## A small consistency issue

Discussion of shapes with curves and holes in various dimensions.

### A small consistency issue

I was just trying to write up a brief introduction to the article, and I wanted a concise, rigorous definition of a rotatope. My first idea was "a shape than can be constructed from spheres of various dimensions and the Cartesian product". If we allow the spheres to vary in radius, that definition would include not just cubes but arbitrary rectangular prisms. We could say instead that all the spheres are the same size, but that means that the diameter of a cylinder must be the same as its length, and it has even weirder implications for toratopes later on. We could try some more sophisticated rules about when the radius is allowed to change, but personally I think we should just bite the bullet and consider any rectangular prism to be a rotatope. What do the rest of you think?

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### Re: A small consistency issue

Well, all rectangular prisms can be expressed as open toratopes -- rectangle as II, cuboid as III etc. You have to allow them, I think.
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### Re: A small consistency issue

The rototope question is interesting.

Mathematically, it's a fairly straight forward comb product, or 'repetition of surface'. That is, we have a copy of the surface of B at each point of the surface of A.

In the case of the torus, we have [(xy)z] = (xy) - [ () z ] comb. Here (xy) represents a circle, the surface of which has a radius in (), and in the space () z we draw a different circle. We can see also that for each point in ()z, the (around the tube thing), you have a full copy of xy.

As long as the brackets are strictly nesting without a tiger )( device, it is possible to construct torotopes out of 'hose and sock' constructions. You take a cylinder (which is the repeated product of two circle-nets), and then join top to bottom. You can do it by way of a hose, where you bend the cylinder around like a straw or garden-hose (hosepipe) thing. This makes for a bigger section, and the bigger section is nested deeper in the tree.

If you roll the top down, like taking off a sock, it will eventually come to where the toe might be, and an elastic band placed outside the sock will come inside the figure. In essence the sock construction covers the original circle with a new surface. Philip might get the idea through imagining the tire of a bicycle is more-or-less covering the rim.

The two are simply left and right operators, or head and tail (or hose and sock), so ABC can be derived by A, covering its surface with a sock B, and then covering the surface of AB with C. Likewise, you can start at C, make a long prism and bend it into the shape of B, and make a prism of BC and bend it into A.

Tigers

The tiger ((ii)(ii)) is still a comb product, ie it is ((wx)(yz)) = (wx)(yz) [() ()] and like the tri-torus has three degrees of freedom. But the key difference is that no new space is added. Instead, one imagines that we are taking the surface of wx and of yz, and making a hedrix in 4d, which we suppose covers a 3d solid. It really does not matter, because we replace the surface of the 3d solid with a circle [() ()] which is orthogonal to the previous surface, but we imagine that the shared axis is because the ()() stage of the tiger is filled with a chorix radiating from the centre, and that the circle shares this radial element, and one more.

But topologically, a tiger is a figure bounded by the cartesian product of three circles, and is thus topologically similar. It just can't be made of hoses and socks.

Extra i's

Adding an extra i into the mix, ups the circle into a sphere, etc, so eg ((ii)i) is a torus, but ((iii)i) turns the circular rim of a torus into a sphere, while ((ii)ii) = (ii) (oii) [where o is a set of brackets reduced to a letter], keeps the circular rim of the wheel, but turns the cross-section of the tube into a sphere.

Removing i's

It probably goes with some hint to explain that a 1d circle is a straight line, and the 1d torus is then a pair of straight lines separated by a distance, eg o------------o => o-----o o------o. In sections of larger toruses, such gives rise to variously, repeated copies or nested shells (the section of a torus ((xy)z) is a pair of circles in the ((x)z) plane, or an analus in the ((xy)) plane.
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### Re: A small consistency issue

The cartesian product inherent in the tiger, eg [{wx}(yz)] really has three degrees of freedom, because we can construct a circle of diameter [] in the xy plane, at the point x={} y=(), r = []. The tiger is then rotated in wx and yz spaces. In maths-speak, one would suppose that x as {} is replaced by x, \alpha, and y by y, \beta (by rotation), and then into wx, and yz by replacing circular with rectangular products.

