## A small consistency issue

Discussion of shapes with curves and holes in various dimensions.

### Re: A small consistency issue

My intention was to try to publish an article about rotatopes and toratopes with help from the forum. As we all know, these include cylinders. If you'd like to publish an article about Wendytopes and include whatever shapes and products you like, that's great, but I think discussion on that should go in another thread.

Besides, it's not true that a cone or a pyramid is a Cartesian product.
Last edited by PWrong on Thu Aug 07, 2014 9:43 am, edited 2 times in total. PWrong
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### Re: A small consistency issue

PWrong wrote:If we're going to publish this we need to stick to a consistent notation, and stay as close as possible to conventional mathematics notation. There definitely is a Cartesian product, it's defined for sets but applies in a straightforward way to Euclidean space. For any set A in R^m and any set B in R^n, there is a unique Cartesian product AxB in R^{m+n}.

Cylinders are useful, interesting objects, and I don't see any reason not to include them. It's true that x is not associative with #, but that's because # is an incredibly restrictive product: the shape on the left must be differentiable, while the shape on the right must be a sphere.

One of the points of the paper is that the products of sets does not have a unique euclidean representation. Where one is going to count the cylinder in the mix, why not count the cone, which is also straight-forward product? There are four different ways you can present a product AB geometrically. The prism (or 'cartesian') product is one, the torus is another, the tegum a third, and the pyramid the fourth.

The relevant sets here do not include content, and the product of A in R^m * B in R^n is producing AB in R^{m+n-1}, because the relevant sets here are not the solid space, but the surface: ie A bounded by R^{m-1} by B bounded by R^{n-1} gives AB bounded by R^{m+n-2}.
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### Re: A small consistency issue

Cone: For each point in set A (a point), for each point in set B (a circle), there is a unique member (line) in the product AB (cone). This is the general process of the draught process.

Surface product: The surface of AB is a set, comprised of members (points or lines), represented by the pairing of each element of the surface of A with each of the elements of the surface of B. For each point of a circle (xy), and each point of the cross-section (oz), there is a unique point AB, at the intersection of the circle at xy (radially to oz), and the point on oz (parallel to the central section).

ie: set <> geometric structure.
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### Re: A small consistency issue

Hmm, well, this was a good thread to start. We seem to have different methods of telling the same story. This means it is utmost important to find a compromise on our symbology and vocabulary, if we're going to achieve this. Too many symbols will cause confusion, not what we want. It should be extremely clear to all readers. I think what we're working on is the first real investigation into finite volume smooth manifolds, from discrete hypersurfaces. I've been searching around for some current methods of description, but it seems to be fairly limited when defining toratopes. This is as close as I can find:

http://en.wikipedia.org/wiki/Finite_topological_space

Which combined with combinatoric topology, helps us explain what it is we're working on. It's a deeper look into algebraic topology. I really wish I knew some of the common terms, so I can describe these shapes. Is there some kind of Idiot's Guide to Algebraic Topology? That's been my attempt with the S1xS2 and such, is defining a build sequence to make a high order toratope. I saw someone else using that terminology, and figured out pretty quick how it worked. That's when I took to translating the notation list into those terms. Some toratopes need a ≥3 tier stacking of fiber bundles, in high enough dimensions. Then, of course, I used [S1*S1] for cartesian surface product and S1xS1 for comb/spherate product. Where, for simplicity, I reduced [S1*S1] to C2.

What I see so far:

• Spheration seems to be the Comb product, also called a Surface Bundle over a Surface
-- There's a few things on vector bundles over a closed manifold
-- The combinations of Clifford tori and n-spheres is called a Frame Bundle (I think!)
-- Spheration of a clifford or n-sphere is called a Fiber Bundle over a Frame

• The discovery of the Tiger is fundamentally new
-- In no cases have I seen a circle bundle over a Clifford torus, anywhere
-- The concept of cartesian surface products also seems to be new, leading to the tiger as first case in 4D
-- The tiger introduces new concepts, which may need some expansion of current methods
-- the tiger leads to an infinite amount of higher-D equivalents, based on surface products alone

• Toratopic notation allows the easy study in just the combinatoric nature of Hypertoric Varieties
-- The notation condenses the implicit equation to as few symbols as possible
-- The cut algorithms derive the exact solutions to hyperplane intercepts --> very noteworthy!
-- The nested circles can derive the build sequence for the Frame bundles
-- The outer circles/spheres derive the comb/spherate products

