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I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-

An engineer, a physicist, and a mathematician are discussing how to visualise four dimensions:

Engineer: I never really get it

Physicist: Oh it's really easy, just imagine three dimensional space over a time- that adds your fourth dimension.

Mathematician: No, it's way easier than that; just imagine $\mathbb{R}^n$ then set n equal to 4.

Now, if you've ever come across anything manifestly four dimensional (as opposed to 3+1 dimensional) like the linking of 2 spheres, it becomes fairly clear that what the physicist is saying doesn't cut the mustard- or, at least, needs some more elaboration as it stands.

The mathematician's answer is abstruse by the design of the joke but, modulo a few charts and bounding 3-folds, it certainly seems to be the dominant perspective- at least in published papers. The situation brings to mind the old Von Neumann quote about "...you never understand things. You just get used to them", and perhaps that really is the best you can do in this situation.

But one of the principal reasons for my interest in geometry is the additional intuition one gets from being in a space a little like one's own and it would be a shame to lose that so sharply, in the way that the engineer does, in going beyond 3 dimensions.

What I am looking for, from this uncountably wise and better experienced than I community of mathematicians, is a crutch- anything that makes it easier to see, for example, the linking of spheres- be that simple tricks, useful articles or esoteric (but, hopefully, ultimately useful) motivational diagrams: anything to help me be better than the engineer.

Community wiki rules apply- one idea per post etc.

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A tangential requirement viz "Visualising functions with a number of independent variables" is asked here on MO.

– ARi – 2013-08-18T16:40:08.8634@KConrad: your joke reminds me of a musical version I witnessed. Most musicians have at some point learned to play triplets against duplets — one of your hands plays 2 evenly spaced notes per beat, the other hand plays 3. 3-against-4 is also not uncommon, and in ≥C20th music, higher divisions also occur. At this particular dinner, one pianist marvelled that another could play a perfectly even 13-against-14 rhythm; a composer present was surprised at the surprise. “But it’s easy, isn’t it? You just set up one hand playing 13, and the other playing 14, and then you put them together!” – Peter LeFanu Lumsdaine – 2014-03-08T21:05:03.760

15"To deal with a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it." -- Geoff Hinton, man who goes directly to third Bayes. – Bella I. – 2014-10-05T06:48:30.940

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@PeterLeFanuLumsdaine Your comment reminded me of this: https://www.youtube.com/watch?v=jG0eKE1LVgE

– Todd Trimble – 2015-01-06T22:09:32.6973I like the idea of "more neighbours" in higher-D euclidean space. Geoff Hinton joked about being in a grocery store buying pizza. Tomato sauce and cheese were near the pizza-dough, but sardines were not. "Unfortunately it's not at 16-dimensional grocery store", because then everything related to pizza-dough could be next to pizza-dough. So I think of ℤ³ as a graph with each vertex having 6 edges. In ℤⁿ each node has 2n edges. – isomorphismes – 2015-03-22T23:19:42.887

8I've always thought the set of mathematicians are finite, and hence countable... :) – Willie Wong – 2010-05-26T10:35:36.293

35... from which I can only draw the conclusion that at least one mathematician has an uncountable amount of wisdom and experience. Kinda makes me jealous, really... – Vaughn Climenhaga – 2010-05-26T11:54:50.767

56Your joke reminds me of Burt Totaro's algebraic topology class, it must have been in 2001. He drew the standard picture of a 2-sphere on the board and wrote '$S^2$' next to it, but quickly started to discuss n-spheres. Someone put up their hand and asked 'What's an n-sphere?' Burt responded 'Oh it's easy, you just do this', erased the '2' on the board, replaced it by an 'n', and carried on talking. – HJRW – 2010-05-26T19:25:03.873

25Not a great joke?!? That's basically the punchline for the funniest math joke I ever heard (usually they are such groaners). A mathematician and engineer go to a physics talk where the speaker discusses 23-dimensional models for spacetime. Afterwards the mathematician says "that talk was great!" and the engineer is shaking his head and is very confused: "The guy was talking about 23-dimensional spaces. How do you picture that?" "Oh," says the mathematician, "it's very easy. Just picture it in n dimensions and set n = 23." Laughed my a** off when I first heard that. – KConrad – 2010-05-27T00:06:29.913

6The joke is on the mathematician, because as everyone knows, the space(-time) is 26-dimensional. Of course, if you are a hammer... – Victor Protsak – 2010-05-27T08:58:06.820

38There's a great joke that I'll repeat (abridged) from the joke topic: "A mathematician who studied a very abstract topic was annoyed that his peers in more applied fields always made fun of him. One day he saw a sign "talk today on the theory of gears", and said to himself "I'll go to that! what could be more practical than a talk on gears?"

He arrives at the talk, eager to learn applicable knowledge. The speaker goes up to the podium and addresses the crowd: "Welcome to this talk on the theory of gear. Today I will be speaking about gears with an irrational number of teeth." – DoubleJay – 2010-05-28T03:40:39.400

@DoubleJay, I prefer the variant where the lecturer begins "Since the theory of gears with a real number of teeth is well known …". – LSpice – 2017-03-23T21:50:41.370