I'm not sure if calling iii a cube is a really good idea here, and i don't see how to make one of it. What would be better is to use (ii) and (iii) as circle things (circle, sphere, etc), and then introduce the 'o' axis, which we replace by a sphere with a shared axis. So, eg the torus is essentially a circle (ii), * a circle (oi), where the (ii) is the rim, and (oi) is the cross-section of the tube, the o opens to () and swallows the rim, ie

(ii). (oi) = ((ii)i).
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### Re: A small consistency issue

Well, the issue is, I think, what exactly counts as a 1D sphere.

In 2D or higher, the term "sphere" can be used interchangeably for either surface or volume. And all closed toratopes are similar: they have a single curved surface that encompasses a volume, and so the same term can designate both.

But a 1D sphere can be either a line segment or just the two points. The shape denoted by (I) is clearly just the two points since its inclusion in toratopic notation causes the figure to fall apart in two separated figures (like in ((I)(I)) which is four circles in vertices of rectangle). But just I, outside any brackets, seems to be a full line so it could satisfy open toratope relationships like having (II)I be a cylinder, while (II)(I) is just two parallel circles.
Basically, my cut algorithm is ambiguous when you're trying to cut a circle (II). Sometimes you get two points (I) and sometimes you get a line segment I. It seems that the first interpretation is used when the circle has another pair of brackets outside of it and the second when it's the outermost pair, so ((I)I) cut of torus ((II)I) is two separated circles while cut of cylinder (II)I is II, a full rectangle. As for the circle itself, it seems to depend on what you want to do with it.

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.

Various rectangles and cuboids stem from the fact that you can multiply various 1D elements together, but a "round" shape will only result when you include at least one 2D or higher.
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### Re: A small consistency issue

I feel rotatopes encompass n-cubes, products of n-balls and/or n-cubes, and toric versions of them. In 5D, we get tori with rotatope crosscuts, like ((II)I)(II) a duocylindric torus, aka cyltorinder. 6D has things like ((II)I)((II)I) as a small duocylinder smeared over large duocylinder margin.We could probably define them as unit sized shapes for all.
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### Re: A small consistency issue

ICN5D wrote:I feel rotatopes encompass n-cubes, products of n-balls and/or n-cubes, and toric versions of them. In 5D, we get tori with rotatope crosscuts, like ((II)I)(II) a duocylindric torus, aka cyltorinder. 6D has things like ((II)I)((II)I) as a small duocylinder smeared over large duocylinder margin.We could probably define them as unit sized shapes for all.

But the toratopes generally don't exist as "unit sized shapes" since they require hierarchy of diameter sizes.
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### Re: A small consistency issue

Good point. Then, maybe just unit solids, combined with toratopic revolutions? Can a unit-distance non intersecting rotation be defined, based off a lower diameter?
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### Re: A small consistency issue

For diameter sizes, I found the next tier down usually follows a 1/2 ratio, to remain non self intersecting. If the minor is set to 1, the next tire up should be in the range of 2, then 4, then 8, etc as a general rule. But, that was already detailed I believe.
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### Re: A small consistency issue

About the 1D sphere, I believe the S^1 is just the 1D edge of a disc. A B^1 , one-ball is a straight line. Whereas a B^2 is the 2-ball , a solid disc with an S^1 for an edge.
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### Re: A small consistency issue

ICN5D wrote:About the 1D sphere, I believe the S^1 is just the 1D edge of a disc. A B^1 , one-ball is a straight line. Whereas a B^2 is the 2-ball , a solid disc with an S^1 for an edge.

Yes, but the toratopic notation doesn't really distinguish these
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### Re: A small consistency issue

No, not in a direct way. Circle by itself (II) can be either S1 or B2. But, products in open toratope term are solid, (II)(II) as B2*B2, (II)(II)(II) is B2*B2*B2. Spherated close term makes ((II)(II)) as S1xC2, also equal to my expression S1x[S1*S1] for spherated product. So, it seems that multi term products are solid when open, hollow when closed.
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### Re: A small consistency issue

ICN5D wrote:No, not in a direct way. Circle by itself (II) can be either S1 or B2. But, products in open toratope term are solid, (II)(II) as B2*B2, (II)(II)(II) is B2*B2*B2. Spherated close term makes ((II)(II)) as S1xC2, also equal to my expression S1x[S1*S1] for spherated product. So, it seems that multi term products are solid when open, hollow when closed.