• Current terms do not include or differentiate between a tiger or ditorus, they are both S1xS1xS1
-- this may require some new expansion, since a tiger is more complex in nature than a sequence of n-spheres
-- we should use both the comb/spherate and cartesian surface products for tigroids, as that's what sets them apart
----- We should agree with some kind of definition between Cartesian Product and the Spherate/Comb
----- If the Comb isn't known very well, then maybe we should just use it, or call it a Fiber Bundle over a Surface

• All toratopes in all dimensions are built by a finite amount of n-sphere bundles over clifford tori frames and/or larger n-spheres
-- Clifford n-tori, C^n, come in a simple sequence of cartesian surface products of n-spheres
-- T^n, S1^n come in a sequence of n-spheres
-- Tigroids, S^n x C^n, come in a variety of n-spheres bundled over a frame sequence of clifford n-tori combined with n-spheres
-- Clifford tori can bundle over larger clifford tori, creating symmetrically nested circles in notation: ((II)(II)) , (((II)I)((II)I)) , ((((II)I)I)(((II)I)I)) , etc

I know a topologist doens't care about distances in the 3-manifolds. But, the tiger is a very different beast to a ditorus, hence my disposition to setting them apart. A clear method should probably exist to define higher order toratopes with surface products, as they are not a combination of n-sphere bundles, and do not make lower intercepts the same way.

I feel keeping the notation as simple as possible, so one can do cuts and figure out the intercepts, is key to success. Figuring out the cuts is very important to the concept, as it is a practical proof system to deriving some sense out of the fiber bundling sequence. The cuts tell you, in the arrays, how the shape is built by rotations, and also sheds light on the finite set of arbitrary bundling sequences for a high order toratope. The cuts are the exact solutions to the enormous polynomials that define high-D shapes. There is some value in that.

Right now, as far as what's agreed on, it seems to be

(II)(II) - clifford torus = C2 = [S1*S1] = (circle x circle)
[(II)(II)] - duocylinder = B2xB2 = (circle,circle)-duoprism
((II)(II)) - tiger = S1xC2 = circle#(circle x circle)

This keeps it fairly simple, and lets us derive equations properly. It also uniquely defines rotatopes apart from toratopes, based entirely on their common (n-2) frame. There's some deeper secrets Marek and I discovered from discussion and exploring functions in the plotter. It's possible to use an entire open toratope as a complex surface of revolution, which spherates the complex margin with a whole toratope. It was our 'A along B' method, that I later found was called a fiber bundle over the frame.

Basically, one can remove Q dimensions out of a toratope, and place a whole rotatope in their place, to inflate the N-Q frame of the rotatope. The starting toratope is the small shape fiber bundle, the rotatope N-Q frame is the large shape revolution bundle. These are worth exploring and detailing, if the paper focuses on the theory. There's a precise way the nested circles can show how they build toratopes, by revolution, products, and spheration. Once we hit 5D, we get into arbitrary construction chains, which I think has confounded categorization and definition in earlier works.
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### Re: A small consistency issue

I'll try to explain what's going on here.

"Spheration" and "Comb Product" are words of my invention. You won't find much references to either in other people's stuff. The two things are utterly different in approach, one is a function, like sin(x), the other is an operator x*y. The degree of overlap here is that the outer brackets of a toratope can be made by either spheration of the surface, or as intended, the repetition of the surface of A with the surface of the section B.

A dumb-bell is a spherated line. It has two big balls at the vertices, and a bar representing the line segment. Spheration lifts elements of any dimension up to solid, and a different rule can be applied for different dimensions. ZomeTools are a set of spherated points and lines.

What makes spheration useful here is that you can explain it as a 'fattening up' a space to solid, such as the hollow shell of a previous figure in the torus chain (sock torus), or the product of surfaces of two circles (eg tiger as 'spherated bi-glomolatric prism'). It's easy to explain to people who don't understand comb products, as a way of pushing them in the right direction.

The attempts to interpret what an unspherated tiger "(ii)(ii)" might look like, then is pretty much folly, since tigers are comb products, not nested spherations. Trying to figure out what (ii)(ii) might mean in ((ii)(ii)) is like trying to figure out what 4 7 means in 4*7. There is no pre-cursor in a product: the actual product is (ii)(ii)(oo), and the comb product requires to ponder a dimension each use, so you have to have two 'o' coordinates. The ordinary chain rule of hoses and socks is (ii)(oi)(oi), but the tiger puts both o's in the same figure. (ii)(ii)(oo).

It's easy to see the process explained like this, but there is a lengthy thread "Does the tiger exist" where it took many of us (me included), to come to grasp the notion that the comb product something different to a chain product.

