[math-fun] Moebius-like torus name?
What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern? z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x Unless I'm fooling myself, this makes a way to wrap the sides of the board around for a game like Life or Langton's Ant, in such a way that the space is locally flat everywhere--without any local kinks or anomalies that would snarl the game rules. I have tried to think of such a thing so many times in the past and failed... Is this consistent? What's it called? --Steve
I don't know how to read sewing patterns, but here's what I think you're describing: Take a square, and put arrows pointing left on the top and bottom sides, and an arrow pointing up on the left and an arrow pointing down on the right If we join the top and bottom so that the arrows match up, we have a cylinder. If we join the left and right edges so that the arrows match up, we have a Mobius strip. If we join both of these pairs of arrows in this way (impossible in three dimensions without self-intersection), we have a Klein bottle. Andy On Sun, May 3, 2020 at 12:46 PM Steve Witham <sw@tiac.net> wrote:
What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern?
z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x
Unless I'm fooling myself, this makes a way to wrap the sides of the board around for a game like Life or Langton's Ant, in such a way that the space is locally flat everywhere--without any local kinks or anomalies that would snarl the game rules. I have tried to think of such a thing so many times in the past and failed...
Is this consistent? What's it called?
--Steve
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Is it not a Klein bottle? -- Gene On Sunday, May 3, 2020, 09:46:10 AM PDT, Steve Witham <sw@tiac.net> wrote: What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern? z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x Unless I'm fooling myself, this makes a way to wrap the sides of the board around for a game like Life or Langton's Ant, in such a way that the space is locally flat everywhere--without any local kinks or anomalies that would snarl the game rules. I have tried to think of such a thing so many times in the past and failed... Is this consistent? What's it called? --Steve _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
=Steve Witham this makes a way to wrap the sides of the board around for a game
(BTW we could curl up hexagonal boards too, and similarly for any even-gon.) Anyway, it seems a finite square board with edges identified would be logically equivalent to an infinite quilt of square patches in a regular board, each of which are initialized with a suitably rotated/reflected copy of the initial configuration. Does this mean that a doubly-twisted "purse" board will behave the same as an untwisted "torus" board?
kinks or anomalies that would snarl the game rules
Indeed this avoids local snarling, but curling up a CA generally mostly seems to introduce additional "global" phenomena that depend sensitively only on arbitrary details, such as the exact diameter of the universe along each axis.
=Keith F. Lynch It would be interesting if one of those was the topology of our universe.
I remember an ancient variant "Space War" that did this. It made shooting photon torpedoes "around the world" interesting -- especially if they interacted with the sun's gravity... A few years ago I saw a pop-sci article somewhere claiming that some cosmologists were trying to investigate this possibility by looking at the largest-scale density fluctuations. I think it may have even suggested something like a dodecahedron with opposite faces identified as being a model that was consistent with observations. Anybody else recall this?
https://arxiv.org/pdf/physics/0509171.pdf
On May 4, 2020, at 7:04 PM, Marc LeBrun <mlb@well.com> wrote:
A few years ago I saw a pop-sci article somewhere claiming that some cosmologists were trying to investigate this possibility by looking at the largest-scale density fluctuations. I think it may have even suggested something like a dodecahedron with opposite faces identified as being a model that was consistent with observations.
Saw this topic a bit late, but just wanted to point out that Golly supports bounded grids with various topologies: plane, torus, Klein bottle, sphere, and cross-surface (aka real projective plane, but Conway preferred the term cross-surface). Details here: http://golly.sourceforge.net/Help/bounded.html If you have Golly, open Patterns/Life/Bounded-Grids and you'll find various examples of gliders and spaceships moving in different topologies. Andrew
participants (6)
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Andrew Trevorrow -
Andy Latto -
Eugene Salamin -
Hans Havermann -
Marc LeBrun -
Steve Witham