RIP John Conway

Was Game of Life really a game? I don’t know. But either way, we lost a brilliant thinker.

My first thought was to confuse this person with Tim Conway, so I clicked on the thread to figure out why this was in hardware and such, then my second thought was ‘hey, who’s that guy in the picture’ so then my third thought was ‘oh yeah, Tim Conway died last year’ so then my fourth thought was to just click somewhere else but then my fifth thought was ‘maybe I should record all this, you know, for posterity’ so here we are.

Man, I loved the game of life back when I was learning to program my Commodore 64. I must have implemented it a half-dozen different times with various display methods.

I had a similar experience BUT: I didn’t remember that Tim Conway had already passed. :( Fortunately I didn’t have enough time with it to feel any pseudo-relief about him not being the subject of this thread. So I have you to thank, which is a small but good thing.

I also programmed Life on multiple platforms. It was also fun to set up various configurations that moved or shot out other elements.

What an awesome tribute:

Words fail me. Except “sliceness”.

Abstract

A knot is said to be slice if it bounds a smooth properly embedded disk in B4. We demonstrate that the Conway knot is not slice. This completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

holy jesus what the fuck

I love the language of mathematics. I can’t make heads or tails of it, but it’s so wonderfully arcane.

In a similar vein, the wikipedia entry on Conway’s contribution to geometry:

“That prototile—will it tile the plane?!”
“I don’t know, Professor. Shall I fetch the Grand Antiprism?”
“At once! And the Leech lattice!”

Playful mathematicians are awesome, and I especially like that Conway’s algebra is described as “off-beat.” RIP, belatedly.

I’ve only studied a bit of traditional knot theory, but from that definition, here’s the idea:

A traditional knot is an embedding of a circle in 3-dimensional space. If you consider a circle in 2-space, all you can have is a circle with an interior & an exterior. (If you fill in the circle’s interior, you get a disk, which we’ll need for the slice definition.) There’s no way for the circle to cross “over” or “under” itself without an extra dimension to move in, and it’s not allowed to directly intersect itself. Tracing an 8 is not allowed because the circle runs into itself.

But if you take that same circle & put it in 3-space, you can now loop it over & under itself before you get back to where you started. It’s like knotting up your shoelace before you tie the ends back together. You trace the circle as you go around the shoelace, but since you’ve knotted it up, there’s no way to untangle it (unless you’ve made an unknot, which fully unravels).

Now if you take that same circle but fill in the inside to form a disk, you can embed it in a bigger space all twisted up, but you need to do it in 4-space. A circle is 1-dimensional & must be put into 3-space to be knotted. A disk is 2-dimensional & must be put into 4-space to be knotted. (B4 tells you what kinds of 4-spaces are feasible for this, but it’s basically just saying you need 4 dimensions for the disk to avoid running into itself.)

Even all twisted up, the boundary of that disk is still a circle, which is also all twisted up, so if you follow the boundary you trace a knot. If you place it just right, the boundary will exist entirely in one 3-dimensional “slice” of the larger 4-dimensional space, hence the name. (There’s probably some theorem about always being able to wiggle it around until the boundary lines up in one 3-dimensional space? I couldn’t tell you.)

Some knots can be found as the boundaries of embedded disks. These are the slice knots. Other knots are twisted up in just the wrong way so that you could never fill their interiors to form a twisted up disk without that disk running into itself, even with 4 dimensions of space to move in. Those knots, like the Conway knot, are not slice.

*Disclaimer that I may be conflating some finer points of the differences between topological sliceness & smooth sliceness. It’s not something I’ve studied. I’m just going off the definition here.

My favorite Conway creation is the surreal numbers, which simultaneously gives a natural construction of the reals & transfinite ordinals.

I appreciate the explanation!

Picture me nodding thoughtfully in two dimensional space but in three dimensional space I’m still confused as fuck.