symbol, means that the radii increase. This allows us to write things like prolate xOxOoOo (x<y=z) and oblate xOoOx (x=y<z), ellipsoids. along with ellipses.

Tegum and Prism products directly reflect into the CD diagram, over the & sign, eg cylinder = x&xOo , cf triangle prism at x&x3o. The dual of a prism is a tegum, so mOo&m&o is the dual of a duocylinder, or bi-circular tegum.

However, it is being considered into providing alternate runes to designate the products separately.

The 'brick' notation for the pyramids, is to suppose { } or curly brackets, represent this product (since we have <T>, [P] and (C)), and because one has to include an altitude as a separate dimension, have ^ for that, so, eg a triangle as a brick might be { ^^ }. In essence, one can regard {} as enclosing a pyramid product, and an empty field as a point, so {^^} is both a 2d figure (two sticks), and a product of three things as {v^v^v}.
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### Re: Elements of Crinds and Tegums

That's some cool stuff, Wendy! I'm going to study it a while, you have a very refined notation I need to learn. Now I understand what the bi-circular tegum is, by the description as the dual of the duocylinder. I never thought about that shape before!

Those are neat tables, too. I guess I'm in the long-division stage of my methods
in search of combinatorial objects of finite extent
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### Re: Elements of Crinds and Tegums

My name for the bi-circular tegum is the duospindle. It would be considered as a fair die due to it having congruent contact regions.
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### Re: Surface calculations

The surface of many of these products can be directly derived from the volume, if a sphere can be shown to touch every face in such a way that every point is on a line tangent to that sphere. Under that case, the volume is simply r \dot S, or a pyramid r high and S of base. Since the volume of a pyramid is the tegum-product of the two, we use this.

ie: T(surface) = T(volume) / radius = 2 T(volume) / diameter.

A crind in 3D, is tangential to a sphere of the same radius, so if the diameter is 1, the radius is 1/2. The square is area 4/pi CL2, and the crind is 4/pi CL3. (ie multiply by 1 CL1). This is 2/3 PL3 or 4 TL3. So we calculate the surface as 4 TL3 / 0.5 TL1 = 8 TL2 = 4 PL2. This means that the surface of a crind is the same surface as the four faces of the square that contains it, in the same way that the sphere is the same surface as the curved wall of the cylinder that contains it.
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