The Comb product applies to polytopes like decagons and icosahedra as well. The product in its raw form multiplies the Euler characteristics of the equation, because as PWrong shows, it is a cartesian product of defined sets. It just does not resolve down to the received cartesian product, so it's not good to use that word. ((10)(10) 3,5) = (10e+10v)*(10e+10v)*(20h+30e+20v) = 2000t+7000c+9200h-5400e+1200v. It's the same sort of arithmetic as used in the regular prism product (cartesian product), but there you put an extra 1 in front of everything, eg the decagon is 1h + 10e + 10v. The letters t.c.h.e.v represent the count of the 4,3,2,1,0 dimensional surtopes. So a decagon*icosahedron prism in 5d is (1h+10e+10v)*(1c*20h*30e+v) = 1p +30t+240c+512h+420e+120v.

Note that in the prism product, we have the extra 1h, 1c representing the interior of the figure. In the comb product, this is absent, because it's a repetition of *surface*, not *content*.

Let's look at Philip's list.

* Spheration seems to be the Comb product, also called ...
-- Spheration is a surface finish, like painting something red. It has nothing to do with combs.
-- The torus-product is a fundementally new product, and can be applied to polytopes.

* The Tiger is fundementally new.
-- It really has nothing to do with clifford-toruses, except the description of the ((ii)(ii)) is easily done by that. Nothing clifford in ((iii)(iii)).
-- The tiger does introduce new concepts.
-- The main impart of the tiger is that you can drain multiple axies from an element of a comb product: the chain rule is broken.
-- The tiger by being (ii)(ii)(oo) drains the same polygon twice. This means that one is "spherating" a circle with a torus!

* The torotopic notation,
-- Much of what is written here is right, but, the
-- Can be extended to include polyhedra, eg ((10)(10) 3,5 ) = "decagon-decagon-icosahedral-tiger"
-- No circle or sphere derives from a comb product. You only get squares and cubes.

* The cartesian product is not involved in any way in forming any of the tigers. They're all done by draining a common element in the comb product.

* Clifford-torii only apply in 4D. The thing is meaningless elsewhere. Saying that the tiger is a spherated clifford-torus is like saying the triangular prism is a digonal cupola. Correct, but accidental.

In short:
Spheration can raise any lesser dimension to solid, and apply different rules for different dimensions. Sph(figure) = Figure, It's one in, one out function.
Comb product is applied to two or more figures, the surface of the product is the product of the surfaces, ie S1 * S1 = S2, means a polygon * polygon = polyhedron.

"Clifford" correctly applies to a class of parallel lines in S3 or circles on S4, which are not coplanar. These do not happen in ((iii)(iii)).
Saying the tiger is a spherated clifford torus is like saying the cube is a square prism. It has little to do either with spherated or clifford torii, except these lead to the same figure in 4D.

When the mathematicians call (((ii)i)i) a tri-torus, they are seeing S3 = S1*S1*S1 (surface 3 etc), so they are actually seeing it as a repetition of surfaces, or comb product. The regular torus ((ii)i) would be 2d space wrapped separately.
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### Re: A small consistency issue

That's definitely more clarifying, thanks wendy!

I should probably point out that the whole C2, C3, C4, C5 is just my little invention, to help describe some common features with how a toratope makes lower-D intercepts. What I noticed over time, is that the number of inner-most nested circles ( or n-spheres) make the overall size and shape of the array. If there are only two nested, it intercepts low enough as a rectangular array, either vertical or horizontal. It will cut, non-empty, no further down than a rectangle array. If the shape has three nested circles, it makes a cuboid array. If there is four, it makes a tesseract array, etc. So, Cn = n-frame of circle^n prism. I named them the Clifford n-tori, to expand the definition a little. This also means Cn cuts all the way down to an n-cube array of 2^n points. These points are the location of a smaller inflating toratope.

In trying to find a good generalization for this, I made a connection around the nested circles being a solid rotatope prism, whose margin had been inflated by a lower toratope. If we swapped out the nested circles with a single dimension, we end up with the toratope that inflated the margin.

For example:

