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I introduced the hypercube (to undergraduate students in the U.S.) in the context of generalizations of the Platonic solids, explained its structure, showed it rotating. I mentioned Alicia Stott, who discovered the $6$ regular polytopes in $\mathbb{R}^4$ (discovered after Schläfli). I sense they largely did not grasp what is the hypercube, let alone the other regular polytopes.

I'd appreciate hearing of techniques for getting students to "grok" the fourth dimension.

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    $\begingroup$ dimensions-math.org/Dim_E.htm $\endgroup$ – Steven Gubkin Apr 6 at 1:40
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    $\begingroup$ I think I first learned about hypercubes from Chapter IV in Gamow's One Two Three ... Infinity (obtained near beginning of 6th grade, 1970), and about a year later I came across a derivation like this (more intuitively explained) for obtaining the number of points, edges, faces, etc. bounding an $n$-cube (was in a popular math survey book in our Junior High library, and finally (January 1974) I learned about (continued) $\endgroup$ – Dave L Renfro Apr 6 at 10:59
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    $\begingroup$ Henry Parker Manning's (editor) The Fourth Dimension Simply Explained from a letter I got from Martin Gardiner. I had written him about some of my higher dimensional speculations, such as how people in higher dimensions would be much smarter than us (square-cube law for packing brain cells) and how in a sufficiently high dimension a cube with $1$ inch edges could have a diagonal more than a light year in length, after which I ordered this book from Dover. $\endgroup$ – Dave L Renfro Apr 6 at 11:05
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    $\begingroup$ A letter from Martin Gardiner---Wow! I hope you saved it. $\endgroup$ – Joseph O'Rourke Apr 6 at 11:37
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    $\begingroup$ Yes, and in fact I happened to come across it in January (3 months ago) and I made a .pdf file of it then. $\endgroup$ – Dave L Renfro Apr 6 at 18:00

11 Answers 11

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I strongly recommend the film Dimensions by Jos Leys, Étienne Ghys and Aurélien Alvarez. It's free! The main tools used by the authors to explain the several dimensions are cross sections and steregraphic projections. The animation is very didactic, building the ideas in 2D and 3D as preparation for 4D. There are dubbed versions in Deutsch, American English, Français, Español, Italiano, 日 本語 and Pусский.

Here are the first four chapters of the film:

Dimension 2: https://youtu.be/6cpTEPT5i0A

enter image description here

Dimension 3: https://youtu.be/AhM9JH5GNiI

enter image description here

Dimension 4: https://youtu.be/nz0ku71x22A

enter image description here

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I would just say you have a square in 2D, and a cube is the similar shape in 3D, then what is the next shape in 4D. Then show them the cube in cube view and cross like fold out.

Don't jump to Schlegel diagrams and the rotating pictures on Wiki so fast...they are confusing. Do like I said instead.

I think showing that transition from 2d to 3D, gets them thinking there must be some thing in 4D. Just doing that makes it simpler than talking about all the Platonic solids. Which themselves most people don't know.

Then next show them the Dali painting: https://en.wikipedia.org/wiki/Crucifixion_(Corpus_Hypercubus) Mention that 4d has been intriguing to arts and writers (mention Flatland and how the sphere comes and explains 3D to the plane figure square. And then when the square says, fine, I bet there are higher dimensions than you, the sphere gets mad and runs off.

The point here is not to be having them list axes of symmetry or character groups or the like. It's to get them comfortable that there's some 4d stuff...and it's weird and hard to think about it. But at least they are in the mood to think about it!

I think you are a super nice guy and wicked smart. But too "hard".

Yes, if it becomes important to show all the polytopes than you will need to backtrack and discuss Platonic solids. Unless you played D&D, you don't know them as much. I mean how many non-chemists know there are C3 axes in the dodecahedron? But start with square-cube-tesseract before doing the Platonic solids. (They're harder than you think, even though all 3D.) Square-cube-tesseract is powerfully intuitive.

P.s. I am so tempted to make the Smith/Wellesly/Radcliff proverb a part of this...

