Manfred Mohr: Cubic Limit
I was on Sci-Fi-O-Rama digging around for the book covers from the Gentlemen Broncos opening credits sequence when I stumbled across this absolutely arresting plotter print by digital art pioneer Manfred Mohr in a post re: the 1984 book Creative Computer Graphics.
Cubic Limit V: Restriction, Manfred Mohr [date unknown to me]
What I was looking at made my brain itch on the inside—not unpleasantly—so I started googling & binging & duckduckgo-ing & brave-ing. Ended up halfway down a rabbit-hole about a bunch of tesseract/hypercube stuff which set me slightly wall-eyed.
Animation showing each individual cube within the B4 Coxeter plane projection of the tesseract
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K5 (top) or K3,3 (bottom) subgraphs
3D projection of a tesseract performing a double rotation about two orthogonal planes in 4-dimensional space. This is something I’m told they often enjoy doing in the wild.
The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space AAAAAAAGH IT’S ALL CONNECTED
Mm-hm, yes yes I see.
The unfolding tesseract on the far right reminds me of a time back in middle school when a friend of mine tried to teach me how to play 4D chess. As I recall, he verbally explained the mapping you see there, instructed me to try and hold that relationship in my mind as reference, and now let’s simply play regular chess, but overlaying these new dimensional relationships atop the standard set of moves. Knights will be particularly fun to use!
Staring at my bishop, I was at a complete loss trying to figure out what even constituted “diagonal” in this sort of slip-space. But even more, I think I wanted to know how that complex wrapping relationship represented time, because hadn’t I heard somewhere that the 4th dimension is supposed to be time? It’s time, right? This wrapping thing didn’t seem to me like it had anything to do with time.
I think the board/set belonged to his father, a professor in one of the sciences. I have no memory of what it looked like.
I think he beat me. How would I know? Maybe he didn’t: he may not have had the heart to finish the game. He’s a good guy.
What I did understand looking at Cubic Limit V: Restriction is that this sort of thing would probably look pretty cool animated, and I could reverse-engineer whatever process was happening in that top plotter image without much fuss using cloner objects. Once I realized that the weirdly-delineated thicker lines’ limits were actually governed by a simple camera-facing “viewport” the same size as a single face of one of these split cubes, the structure was clear and I was able to crank out the animation below:
P-197 N/R 801 Cubic Limit II (1977) replica animation [after Mohr]
So much quicker & easier for me in Cinema 4D than it must have been in Fortran IV (or whatever Mohr was using)! My right/left split on each cube in the matrix is simply unrelated, randomized rotation. I couldn’t say whether Mohr’s cube-splits are random as well or if they have some sort of mathematical L-R relationship I’m unaware of. I wouldn’t be surprised if it were the latter; love to find out if that’s so. I’ll probably update this animation to be longer & looping, massage the timing a bit. Nice to take a moment to play with something render-light yet elegant.
Here are a few more Mohr pieces (for those who want more Mohr): a couple views of an inverted serigraph on paper version of his plotter work, then Cubic Limit (probably his best-known work), and finally Complimentary Cubes:
P-197 Cubic Limit II (from the Artiste et Ordinateur [Artist and Computer] Portfolio), via the Anne and Michael Spalter Digital Art Collection database
P-197 Cubic Limit II clearer, more crisp scan (or maybe a reproduction) I dug up on Reddit