- Take ((II)(II)), replace 2 nested circles, and we get (II), as S1xC2, cuts to ((I)(I)) : 2x2 array of 4 (II)
- ((II)(II)I), replace 2 circles, and we get (III), as S2xC2, cuts to ((I)(I)I) : 2x2 square of 4 (III)
- ((II)(II)II), replace 2 circles, and we get (IIII), as S3xC2, cuts to ((I)(I)II) : 2x2 square of 4 (IIII)
- (((II)(II))I), replace 2 circles, and we get ((II)I), as T2xC2, cuts to (((I)(I))I) : 2x2 square of 4 ((II)I)
- ((((II)(II))I)I), replace 2 circles, and we get (((II)I)I), as T3xC2, cuts to ((((I)(I))I)I) : 2x2 square of 4 (((II)I)I)
- (((II)I)((II)I)), replace 2 circles, and we get ((II)(II)), as S1xC2xC2, cuts to (((I)I)((I)I)) : 2x2 square of 4 ((II)(II))
- ((II)(II)(II)), replace 3 circles, and we get (III), as S2xC3, cuts to ((I)(I)(I)) : 2x2x2 cube of 8 (III)
- (((II)(II))(II)), replace 3 circles, and we get ((II)I), as T2xC3, cuts to (((I)(I))(I)) : 2x2x2 cube of 8 ((II)I)
- (((II)(II)(II))(II)), replace 4 circles, and we get ((III)I) as S1xS2xC4, cuts to (((I)(I)(I))(I)) : 2x2x2x2 tess array of 16 ((III)I)
- (((II)(II))(II)(II)), replace 4 circles, and we get ((II)II) as S2xS1xC4, cuts to (((I)(I))(I)(I)) : 2x2x2x2 tess array of 16 ((II)II)
- ((((II)(II))(II))(II)), replace 4 circles, and we get (((II)I)I) as T3xC4, cuts to ((((I)(I))(I))(I)) : 2x2x2x2 tess array of 16 (((II)I)I)

and so forth.

By combining this with the " A along B " method of decomposing toratopes, we can get a chain of Sn, Tn and Cn. I saw someone else, and read some examples, of using S1xS2 for a torisphere, and S2xS1 for spheritorus. So, in expanding that idea, I took to chaining the n-spheres, n-tori and Cn together to help describe the build sequence of toratopes.

I came across the Clifford torus definition, here: http://en.wikipedia.org/wiki/Duocylinder#The_ridge . So, if this is correct, what I did was expand the definition, to include the ridge of a circle^3 prism, as a C3, and a circle^4 prism as C4, etc. If the tiger is an inflated Clifford torus ( duocyl ridge), in which it seems to be, then the tiger cuts down to a square of four circles. Here, I made the connection that a square intercept probably comes from inflating a Clifford torus at some point. In cutting toratopes containing two nested circles, this generalization seems to agree with how it makes intercepts. If cutting a three nested circle toratope, it makes no lower than a cubic array.

This also plays a role in the small shape along large shape breakdown. We've all come understand a torisphere as a small circle stretched along a large sphere, ((III)I), and spheritorus as small sphere along big circle ((II)II). So, what I wanted to do was to broaden this technique with the use of Tn, Cn, and Sn. If describing a (((II)(II))(II)), we can understand it easier with a translation to T2xC3, which tells us it's a small-shape 2-torus ((II)I) stretched over a large-shape 3-frame of circle^3 prism, the C3. What we get in 3D is the cut (((I)(I))(I)) , a 2x2x2 cube array of 8 small tori.

One thing I noticed with discussion in this post was that the shape (((II)I)((II)I)) is a tiger stretched over a clifford torus, making (((I)I)((I)I)) : four tigers in a square in 4D. It fully cuts down to a 4x4 square array of 16 circles. This 4x4 array can be understood as a small shape C2 stretched over a larger C2, making a 2x2 array of smaller 2x2 arrays. This is why I use S1xC2xC2 to describe the build sequence of (((II)I)((II)I)), starting with the smallest and inflating larger as we go down the chain. And, for a (((II)I)((II)I)((II)I)), it cuts down to (((I))((I))((I))) : a 4x4x4 array of 64 spheres. It's a small-shape 2-sphere inflating a medium shape C3, making triger ((II)(II)(II)) , infl largest shape C3, making a 2x2x2 array of smaller 2x2x2 arrays of spheres. That's 8 smaller trigers in a larger cubic array. Which makes the (((II)I)((II)I)((II)I)) easier to understand as S2xC3xC3.

Hope that clarifies some of the things I use and say -- Philip
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### Re: A small consistency issue

The trouble i find with 'expanding' words, is that you can miss the original intention in the process. It's the eskimo 'snow' thing. You point at a batch of snow, and they say 'building material' or whatever. This is dutifully recorded as a word for snow. In practice, the eskimos have just two words for snow.

"Clifford" is horribly overloaded because he wrote the necessary maths to do rotations in N dimensions, like S(3) and S(4). He also found the non-planar equidistants in S3, which is the basis of the clifford torus in 4D. I'm still having reservations in using it elsewhere, but i am happy to let you run with it, since any of the simple tigers ((ii...)(ii...)....), do enclose cifford parallels.

