Visualizing higher dimensions

The meme is

> attend a string theory conference
> speaker keeps mentioning 10d space
> can't visualize and am completely lost
> mathematician beside me seems to understand
> ask him, how do you visualize 10d space?
> it's easy anon, first picture N-dimensions and then set N = 10 pic.twitter.com/Sg87emdUuY

— ricky (@rickyflowsinyou) July 22, 2023

A mathematician’s reply is

Most mathematicians can visualize N dimensions for N a positive integer, or infinite, and in some cases a positive fraction.

I can do it for N integer but negative. https://t.co/6WnxMZVT6J

— Algebraic Geometer (@BarbaraFantechi) July 23, 2023

What I do is simply follow

• step 0: Picture 1, 2, 3 dimensional vector spaces
• step 1: Write $n$ instead of 1, 2 or 3 (imagine n=1,2 or 3 but don’t write it down, only use n) whenever I am generalizing. Note if we never use the fact that $n=2$ or $3$, i.e. we write $e_1,e_2,...e_n$ all the time etc, pretend we’re working with $n\in \mathbb{N}$ a natural number but visualize only 2 or 3 dimensions then our work is done: we have “visualized dim $=n$”, by simply forgetting $n=2$ or $3$, yay!

• step 2: put $n=$ whatever natural number you like!

That’s what a first semester course on linear algebra is supposed to do initially (after defining the “dimension” of a vector space). This is the algebraic way to visualize things: using symbols!

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There’s a next step for doing functional analysis type stuff:

• step 3: Don’t even think about n being a natural number anymore, it can be infinite.

This doesn’t help much working with infinite dimensional vector spaces, but alright!

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We can generalize “dimension” and “spaces” to things beyond vector spaces: they are called ”manifolds”. There are many “different” manifolds of a fixed dimension even: a circle and $\mathbb{R}$ (real line) are both manifolds of dimension 1, or “1-manifolds”.

2-manifolds are just surfaces: 2-sphere, torus, etc. ${\mathbb{R}}^3$ is a 3-manifold, so is the “solid ball” 3-ball.

i think that genus 1 is mischievous pic.twitter.com/FWjsJYcRyD

— chiara travesset (she/her) (@chairtraveler) February 10, 2023

We can’t even all 3-manifolds like the 3-torus, 3-sphere (the sphere that lives in ${\mathbb{R}}^4$) etc.

But I can visualize the 3-sphere upto removal of just a tiny a point! The joke/reason being 3-sphere minus a point is just ${\mathbb{R}}^3$ or 3-ball (just like how the usual 2-sphere minus a point is just the disk or “2-ball”).

OH GOD WHO GAVE HIM A KNIFE pic.twitter.com/niGRQjRDvs

— chiara travesset (she/her) (@chairtraveler) February 10, 2023