May 2020 - mathfun@mailman.xmission.com

Re: [math-fun] Complex modulo function?
by Keith F. Lynch 16 May '20

16 May '20

Kerry Mitchell <lkmitch(a)gmail.com> wrote: > Is there a standard, or at least, reasonable, definition of the > modulo function for complex numbers? I don't know if it's any kind of standard, but it has occurred to me that, modulo n^2+1, -1 has a real square root. For instance -1 is congruent to 9 mod 10, so i is 3 mod 10, and -i is 7 mod 10. (Or vice versa if you prefer.) Hence i+1 is 4 mod 10, etc. I don't know if this is useful for anything, but it doesn't seem to lead to any inconsistencies. Everyone's first thought when they learn about i is to ask what it *is*. To which I can answer that at least I know its last digit, which is 3. :-) Since no other power of 10 is 1 plus a square, I unfortunately can't find any other digits of i. The same is true in all other bases, thanks to Mihailescu's theorem (better known as Catalan's conjecture).

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[math-fun] Canway
by Bernie Cosell 16 May '20

16 May '20

Article in today's NYT: Travels with John Conway, in 258 Septillion Dimensions The Princeton mathemagician, who died in April, left an engaging legacy of numerical gamesmanship. https://www.nytimes.com/2020/05/16/science/john-conway-math.html /B\ Bernie Cosell bernie(a)fantasyfarm.com -- Too many people; too few sheep --

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Re: [math-fun] Extension of Game of Life
by Henry Baker 15 May '20

15 May '20

I really like the idea of a "quantum" version of Life! Has there been *any* research on Life played on a 2D surface of qubits? Perhaps Roger Penrose has considered these? At 04:53 PM 5/15/2020, Allan Wechsler wrote: >I just now convinced myself that the spooky blinker is a valid oscillator. > >On Tue, Apr 21, 2020 at 7:10 PM Allan Wechsler <acwacw(a)gmail.com> wrote: >> While musing about searching for Life patterns with particular properties, >> I had an idea for a Lifelike cellular automaton rule that surely must have >> been invented before. So I'm hoping somebody can give me references. >> >> The rule has three states. Two are Life's existing "dead" and "alive", and >> any situation involving only these states evolves exactly as in Life. >> >> The third state is like a quantum superposition of "dead" and "alive". >> Call it "spooky". >> >> The principle is this: a pattern P with spooky cells corresponds to a set >> L(P) of ordinary Life patterns. A Life pattern belongs to the set if it >> agrees with P on all non-spooky cells. >> >> A cell is alive in P', the successor of P, if for all patterns Q in L(P), >> it is alive in Q'. >> >> Similarly, it is dead in P' if it is always dead in Q' for all Q in L(P). >> >> A cell is spooky in P' in all other cases, that is, if it is sometimes >> alive and sometimes dead in Q' depending on the choice of Q from L(P). >> >> Spookiness is ignorance. Sometimes it disappears completely -- an isolated >> spooky cell in empty space dies is one step. >> >> This is a well-defined automaton, and its rules can be completely stated; >> an example is "a live cell with two live neighbors and one spooky neighbor >> will stay alive", an "a dead cell with three or more spooky neighbors and >> no live neighbors becomes spooky". >> >> Has anybody implemented this rule for Golly? (I am a Golly tyro and don't >> know the rule-specification language yet.) >> >> Does this rule have any non-trivial interesting patterns? Like -- does it >> have any oscillators involving spooky cells? Any spaceships? I know it has >> at least one still-life: the spooky block is stable.

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[math-fun] Extension of Game of Life
by Allan Wechsler 15 May '20

15 May '20

While musing about searching for Life patterns with particular properties, I had an idea for a Lifelike cellular automaton rule that surely must have been invented before. So I'm hoping somebody can give me references. The rule has three states. Two are Life's existing "dead" and "alive", and any situation involving only these states evolves exactly as in Life. The third state is like a quantum superposition of "dead" and "alive". Call it "spooky". The principle is this: a pattern P with spooky cells corresponds to a set L(P) of ordinary Life patterns. A Life pattern belongs to the set if it agrees with P on all non-spooky cells. A cell is alive in P', the successor of P, if for all patterns Q in L(P), it is alive in Q'. Similarly, it is dead in P' if it is always dead in Q' for all Q in L(P). A cell is spooky in P' in all other cases, that is, if it is sometimes alive and sometimes dead in Q' depending on the choice of Q from L(P). Spookiness is ignorance. Sometimes it disappears completely -- an isolated spooky cell in empty space dies is one step. This is a well-defined automaton, and its rules can be completely stated; an example is "a live cell with two live neighbors and one spooky neighbor will stay alive", an "a dead cell with three or more spooky neighbors and no live neighbors becomes spooky". Has anybody implemented this rule for Golly? (I am a Golly tyro and don't know the rule-specification language yet.) Does this rule have any non-trivial interesting patterns? Like -- does it have any oscillators involving spooky cells? Any spaceships? I know it has at least one still-life: the spooky block is stable.