The rectangular array that turns up in sections come from what i call 'legs'. If you look in my proposed second paper, you see that i use a pre-cursor notation like (ii)(ii)(oo) for the tiger, before opening the o's into (). This is because the comb product absorbs a dimension every use, and the o's are where the dimension is absorbed. The brackets then enclose proper figures, eg (iii)(iii)(ooi) is a tri-spheric tiger-torus.

The array comes from that not all brackets occur at any one place. These are the 'legs' of something like (oo), which descend inwards to different axies. The rectangular array comes from the fact that every pair of i's are orthogonal themselves.

You did see my note on how to find the outer section when there are no i's in the bracket, eg ((ii)()) -> circles.
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### Re: A small consistency issue

Noted. No word expansion. Though, the others in higher-D are a lot like the clifford torus! They have the same characteristic as smooth manifolds that curve in the same way. I didn't know you wrote a paper about it, I'll have to give it a read. So, in your own words,

• How would you define the basic sequence of n-cube arrays, made from the cut down frames of multicircular prisms?
• How would you describe why smaller toratopes inflating the points of those n-cube arrays?
• How about the 3-frame of ((II)I)(II)? The comb product would be draining the elements of a duocylindric torus, making 8 legs in 2D.
• What do you think of the whole surface bundle thing? Does it help with reading the notation, or visualizing the shape?
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### Re: A small consistency issue

I've been thinking of the torotopic notation, and how one might derive names to it.

In terms of the comb product, each set of brackets at the level of some 'I' counts as a 'drained' dimension. You still see the full dimensions of each part, but an axis is shared. For example, the ordinary torus is ((II)I), but it's actually a product of two circles (eg the 28-inch rim and the 2-inch tire). Since both of these can be replaced by actual polygons etc, so you have a wheel of 80 segments, and the wheel of 6 segments, giving a surface made of 6*80 squares. So we need to be able to show the full dimensionality at each level.

This is what i was trying to do with the ((II)oI) thing. There are two symbols inside each bracket, so we see we can write eighty-shot hexagonal torus, ie ((80)6).

Then i figured out that each 'o' governs the previous set of un-used brackets, and so one closes a set before going back to the next 'o'. So if one can create a notation based on how many o's a torus has, that is sufficient to name the comb products with the tree structure.

The dyad (or one-circle), produces both shelled and repeated toruses. Like all toruses, it depends pretty much on where it appears. A cube of spheres, then is ((i)(i)(i)ooo), would be a tri-dyadic sphere tri-tiger. If you put dodecahedra there, it's a tri-dyadic dodecahedra tri-tiger.

For example, in ((((i))i)) we see the drained axies are ((((i)o)oi)o), which is a dyad-dyad-circle-dyad torus. It's an array of 4*1 annules, the annulus is caused by the dyad after the circle, the row is caused by the dyads before the circle as 2^2=4. In simple terms, you put an o after every close-bracket, except the last, and then you can shuffle o's across matched opening and closed brackets, eg tiger = ((ii)o(ii)o) = ((ii)(ii)oo).

With the (((ii)i)(ii)), the draining occurs at the extra o's, visible in the cross-section, but adding no extra dimension to the figure, so the undrained form is ((ii)oi)(ii)oo), which is a product of four circles. The four circles are all clearly visible, but three of the eight dimensions are shown as o, because they are shared with, and subordinate to (ie appear as a +/- term like the annulus.), the i axies.

When you pop this down to a grid of two-by-four squares (ie (((Ii)oI)(Ii)(oo)), It has a name: (((I)oI)(I)(oo) = "dyad-circle torus dyad circle tiger", written, i am afraid directly from the symbol! When you are doing sections, you are sectioning the elements of the product, so eg (ii) becomes (i). The full name of the unreduced figure is to replace 'dyad' with 'circle'.

Likewise, adding extra i's into the brackets, will cause the dimensions to increase. Three letters at the same level make a sphere, four a glome, so something like ((ii)oi)(ii)ooi) is circle-circle-torus-circle-sphere tiger.

The "legs" are the number of different o's at a level, and tell you whether you have a torus or a tiger or a tri-tiger etc for that name. Each o belongs to a single set of brackets, and you can use ordinary bracketing rules to complete right-to-left, one leg before moving onto the next. It's sort of working a tree over one branch at a time.

Surface-bundling, or spheration is handy to explain to someone how one can turn something like a circle into a torus, and how it goes onwards. But spheration really is a 'sock' process, which creates the outermost set of brackets. But it only really works with round things, and not polytopes, whereas the comb works well with polytopes too. Finding an alternative way to generate the tiger while showing its bicylinderic symmetry is a tough call, though.
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### Re: A small consistency issue

I find the cuts easier to read by lack of symbols, instead of extra ones.

