Q: How can we have any idea what a 4D hypercube or any n-D object “looks like”? What is the process of developing a picture of a higher dimensional object?

Physicist: Math.  Math all over.

A picture of a 3D object is a “projection” of that object onto a 2D page.  Projection to an artist means taking a picture or drawing a picture.  To a mathematician it means keeping some dimensions and “pancaking” others.

So when you take a picture the “up/down” and “left/right” dimensions are retained, but the “forward/back” dimension is flattened.  Mathematicians, being clever, have formalized this into a form that is independent of dimension.  That is, you can take an object in any number of dimensions and “project out” any number of dimensions, until it’s something we can picture (3 or fewer dimensions).

Top: An object in 3 dimensions. To see it, cross your eyes by looking “through” the screen until the two images line up. Middle: By “projecting out” the z axis (toward/away) the object is collapsed into two dimensions. This is what cameras do. Bottom: By projecting out the y axis (up/down) the object is collapsed again into 1 dimension. This is akin to what a 2D camera would see, photographing from below.

We’re used to a 3D-to-2D projection (it’s what our eyeballs do).  A 4D-to-2D projection, like in the picture above, would involve 2 “camera/eyeball like” projections, so it’s not as simple as “seeing” a 4D object.

As for knowing what a 4D, 5D, … shape is, we just describe its properties mathematically, and solve.  It’s necessary to use math to describe things that can’t be otherwise pictured or understood directly.  If we had to completely understand modern physics to use it, we’d be up shit creek.  However, by describing things mathematically, and then following the calculations to their conclusions, we can get a lot farther than our puny minds might otherwise allow.

Lines, squares, cubes, hyper-cubes, hyper-hyper-cubes, etc. all follow from each other pretty naturally.  The 4D picture (being 4D) should be difficult to understand.

For example, to describe a hypercube you start with a line (all shapes are lines in 1D).

To go to 2D, you’d slide the line in a new direction (the 2nd dimension) and pick up all the points the line covers.  Now you’ve got a square.

To go to 3D, you’d slide the square in a new direction (the 3rd dimension) and pick up all the points the square covers.  Cube!

To go to 4D, same thing: slide the cube in the new (4th) direction.  The only difference between this and all the previous times is that we can no longer picture the process.  However, mathematically speaking, it’s nothing special.


Answer gravy: This isn’t more of an answer, it’s just an example of how, starting from a pattern in lower dimensions, you can talk about the properties of something in higher dimensions.  In this case, the number of lines, faces, etc. that a hyper-cube will have in more than 3 dimensions.

Define e_N as an N dimensional “surface”.  So, e_0 is a point, e_1 is a line, e_2 is a square,e_3 is a cube, and so on.

Now define e_N(D) as the number of N-dimensional surfaces in a D-dimensional cube.

For example, by looking at the square (picture above) you’ll notice that e_0(2)=4, e_1(2)=4, and e_2(2)=1.  That is, a square (2D cube) has four corners, four edges, and one square.

The “slide, connect, and fill in” technique can be though of like this: when you slide a point it creates a line, when you slide a line it creates a square, when you slide a square it creates a cube, etc.  Also, you find that you’ll have two copies of the original shape (picture above).

So, if you want to figure out how many “square pieces” you have in a D-dimensional cube you’d take the number of squares in a D-1 dimensional cube, double it (2 copies), and then add the number of lines in a D-1 dimensional cube (from sliding).

e_N(D) = 2e_N(D-1) + e_{N-1}(D-1).  Starting with a 0 dimensional cube (a point) you can safely define e_0(0) = 1.

The values of e_N(D) arranged to make the pattern clearer. You can use the pattern to accurately predict what the cube in the next dimension will be like.

It’s neither obvious nor interesting how, but with a little mathing you’ll find that e_N(D) = {D \choose N} 2^{D-N} = \frac{D! \,2^D}{N!(D-N)! \,2^N}, where “!” means factorial.  So, without ever having seen a hypercube, you can confidently talk about its properties!  For example; a hypercube has 8 cubic “faces”, 24 square faces, 32 edges, and 16 corners.

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10 Responses to Q: How can we have any idea what a 4D hypercube or any n-D object “looks like”? What is the process of developing a picture of a higher dimensional object?