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Re: [math-fun] Graph question
by Keith F. Lynch 15 May '20

15 May '20

Dan Asimov <dasimov(a)earthlink.net> wrote: > Find a connected graph G on a countably infinite set of vertices such that > * each vertex has infinite valence > * each vertex is non-adjacent to exactly 2 vertices > * any isomorphism of a subgraph H_1 to a subgraph H_2 extends to an > isomorphism of the entire graph G to itself. > Note: By a "subgraph" here is mean any collection of the vertices > and any collection of the edges which together form a connected graph. If I understand you correctly, I think the graph that links each integer n to every other integer except n-1 and n+1 would be a solution.

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Re: [math-fun] Graph question
by Dan Asimov 15 May '20

15 May '20

Yup, I think so, too. I haven't sat down to prove that's homogeneous in the sense I described, but it appears so. (The correct term in graph theory turns out to be "connected-homogeneous" or for short, "C-homogeneous", since the subgraphs H_1 and H_2 below are assumed to be connected.) Another definition of an isomorphic graph G is this, I think: Let Q/Z denote the group of rationals (mod 1). Let G be the graph having Q/Z as vertices with the rational q (mod 1) connected to all other rationals (mod 1) except for q ± 1/3 (mod 1), It turns out that all connected, countable C-homogeneous graphs were classified in 2010 by R. Gray and D. Macpherson in <https://www.sciencedirect.com/science/article/pii/S009589560900046X>. —Dan Keith Lynch wrote: ----- Dan Asimov <dasimov(a)earthlink.net> wrote: > Find a connected graph G on a countably infinite set of vertices such that > * each vertex has infinite valence > * each vertex is non-adjacent to exactly 2 vertices > * any isomorphism of a subgraph H_1 to a subgraph H_2 extends to an > isomorphism of the entire graph G to itself. > Note: By a "subgraph" here is mean any collection of the vertices > and any collection of the edges which together form a connected graph. If I understand you correctly, I think the graph that links each integer n to every other integer except n-1 and n+1 would be a solution. -----

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[math-fun] Dodecahedron packing conjecture
by Bill Gosper 14 May '20

14 May '20

Ages ago I mentioned that regular dodex can interlock in an "airtight" sheet <http://gosper.org/dodex.gif>, analogous to a sheet of cubes ("Martin's Marbles") <http://gosper.org/martinsmarbles.png>. Isn't this just a plane section perpendicular to (1,1,1) through the 3D endo-dodec checkerboard <http://gosper.org/Endo-dodecahedron_honeycomb_1.png> (absent the endos)? I'd like to see that plane sliding along the major diagonal, like the Menger Sponge movie <https://www.youtube.com/watch?v=fWsmq9E4YC0>. —rwg

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Re: [math-fun] Graph question
by Dan Asimov 14 May '20

14 May '20

I just thought of one, so I'll make it a puzzle: Find a connected graph G on a countably infinite set of vertices such that * each vertex has infinite valence * each vertex is non-adjacent to exactly 2 vertices * any isomorphism of a subgraph H_1 to a subgraph H_2 extends to an isomorphism of the entire graph G to itself. Note: By a "subgraph" here is mean any collection of the vertices and any collection of the edges which together form a connected graph. —Dan

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Re: [math-fun] Graph question
by Dan Asimov 14 May '20

14 May '20

Fascinating! Thanks, Adam. Now I wonder the same thing if n and k are allowed to be infinite. If one or both are countable, that would go a long way. How about countable valence & every vertex non-adjacent to exactly 2 vertices? Dan ----- Wikipedia gives a complete classification of these graphs: https://en.wikipedia.org/wiki/Homogeneous_graph#Classification which implies that the answer to your question is 'precisely when (n, k) is one of the following tuples': (0, 0); (2, 2); (4, 4); (a-1, ab) where a and b are positive integers; (ab, a-1) where a and b are positive integers. > Sent: Thursday, May 14, 2020 at 5:48 PM > From: "Andy Latto" <andy.latto(a)pobox.com> > To: "Dan Asimov" <dasimov(a)earthlink.net>, "math-fun" <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Graph question > > On Thu, May 14, 2020 at 11:02 AM Dan Asimov <dasimov(a)earthlink.net> wrote: > > > Question: > > --------- > > Given a pair of positive integers (n,k), when does there exist a homogeneous graph G such that > > > > * every vertex has valence n, and > > > > * every vertex has just k vertices that are not adjacent to it > > > > ??? -----

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[math-fun] Graph question
by Dan Asimov 14 May '20

14 May '20

Suppose a graph G means: * set V of points called vertices; * a collection E of unordered pairs of vertices called edges; * any two vertices have a path of edges connecting them: G is connected. (No edges from a vertex to itself, and no multiple edges between two vertices.) A subgraph H of G is a graph H whose vertices and edges belong to G. Call a graph G "homogeneous" if every isomorphism between subgraphs of G extends to an isomorphism of G with itself. Question: --------- Given a pair of positive integers (n,k), when does there exist a homogeneous graph G such that * every vertex has valence n, and * every vertex has just k vertices that are not adjacent to it ??? —Dan

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