Wikipedia:Reference desk/Archives/Mathematics/2023 July 3

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

July 3

Clifford torus

Where is the hole in the Clifford torus? The 3D projection of the torus rotating on that page clearly shows something with a hole in the middle, but from what I can picture mentally using the equation x²+y²=z²+w²=1/2, I should be able to inscribe a side √2 tesseract in it, just as I can inscribe a side √2 square in each of the circles whose product it is. 93.137.57.104 (talk) 13:44, 3 July 2023 (UTC)

The "hole" in a torus is a property of its embedding in 3-space. You can draw a loop through the hole, and you won't be able to deform the loop so that it no longer goes through the hole. That's kind of what defines it as a hole, at least topologically. The Clifford torus is embedded in 4-space, and there is no such loop. The Clifford torus is a torus in the sense that it's homeomorphic to the familiar torus embedded in 3-space, but you shouldn't confuse the space with it's embedding. Another description of a torus is that it's what you get when you identify opposite sides of a square; in that case there's no embedding at all and no "holes". (Some video games used this. For example in Asteroids if you drove your ship past the edge of the screen it would reappear at the other side.) In general, visualizing geometric figures in 4-space is not for mere mortals. You can do computations and find interesting facts by doing the math, but actually seeing it correctly in your mind is a different story. --RDBury (talk) 14:38, 3 July 2023 (UTC)

PS. According to my calculations the circles used in the product have radius 1/√2 and the inscribed squares have side 1, so the tesseract would have side 1. That's not important here though. --RDBury (talk) 14:51, 3 July 2023 (UTC)

Yeah, you're right about the radius. I figured the Clifford torus is a subset of the points that form a 3-sphere, but forgot that the unit tesseract is circumscribed by a unit 3-sphere and did the calculation in my head and did it wrong. It's just really surprising to me that a 3D projection of a loopless 4D object has a loop. Is there a 3D=>2D analogy to this? Is this a curiosity of 4D=>3D or maybe just because the projection was stereographic? I can think of 4D=>3D/3D=>2D analogies for handedness, self-intersection and such (Klein bottle vs. knots for example), and ofc the opposite situation (a boring old 3D torus viewed from the side), but not for this. 93.137.57.104 (talk) 18:13, 3 July 2023 (UTC)

The set of points $\{({\tfrac {1}{2}}{\sqrt {2}}\cos \theta ,{\tfrac {1}{2}}{\sqrt {2}}\sin \theta ,1,0):\,0\leq \theta <2\pi \}$ forms a loop (in fact, a circle).  --Lambiam 23:22, 3 July 2023 (UTC)

True, but that's not simply connected in 4D either. Thanks tho, it did help me come up with some 3D simply connected objects that can be projected into non-simply connected shapes in 2D, but none are nearly as symmetrical as the Clifford torus. (After thinking some more I also grok how a torus can be developable in 4D :) 93.137.59.16 (talk) 01:07, 4 July 2023 (UTC)

The 16 points $\{(x_{1},x_{2},x_{3},x_{4})\in \mathbb {R} ^{4}:\,|x_{i}|={\tfrac {1}{2}}\}$ lie on your Clifford torus and form the vertices of a tesseract. By an obvious symmetry, this tesseract is also an inscribed tesseract of five other Clifford tori.  --Lambiam 18:03, 3 July 2023 (UTC)

Where is the interior of the Equator? —Tamfang (talk) 15:18, 10 July 2023 (UTC)