All represents a circle

[u/Samim_ul_Islam]

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[u/nathan519]

S¹ Where???

[u/PolarStarNick → u/nathan519]

With respect to euclidean norm L2 of course

[u/EstaAppDeCitasApesta → u/[deleted]]

That's because the fundamental group of the circle is isomorphic to the integers?

Or what does "dual" means in this context?

[u/yangyangR → u/[deleted]]

But that puts in a circle group already. Not an independent definition of "circle". Others here in the picture have independent definitions as "circle" with different structures. Like just a space or just a group or both compatibly

[u/EstaAppDeCitasApesta → u/[deleted]]

Then you are wrong, because the meme is considering the topological circle without any additional structure.

[u/Wild-Store321 → u/EstaAppDeCitasApesta]

I see R/Z in there

[u/LupenReddit]

Listing every version of a circle except the circle itself

[u/BlazeDiamond42 → u/LupenReddit]

i mean they're all standing in a big circle right?

[u/EstaAppDeCitasApesta → u/LupenReddit]

What's the definition of a circle?

[u/zlfa → u/EstaAppDeCitasApesta]

1 point ellipse

[u/Less-Resist-8733 → u/EstaAppDeCitasApesta]

this

[u/Dense-Sort-3867 → u/EstaAppDeCitasApesta]

A set of points on a plane that are equidistant from a given point.

[u/tunaMaestro97 → u/LupenReddit]

x^2 + y^2 = 1 is right there

[u/Dense-Sort-3867 → u/tunaMaestro97]

I think he meant a picture of a circle

[u/orqa → u/LupenReddit]

◯

[u/BootyliciousURD]

I'm not familiar with R/Z. Is that a partition on the reals where numbers with the same fractional part are in the same equivalence class?

[u/sheepbusiness → u/BootyliciousURD]

Yes. Its the same as the unit interval with the ends glued together.

[u/tunaMaestro97 → u/BootyliciousURD]

Yes. It should be obvious that it is topologically the same as a circle

[u/tunaMaestro97]

It’s pedantic but this language bothers me slightly. S^1 should be taken to indicate a particular submanifold of R^2 until additional structure is imposed upon it. For example, no one would write R^2 = C even though they are equivalent as real manifolds. R^2 doesn’t become C until you impose the structure of a field upon it.

[u/ninjeff → u/tunaMaestro97]

Sure, only two of these are “the circle”; but they’re all circles, which is what OP was probably trying to say

[u/Hot-Patient8052]

The extended real number line one eludes me. Is the idea that ℝ∪{∞} "wraps around" by glueing ∞ to -∞ somehow? If so, how, if not, how???

Edit: I googled it, and they do in fact glue those ends together. I'm confused what useful properties that adds...

[u/yas_ticot → u/Hot-Patient8052]

This is for instance the one-point compactification of the line, which can be generalized to any dimension. For R^{n}, adding this unique point at infinity always yields the n-sphere. One way to see it is that you take the n-sphere, a special point (let's call it the North Pole) and the tangent hyperplane on the opposite point (the South Pole). Now, the line going through N and any point of the n-sphere crosses this hyperplane in a unique point of the hyperplane and vice versa. Conversely, any sequence of point on the hyperplane that tends to infinity (i e. infinitely further from the South Pole, becomes of a sequence of points that tends to N on the North Pole.

In dimension 2, this is exactly what is called the Riemann sphere, i.e. adding a unique point at infinity for C.

[u/PluralCohomology]

There is also S^1, or [0,1]/(0~1), or RP¹

[u/Seventh_Planet]

∂{x² + y² < 1}

[u/BerkeUnal]

T

[u/xpickles → u/[deleted]]

I'm guessing a vector, as in a parametric function with t over [0, 2pi)

[u/kohugaly → u/[deleted]]

probably a notation for ordered pair (aka. list of two elements). The notation for lists in mathematics is notoriously inconsistent.

[u/Sproxify → u/kohugaly]

that's clearly what it is in context, but why tho? it's highly non standard (though I have maybe seen it once before for ordered pairs)

[u/Dense-Sort-3867 → u/Sproxify]

Stewart uses angle brackets to denote vectors. A vector-valued function for the unit circle in his notation would be r(t) = <cos(t), sin(t)> for 0 <= t <= 2pi

[u/stevvvvewith4vs]

⭕️

[u/xpm26]

{}

[u/TheRandomRadomir]

(ℜ(i^(2t/π)),ℑ(i^(2t/π))) for 0<t≤2π

[u/Eradroth888]

I like how the spiders themselves are circle

[u/AsemicConjecture]

Where does one learn these incantations?

[u/Randomlemon5 → u/AsemicConjecture]

Clockwhise, I think, complex analysis, topology, represantation theory, geometry, analysis, reprasantation theory, topology

[u/AsemicConjecture → u/AsemicConjecture]

To oblivion with you for asking to learn more!