Torus

Do you stick the sword, the wand, or the cup into your torus first?

>Mishen discourses the Klein Bottle and Ash of Temperature in Black-body Radiation as it devours the Torus.

youtube.com/watch?v=iA4iKl9UArE

Their videos leave a lot to be desired. Many are misleading and don't really explain things correctly. I was especially disappointed with the quantum field theory videos.

It's defined as the direct product of two circles: [eqn]\mathbb{T}^2:=\mathbb{S}^1 \times mathbb{S}^1[/eqn] and you can equip it with the product topology to give it a topological space structure. Then define a smooth diffeomorphism [eqn]x : \mathbb{T}^2 \rightarrow \mathbb{R}^2[/eqn] and you have the structure needed for a two dimensional smooth manifold.

From there on, one could define the generalized n-Torus as
[eqn]\mathbb{T}^2:= \mathbb{S}^1 \times \cdots \times mathbb{S}^1 [/eqn]

Also, other than being a smooth manifold, the Torus has some very interesting algebraic properties.

you messed up! dummy! idiot!

Here you go
[eqn]\mathbb{S}^1[/eqn]

>the Torus has some very interesting algebraic properties.
Please expand on this. How can a shape have algebraic properties, let alone interesting ones?

there are lots of shapes with interesting algebraic properties
example:
>elliptic curves

Another way of defining it, is through the quotient of the Euclidean plane by the pairs of integers: [eqn]\mathbb{T}^2 := \mathbb{R}^2 / \mathbb{Z}^2[/eqn]

Now here, you can view [math]\mathbb{R}^2[/math] and \mathbb{Z}^2[/math] as abelian groups, so you have a quotient topological space, with strict algebraic structure.

It means yo AC is working.

I recently had to calculate the volume of a torus
Felt pretty smart after I did that desu

how tf do you do that

First you have to parameterise it:
A Torus is basically a cicle that moves along a circlular path you can use that to parameterise it

That parametrisation basically allows you to transform a cube into a torus
Then you need to calculate the Determinant of the Jacobian of the parameterisation
With that you can use the transformation rule and integrate it fairly easily

ah yes of course

the pentacle

welcome varg! this thread needs you

...

Have seen this pattern before. It would be cool to learn the mathematical definitions/ implications of it.

It makes up all closed compact 2-dimensional manifolds.
>mfw Dyck's theorem

what Youtube video is this from?