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Re: [math-fun] Torus tiling problem
by Dan Asimov 02 Feb '17

02 Feb '17

This problem was inspired by having read some cool theorems about tiling higher-dimensional things with simplices (n-dimensional polytopes that are isometric to the convex hull of n+1 points in general position in n-space — the natural generalization of a triangle). Definitions: * An n-simplex is "acute" if all of its dihedral angles (there are n(n+1)/2 of them, one for every pair of (n-1)-dimensional faces) is acute. * A "flat" n-manifold is one such that every point has a neighborhood that is isometric with some open ball in n-dimensional Euclidean space. * An "acute triangulation" of a flat manifold X is a way of expressing X as the union of acute simplices with disjoint interiors such that any (n-1)-face of any simplex belongs to exactly two simplices. * The cubical n-torus is R^n / Z^n, the result of identifying opposite (n-1)-faces of the unit n-cube. Theorem: -------- There is an acute triangulation of the 3-cube having 1370 tetrahedra. (This is apparently the fewest known.) Theorem: -------- There exists no acute triangulation of n-space for n >= 5. Theorem: -------- There exists no acute triangulation of the cubical n-torus for n >= 4. (It is unknown whether there exists any acute triangulation of 4-space.) Theorem: -------- If there exists an acute triangulation of 4-space, then the supremum of its set of dihedral angles must be pi/2. –Dan > On Jan 31, 2017, at 6:16 AM, Veit Elser <ve10(a)cornell.edu> wrote: > > By Euler the average degree of a vertex is 6, and the number of triangles is twice the number of vertices. I can construct a tiling from 3 vertices, for 6 triangles (first slice Z^2 on square diagonals to make strips, then subdivide strips). Ruling out a tiling with just one vertex looks doable, but the two vertex case might be tedious … > > Nice problem. > > -Veit > >> On Jan 30, 2017, at 7:43 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> >> ----- >> QUESTION: Given the square torus T^2, what is the smallest number of triangles in a tiling of the torus by triangles such that they are all acute? >> >> I need to define a few terms for this particular question: >> >> * The "square torus" T^2 is the result of identifying opposite edges of a square. (Equivalently, it is the metric space obtained from the quotient of the plane R^2 by its integer subgroup Z^2.) >> >> * A "tiling of the torus by triangles", for the purposes of this question, is any union of triangles that equals the torus, such that >> >> a) Any two triangles have disjoint interiors, >> >> and >> AAAA >> b) No triangle's vertex lies on the interior of any triangle's edge. >> ----- >> >> It's well known that the square can be tiled in this sense with 8 acute triangles and no fewer. The standard example (https://www.ics.uci.edu/~eppstein/junkyard/acute-square/8-square.gif) gives, by identifying opposite edges of the square, a tiling of the square torus using 8 acute triangles. >> >> Note: The most general flat torus is obtained by identifying opposite edges of a parallelogram. Clearly, any non-rectangular flat torus can be tiled in the sense of this question using only 2 acute triangles. >> >> Can the upper bound of 8 for the square torus be improved on? And what about a non-square rectangular torus? >> >> —Dan >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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[math-fun] Algebra question
by David Wilson 31 Jan '17

31 Jan '17

Consider the set S of sequences where the nth element is a residue of the nth prime. Natural elementwise addition and multiplication operations can be defined on S. With respect to these operations, S has unique additive and multiplicative identities (0 and 1), as well as additive and multiplicative inverses. S also has a subset isomorphic to the integer, where integer k corresponds to the sequence whose nth element is k modulo the nth prime. Not being a number theorist, I was wondering about the algebraic structure of this set.

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[math-fun] Should ET know number theory?
by Henry Baker 31 Jan '17

31 Jan '17

In trying to figure out how mankind might talk with an extra-terrestrial, it seems to be a given that the extra-terrestrial would have to know about prime numbers and divisibility. Is there a simple short argument for why that should be the case? After all, humans seem to have gotten along quite a while w/o knowing anything about prime numbers, and primes only seem to have forced themselves into engineering relatively recently. There's also a potentially serious problem about the representation of numbers. We love our positional notation, but where's the proof that there aren't more interesting representations? After all, the Egyptions used (or at least some small fraction (!) of them did) used "Egyptian fractions", which is similar to partial fraction decomposition. There may be a civilization that used some sort of floating point notation (with the exponent & mantissa represented in some unspecified form) from the very start, and may never have cared at all about "integers". Of course, if ET lives in the same universe, then ET would have to know about quantum theory, so how much of quantum theory is understandable w/o any serious knowledge of primes & divisibility?

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Re: [math-fun] no politics on math-fun, please
by Henry Baker 31 Jan '17

31 Jan '17

I wasn't involved (this time)... At 01:19 PM 1/31/2017, you wrote: >I strongly discourage political discussion on Math-Fun. >People (Henry?) are free to contact Fred to help organize a new group >dedicated to the forbidden topics, and may announce it here once.

