2D 3D 4D 5D 6D 7D
(((II)I)II)
((III)I) (((II)II)I) (((II)(II)I)I)
((II)I) ((II)(II)) (((II)I)(II)) ((((II)I)(II))I)
(II) ((((II)I)(II)I)I)
(III) ((II)II) (((II)(II))I) (((II)I)(II)I)
(((II)I)I) ((II)(II)I) ((((II)I)II)I)
((((II)I)I)I)
I think this is caused by our differing approaches: you see the toratopes from their minor elements (minor element along a major element) while I see them from major elements (major element inflated by minor element). Both are correct, but if you'll read my 7D file, you might get confused by me marking some dimensions and diameters as major while you wouldn't do so.
So, for me, "major" dimensions are those that can be freely increased. Any subordinate dimension cannot be increased freely since the shape would eventually become self-intersecting, but the major dimensions can be arbitrarily large.
ICN5D wrote:All right, so what's the deal, here? I've google searched just about all forms of " multidimensional torus" or "toroid cross section", but I can't find anything like our discussions. No references to A000669, or the implicit functions, or multitangent Villarceau sections. What little there is stops at 4D, like a few youtube vids of the ditorus cut of displaced torii or a computer microchip. There's nothing even close to what this thread turned into, even in the mathematical PDF papers. And, there's a lot of them. Is this some sort of unexplored field of research? Are these GIFs and pics I make something of a novelty? It's actually kind of encouraging. Something new for the world, right?
ICN5D wrote:As in,
S2xS1xC2 = S2x[T2*S1] = S2x[(S1xS1)*S1] = S2x[S1*S1]xS1 = S2xS1x[S1*S1] ?
They're all equivalent. It depends on inflation sequence. This system actually follows my notation for toratopes, strangely enough. But, I wouldn't have known about the commutative property for a while. It happens with higher order Clifford tori, as in the tori or other inflations of the original duocylinder margin. So, for a duoring torus: Circle along circle times circle, [(S1xS1)*S1] equals circle times circle along circle, [S1*S1]xS1 the 3-manifold edge of a cyltorinder, aka duocylinder torus.
Edit:
But, if you mean S1xS2xS3 , then no it will not equal S1xS3xS2 or S2xS1xS3 or S3xS1xS2 or S3xS2xS1. An n-sphere bundle can only commute in and out of clifford manifold products, the edges of inflation.
ICN5D wrote:It seems that the order of S1xS2 shows the toratopic dual relationships, when reversed. Is that what you mean? What exactly does 'topological equivalent' mean?
ICN5D wrote:Okay, I see how that works. Other instances also seem to relate to a commuting of Sn, Tn in and out of a Clifford manifold.
ICN5D wrote:I would agree with that. That's what I like about this system, as it shows up quite well. I'm inclined to enumerate 8D this evening.
ICN5D wrote:Well, actually, now that I've thought about it, wouldn't [S1*S1] be different from S1xS1 ? Both are products of two circles, but they differ in orientation. The [S1*S1] is the duocylinder margin C2 from [B2*B2] ( Bn = n-ball); and S1xS1 is the torus T2. But, I think I remember seeing somewhere that a ditorus T3 and tiger S1xC2 are topological equivalents. Both are a product of three circles, but they differ in orientation. Where if on the surface of a ditorus, wouldn't we see the effects of the medium diameter, which is not present on a tiger? If travelling around the diameters of the 3-manifold, we could trace out three different distances that repeat, in either X, Y, or Z. If measuring the diameters on the 3-manifold of a tiger, we would record two equal large and one small repeating lengths, in X, Y, or Z.
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