When I put squares next to each other, I’m tiling. When I put cubes next to each other, I’m stacking. When I put tesseracts next to each other, what’s the verb?
A square, having four edges, can be surrounded by at most four other squares. One on an edge has three neighbors. One on a corner has two. The edges touch.
A cube, having six faces, can be surrounded by at most six other cubes. One on a face has five neighbors. One on an edge has four neighbors. One on a corner has three. The faces touch.
Where does the +2 from sharing 4 edges to sharing 6 faces come from? I’m adding 1 dimension, but creating 2 additional touch “points”.
A tesseract is bounded by eight cubes (its cells). Each cube shares each of its faces with another cube. Three cubes and three squares meet at each edge. Four cubes, six squares, and four edges meet at every vertex. Collectively, the tesseract consists of eight cubes, twenty-four squares, thirty-two edges, and sixteen vertices.
That seems to imply that hyper-stacking them would result in at most eight other tesseracts. One on cube has seven neighbors. One on a face has six neighbors. Can there be one on an edge?
How does this work?
This is the umpteenth attempt to figure this out. I didn’t post the bad ones. I’ve also changed all the names to be xD something.
For example, an edge is how ever many dimensions we want it to be. The normal English “edge” is a 2D-edge. A “face” is a 3D-edge. The has-no-name cube-ish thing is a 4D-edge. Etc… I’m using “axises” instead of “axes” because “axes” are weapons in this book and it will be even more confusing.
Digression: Oddly enough indecies and appendicies are both flagged by spellcheck. It likes “indexes” and “appendixes”, yet axes is preferred over axises and both verticies and vertexes are OK. English is weird.
Tiling 2D-squares:
- Put 1 down.
- Put another on each 2D-edge (4).
- This creates two axises of three, with the center one counted 2 times. Two axises (1 pair) define a 2D-space (aka plane). 2 items pair uniquely only 1 way.
- This step doesn’t exist in 2D. (Fill out each 1D-space …)
- One more at each 2D-vertex (4) fills the 2D space.
- 1+4+4=9=3×3.
Tiling 3D-squares (a pile of dice would have been helpful!):
- Put 1 down.
- Put another on each 3D-edge (6).
- This creates three axes of three, with the center one counted 3 times. Two axises define a 2D-space. 3 axises pair uniquely 3 ways.
- Fill out each 2D-space as if tiling 2D-squares,
- Put one at each 2D-vertex (3 2D-spaces at 4 2D-vertexes each is 12) to fill each 2D-space.
- That triplet of 2D-spaces defines 1 3D-space (aka 3 axises means 3D)
- One more 3D-square at each 3D-vertex (8) fills the space.
- 1+6+(3×4=12)+8=27=3x3x3.
Words begin to fail here, which is why everything xD above: To get used to it.
For tesseracts, we want to end up with =81=3x3x3x3. Given how long 27 took me to reach, this is not promising.
Tiling 4D-squares:
- Put 1 down.
- Put another on each 4D edge (8).
- This creates four axises of three, with the center one counted 4 times. Each triplet of axes defines one 3D space
Research: Four items can be arranged in how many unique groups of three? Writing math is hard. C(n,r) [n=4,r=3 for this] = n-factorial divided by (r-factorial times (n minus r)-factorial). Thank you AI for that math I had long since forgotten. Let’s double check that with 3 and 2: 3!/(2!(1!)) = 6/2(1) = 3. Yay. 4!/(3!(4-3!) = 24/6 = 4 3D-spaces. 4!/(2!(4-2)!) = 24/4 = 8 2D-spaces – NOT 12 that the 4 3D-spaces above times 3 each implies!
- for a total of 4 3D-spaces and 8 2D spaces.
- This cannot be recursive. It must be linear.
- Put one at each 2D-vertex (8 2D-spaces at 4 2D-vertexes each is 32) to fill each 2D space.
- Put one at each 3D-vertex (4 3D-spaces at 8 3D-vertexes each is 32) to fill each 3D space.
- That quadruplet of 3D-spaces defines 1 4D-space (i.e. 4 axises)
- One more 4D-square at each 4D-vertex (16) fills the 4D-space.
- 1+8+32+32+16=89, which is not 81.
Something is overlapping, but what?
Screw it. They’re not building it from the center out. They’re building it row by row across how ever many dimensions they want. One, two, three. Repeat as needed.
This is too frustrating without the math background to understand the answers to my questions. Is a 5D space holding a 4D jungle gym really that much to ask from an AI?
Rejected image: The wonky face. I like the rest of it.