P.s.s. https://en.wikipedia.org/wiki/%22%E2%80%94And_He_Built_a_Crooked_House%E2%80%94%22 "Stating that it "was, for many readers, the first introduction to four-dimensional geometry that held any promise of comprehensibility", Carl Sagan in 1978 listed "—And He Built a Crooked House—" as an example of how science fiction "can convey bits and pieces, hints and phrases, of knowledge unknown or inaccessible to the reader"."

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  • $\begingroup$ Thanks for your ideas. I actually showed Dali's Corpus Hypercubus painting. :-) See also this MO posting on unfolding the hypercube. $\endgroup$ – Joseph O'Rourke Apr 6 at 0:54
  • $\begingroup$ Good work. Thanks for putting up with my mischief. ;-) $\endgroup$ – guest Apr 6 at 1:32
  • $\begingroup$ In order to see more of the progression before the great leap of abstraction, no reason not to start with 1D and then 2D and 3D! $\endgroup$ – AviFS Apr 12 at 4:00
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    $\begingroup$ If you are joking, then fine, haha. If you are sort of serious, I would say no. Because the 1d to 2d is not obvious. The whole point is very intuitive visual pedagogy. Not mathiness, not matrices, not character tables. That's why I said to eschew unfamiliar 3d objects (Platonic solids), at least for now, and concentrate on square-cube-tesseract. KISS. (Keep it simple, stupid.) [Moderators, that's not an ad hom, it's a Navy motto, Google it...you'll see.] $\endgroup$ – guest Apr 12 at 12:12
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    $\begingroup$ This is also, why I said to show the "cube in cube" first. Not the unfolded cross. Not the rotating Wiki stuff. That can come a little later. But for now, emphasis should be on motivation, not derivation. Even if imperfect...like looking at the cube in cube and thinking about the actual 3D object shown. It's still a graspable motivation, that is moving the student PSYCHOLOGICALLY (vice rationally) down the road to "what is the fourth dimension" . Teachers need to think about the imperfect humans in the loop and how to "skin that cat"...not just the math logic. $\endgroup$ – guest Apr 12 at 12:42
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My suggestion would be to demystify the concept and try to disassociate it from spatial interpretations at the first approach.

Having $n$ dimensions is just having $n$ variables. As one professor I had liked to repeat, a grocery store owner who is trying to maximize gains by selling oranges, apples, bananas and peaches with some constraint in his storage is attempting to solve a $4$-dimensional problem without even thinking about it. This is much easier to use for motivation. Then, from there, if it is useful/needed/appreciated, one can introduce spatial visualizations like projections into space/plane etc and geometric structures like the inner product etc.

I've found (personally) that relying on "materializing" higher dimensions as an initial approach often backfires, both for the understanding of students (those who get interested frequently veer into philosophical aspects rather than mathematical ones) and their motivation (those who do not get interested will just act as if it is useless and/or nonsensical and/or obscure).

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  • $\begingroup$ Often I find coordinates evil (they are an extra burden that is UNRELATED to the shape). Ie, solving a vector problem one can get a lot more insight by leaving out whatever basis you use for the vector but instead view it as an "arrow" with a direction and length in an N dimensional space and then solve the problem exclusively using dot and cross products. This way of course, you can't think of the vectors as N numbers. Introducing this concept with n variables (aka, number of peers and appels, etc) might lead eventually to stagnation and misunderstanding when you want to go to higher abstract $\endgroup$ – Carlo Wood Apr 7 at 10:33
  • $\begingroup$ I do not disgree with your answer. I just wanted to add that coordinates is a limitation that only might get in the way at some point. One of the main reasons I hate them is because they introduce the number of dimensions to begin with, which only limits you. Without the number of dimensions you can easily talk about higher dimensions ;). The other reason is of course that they force you to chose a basis (which in generally isn't even relevant!) $\endgroup$ – Carlo Wood Apr 7 at 10:36
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I highly recommend "Flatland The Movie." Your institution should be able to purchase it. You can find a free trailer on the internet.

When I was young, I read the book "Flatland: A Romance of Many Dimensions," probably in high school, and it made me "grok" the fourth dimension.

