
Clifford torus - elierotenberg
https://en.wikipedia.org/wiki/Clifford_torus
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virtuous_signal
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.

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swiley
Tori in 4 dimensions are just nicer in general.

~~~
virtuous_signal
>Tori in 4 dimensions are just nicer in general.

[https://en.wikipedia.org/wiki/Clifford_torus#Still_more_gene...](https://en.wikipedia.org/wiki/Clifford_torus#Still_more_general_definition_of_Clifford_tori_in_higher_dimensions)

Torus (as in 2-dimensional torus) is nice in R^4, and in general the
n-dimensional torus is nice in R^{2n}

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

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Frost1x
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.

~~~
fanf2
I first encountered the idea of wrapping coordinates being equivalent to a
torus when playing around with the Game of Life. I assumed that it was a
topological equivalence as illustrated by
[https://commons.m.wikimedia.org/wiki/File:Torus_from_rectang...](https://commons.m.wikimedia.org/wiki/File:Torus_from_rectangle.gif)
\- I did not know there could be geometrically flat toruses!

~~~
dunkelheit
Yes, it is common for 2D lattice simulations to use this topology. The
rationale is that boundary effects get smeared across the whole lattice and
thus are much smaller than if they were concentrated near, well, boundary.

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

~~~
fenwick67
Totally agree. I understand geometry better than 99% of the population.
There's no reason this article should be this indecipherable. Meanwhile, I can
read the article on Archaea (having never taken a biology class past high-
school) and pretty well grasp what they are.

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4gotunameagain
Topology never fails to boggle up my mind.

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ur-whale
Does this have any practical application?