Which seems somehow arbitrary.
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### Re: A small consistency issue

Yeah, its probably not the most important distinction. But it does help when figuring how a margin gets spherated, by placing an open toratope within a closed. By removing a dimension per parameter, we can stick an open in their places to spherate its margin, even with a higher closed toratope. Like using a small shape torisphere ((III)I) , and spherating a large shape cyltorinder margin ((II)I)(II). We use ((iiI)I) and inflate a ((I))(I) to get ((((II)I)(II)I)I), which cuts to ((((I))(I)I)I) , a 4x2 array of small torispheres, as embedded in the 3-frame of a large cyltorinder.
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### Re: A small consistency issue

But just I, outside any brackets, seems to be a full line so it could satisfy open toratope relationships like having (II)I be a cylinder, while (II)(I) is just two parallel circles.

Well, if the idea that open toratopes are solids, then we need a different interpretation of the cut algorithm. Considering solid shapes, a (II)(I) is a cylinder with circular height symmetry. The (I) at the end describes the evolution of moving out, as a cylinder that collapses to a circle, then vanishes. This holds up well with ((II)(I)) , the spherated margin of a cylinder with circular height, the tiger dance! The two tori in the column correspond exactly as the edges of the cylinder.

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.

• The terms (n) makes shapes hollow, with a set of points equidistant from the margin

• A ' I ' is a solid line , (I) is a hollow line as two points. The line has no margin, by default

• For all closed toratopes, using (I) stacks the whole toratope by 2 in that dimension.

• For all open toratopes, using (I) is a product with a line, with circular symmetry. Meaning, moving out will collapse that line-product to zero.

Take cylinder (II)I:

(Ii)I is square with circular height, collapses to line
(II)i is circle of constant size

Take spherinder (III)I:

(IIi)I is cylinder with circular diameter, collapses to line
(III)i is sphere of constant size

Cubinder (II)II :

(Ii)II is cube with circular height, collapses to square
(II)Ii is cylinder of constant size

Duocylindric Torus ((II)I)(II), which uses BOTH rules, as having both open and closed products:

((Ii)I)(II) is 2 duocylinders side by side, merges together
((II)i)(II) is torinder, from cutting torus in half different way, leaving the main diameter untouched
((II)I)(Ii) is torinder with circular height, collapses to torus
((Ii)i)(II) is two displaced cylinders laying on rolling side, with circular height
((II)i)(Ii) is square torus, having cut both circle parts of duocylinder, leaving main diameter of torus intact
((Ii)I)(Ii) is two displaced cylinders standing up, with circular height, can collapse height or merge in either direction of cut array

Of course, the interpretation with open notation changes majorly, but has its own rewards. I don't think a square torus has yet been discovered to exist in the notation, but there it is! You may find it easier to go back and forth with the closed cuts, to see how open cuts work. The main thing is when getting 'concentric brackets' , it means a major diameter was left alone, and only the minor-shape was sliced.

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### Re: A small consistency issue

On another note, I also figured out that if a ((II))(I) is a square torus, then making another cut will result in ((II)) an annulus, or gasket, then ((I))(I) , a hollow tube. But, of course all this overlapping notation will just cause endless confusion. Maybe it should just be in [] ? So, [[II]][I] is square torus, which cuts to [[II]] for gasket, and [[I]][I] for hollow tube. Not sure how I feel about that. Could just be trivial.
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### Re: A small consistency issue

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?)

This should be included in the conditions for the spheration product. A#B is admissible if and only if A has a continuous tangent space and normal space at every point and B is a sphere of some dimension. There are probably some equivalent definitions.

There was a thread where we had a pretty effective notation for "frames" and "cells" of shapes. I'd prefer to stick to that rather than treat objects like "a pair of circles", "a tube with two disk caps" and "a solid cylinder" as separate and unrelated objects. They are the 1-frame, 2-frame and 3-frame of the same shape. So for that reason I don't like this 'I' v.s. '(I)' notation.

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.

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### Re: A small consistency issue

Actually, the 1-sphere being a point-pair is exactly the sort of thing i use to read some of these piccies that Philip et al have been feeding us.

1-spheres exist with holes, or they can present as concentric sets, rather like the torus gives variously

((ii)) = two circles
((i)i) = annalus or hollow circle.

Putting more brackets between the two i's gives a power of 2 of rings, and brackets around the i gives copies in line.
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### Re: A small consistency issue

wendy wrote:Actually, the 1-sphere being a point-pair is exactly the sort of thing i use to read some of these piccies that Philip et al have been feeding us.