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  • $\begingroup$ The Flatland book was fun, but awfully sexist by modern standards. $\endgroup$ – Ben Crowell Apr 6 at 2:16
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    $\begingroup$ @BenCrowell The flatland book satirises Victorian Era sexism, just like how the book satirises the Victorian class society. You might as well criticise Atwoods Handmaid's Tale for sexism, or any other dystopian story in which bad things happen. The author explicitly defends against this criticism in a foreword to later editions and this point is discussed in The Annotated Flatland. $\endgroup$ – gerrit Apr 6 at 8:54
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That reminds me of my first college programming course where they drew a square picture of a 2D array, a cube for a 3D array, and then said 4D arrays were very hard to understand. But I'd already made 4D arrays that were fine, since they weren't representing points in 4d space. I'd played a computer dungeon game where you had continent, province, dungeon, and floor. 4D. They're not orthogonal to each other, exactly, but the general concept of N dimensions is simple enough, that way.

A questionaire on a bad dating site could be: True/False, you enjoy: Hiking, Cooking, Dancing, Travel. That has the properties of a 4D cube, right? Each set of answers is like a corner, with Hamming distance of a most 4. Adding more questions increases the dimension. The number of corners and edges blow up, but conceptually, a dozen dimensions is simple.

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The best explanation of the general concept I've encountered so far is the introduction to a 4D game, Miegakure.

The idea of extra dimensions is described in the jump from 2D to 3D first, which makes it much easier to visualize and extrapolate.

https://www.youtube.com/watch?v=9yW--eQaA2I

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    $\begingroup$ There's a related interactive demonstration/explanation in the same creator's "4D Toys" for iOS and PC. It has a corresponding Youtube video as well, at youtu.be/0t4aKJuKP0Q $\endgroup$ – Mark S. Apr 10 at 11:38
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For more than four dimensions, I would consider treating each pixel in a gray scale image as a "dimension" and its brightness as the value of the corresponding "coordinate". Then, ℝm x n is just the set of all images (including photos) of m pixels by n pixels.

There is a video on YouTube which explores this approach (and a few geometric interesting aspects of these high dimension spaces): My understanding of the Manifold Hypothesis | Machine learning

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  • $\begingroup$ Ah, in other words, a 6x9 image has 54 independent pixels, which is (p0, p1, ... p53), which is standard vector representation is a 54 dimensional object. $\endgroup$ – Owen Reynolds Apr 7 at 17:50
  • $\begingroup$ ...oops, after a failed edit attempt: but having an NxM image size seems to add a needless complication. 49 dimensions feels better than 47 (a 7x7 image vs. a 1x47 line) and 54 dimensions is now ambiguous as an image. $\endgroup$ – Owen Reynolds Apr 7 at 18:06
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Here's how i understood it, am 18 so correct me if I'm wrong

This site was cool --> https://4dtoys.com/

I think its important to have a physical understanding/visualization of the dimensions.

1D point would be like a bead on a string. 2D shape is like a hockey puck on a hockey table, the plane. 3D shape is something like an apple or a cube in a room.

note how a 3d shape like an apple is made up of many 2d slices/apple crossections, and how a 2d shape is made out of many lines, and that lines are a bunch of points.

A super important part of these dimensions is that when a 1D shape is stuck along one dimension, another shape cannot exist at the same point in that one dimension without overlap. Basically two beads cannot both be at the 1 inch mark, 2 hockey pucks cannot both lay in the same place while still laying flat on their plane, and two people cannot take up the same space.

The only way to have two objects lay in the same x dimensions is to add another dimension. Like put the bead on another string, or stack the puck on top of the other. From a certain perspective, these two objects are now totally overlapping, when in reality they are separate thanks to the additional dimension

But what about 3D objects? Like the apple and the cube?

Simply pick up the cube, and put the apple where the cube was. Now they have taken up the same overlapping space without actually overlapping. The 4th dimension here is time. If you "remove" that dimension then the apple and cube would overlap.

oh, and stuff can disappear if it escapes your observable dimension. like how the apple slices get bigger and smaller 3d stuff can get bigger, smaller, and disappear in this dimension, but really still exist in another.

And in the same way you can slice a cube in different ways to create shapes other than more rectangular prisms, these 3D slices of 4D things can be more than cubes.

Idk, this stuff really is wack. But the idea is there.

