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?