1-spheres exist with holes, or they can present as concentric sets, rather like the torus gives variously

((ii)) = two circles
((i)i) = annalus or hollow circle.

Putting more brackets between the two i's gives a power of 2 of rings, and brackets around the i gives copies in line.

Shouldn't it be the other way around? ((II)) is hollow circle and ((I)I) is two circles.
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### Re: A small consistency issue

Marek wrote:Shouldn't it be the other way around? ((II)) is hollow circle and ((I)I) is two circles.

I think that's what wendy meant

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.

I've thought some more, about cutting solids. It could be just as easy as using [] around a normal notation sequence for opens. The [] has a rigid, " solid" feeling, as filled in. Whereas, using () around an open notation makes hollow, spherated hypertori. This makes original open ((II)I)(II) just a margin, which we cut down to ((I))(I) , a 4x2 array of points. But, if we use [] around it, now we have a solid open toratope [((II)I)(II)] , as compared to (((II)I)(II)) for a spherated margin.

This allows [((I))(I)] to be two displaced squares, and (((I))(I)) to be a spherated margin ( the corners) of those square cuts, making a 4x2 array of circles! I like this, it actually works and helps distinguish apart without getting into too much additional brackets. Ultimately, it's just the closing brackets that differentiate them apart, all else within is the same. And, in using this, we can describe the margin itself, which is quite useful. How about it?
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### Re: A small consistency issue

I've thought some more, about cutting solids. It could be just as easy as using [] around a normal notation sequence for opens. The [] has a rigid, " solid" feeling, as filled in. Whereas, using () around an open notation makes hollow, spherated hypertori. This makes original open ((II)I)(II) just a margin, which we cut down to ((I))(I) , a 4x2 array of points. But, if we use [] around it, now we have a solid open toratope [((II)I)(II)] , as compared to (((II)I)(II)) for a spherated margin.

That seems reasonable, as long as there aren't any square brackets on the inside.

Do we really need the notation to cover arbitrary arrays of things? Arrays of known objects are not very interesting. It's one thing to include cubes and cylinders, but a 4x2 array of points in the plane? This would lead to an infinite number of shapes in the plane, but it would still be only a very small class of shapes (you can't talk about triangles for example). I get that it's useful for describing cuts, but as a way of describing shapes it will just bore everyone.

Also, isn't using the notation this way a bit redundant? You say ((II)) is two circles, but that's just the 1-frame of the cylinder 21.

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### Re: A small consistency issue

Well, I think it's more of an artifact of expanding the notation. Originally, it was a cut of the open toratope solid. But, by setting the two apart with closing brackets, both open and closed can be shown how they relate, by lack of brakets. The 4x2 array of points ((I))(I) is the common ingredient between 2 squares in a row [((I))(I)] and a 4x2 array of circles (((I))(I)). It's all a matter of how the shape was filled in from the margin, it seems.

So, if the 3-frame of [((II)I)(II)] cuts down to ((I))(I), it makes the equation:

(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 = 0

And, if we spherate with () , we get (((II)I)(II)), making the equation:

(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 - R3^2 = 0

or, if we fill in edges to make a hypersolid, it becomes some equation I haven't learned yet. Is there an implicit form for a duocylinder that closes off the shape? I think I saw something like an absolute value around a dimension or number for prisms.

Would it be

(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 - abs(R3) = 0 , or something like that?

It probably isn't necessary to go into that much detail in the large variety of things that can come from the notation. But, maybe a brief introduction into the relation between the open and closed? The use of a 4x2 array of points is mostly for exploring the theory, and how it ties together both. The cut algorithm can be used across all three classes of open, closed, and margins. If speaking in the terms of vector bundles,

((II)I)(II) - clifford torus bundle over the circle
(((II)I)(II)) - tiger bundle over the circle
[((II)I)(II)] - duocylinder bundle over the circle

Where, the clifford torus is the margin of the duocylinder, which spherates to a tiger, while bundled over a circle.

Also, isn't using the notation this way a bit redundant? You say ((II)) is two circles, but that's just the 1-frame of the cylinder 21.

