ICN5D wrote:Probably not imagining them, but cutting them is very mathematical It is still performing calculations using a notation and algorithms. At least that's how I look at it!
As for the bitangent plane:
https://www.youtube.com/watch?v=iYy-26JbIv0
I know you will like this video, if you haven't seen it before.
I'll bet that a torisphere can make Villarso spheres overlapping in the same way.
Marek14 wrote:Here's a question: if we cut a toratope with arbitrary 3D hyperplane, how many distinct topological shapes can we get?
Seeing your mentions of horizontal columns, I think we might need to standardize our terms there. You call them "horizontal" since they are oriented in a direction of the original shape you chose to call "horizontal" -- but you don't explicitly mention this assumption.
You'll also notice that I started to use the general term "array" for arrangement of toratopes. We don't need a new word for every dimension and calling the 4x4x4 array of toruses a "cube" is not technically correct -- sure, we will visualize them like that, but you must remember that torus square actually has eight different diameters and the 64 identical toruses only have two. Where does the remaining six go? Well, they go into specific distances between toruses, in each of the three dimensions there are two diameters that control the separation; while the cubic arrangement is possible, it's far from the only one.
Besides, the reduction to 3D, as you might notice, will only work with toratopes that have at most three "legs" (i.e. lowest levels of parenthesis).
Actually, every species of toratope can be uniquely described by its "footprint", i.e. by its lowest-dimensional arrangement cut. For example, the "tiger squared", a 16-dimensional toratope of form
((((II)(II))((II)(II)))(((II)(II))((II)(II))))
has an 8-dimensional lowest cut
((((I)(I))((I)(I)))(((I)(I))((I)(I))))
which is an arrangement of 256 tiger/tiger tigers (tora tora tora!) (((II)(II))((II)(II))) in 2x2x2x2x2x2x2x2 array.
ICN5D wrote:Marek14 wrote:Here's a question: if we cut a toratope with arbitrary 3D hyperplane, how many distinct topological shapes can we get?
I think that only a sphere or torus will appear. No matter how many dimensions, there must always be three markers. The only way this can be, is inside of many overlapping brackets. If there is an offset of the quantity of brackets around less than the three, we have a torus of some kind, and thus, concentric circles. If there are no extra enclosing brackets within, we have a sphere. I suppose that the minimum number of distinct shapes are the total amount available in the respective dimension.
ICN5D wrote:Oh, my bad. Is that all of them, just the three? Would there be any more?
ICN5D wrote:Yes, the minor radius of the toruses do the cassini ovals, I knew about that one. I remember in some previous post how you said the major radius also would do a cassini oval deforming, but I guess it doesn't. I want to show the merge evolution of the concentric spheres / cocircular toruses next. It's as easy as restricting the bounding box so it cuts the shape in half. I'm taking a stab at 5D tonight.
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