Also, wouldn't a circle be bundled, in a ways, over the tiger frame, aka duocylinder margin? In the same way you would do this to a T2 to make T3, or an S2 to make S1xS2? The duocylinder margin is just a closed 2-D sheet, that curves into 3 and 4D. It cuts into 3D as the vertical stack of parallel 1-D disk edges. And, cutting further, can make 4 points in a square. If one were to surface bundle a circle over this sheet, identical to T2 , it would inflate the 1D disk edges to make tori in ((II)(I)), and the 4 points into 4 circles ((I)(I)). This can be written as S1xC2 , where C2 is the duocyl margin. Anything bundled over a C2 can make 4 shapes in a square, when cut properly.
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### Re: A small consistency issue

I have some suggestions for a more consistent toratope notation.

1) For any object A, the string "(A)" means the set of all points at a certain distance from A, and "[A]" means the set of all points within a certain distance from A. In general, (A) is the boundary of the solid [A].

2) For any objects A and B, concatenation "AB" denotes the Cartesian product, assuming A to be completely perpendicular / independent of B.

3) The string "I" denotes a point in one dimension.

I'll provide some examples. I think this is the right idea:

Marek14 wrote: In the same way, an empty string corresponds to a point (it can be added to anything to denote cartesian product by a single point, i.e. identity), while () is "surface of point", i.e. an empty set.

empty string = a point in 0D
() = nothing in 0D (the set of all points at a non-zero distance from the origin; doesn't include the origin)
[] = a point in 0D (the set of all points within some distance from the origin; includes the origin)

I = point in 1D
(I) = pair of points in 1D (the set of all points with x^2 = a^2)
[l] = line segment (the set of all points with x^2 <= a^2)
((I)) = four points in 1D
(((I))) = eight points in 1D
[(I)] = pair of line segments in 1D
([l]) = pair of points in 1D (the set of all points at a certain distance from a line segment)

II = point in 2D
(I)I = pair of points on x-axis
I(I) = pair of points on y-axis
[l]I = line segment on x-axis
I[l] = line segment on y-axis
(I)(I) = four vertices of a rectangle
[l][l] = rectangle (interior)
[l](I) = pair of parallel line segments
(II) = circle (boundary; x^2+y^2 = a^2)
[II] = disk (interior; x^2+y^2 <= a^2)
((II)) = concentric circles
[(II)] = annulus
([II]) = circle (the set of all points at a certain distance from a disk)
[[II]] = disk (the set of all points within a certain distance from a disk)
((I)I) = pair of circles on x-axis
(I(I)) = pair of circles on y-axis
([l][l]) = boundary of a rounded rectangle (the set of all points at a certain distance from a rectangle)

III = point in 3D
(III) = sphere (boundary)
[III] = ball (interior)
(I)(I)(I) = eight vertices of a box
[l][l][l] = box (interior)
[l][l](I) = pair of rectangles parallel to the xy-plane, displaced along the z-axis (non-coplanar)
[II](I) = pair of disks parallel to the xy-plane
[II][l] = solid cylinder
(II)[l] = lateral surface of cylinder
(II)(I) = pair of circles parallel to the xy-plane
(II)I = circle in the xy-plane
((II)I) = torus (boundary)
[(II)I] = solid torus
[II]I = disk in the xy-plane
([II]I) = hollow pancake shape (a pair of xy-disks z-displaced, and the outer part of a torus connecting them)
[[II]I] = solid pancake shape

IIII = point in 4D
(IIII) = glome
(II)(II) = Clifford torus
[II][II] = duocylinder
((II)(II)) = tiger (boundary of a neighbourhood of a Clifford torus)
[(II)(II)] = solid tiger (neighbourhood of a Clifford torus)
etc.

Some of these objects are ugly, like ([l][l]) and [[II]I]; in general, these can be avoided by applying [] only at the higher levels, not inside of another [] or ().

ICN5D wrote: Though, now I see the issues with arbitrary rules. If (I) is supposed to be two points as cut of hollow circle, then it won't hold up with (II)(I). The cut of duocylinder (II)(I) has three dimensions, so it should be a 3D cylinder. It becomes well defined when, and only when, we spherate with closing brackets to ((II)(I)). So, maybe to overcome this little obstacle, if necessary for clarity sake, would be to introduce new brackets [] for solid open prisms. But, maybe establishing more clear rules with what (I) means is the key, without new notations. If so, the ' I ' is a line , and (I) is spherating the line, making two points.

With the changes I suggested, (II)(II) is not a duocylinder but a Clifford torus. Its 3D cut (II)(I) is not a cylinder (solid nor surface) but a parallel pair of circles.