I don't think it's possible to represent the 1-frame in the notation. ((II)) should be the cut of a torus, making 2 concentric circles. But, if using the brackets to fill in a solid shape, then (II)(I) could be the 1-frame, as a column of two 1D disk edges. This makes,

(I) - two points
(II) - 1D disk edge, S^1
(II)(I) - vertical column of two 1D disk edges
[(II)(I)] - cylinder with circular height, cut of duocylinder, filling in the edges
((II)(I)) - column of two tori, cut of tiger, spherating the 2 disks

The open toratopes haven't been explored very much until now, really. I wasn't too sure about how to interpret the cuts for a while, since it overlapped with closed. I knew that both were closely related to the common ingredient as the margin. But, the expansion of defining them with closing brackets should do the trick, and allow a third member ( the margin ) to join the two. But, it also creates two more cut interpretations, for the opens and margins. No big deal, I can do those lists. The particulars are pretty straightforward.
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### Re: A small consistency issue

I'm trying to match up these new shapes with the spheration and Cartesian products.

S^0 x S^0 = II = square
S^0 # S^0 = ((I)) = four points in a line
S^1 x S^0 = (II)I = cylinder
S^1 # S^0 = ((II)) = annulus

All of these have various frames available. I'll try to do a table with some images later.

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### Re: A small consistency issue

Looks right to me! It also sheds light on the spheration vs cartesian you're using. It's very similar to the bundling term, with some additions.
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### Re: A small consistency issue

I just made this grid in Mathematica 10. It contains a lot of the shapes up to 3D with all their frames. It doesn't include everything because you can have an arbitrary number of brackets. I'm also not sure if there's any natural order to put them in.

https://dl.dropboxusercontent.com/u/12660336/Maths/Toratopes/Toratope%20grid.pdf

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### Re: A small consistency issue

We need a name for these shapes that include annuluses and arrays of objects. How about annuloids?

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### Re: A small consistency issue

The real issue in here is that people are seeing a repetition-product, and poking in a cartesian product. There is no cartesian product. Something like III is not resolvable in this sense.

A repetition product produces for each point in A and each point in B, a point in AB. That is, there is a full copy of A for each point in B, and vice verca. The difference here is that the cartesian product applies to all-space, including the content, whereas the comb product only produces a surface of AB that is the repeated-product of surfaces of A and B.

A product is an operation that reduces to an algebraic product P(A×B) = P(A)P(B), would make whatever is represented by × into a product. Here P() represents the Euler Characteristic of the surface (ie the surtope equation, like 6x^2 + 12x^1 + 8x^0 for the cube.). The difference is that the prism or cartesian product adds in a content term, 1x^3+6x^2+12x^1+8x^0 = (x^1+2x^0)^3, where the comb product does not.

Instead, the polygon p looks like px^1+px^0, and the torus formed by p×q is pqx^2+2pqx^1+pqx^0. This property differs from the cartesian product in that we do not add a term x^2 for the interior of the polygon: instead, we would restore the polyhedral nature by adding an x^3 to the finished product.

Such a polyhedron could be made from the cartesian product of the nets (a net folds up to be a surface). One takes a p×q rectangle, folds it up into a tube q round, and then connects the ends of the tube into a loop p round.

One can relax the condition of flat-surface, and still let the product work. A polygon thus includes a circle, and a polyhedron includes any three-dimensional solid, whose surface might become polygons of various kind. A cartesian product of polygons gives a polychoron or 4-solid, while a comb-product of polygons (bounded by 1-fabric), gives a polyhedron (bounded by 2-fabric).

The comb product looses a dimension on every application, and it is the loss of extra dimensions that makes the large variety seen in the higher dimensions. If you look at the cross-section of a torus, you will see two circles, separated by a distance. The planes of the two circles contain a shared axis, and it is this shared axis is "lost" to the product.

[b]A Wheel and a ring [/b]

The 'z' axis represents "up". The 'y' axis represents "forward". So the notional wheel in any dimension, has a rotation in the up and forward space. This is the large axis of the wheel, represented by a circle, say [zy]. There is a smaller section represented by the tyre. This has a radial component in the zy plane, and an "across" section in the x plane, so the section of the tyre is (ox). We can make [] a diameter of 26 inches, and () a diameter of 2 inches. The brackets are parametric, and meant to be identical in style, the different forms are simply to refer to different circles.

The wheel, is then the comb product of the rim circle [] and the section circle (). One could draw red lines parallel to [] and blue lines around each section, and each red circle would cross every blue circle, just once, and vice versa. The wheel is then (zy)(ox), say.

A ring is intended to go around a (linear) finger, so we might suppose that it is constraining the z-axis in the plane z=0. So we need a large circle to completely surround a point in z=0 (ie an n-1 sphere), and take the comb product of that with a circular section in the plane (oz). So we get a figure [yx..](oz).