  1. Neal says:

    In order to actually do meaningful math in dimensions higher than three, you have to make your peace with being entirely unable to do more than (a) draw crude two- or three-dimensional visualizations or (b) draw vague scribbly conceptual pictures.

    (It’s actually kind of funny to watch topologists “prove”* things. Spaces (even spaces where “dimension” doesn’t even make sense) become boxes, or lines if you’re taking products or suspending. Embeddings become squiggles in blobs. Suspensions? Pointy spheres. Taking a quotient? Scribble in the subspace being identified. It goes on and on.)

    * These things all have precise definitions. The pictures merely suggest the actual proofs, which of course are always left as an exercise.

  2. Johnny says:

    On the hypercube, are the angles also 90 degrees, like in a regular cube and in a square?

    Assuming yes: so there are 4 sides which all are perpendicular to each other?
    Assuming no: what are the angles between the sides? can they be measured?

  3. The Physicist The Physicist says:

    Yes.
    In an N-dimensional cube each corner is the intersection of N mutually perpendicular sides.

  4. lifebiomedguru says:

    The fourth dimension is not at all hard to conceptualize from the drawing. If one is used to seeing three dimensional objects move in any of the three dimensions, and can remember that at time t the 3D object was here, and now at time t+1, the 3D object is now THERE, you basically have it.

    Intuitively conceptualizing the fifth dimension from the first four is the tricky part because we have no intuitive frame of reference. I imagine it would be akin to watching a 3D cube move in one direction, while its 6 shadows on the sides of a larger box containing it from the six coordinate light sources also move, and you would see the lines of the shadow persisting from each opaque line itself to the walls of the cube containing the 3D cube.

    Of course, this is mere intuition.

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  6. FA says:

    hi : i think that the hypercube above is just an other 3D shape , and i believe that in 3d world we can assume how 4D SHAPE will look like , if we take on consedairation other condition , …..
    thanks

  7. some guy says:

    What do You mean by N-Dimensional?

  8. The Physicist The Physicist says:

    “N-dimensional” is just a short hand way of saying “any dimension”. Just replace “N” with any number.

  9. I had the following problem in Dr. Strang’s Linear Algebra book:

    Problem: “How many corners does a cube have in 4 dimensions? How many 3D faces? How many edges? A typical corner is (0,0,1,0). A typical edge goes to (0,1,0,0).”

    I looked at the problem for 60 minutes and only solved the number of corners. Although I have taken much math as a chemical engineer, problems such as this, fundamental problems in math, have always given me great difficulty. I guess that is why I earned a BS in chemical engineering instead of a PhD.

    In the back of Dr. Strang’s book, the answer to the problem says the number of 3D faces is “4 x 2 = 8.” I have no idea where he came up with that solution. From looking at hypercubes on the internet, something I could not visualize myself, I see that a 4D cube is often drawn as 2 3D cubes. Still, I don’t believe that is where the “2″ came from if considering the “4″ coming from 4 dimensions.

    To be honest, I don’t see how you got your faces and edges either. I do know that I would have never come up with your mathematical explanation as an answer to Dr. Strang’s problem.

    I wish I had such skills.

  10. Hello,

    I see some of the patterns now as I look at your well organized data. I have not done math in a while, but I see that my organizational skills have gone far south. Sadly, I now suffer from 1991 Gulf War Illness and schizoaffective disorder and each causes cognitive problems. Stress affects me horribly too.

    As I look at your data, I can see that I would have found the pattern that shows why Dr. Strang gave a solution of “4 x 2 = 8″. Also, I had already figured out the number of corners. I believe I see the pattern that allows one to predict 24, 2D faces as well. I see the “diagonal” patterns in your data organization, but I don’t know if I “see” the patters because I already know the answer.

    It frustrates me to do so poorly on a problem. Like I said, it is likely why I am a BS chemical engineer than a PhD. By the way, I actually did well in my math classes and averaged A’s. I cannot explain it, but problems like this always kicked my butt. I believe it is because I have always had an organizational problem. Also, I now have a problem with concrete thinking, which is a cognitive symptom of schizophrenia.

    Thanks for the education. The only possible pattern that I do not see is the one that produces 32. That is assuming my other recognized patterns are correct.

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