A spherated line cannot be a pair of points; the line is infinite and fills all of 1D, so there are no points at any distance from it. This is one inconsistency that would be solved by redefining "I". Likewise, a spherated square would not be a circle, but a rounded square, with straight edges and curved corners; this would also be solved by redefining "II".

PWrong wrote: So the main issue with the 1D sphere is that it's disconnected. This leads to other problems, like the fact that a square is not differentiable (note that the parametric equations for a square are generally piecewise), so its tangent space and normal space are discontinuous. This is why we don't usually allow square # circle but we do allow duocylinder # circle (are we still using # for the spheration product or is there something new?)

The boundary of a square cannot be obtained with my notation. The interior of a square is differentiable.

PWrong wrote: I would also prefer to stay away from multiple types of brackets if we can help it. The main reason for this is that if we want to do mathematical operations on these shapes, we'll have to use brackets. For example if we have some function f that acts on toratopes (like volume, tangent spaces, homology groups or whatever), then it would be less confusing to write f[A] than f(A). Otherwise, you get readers wondering "does f(22) mean f(duocylinder) or f(tiger)?" The curly brackets { and } are generally used for set theory.

Yes, that is a problem. It could be solved by simply not making [] solids, or by using more standard notation, like replacing "AB" with "AxB". There should be some named function for "the neighbourhood of A within distance b", so that "[A]" could be replaced with "N_b(A)". The standard notation for "boundary" is the "curly d" used for partial derivatives, so "(A)" could be replaced with "[curly d]N_b(A)". Of course, this would make the expressions longer and harder to read.

Perhaps you could just name the object "A", then apply the function "f" without parentheses: "fA".

wendy wrote: One of the points of the paper is that the products of sets does not have a unique euclidean representation. Where one is going to count the cylinder in the mix, why not count the cone, which is also straight-forward product? There are four different ways you can present a product AB geometrically. The prism (or 'cartesian') product is one, the torus is another, the tegum a third, and the pyramid the fourth.

The relevant sets here do not include content, and the product of A in R^m * B in R^n is producing AB in R^{m+n-1}, because the relevant sets here are not the solid space, but the surface: ie A bounded by R^{m-1} by B bounded by R^{n-1} gives AB bounded by R^{m+n-2}.

Cone: For each point in set A (a point), for each point in set B (a circle), there is a unique member (line) in the product AB (cone). This is the general process of the draught process.

Points, not lines, are elements of the cone. The Cartesian product of a point and a circle is a circle.

If A is in R^m, and B is in R^n, the Cartesian product treats the two spaces as completely independent, so the result is in R^(m+n) (though its intrinsic dimension may be lower; for example, a Clifford torus is a 2D subset of 4D space.) You may define other products that work differently, such as that which produces a standard torus (in 3D) from two circles. This is not a Cartesian product.

How does your pyramid product work? Here's my guess: Assume that A and B are in the same space, then take C to be the set of all points c = (1-t)a + tb, where a is in A, b is in B, and t is between 0 and 1. This is a linear interpolation between A and B. A different possibility is the convex hull of A and B.

I'm new here, so I have no idea what you mean by "tegum" or "crind" products. (The wiki doesn't explain them.)
Last edited by mr_e_man on Wed Sep 19, 2018 4:32 pm, edited 1 time in total.
mr_e_man
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### Re: A small consistency issue

The regular products give rise to the regular solids in every dimension. The difference is that we have moved them from an iteration by dimension to a general product.

The prism, crind and tegum products are radiant products. That is, one supposes that a polytope is defined as a direction-function, where the surface is 1. Any point less than 1 is inside, while outside it is greater than 1. If several different solids are set orthogonal around a point 0, then the general coordinate x1,x2,x3,...,y1,y2,y2, ... z1,z2,z3,..., will, for the base polytopes, produce a radiant function, X,Y,Z,... The product is then a simple function of these, to define the radial value of the point, in terms of these values

prism = max(x,y,z,...) prism (n @ lines) = n-measure polytope
tegum = sum(x,y,z,...) tegum (n@ lines) = n- orthotope
crind = rss(x,y,z,...) crind (n@ lines) = n-sphere

Since the notional shape for these are [prism], <tegum> and (crind), this is the source of these brackets.

The pyramid product is the draught of content, in that between two orthogonal items of content, for every point in each, there is a line drawn to connect them. So the section against any line is a prism xA * (1-x) B, the map against the altitude is the tegum product A B.

simplex = pyramid product (n+1 @ vertices)

The comb product is a pondering product, in that M*N dimensions, gives a figure whose surface is the product of surfaces. So you get a torus = polygon * polygon = polyhedron .

cubic aperitope = comb product (n-1 @ aperigon)

The aperigon is the tiling of line segments in a euclidean line, ie the surface of a 2d polytope. The 3d polytope is the comb product of two 2d polytopes, being their surfaces, gives a square lattice. The 4d polytope or polychoron is the cubic tiling, &c.
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### Re: A small consistency issue

I think I understand the three "radiant products" now.