In four dimensions, we add an extra axis 'w'. Of the wheel, we do not want the cabin of the vehicle to be rotating, so we want everything other than [zy] to be fixed. So the extra term is (oxw), which gives the wheel a spherical cross-section, and the large shape is round.

For the ring, the concern is now for the point escaping a closed surface in the yxw plane. So we need a sphere here [yxw] and (oz) remains the same. We have produced two topologically different figures, whose surface is identical in shape.

One can make a tri-torus, by supposing that the centre of the ring is actually to be contained in a torus in the zyx space, and the finger is pointing in the 'w' direction. This gives rise to a torus say [zy](ox), and we wrap the surface up in the space ow, to get a tritorus [zy](ox){ow}. Because this is the repetition-product of three circle-surfaces, it represents a cyclic portion of the cubic space, and is called thus a 'tri-torus'.

This turns up more often than not in sections, so it is best to discuss it. A one-dimensional polytope is a line-section. A one-dimensional circle is the space between +r and -r, say (z).

When we multiply two one-dimensional figures, with an euler characteristic of (2x^0). we get eg (4x^0), or two line-segments. It is the repeated application of this that gives rise to ever-increasing powers of two.

A simple 1D torus is then [z](o). It is represented by two line segments in z, separated at their centres by [], and the lengths are (). A quad-line is then [z](o){o}, and so forth. It consists variously of a [z](o)'s surface (end-points) used to centre line segments {o}, or a pair of (o){o} 1-torus separated by a length [z].

When we want to convert this into 2D or 3D, we add extra letters y, x, which do not replace the o, but just as extra segments. The effect of adding extra letters means that the linear parameters are seen in the additional axies. For example, a torus [zy](ox), one can see the full size of the wheel and the tyre in the z and in the y axis, but only the thickness of the tyre in the x axis.

In the [zy](o) space, the bi-segment [z](o) is rotated around the y axis, so the imprint on the y axis is [y](o). This is an analus, representing here the full extent of the rim and the linear thickness of the tube.

In the zx space, we see [z](ox). This produces two circles separated by [] in the z-axis, the tube radius is seen in the z and x sections. But x does not see any of the size of the rim, so [](ox) is only () thick.

Linkages of chains in general, follow the solid comb product of two spheres. If A and B are spheres, and the comb product A×B is solid in a given dimension, then a chain might be made of alternating links A×B and B×A. Because the sum of surfaces adds up to N-1. one sees that the dimension of the bodies of the links adds up to N+1, eg circle + circle = 2+2 = 4 in 3d, but for 4d chains, you need 2+3 = 5 to make this link.

[b]Of Hoses and socks[/b]

In the form [zy](ox){ow}... one can suppose that the torus product can be made in two different ways. The way to add a term to the front is to suppose the circle [zy] is actually a cross-section of a long tube, which one joins the end as one might a garden-hose. This happens by supposing a space az, the loop is an even larger circle <az>[oy]... The z is transferred and in the [] section, the z there becomes a radial section from the directions in the az-space.

The sock section adds a tail segment <ov>, smaller than {ow}. The surface of what proceeds is covered by the new circle <ov> which is rolled down as one might take off a sock. An elastic band on the sock, is covered by the sock, and likewise, the surface of ...{ow} is covered by <ov>.

The two process produce identical results. Starting off with {zw}, and repeatedly applying hose-sections, gives (zx){ow} and then [zy](ox){ow}. The sock process applied to [zy] gives [zy](ox) and then [zy](ox){ow}.

[b]Of Tigers[/b]

{One needs to explain the pun involving japanish here. I don't know it outside the existence.}

One can use a polytope in the torus product as the reduction-axis for more than one polygon. That is [zy](xw){oo} makes some sort of sense, since one starts off eg with a circle {yx}, and then hose-fits the z axis to get [zy]{ox}, a torus. But now a second hose function might be made to the height or thickness of a tube, where the letter {x} is transferred to (xw){o} to give eg [zy](xw){oo}.

This is a tri-parametric figure too, but there is no intersection between zy and wx, these can be set to anything as long as they are larger than {}. One can suppose a circle centred at [] and () in the yx space, of radius {}. One rotates this circle in the zy space, to give a torus in the zy plane, and then rotates the result in the xw space.

