Hey, great read as always. I've realy been fascinated by how tesseracts are rendered. Sometimes I feel like I'm trying to fold myself into a hypercube in Pilates, so this breakdown just resonated. Thanks for making such a complex topic so clear!
This was a fun read, thanks! By the way, have you considered using time as the fourth dimension? Then the visualizations are, similarly, the analog of taking subsequent slices through a cube, and you can get into how to get equivalent time durations to a spatial interval to make the sides all the same length (probably using the speed of light). I would be interested to hear if there's any analogy in that scenario to the illusions due to physically impossible axis positions!
I did think about it! I think that a naive slicing of a hypercube into 3D slices would be boring to watch because it would be just a cube that pops in and out of existence? There could be some 4D rotated version of it that produces a series of more interesting visuals, I might give it a try.
The main issue with the "fourth dimension is time" approach is that time has no defined relationship to spatial dimensions. In contrast, the fourth spatial dimension has (it's orthogonal, so you can do rotations and projections that "mean" something). In contrast, time is notionally a separate number line on a separate piece of paper. Although I guess in relativity, there is a specific relationship between t and x,y,z, it's just fairly wonky.
At one point I worked for a computer company that was building supercomputers with 4 communication channels per node. If each node was one vertex of the cube, a regular 3-d cube (tinker toy model) clearly has 3 lines or cables connecting each vertex. The fourth cable means you can build an @ dimension with only a depth of 2, but with a 4th cable connecting @1 and @2 together. I don't know if anyone tried making visualizations for anything larger than a 8 node system. I guess my point is that in this case, a time-as-4D doesn't help me build a wiring diagram to tell me how to connect the cables.
OTOH, with limited/discrete interpretations of the @ dimension (i.e. here we have only 2 possible values), perhaps color could help one discern the difference between the two?
PS. MATLAB code used to generate animation frames is here: https://lcamtuf.coredump.cx/blog/hypercube.m
Hey, great read as always. I've realy been fascinated by how tesseracts are rendered. Sometimes I feel like I'm trying to fold myself into a hypercube in Pilates, so this breakdown just resonated. Thanks for making such a complex topic so clear!
This was a fun read, thanks! By the way, have you considered using time as the fourth dimension? Then the visualizations are, similarly, the analog of taking subsequent slices through a cube, and you can get into how to get equivalent time durations to a spatial interval to make the sides all the same length (probably using the speed of light). I would be interested to hear if there's any analogy in that scenario to the illusions due to physically impossible axis positions!
I did think about it! I think that a naive slicing of a hypercube into 3D slices would be boring to watch because it would be just a cube that pops in and out of existence? There could be some 4D rotated version of it that produces a series of more interesting visuals, I might give it a try.
The main issue with the "fourth dimension is time" approach is that time has no defined relationship to spatial dimensions. In contrast, the fourth spatial dimension has (it's orthogonal, so you can do rotations and projections that "mean" something). In contrast, time is notionally a separate number line on a separate piece of paper. Although I guess in relativity, there is a specific relationship between t and x,y,z, it's just fairly wonky.
At one point I worked for a computer company that was building supercomputers with 4 communication channels per node. If each node was one vertex of the cube, a regular 3-d cube (tinker toy model) clearly has 3 lines or cables connecting each vertex. The fourth cable means you can build an @ dimension with only a depth of 2, but with a 4th cable connecting @1 and @2 together. I don't know if anyone tried making visualizations for anything larger than a 8 node system. I guess my point is that in this case, a time-as-4D doesn't help me build a wiring diagram to tell me how to connect the cables.
OTOH, with limited/discrete interpretations of the @ dimension (i.e. here we have only 2 possible values), perhaps color could help one discern the difference between the two?