Jumping out of our story for a little conversation…
I find this technique for visualizing higher dimensions — referred to as “Ariel’s design” in the story — to be very useful for my own understanding of 4D shapes. I am now much better able to, say, think about the parts of yesterday’s shape.
By the way, that shape is actually called a 16-cell, because it has sixteen three dimensional “sides”. It’s also sometimes called a hexadecachoron, which is just a fancy Greek way of saying “sixteen sided space”.
Using this approach, I can now understand the more complicated 24-cell much better than I could ten days ago (when I first wrote about it here).
Still, Trey’s question haunts me. To put the question simply: It is clear that a two dimensional square contains area. And it is clear that a three dimensional cube contains volume. But what sort of stuff does a four dimensional hypercube contain? What does it feel like to be ‘inside’ a hypercube? Is this something our poor human brains could ever get a feel for?
My apologies for this unrelated comment. I just stumbled in your amazing blog because I was thinking of a catchy domain. So I typed in my browser “somethingcatchy.com”. Unfortunately, it’s already taken by you. Hehe.
Great blog btw. 🙂
Consider a 3m cube that appears abruptly, and three minutes later disappears. It occupies some space-time 4D volume. We can measure it: it is 3m by 3m by 3m by 3minutes. Now lets hollow out that 4D volume and surround it with eight bounding 3D cubes. Two are easy: a cube appears and immediately disappears right before the 3 minute window, and another appears and disappears right at the end. Each of these is a 3m by 3m by 3m cube. But what about the remaining six 3D faces? Well each of these have 2 dimensions corresponding to space and 1 corresponding to time. So these are 2D square membranes that appear at the top, bottom, and sides of the hollowed-out volume at the beginning of the 3 minute window, and disappear at the end. So each of these six is a 3m by 3m by 3minute cube.
What it feels like to be inside this 4D volume is to be unable to escape by moving in 3D space, and also be confined in time by the instant cubes at the start and end of the window.
OK, I buy that. But now try rotating it. 🙂
“But now try rotating it”, you think that is a gotcha but actually it is easy! Or at least, just as easy as in lower dimensions where our intuition immediately applies.
Let’s explain to a being living in an n-dimensional world (plus time), where n=1, what being in the interior of a rotated square feels like. It starts with a single point, which explodes into a handful (well, into n+1=2) n-1=0 dimensional flat membranes (i.e., points), each of which moves linearly with time. Each such membrane is, after a while, occluded by another membrane, also moving linearly. These membranes completely surround you, enclosing you, until after a while they converge at a point, squishing you into a singularity.
This holds not just for n=1, or for n-cubes, but is a general description of what it feels like for an n-dimensional being to be in the interior of an n+1 dimensional non-axis-aligned polytope. Different polytopes correspond to different numbers of membranes and different linear (well, affine to be precise) relationships between the membranes. For a 4D-cube, which is a sphere with an L_infty norm, there are 2*d=8 membranes. For a 4D-octohedron, which is a sphere with an L_1 norm, there are 2^d=16 membranes. For a d-dimensional tetrahedron, there are choose(d,d-1)+1=d+1 membranes.