The notation can get quite complex quite quickly, because one might have difficulty following

The fix is to turn the o's into brackets, and substitute what the o's are radial to. that is eg [zy](xw){oo} gives {[zy](wx}}. There is enormous profit in this, since one can see that the parameters represented by {}, [] and () are visible into an axis, only if the letter is inside that set of braces. Likewise, an inner set of brackets represents a length that is at least large enough to contain all of the brackets that contain it. So we see that [] must be bigger than {}, but not necessarily ().

One looses the simple dimensionality inherent in {oo} = circle, because you have, eg {[....](.....)}. A bracket set represents an additional dimension in the cross-section.

[b]Torotopic Notation[/b]

When one makes these notes, it is possible to reduce all letters to I, and all brackets to (). So the tiger is ((II)(II)). It has three opening brackets, so is tri-parametric. It has four verticals, so is four-dimensional.

[list]
[*]Wheel: [zy](oxw) = ([zy]xw) = ((ii)ii)
[*]Ring: [yxw] (oz) = ([yxw]z) = ((iii)i)
[*]Tritorus: [zy](ox){ow} = {([zy]x)w} = (((ii)i)i)
[*]Tiger: [zx](yw){oo} = {[zx](yw)} = ((ii)(ii))
[/list]
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wendy
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### Re: A small consistency issue

For the annuluses and arrays of objects, these are simply 'dyadic toruses', since they are toric products that involved dyads (or line-segments).

I would not include the cartesian product at all, (ie cylinders etc), since it is not associative with the comb product. The cartesian product always involves surface, and produces a sum of face-numbers, while the comb product never does, and produces a product of face-numbers.

You don't need spheration and cartesian product. The comb product handles both of these well. What happens in the case of the tiger, ((zy)(wx)), one is supposing that the construction must lead through constructing (zy) and (wx), take the cartesian product, and then spherate the surface to get the outer brackets. These are simply devices because the comb product is not commonly known, and the fastest way to construct the tiger is to construct a shell, and flesh it out.

Actually the comb product starts off with (yx), and produces a torus ((zy)x), and then uses the height or tube-thickness to make the second ring, ie ((zy)(xw))

It is also meaningful to add letters into the brackets, without creating brackets. (zy) circle is a cross-section of (zyx) sphere. Adding a letter into a set of brackets adds a dome at that circle (and everything outside of it), that turns the circle into a sphere. So ((zy)x) into ((zy)wx), applies w only to the outer set of brackets, not the inner one.

The transcept across a given axis is found by dropping all letters than what the axis-system calls for. Normally this is evaluated at 0, so if you end up with (), you get nothing. But it is possible to evaluate it at the maximum section, by simply ignoring the blank ().

So, eg the section at xw of ((zy)xw) is (()xw), which tells us there is no tube at the centre of the circle, but the maximum extent of the figure is displaced by (), as (wx) = circle.

So since the use of nested and concurrent brackets is sufficient to make the full range of right-comb products, the notation is necessary and complete. In other words, there is no right torus not constructed by the torotope notation.
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wendy
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### Re: A small consistency issue

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.

PWrong
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### Re: A small consistency issue

The cylinder might be interesting, but it is made by the 'prism' product. It's a circular prism.

What's happening here is that while both the prism and comb products are repetition products, they are implemented differently in geometry, and cartesian product is associated with the prism product, not the comb product.

A graphic, showing the torus as a partly constructed grid of squares, with some full circles and some full rims would help explain. The volume is not involved in the product-set. That's the difference: it is not a cartesian-product as it occurs in geometry. It's a repetition product, where an element of AB exists for each element of A paired with each element of B.

While one might construct the tiger as a 'spherated bi-circular prism', this is more a device to make a tiger work and show its full symmetry. Spheration is a kind of spray-paint you can do to any non-solid thing. The http://www.atomium.be/ is a spherated edge-frame. It's not a kind of product. What the tiger is, in terms of the hose-and-sock analogy, is to make the torus into a cross-section, and use the tube-height to make the next hose.

On the other hand, by way of the product of cartesian-set theory, even the pyramids and octahedra are cartesian products, because all the product needs is something in AB to represent the product of every member of A and every member of B. So really, if you are going to include the cylinder into the mix, why not the cone (where the line connecting a to b is the element ab). The cartesian product is loaded in geometric terms, and it is best to avoid it all together.
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