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[math-fun] no politics on math-fun, please
by rcs＠xmission.com 31 Jan '17

31 Jan '17

I strongly discourage political discussion on Math-Fun. People (Henry?) are free to contact Fred to help organize a new group dedicated to the forbidden topics, and may announce it here once. I will be mulling about the possibility of a tetrahedron having a 27-point sphere. Rich

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[math-fun] Torus tiling problem
by Dan Asimov 31 Jan '17

31 Jan '17

----- QUESTION: Given the square torus T^2, what is the smallest number of triangles in a tiling of the torus by triangles such that they are all acute? I need to define a few terms for this particular question: * The "square torus" T^2 is the result of identifying opposite edges of a square. (Equivalently, it is the metric space obtained from the quotient of the plane R^2 by its integer subgroup Z^2.) * A "tiling of the torus by triangles", for the purposes of this question, is any union of triangles that equals the torus, such that a) Any two triangles have disjoint interiors, and AAAA b) No triangle's vertex lies on the interior of any triangle's edge. ----- It's well known that the square can be tiled in this sense with 8 acute triangles and no fewer. The standard example (https://www.ics.uci.edu/~eppstein/junkyard/acute-square/8-square.gif) gives, by identifying opposite edges of the square, a tiling of the square torus using 8 acute triangles. Note: The most general flat torus is obtained by identifying opposite edges of a parallelogram. Clearly, any non-rectangular flat torus can be tiled in the sense of this question using only 2 acute triangles. Can the upper bound of 8 for the square torus be improved on? And what about a non-square rectangular torus? —Dan

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[math-fun] First they came ...
by Fred Lunnon 31 Jan '17

31 Jan '17

https://terrytao.wordpress.com/2017/01/31/open-thread-for-mathematicians-on… << Of the 774 posts on this blog, I estimate that about 99% of the posts indeed relate to mathematics, mathematicians, or the administration of this mathematical blog, and only about 1% are not related to mathematics or the community of mathematicians in any significant fashion. This is not one of the 1%. >> https://en.wikipedia.org/wiki/First_they_came_... WFL

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Re: [math-fun] Torus tiling problem
by William R Somsky 31 Jan '17

31 Jan '17

Or simpler yet for the rectangle. Rotate so it's taller than wide. Draw the top edge, bottom edge, horizontal bisector and both diagonals. On Jan 31, 2017 1:43 AM, "William R Somsky" <wrsomsky(a)gmail.com> wrote: Rectangular non-square can be done w/ four. Eg on rectangle ([0,2],[0,4]) draw (0,0)-(2,0) (0,2)-(2,2) (0,4)-(2,4) (0,2)-(1,0)-(2,2) (0,2)-(1,4)-(2,2) On Jan 31, 2017 1:19 AM, "William R Somsky" <wrsomsky(a)gmail.com> wrote: > Thinking more on this, I don't think the "clearly all non-rectangular > Tori" is so clear. What about for very skew parallelograms with large > aspect ratios? Neither diagonal splits them into acute triangles. (Think of > a parallelogram with sides of 1 and 4 and angles of 45 and 135 degrees) > > On Jan 31, 2017 1:02 AM, "William R Somsky" <wrsomsky(a)gmail.com> wrote: > >> What I meant was: take your torus, cut and unroll it to be your square >> ABCD. Now, as a square, clip off BCE and glue it to the other side with BC >> on AD to make a parallelogram. Divide it on its diagonal as roll it back >> into a torus. But now, spelling it out, I see my flaw: identifying the >> edges of the new parallelogram, CD wouldn't get pasted back onto BA, but >> into the shifted EABE. Yeah, I think it's the same as you spelled out or >> maybe a "dual" where I don't violate the triangle rule but end up creating >> a different torus. >> >> On Jan 30, 2017 21:37, "Dan Asimov" <dasimov(a)earthlink.net> wrote: >> >> WRS, I don't know if I'm understanding this from your description. >> >> I'm thinking of the square being say ABCD (ccw starting from upper right). >> >> B————E————A >> | / \ | >> | / \ | >> | / \ | >> |/ \| >> C—————————D >> >> Then say let E be the midpoint of AB and F be the midpoint of CD. Finally >> one triangle is CED and the other is what remains: the triangles EBC >> and AED joined along their common edge BC = AD by the identification of >> the square to make a torus. >> >> IF that is the same idea as you are suggesting, it has the flaw that the >> vertex E (of triangle CED) is identified with an interior point of the >> edge CD. >> >> Which violates condition b) below. >> >> Or am I misunderstanding? >> >> —Dan >> >> >> > On Jan 30, 2017, at 8:13 PM, William R Somsky <wrsomsky(a)gmail.com> >> wrote: >> > >> > Two, I believe. Unless I'm mistaken, you just trim a triangle off the >> left >> > edge of the square, move it to the right side matching the identified >> > edges, and you have a parallelogram, which is trivially >> > two-acute-triangularizable. >> > >> > On Jan 30, 2017 16:44, "Dan Asimov" <dasimov(a)earthlink.net> wrote: >> > >> >> ----- >> >> QUESTION: Given the square torus T^2, what is the smallest number of >> >> triangles in a tiling of the torus by triangles such that they are all >> >> acute? >> >> >> >> I need to define a few terms for this particular question: >> >> >> >> * The "square torus" T^2 is the result of identifying opposite edges >> of >> >> a square. (Equivalently, it is the metric space obtained from the >> quotient >> >> of the plane R^2 by its integer subgroup Z^2.) >> >> >> >> * A "tiling of the torus by triangles", for the purposes of this >> >> question, is any union of triangles that equals the torus, such that >> >> >> >> a) Any two triangles have disjoint interiors, >> >> >> >> and >> >> >> >> b) No triangle's vertex lies on the interior of any triangle's edge. >> >> ----- >> >> >> >> It's well known that the square can be tiled in this sense with 8 acute >> >> triangles and no fewer. The standard example ( >> https://www.ics.uci.edu/~ >> >> eppstein/junkyard/acute-square/8-square.gif) gives, by identifying >> >> opposite edges of the square, a tiling of the square torus using 8 >> acute >> >> triangles. >> >> >> >> Note: The most general flat torus is obtained by identifying opposite >> >> edges of a parallelogram. Clearly, any non-rectangular flat torus can >> be >> >> tiled in the sense of this question using only 2 acute triangles. >> >> >> >> Can the upper bound of 8 for the square torus be improved on? And what >> >> about a non-square rectangular torus? >> >> >> >> —Dan >> >> >> >> _______________________________________________ >> >> math-fun mailing list >> >> math-fun(a)mailman.xmission.com >> >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >> >> > _______________________________________________ >> > math-fun mailing list >> > math-fun(a)mailman.xmission.com >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >> >>

