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So I recently saw Visualizing the 4th dimension. and thought it was a very good question. But I already know how to see a 4D shape in 3D space. However, I was reading through this Wikipedia page and, being Wikipedia, it was written using language beyond my scope.

So could someone please explain how to see a 5D equivalent of a cube?

  • The diagrams in the Wiki article you linked to can be thought of as "shadows" of the corresponding 5D objects. It's completely analogous to the answer by Andrew in the thread you linked to. – J. M. ain't a mathematician May 19 '17 at 06:50
  • Not sure if you can visualize it, but if you think of it as a 5-tuple that can help. So for instance, assume some object is defined by 5 independent properties. Then you can "visualize" a 5d space where each point corresponds to a unique object. So, if the dimensions were -- length, width, height, color (single frequency), and density for example -- then, if all the dimensions were the same but the color dimension was slightly different, you'd have two boxes of same dimensions and density but slightly different colors. – SilverSlash May 19 '17 at 06:50
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    The old joke is that you first visualize an infinite-dimensional space, then cut it down to five dimensions. – Gerry Myerson May 19 '17 at 07:15
  • I have heard the joke that you visualize 3 dimensional space while thinking "five" really loudly – TomKern Dec 03 '21 at 21:41
  • @GerryMyerson Or an $n$-dimensional space where you set $n=5$. – J.G. Dec 03 '21 at 22:58

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A point is zero dimensional. If you extend the point by smearing it through one dimension, you get a line segment. The line segment is the one-dimensional version of the zero-dimensional point.

Now, take that line segment, and smear/spread it in a different dimension, orthogonal to (i.e. at a right angle to) the first dimension. Then you have a square, the two-dimensional analog of the one-dimensional line.

Luckily you live in three dimensions, so you can now smear your square in a direction at a right angle to both of the original smearings you did. When you do that, you get a three dimensional cube.

Now is when it starts getting tricky. Smearing your cube into the fourth dimension is not easy for people that live in three dimensions. But to make a drawing of it that gives you some basic idea of what it "looks like", you can smear your cube to make a hypercube, the 4-dimensional analog of a 3-dimensional cube. You can do this by taking your drawing of a cube, drawing another cube next to it, and drawing connecting lines between the corresponding vertices.

Nothing but prudence and a desire to remain sane prevents you from going on! You can now draw two hypercubes side by side, and then connect their corresponding vertices with line segments, to draw the 5-dimensional analog of your four-dimensional hypercube.

Dimension N+1 can be thought of as dimension N smeared or spread through a new orthogonal dimension. Each of the drawing steps we did is the same--first you draw two points next to each other and connect them, giving you the one-dimensional line segment. Then you draw a copy of that line segment and connect the corresponding vertices to get the square. Draw a copy of the square and connect the corresponding vertices to get a cube. Draw a copy of the cube and connect the vertices to make your hypercube. (Look at the pictures of the hypercubes from the linked question, and you can see that it is just two cubes with their vertices connected.) In each case, the new object is just a smeared version of the old object.

There are good resources in the previous question to help you understand why visualizing dimensions higher than the set your body was born into is exceedingly difficult, if not essentially impossible. But it's fun to try! And we can get some level of understanding of it by trying--not to mention the interesting math that comes out of the exercise.

msouth
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  • Note to future editors--I'm not in a position to put illustrations in here right now, but I hope you won't be shy about editing my answer to include them if you are so inclined. I'm putting this comment here explicitly inviting you to do it so that you don't feel that it would be an intrusion. I am always at least a bit hesitant to make a major change to someone else's answer, so I'm trying to preempt that hesitation and just telling you to go for it! – msouth May 20 '17 at 13:07
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    I would try to draw these but I have a desire to remain sane! – caird coinheringaahing May 20 '17 at 13:28