I think stuff like the Klein bottle needs this 4th dimension to exist since it overlaps itself in the 3rd dimension, and basically the parts that would've overlapped exist in different dimensions?

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  • $\begingroup$ I like your emphasis on slices: Slicing a hypercube by a 3-flat. $\endgroup$ – Joseph O'Rourke Apr 7 at 23:10
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I would posit it's impossible for most humans to truly grok higher dimensions. I've heard of extremely rare exceptions but I'm going to ignore them here.

Having an intuitive understanding of some fact about geometry requires having a mental image of what is going on. You can't have 4D mental images. Can you mentally "see" a tesseract? Not the common 3D bastardization of it but the real thing?

If anything, trying to represent 4D objects for 3D brains on a 2D medium seems counter-productive. The classical 3D representation of a tesseract is actually misleading. A better way to put it would be to film a cube changing colors over time, or to show 2 cubes while saying that there is a unit distance in the 4th dimension between the "same" vertices of the two cubes.

To me, complete visual representations are just the wrong tool for this. You might sort of get something out of it for the 4th dimension, but it becomes hopeless beyond that.

For inspiration, here's a 3blue1brown video which did an OK job giving intuition about a higher dimension problem, though a completely different one: https://youtu.be/zwAD6dRSVyI

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Use arrays to store the values of a place:

0 dimensions:

a = 1
v = a

1 dimension:

a = [1, 2, 3, ...]
v = a[x]

2 dimensions:

a = 
[
    [1, 2, 3, ...],
    [1, 2, 3, ...],
    [1, 2, 3, ...],
    ...
]
v = a[x][y]

3 dimensions:

a = 
[
    [
        [1, 2, 3, ...],
        [1, 2, 3, ...],
        [1, 2, 3, ...],
        ...
    ],
    [
        [1, 2, 3, ...],
        [1, 2, 3, ...],
        [1, 2, 3, ...],
        ...
    ],
    [
        [1, 2, 3, ...],
        [1, 2, 3, ...],
        [1, 2, 3, ...],
        ...
    ],
    ...
]
v = a[x][y][z]

etc. Note the self-similarity.

When you scale the dimensional axes, the amount of data changes by the power of the number of axes. Doubling a 1D array doubles the data (21 = 2), doubling a 2D array squares the data (22 = 4), doubling a 3D array cubes the data (23 = 8), and so on. To save data, one could simply store the coordinates on those axes, as a dimension is "a measurable extent of a particular kind, such as length, breadth, depth, or height" according to the dictionary.

Other self-similar things are fractals, so named because they have a fractional dimension. Consider a Sierpiński triangle. Doubling its size leads to a data increase of 3, so 2D = 3, so its dimension is log2(3) = 1.58496250072.

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I invented higher dimensions at an early age and have scored "off the charts" in spatial insight on any test. Since I've made numerous "infinite dimensional" puzzles (because 3 dimensional ones were too boring).

This is how I grasp the concept of a hypercube:

Each hyper cube of N dimensions exists of 2^N points. Half of those points (aka 2^(N-1)) form a N-1 dimensional hypercube, as does the other half. There a N ways to pick such a pair *). One such half is a copy of the other, merely translated the side L into a dimension perpendicular to the N-1 dimensions that those hypercubes exist in.

*) Each corner point has (for example) coordinates 0 or 1 for each dimension: each point is represented by a vector like [0,1,1,0,0,1,0,0,0,1] where every permutation of 0 and 1's occur (leading to the 2^N points). Chose any coordinate and separate the points into two groups: one where that coordinate is 0 and one where that coordinate is 1. Hence, N choices. The remaining N-1 coordinates are again a vector of 0 and 1's that contain every permutation; so they are obviously also hypercubes, of one dimension less.

Hence you can "build up" a hypercube from lower dimensions as follows: start with a point. Translate this point over a distance L. Note how it doesn't matter in WHICH direction, even though you have 3 dimensions to pick from (when restricting yourself still to 3D space). The point "draws" a line while being translated, giving you a line piece. The number of points have doubled: from 1 point to 2 points. Now you have a 1D hypercube.

