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I was just wondering how a "cube" would look in 4-D. I know that in 1-D it is a line, in 2-D it is a square, in 3-D it is a cube.

Is it possible to envision it? If it is, how would the axes be defined? (i.e: 3-D as the x,y, and z axes)

P.S.: Not sure what tag this would go under. Geometry maybe?

Sujaan Kunalan
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4 Answers4

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Generally speaking, I imagine the fourth-dimension as the third-dimension with a grey-scale. Given a point $(x,y,z,w)$, first find $(x,y,z)$ in the third-dimension. Next, think about $w$ as looking whitish if it's very small (i.e., approaches white as $w \rightarrow -\infty$) and black if it's very large (i.e., approaches black as $w \rightarrow \infty$).

For example, the point $(0,0,0,0)$ would be at the origin in three-space and be a middling grey. Comparatively, the point $(0,0,0,37912739217)$ would be at the origin in three-space and be a much darker shade of grey.

I will leave it to you to think about what the four-dimension unit hypercube would look like with this grey-scale model underlying it.

Benjamin Dickman
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    How would you represent the two points (x,y,z,v) and (x,y,z,w) in your grey-scale representation? – Johannes Jul 07 '13 at 12:36
  • I fix $(x,y,z)$ in my mind, and then imagine two different colorings at this single point in 3 space: one for $v$ and one for $w$. More generally, I find a point in four-space by navigating to the projection $(x,y,z,w) \mapsto (x,y,z)$ and then "warping" along a grey-scale axis to $w$. – Benjamin Dickman Jul 07 '13 at 12:43
  • I find myself unable to "warp" to a pair of different $w$ values simultaneously, while it is easy to envision a pair of distinct points in $3$-space. This throws off the balance of the dimensions for me. – Karl Kroningfeld Jul 07 '13 at 12:51
  • @user1 Simultaneous warping can certainly throw off one's balance. – Benjamin Dickman Jul 07 '13 at 13:05
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    I`ve never thought of it like that. Very helpful! Thanks. – Sujaan Kunalan Jul 08 '13 at 03:44
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For a novelist's view of the hypercube (4-D cube, or tesseract), I strongly recommend reading

"—And He Built a Crooked House—" by Robert Heinlein,

in which a Californian architect builds a house in the shape of the 3-D unfolding of the hypercube. The full text (pdf) of the short story is available.

enter image description here (Image from Wikipedia.)

Note also that the version of the short story I linked to is part of the curriculum of a Maths course whose goal is to "gain an understanding of the techniques involved in analyzing objects in four dimensions and higher".

Andreas Caranti
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you can draw four axes on your sheet of paper, like you draw three of them when you want to represent a cube. Here are some pictures:

https://en.wikipedia.org/wiki/Hypercube

Emanuele Paolini
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This is the sort of questions that may receive completely different answers under different tags. Since you put a geometry tag to it, it is not surprising to see things like tesseracts and hypercubes come up, which are natural generalizations of the cubes in 3D in the geometric direction (for instance, the combinatoric relation between the number of vertices, faces and bodies, as well as the connectivity when you try to go from one vertices to others).

However, if you ask people doing analysis, they will definitely give a different answer. To them, a cube is a basic unit when you tried to measure things. For instance, in 1D, a cube is just a unit interval, which can be taken to be the unit of length. If you have a line segment, you can just compare it to a unit interval and this gives you its length.

In 2D, the cube becomes the unit square, which is the unit for area. When I was in elementary school, I learnt about the concept of area by counting the number of small squares inside some random shapes.

In 3D, the cube is the unit cube and it is our unit for volume.

So generally, in $d$ dimensional space, a cube is the unit of measurement. This is the basic intuition behind real analysis in Euclidean spaces.

Hui Yu
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