Clifford torus (opens in new tab)

For anyone wondering about the significance of this, the following is my loose explanation: you probably know that a torus is the surface of a doughnut. Better yet, it can be thought of as a square with edges identified (like the videogame screen when you play asteroids: when you reach the top edge you come out the bottom edge at the corresponding place; same for left and right). The latter representation is "better" than the doughnut in the sense that it's symmetric: the horizontal and vertical directions are treated the same. But it has the downside that it requires "identifications"; corresponding points at the top and the bottom represent the "same" point while points in the middle of the screen are unique.

The doughnut representation satisfies uniqueness everywhere, but isn't symmetric: the circles in the vertical direction are all the same size; but if you consider circles in the other direction, they depend on the angle; for instance circles at the top of the doughnut are bigger than the circle encircling the doughnut hole, and smaller than the circle encircling the entire doughnut. You can visualize how this asymmetry happens by turning the square into a doughnut: start with the square above; make it into a roll so that the left and right sides are identified, and then now bend the roll to identify the top and bottom edges: if you bent towards you, then the backside gets stretched a lot more than the front side on the way to forming the full doughnut.

Clifford torus is a way to satisfy both properties (symmetry and uniqueness) at the same time and it's a theorem that one needs to be in 4 dimensions for this to happen.

I ran across this recently when a 2D based simulator described its world wrapping topology as being a torus. Turns out I took their metaphor a bit too far, assuming they actually considered geometry implications for world wrapping.

Spent awhile digging around until I found a euclidean 2-torus: the Clifford torus which can provide foundations to make the world wrapping metaphor a bit more accurate for flat 2D systems. It's still unknown to me if the authors actually had this in mind when they wrote the description of the wrapping system but it was certainly a fun side trek.

another perfect example of Wikipedia's mathematics pages being unreadable.

Ooo, that footnote: "Flat tori in three-dimensional space and convex integration" https://www.pnas.org/content/109/19/7218

Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration", Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi:10.1073/pnas.1118478109, PMC 3358891, PMID 22523238

> It was a long-held belief that ... no isometric embedding of the square flat torus—a differentiable injective map that preserves distances—could exist into three-dimensional space. In the mid 1950s Nash (1) and Kuiper (2) amazed the world mathematical community by showing that such an embedding actually exists.

> ...

> In this article, we convert convex integration theory into an explicit algorithm. We then provide an implementation leading to images of an embedded square flat torus in three-dimensional space. This visualization has led us in turn to discover a unique geometric structure. This structure, described in the corrugation theorem below, reveals a remarkable property: The normal vector exhibits a fractal behavior.

Topology never fails to boggle up my mind.

Does this have any practical application?