Next translate this line piece (1D hypercube) in any direction perpendicular to the previous used direction (even in 3D space this still allows choice, but which choice you make doesn't matter: all not used dimensions are equivalent), over a distance L. This doubles the points again, and each point draws a line again while being translated (in the end ask the students to find the formula for the number of lines as function of N). Next translate the resulting 2D hypercube (the square) over a distance L perpendicular to the square. This draws four more lines and doubles the number of points from 4 (one square) to 8 (original square plus copy).

Next, translate the 3D hypercube over a distance L in a direction perpendicular to all previously used 3 dimensions. Note that there are infinite dimensions, but which direction you choose is not important, as long as it is per perpendicular to the used dimensions. The result of that is that new lines that are being drawn during the translation of the copy all are perpendicular to the orginal hypercube and thus all make an angle of 90 degrees with every previous drawn line.

And so on: make a copy of the N-dimensional hypercube, translate it over a distance L perpendicular to all previous used dimensions, making all 2^N points draw 2^N extra lines.

Note how every dimension is symmetrical: there are N axis, on each axis there are two opposite N-1 dimensional hypercubes: the "outsides" that limit the hypercube on that dimension (aka there are 2N outsides).

Some students will grasp it. Let them form groups were students that got it explain in their own words to other students how they see it and how they grasped it. It can help to have someone else explain it (in different words).

Here is a puzzle that I made:

Given a hypercube of N dimension in an N dimensional space. If you paint the 2N outsides of the hypercube from a pallet of k colors, how many under rotation different permutations can you make? For example, N=2, k=2 gives: AAAA, AAAB, AABB, ABAB, ABBB and BBBB, so 6 different permutations (rotation of the squares is rotations of the strings here). N=2, k=3 gives 24 different permutations. What is the general formula? Don't look it up cause I have it published on the net somewhere :p

Edit:

More abstract, but certainly important, are the coordinate vectors with all permutations of 0's and 1's. You could explain that if you add more zero's but never change those zero's - then they don't matter. Aka:

0,0,0,0,0,0
0,0,0,1,0,0
0,0,1,0,0,0
0,0,1,1,0,0
0,1,0,0,0,0
0,1,0,1,0,0
0,1,1,0,0,0
0,1,1,1,0,0

spans a 3D cube (in 6D space, but that doesn't matter at all).

Likewise you could keep a coordinate at 1 (or whatever) as long as it doesn't change, it isn't used.

Making a copy then is easy: copy the table and change one of the unused 0's into a 1. Both are 3D cubes as explained before, but they are translated by a distance 0,0,0,0,0,1 (or whatever coordinate you changed), and together now form a 4D hypercube.

Question for the class: what if you correlate the coordinates? Ie, you pick two columns and only use 0,1 or 1,0 and never 0,0 or 1,1. Then that one column counts as 1 bit. This way you can ALSO make 2^N vectors of every "permutation", but using more than N (changing) coordinates (answer: a hyperblock; unless you only use pairs, for example,

0,1,0,1,0,1
0,1,0,1,1,0
0,1,1,0,0,1
0,1,1,0,1,0
1,0,0,1,0,1
1,0,0,1,1,0
1,0,1,0,0,1
1,0,1,0,1,0

is a perfect 3D cube, in 6D space).

EDIT 2

unrelated maybe, but a neat invention of me:

0,0,0,0,0,1
0,0,0,0,1,0
0,0,0,1,0,0
0,0,1,0,0,0
0,1,0,0,0,0
1,0,0,0,0,0

Is an N-dimensional hyper tetrahedron in N+1 dimensions. Isn't it amazing how simple the coordinates become if you add one dimension?! Try to write the coordinates down using only N dimensions :p (if at all possible!).

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  • $\begingroup$ These are good examples, but I'd refrain from calling them your own inventions; your last example is the standard N-simplex: en.wikipedia.org/wiki/Simplex#The_standard_simplex $\endgroup$ – Opal E Jun 29 at 18:25
  • $\begingroup$ I didn't know that. I still invented it myself even when someone else already did before me and I never heard about that :p. I have invented/discovered loads of things that later already turned out to be discovered by someone else before I was born. Great names too. Seems to me that it IS something to be proud of. $\endgroup$ – Carlo Wood Jul 1 at 1:05

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