I A way to comprehend the geometry of a hypercube

[Warp]

It's extremely hard, if not outright impossible, for our limited brain to visualize 4D objects. 3D objects are fine, but 4D is just impossible (except, maybe, to some people). However, at one point I figured out a way to comprehend (in some ways "visualize") the 4-dimensional hypercube, which allows deducing facts about its geometry. This is probably nothing new nor innovative and probably lots of other people have thought of this way, but this is something I came up with myself.

1. Take a point. It can be considered a 0-dimensional shape (in this context, a sort of "0-dimensional cube" of sorts.)
2. Now duplicate the point and move this copy one unit towards the first dimension (ie. the positive x-axis), while keeping the two points connected with an "edge". This is now a 1-dimensional shape (a "1-dimensional cube" of sorts.)
3. Now duplicate this shape and move the copy one unit towards the second dimension (ie. the positive y-axis), while keeping the corresponding points connected with edges. This is now a 2-dimensional shape (a "2-dimensional cube" of sorts.)
4. Now duplicate this shape and move copy one unit towards the third dimension (ie. the positive z-axis), while keeping the corresponding points connected with edges. This is now your regular old 3-dimensional cube.
5. So, just do the same process one more time: Duplicate this cube and move the copy one unit towards the "fourth dimension", while keeping the corresponding points connected with edges.

You don't need to be able to exactly visualize in your mind where this "fourth dimension" might be. Just conceptualize (even if you can't visualize it) that it's an additional dimension that's perpendicular to all the other three dimensions, and you are thus moving the copy of the cube away from the original cube in this extra dimension.

This ought to help understanding the geometry of the hypercube and, for example, deduce geometric facts about it. Simple example: Notice how on each step we doubled the amount of vertices (or "points"). Which means, without having to think at all about it (nor having to visualize anything), that we can easily deduce that a hypercube has double the vertices as a normal cube, ie. 16 (since the cube has 8).

Likewise at each step the number of edges became double of those of the previous shape plus an additional edge for each original vertex (because we kept copied vertices connected with new edges as we moved the copy). 0 edges became 0x2+1 = 1 edge. Then 1 edge became 2x1+2 = 4 edges. Then 4 edges became 2x4+4 = 12 edges. Thus, logically, in a hypercube there will be 2x12+8 = 32 edges.

Exercise for the reader: Using the same logic, how many faces does a hypercube have? How many edges are connected to each vertex?

 

[Vanadium 50]

is just impossible (except, maybe, to some people).

So your statement is that its impossible, except when it isn't. I guess I can't argue with thatt.

Let;s see if your idea is any good. In 2-space, we have an infinite number of regular polygons. Inn 3-space we have five corresponding polyhedra - the Platonic solids. How many do we have in 4D? 5D?

I'm not asking you to describe them. Just tell me how many there are.

 

[jedishrfu]

Thanks for sharing your method. It is useful to note that there are many examples of people doing the same or similar step by step algorithm on Youtube that you can check out.