Math I Think Is Pretty, Fun, or Pretty Fun

Best equation, according to John Lund: $\displaystyle\frac{e^{i\pi}+1}{2} = 0$

An implicit equation for 3D plotting a torus: $\displaystyle\left(\sqrt{x^2+y^2}-R\right)^2+z^2 = r^2$ where $R$ is the radius from the donut hole to the middle of the dough, and $r$ is the radius from the middle of the dough to the frosting.

An implicit equation for 3D plotting an $n$ torus: $\displaystyle\left(\left(\prod_1^n (x-(i-1))(x-i)\right)+y^2\right)^2+z^2 = r^2$ where $r$ is small. Credit to user856.

A theorem I didn't prove

Different Ways of Looking at a Circle

(Some math peoples') Definitions of the circle, $\mathbb{S}^1$:

$\{(x,y)\in \mathbb{R}^2\mid x^2+y^2=1\}$,
or $\{z\in \mathbb{C}\mid |z|=1\} = \{e^{i\theta}\mid \theta \in \mathbb{R}\}$,
or $\mathbb{R}/\mathbb{Z}$,
or $[0,1]/0\sim 1$,
or $\mathbb{RP}^1$, or $U(1)$, or $SO(2)$,
or the coequalizer of the diagram $\Delta[0]\rightrightarrows\Delta[1]$,
or is a $K(\mathbb{Z},1)$,
or $\mathsf{Gr}_1(\mathbb{R}^2)$
or this $\to$${\Huge \circ}$.

An example of a shape with constant diameter that is not a (usual) circle:

(Reuleaux) Curve of constant width.

Spectra Spectra Spectra

Some people write $\mathbb{S}^1$ to mean the suspension of the sphere spectrum, $\mathbb{S}^0.$

Note that spectrum (or other variations of this word) could mean spectrum, spectrum, spectrum, or spectr...al sequence.

Here are all of Wikipedia's versions of the word spectrum.

While spectral sequences may put some people to sleep, you could read this picture book by Ravi Vakil if you want to get excited about spectral sequences.

The association of LGBT mathematicians operate under the name Spectra.

Topologists Want a Different Job

These are all real things. I'm not kidding.

Knots
Plumbing
Surgery
Soldering
Tractor Bundle (this one is actually conformal geometry)

Miscellaneous Pretty math from the internet

Media Recommendations

Media about math/work philosophy that made me think hard. I do think they are thought-provoking. They are in no particular order.

Advice I've read on how to do research. Maybe I have taken it, maybe I haven't. It depends.

YouTube Recommendations

If you like math-y,riddle-y problems, I recommend these YouTube channels

If you like people who are very good at talking about math, I recommend these YouTube Channels:

Richard E. BORCHERDS: Lectures on important graduate-level math.
3Blue1Brown: Great explainer and math visualizer.
Aleph0
Kinertia

These YouTubers don't talk about specifically math, but they are in the same sector of YouTube:

Critiques of how math is done. Interesting.

Rebellion against traditional exposition.

Scotty Tilton