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Re: [math-fun] ITS (PDP-10) Lives! -- How about the PDP-1?
by rcs＠xmission.com 30 Jan '17

30 Jan '17

I don't know of anyplace that reads old tapes. I got a tape from Bill Plummer long ago, and I have a bunch of my own too. I may still have some old paper tapes, but no system stuff. I did do a PDP1X simulator in Lisp c. 1985, but it's also on large tape somewhere. --Rich ------- [forwarded from WBA. --Rich] Date: Mon, 30 Jan 2017 23:03:36 -0500 Subject: Re: [math-fun] ITS (PDP-10) Lives! -- How about the PDP-1? From: wba <wbackerman(a)gmail.com> The discussion of the PDP-10 reminds me of another problem that's been sitting in my living room for decades. The MIT RLE PDP-1 timesharing system was the world's first, and for a long time the oldest timesharing system still running. It was the original "spacewar" machine. Some time ago, with the assistance of David Moon, we saved all the system stuff onto tape ("macrotapes" we called them then). Both on the ITS "MC" machine, which had "modern" 9-track tape drives, as opposed to the 7-track drives of AI, ML, and DM, and onto TOPS-20 drives from machine mit-xx. You're supposed to copy stuff over onto modern technology every 10 years or so to prevent disasters like this. But we managed to skip 8-inch floppies, 5-inch floppies, 10-cm floppies, various tape cassettes, CD's, DVD's, and a whole lot of other stuff. Of course I should have attended to this stuff decades ago. Does anyone know of any place on the planet where these tapes could be read? Rich: I believe you got some copies of this when you came to visit a long time ago. You didn't do anything with it by any chance? Bill Ackerman

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[math-fun] social networking cellular computations
by Henry Baker 30 Jan '17

30 Jan '17

After attending CES2017 (Consumer Electronics Show) in Las Vegas this month, I concluded that the following device is inevitable, so I started thinking about what sorts of computations & patterns that might be obtained. One of the interesting things is the progress of "wearable computers", in which computation is embedded in socks, gloves, shirts, glasses, etc. I haven't seen a fully-electronic T-shirt yet, but it's inevitable. By "electronic T-shirt", I mean one with a (bit-mapped?) display on the front & back which can display any stupid thing that your heart desires -- e.g., "I'm with stupid", etc. If there is a display on your sleeve, you could literally wear year heart on your sleeve. Now such a stand-alone display is interesting, but would get boring after perhaps 5 minutes. Far more interesting is a display *that senses & reacts to other people's displays* -- probably by using Bluetooth LE, ANT+, or some other short-range wireless technology. You've probably heard the phrase "he was swayed by the last person he talked with"; an old joke recognizes this tendency and suggests that such information simply be displayed on his forehead. Well, an electronic T-shirt could do exactly that: indicate some amalgam of the electronic T-shirts that our target T-shirt had been recently close to. So, what if each such T-shirt had the same finite set of states, and all the T-shirts had the same rule for combining the states of the N closest T-shirts in order to compute this T-shirt's next state ? What kinds of computations & patterns could be obtained from such a cellular model? One could implement such a device today using Bluetooth smartwatches or smartrings (yes, they were at CES2017) communicating amongst themselves; of course, displaying the results on a T-shirt would be a lot more dramatic. BTW, this sort of model isn't 100% novel; I referred to a similar model here last November: SCHELLING, T. 1971. "Dynamic Models of Segregation". Journal of Mathematical Sociology 1: 143-186. https://www.stat.berkeley.edu/~aldous/157/Papers/Schelling_Seg_Models.pdf

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