It looks related to this and this. An object A determines a norm NA, which has NA(x) <= 1 for any point x in A. The tegum product of two objects <AB> is the set of all points (x,y) satisfying |NA(x)|+|NB(y)| <= 1, where x is in the space of A and y is in the space of B.

Is this correct?

Klitzing describes these somewhat differently (though they may produce the same results). https://bendwavy.org/klitzing/explain/product.htm
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mr_e_man
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### Re: A small consistency issue

That's an interesting addition, and I have some thoughts:

* The difference between ' [II][l] = solid cylinder ' and ' [[II]I] = solid pancake shape '

- Seems like the two are equally solid cylinders but with heights aligned to either the x,y, or z-axes

mr_e_man wrote:([II]) = circle (the set of all points at a certain distance from a disk)
[[II]] = disk (the set of all points within a certain distance from a disk)

Can you clarify these a little more? Are they both solid disks?

Also, If [] is a solid, and () is the surface of a hollow object, then it seems like ([]) would just simplify into (). So, ([II]) means the surface of a solid disk, the circle (II).
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ICN5D
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### Re: A small consistency issue

Anyway, the cylinder [ll][l] is described by the inequalities

x2+y2 <= a2,
and
|z| <= b

And the [[ll]l] is the union of a (solid) torus and cylinder:

(sqrt{x2+y2} - a)2 + z2 <= b2,
or
x2+y2 <= a2, and |z| <= b

The other two things are based on the disk x2+y2 <= a2. ([ll]) is indeed equivalent to (ll):

x2+y2 = (a+b)2

while [[ll]] satisfies the corresponding inequality

x2+y2 <= (a+b)2
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mr_e_man
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### Re: A small consistency issue

Oh dear, sorry for taking a month to respond, lol. To upload pics, you can ask Keiji to give you permissions for this site, or create an Imgur account, and paste the direct image url link on your forum posts using the 'Img' tag.

As for your notation, I'd like to see the whole 4D list, if you've made it yet. Also, can you show how all the slices work in your notation? I.E., take a full shape expressed by symbols, remove the dimension markers ' I ' , and describe what that slice is? Similar to what I'm doing here.

And by all means, go ahead and invent all the notations you want, lol. That's what we do here, find new ways of defining shapes in some hopefully consistent manner.
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ICN5D
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### Re: A small consistency issue

The whole 4D list is infinite; I haven't made it yet . We can always apply another () to get something else. Do you mean only the objects that are self-connected? That would exclude, for example, the torus cut ((I)I), a disconnected pair of circles.

I think my notation is about the same as yours, except for [] solids. Multiplication by I just adds a dimension to the surroundings (not changing the object itself); and applying [] or () is the same as adding a dimension, then applying it, then removing that dimension; so slices work in the obvious way.

[(II)I] = solid torus
[(I)I] = pair of disks in xz- or yz-plane
[(II)] = annulus in xy-plane
[()I] = nothing on z-axis
[(I)] = pair of segments on x- or y-axis
[()] = nothing at the centre

[ll][ll] = duocylinder
[ll][l] = solid cylinder; xy-disk times z- or w-segment
[l][ll] = solid cylinder; x- or y-segment times zw-disk
[l][l] = square
[ll][] = [ll] = xy-disk
[][ll] = [ll] = zw-disk
[l] = segment
[] = point in the centre

Note that [ll] doesn't tell us what plane it's in; if you want to keep that information, you could insert a 0 or something in place of the removed I.

([lll]l) = hollow 4D pancake shape
([lll]) = sphere
([ll]l) = hollow 3D pancake shape; convexified torus
([ll]) = circle
([l]l) = elongated circle; two semicircles connected by segments
([l]) = pair of points
([]l) = (l) = pair of points
([]) = () = nothing at the centre
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### Re: A small consistency issue

Here are the pictures.
Attachments toratopeNotation.png
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mr_e_man
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### Re: A small consistency issue

Viewing those pictures in a different browser, I see that the background is black, not contrasting enough. I'll try to fix that now: toratopeNotation.png

Also my "ells" (l), used instead of "pipes" (|) because they're too tall, and instead of "eyes" (I) because they make italics, are more visibly different.
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mr_e_man
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### Re: A small consistency issue

Thanks for the detailed explanations. I will deal with this task. Well, that caught my eye.
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