August 2013 - mathfun@mailman.xmission.com

Re: [math-fun] Lego aperiodic monotile (AP Goucher)
by Warren D Smith 24 Aug '13

24 Aug '13

> WDS: > Recently a new phase of matter was discovered. It supposedly appears amorphous > to Xray crystallography, but nevertheless exact chemical-composition > ratios and "nucleation+growth" behavior are seen. I suppose your sort > of lego is one possible explanation of that. APG: X-ray crystallography would probably detect the `layered' aspect of the tiling from certain angles, but appear amorphous from other angles. Doesn't glass have an exact chemical composition (SiO2, empirically) coupled with an amorphous structure (the crystalline phase being quartz)? Aperiodic tilings are observed in quasicrystals. I seem to recall that certain alloys have local icosahedral symmetry, and generalise the Penrose tiling to three dimensions. --WDS: the way they discovered it, it is some metal-mixture. Inside this metal mixture when it solidifies are these "crystal grains" which actually are not crystals, they are something with no name, and these grains form via nucleation & growth and can be shown to have fixed atomic composition ratios using electron microprobe, but they are not crystals. Anyhow, these "grains" all are randomly oriented inside the solid. Now if you were to do Xray diffraction on a normal polycrystal or powdered crystal with random grain orientations, monochromatic Xrays, you would see sharp "rings" in your Xray photo at the correct cone-angles. Apparently they do not. Hmmm. So, yes, I think you (APG) are correct; your legos would have shown Xray rings due to the "layering." So good. We deduce this new state of matter is not layered, it is something else. I dug up the paper on it: Phys Rev Lett. 2013 Jul 5;111(1):015502. Highly ordered noncrystalline metallic phase. Long GG, Chapman KW, Chupas PJ, Bendersky LA, Levine LE, Mompiou F, Stalick JK, Cahn JW. Abstract We report the characterization of a unique metallic glass that, during rapid cooling of an Al-Fe-Si melt, forms by nucleation, followed by growth normal to a moving interface between the solid and melt with partitioning of the chemical elements. We determine experimentally that this is not a polycrystalline composite with nanometer-sized grains, and conclude that this may be a new kind of structure: an atomically ordered, isotropic, noncrystalline solid, possessing no long-range translational symmetry. > Subject: Re: [math-fun] Lego aperiodic monotile (AP Goucher) > Message-ID: <20130823204841.30000(a)gmx.com> > Content-Type: text/plain; charset="utf-8" > >> Warren Smith wrote: >> >> Brilliantly simple! Lovable! > > Thank you! > > >> Recently a new phase of matter was discovered. It supposedly appears >> amorphous >> to Xray crystallography, but nevertheless exact chemical-composition >> ratios and "nucleation+growth" behavior are seen. I suppose your sort >> of lego is one possible explanation of that. > > X-ray crystallography would probably detect the `layered' aspect of the > tiling from certain angles, but appear amorphous from other angles. Doesn't > glass have an exact chemical composition (SiO2, empirically) coupled with an > amorphous structure (the crystalline phase being quartz)? Aperiodic tilings > are observed in quasicrystals. I seem to recall that certain alloys have > local icosahedral symmetry, and generalise the Penrose tiling to three > dimensions. > > >> (PS. How'd you make the cool graphics?) > > Wolfram Mathematica 8. I actually cheated by not including the indentations > on the bottom (since they wouldn't be visible from that aspect): > > Module[{block = {Cuboid[{-5, -5, 0}, {5, 5, 3}], > ? ?GeometricTransformation[ > ? ? Table[If[Abs[i] + Abs[j] <= 3, > ? ? ? Cylinder[{{2 i, 2 j, 3}, {2 i, 2 j, 3.5}}, 0.6], {}], {i, -3, > ? ? ? 3}, {j, -3, 3}], {{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}]}}, > ?Graphics3D[{Red, block, Green, Translate[block, {10, 0, 0}], Purple, > ? ?Translate[block, {10, 10, 0}], Yellow, > ? GeometricTransformation[{block, Orange, > ? ? GeometricTransformation[ > ? ? ?block, {{{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}, {0, 0, > ? ? ? ?3}}]}, {{{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}, {10, 0, > ? ? ?3}}]}, Boxed -> False]] > > I think that Mathematica might be able to export 3D models, although it > would be far preferable to rebuild the block in a dedicated CAD package. > > > Sincerely, > > > Adam P. Goucher > > > > ------------------------------ > > Message: 3 > Date: Fri, 23 Aug 2013 16:54:18 -0400 > From: Mike Speciner <ms(a)alum.mit.edu> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] absolute value (branchless) > Message-ID: <5217CBFA.6020206(a)alum.mit.edu> > Content-Type: text/plain; charset=ISO-8859-1; format=flowed > > well, let's try that... > x = 1 > y = 0 > x^y = 1 > x^y - y = 0 > > On 2013-08-23 16:45, J.P. Grossman wrote: >> (typo: that should read 'absx', not 'negx') >> >> >> On Fri, Aug 23, 2013 at 4:44 PM, J.P. Grossman <jpg(a)alum.mit.edu> wrote: >> >>> Assuming x is a 32-bit integer, here's one way that uses 3 instructions: >>> >>> int y = x >> 31; >>> int negx = (x ^ y) - y; >>> >>> J.P. >>> >>> >>> On Fri, Aug 23, 2013 at 4:25 PM, Warren D Smith >>> <warren.wds(a)gmail.com>wrote: >>> >>>> suppose my computer wants |x| >>>> but it refuses to perform an "if" statement. >>>> >>>> What to do? >>>> >>>> _______________________________________________ >>>> math-fun mailing list >>>> math-fun(a)mailman.xmission.com >>>> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>>> >>> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > > > ------------------------------ > > Message: 4 > Date: Fri, 23 Aug 2013 22:58:13 +0200 > From: "Adam P. Goucher" <apgoucher(a)gmx.com> > To: ms(a)alum.mit.edu,"math-fun" <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] absolute value (branchless) > Message-ID: <20130823205813.29990(a)gmx.com> > Content-Type: text/plain; charset="utf-8" > > > >> well, let's try that... >> x = 1 >> y = 0 >> x^y = 1 >> x^y - y = 0 > > Take a look at your fourth computation. (Also, ^ is bitwise XOR, not > exponentiation, although in this example they return the same result.) > > > Sincerely, > > > Adam P. Goucher > > http://cp4space.wordpress.com > >> On 2013-08-23 16:45, J.P. Grossman wrote: >> > (typo: that should read 'absx', not 'negx') >> > >> > >> > On Fri, Aug 23, 2013 at 4:44 PM, J.P. Grossman <jpg(a)alum.mit.edu> >> > wrote: >> > >> >> Assuming x is a 32-bit integer, here's one way that uses 3 >> >> instructions: >> >> >> >> int y = x >> 31; >> >> int negx = (x ^ y) - y; >> >> >> >> J.P. >> >> >> >> >> >> On Fri, Aug 23, 2013 at 4:25 PM, Warren D Smith >> >> <warren.wds(a)gmail.com>wrote: >> >> >> >>> suppose my computer wants |x| >> >>> but it refuses to perform an "if" statement. >> >>> >> >>> What to do? >> >>> >> >>> _______________________________________________ >> >>> math-fun mailing list >> >>> math-fun(a)mailman.xmission.com >> >>> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >>> >> >> >> > _______________________________________________ >> > math-fun mailing list >> > math-fun(a)mailman.xmission.com >> > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > ------------------------------ > > Message: 5 > Date: Fri, 23 Aug 2013 16:59:45 -0400 > From: Mike Speciner <ms(a)alum.mit.edu> > To: ms(a)alum.mit.edu, math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] absolute value (branchless) > Message-ID: <5217CD41.9060505(a)alum.mit.edu> > Content-Type: text/plain; charset=ISO-8859-1; format=flowed > > OOPS. Last line should clearly be x^y-y=1 > So, it seems to work. > Negatives do a 1-s complement and then add 1. > Positives don't change. > SOunds good to me. > > > --ms > > On 2013-08-23 16:54, Mike Speciner wrote: >> well, let's try that... >> x = 1 >> y = 0 >> x^y = 1 >> x^y - y = 0 >> >> On 2013-08-23 16:45, J.P. Grossman wrote: >>> (typo: that should read 'absx', not 'negx') >>> >>> >>> On Fri, Aug 23, 2013 at 4:44 PM, J.P. Grossman <jpg(a)alum.mit.edu> wrote: >>> >>>> Assuming x is a 32-bit integer, here's one way that uses 3 >>>> instructions: >>>> >>>> int y = x >> 31; >>>> int negx = (x ^ y) - y; >>>> >>>> J.P. >>>> >>>> >>>> On Fri, Aug 23, 2013 at 4:25 PM, Warren D Smith >>>> <warren.wds(a)gmail.com>wrote: >>>> >>>>> suppose my computer wants |x| >>>>> but it refuses to perform an "if" statement. >>>>> >>>>> What to do? >>>>> >>>>> _______________________________________________ >>>>> math-fun mailing list >>>>> math-fun(a)mailman.xmission.com >>>>> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>>>> >>>> >>> _______________________________________________ >>> math-fun mailing list >>> math-fun(a)mailman.xmission.com >>> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > > > ------------------------------ > > Message: 6 > Date: Fri, 23 Aug 2013 14:03:58 -0700 > From: Tom Karzes <karzes(a)sonic.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] absolute value (branchless) > Message-ID: <21015.52798.19613.694138(a)gargle.gargle.HOWL> > Content-Type: text/plain; charset=us-ascii > > The last one is wrong: > > (x^y) - y > = (1^0) - 0 > = 1 - 0 > = 1 > > Tom > > Mike Speciner writes: > > well, let's try that... > > x = 1 > > y = 0 > > x^y = 1 > > x^y - y = 0 > > > > On 2013-08-23 16:45, J.P. Grossman wrote: > > > (typo: that should read 'absx', not 'negx') > > > > > > > > > On Fri, Aug 23, 2013 at 4:44 PM, J.P. Grossman <jpg(a)alum.mit.edu> > wrote: > > > > > >> Assuming x is a 32-bit integer, here's one way that uses 3 > instructions: > > >> > > >> int y = x >> 31; > > >> int negx = (x ^ y) - y; > > >> > > >> J.P. > > >> > > >> > > >> On Fri, Aug 23, 2013 at 4:25 PM, Warren D Smith > <warren.wds(a)gmail.com>wrote: > > >> > > >>> suppose my computer wants |x| > > >>> but it refuses to perform an "if" statement. > > >>> > > >>> What to do? > > >>> > > > > ------------------------------ > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > End of math-fun Digest, Vol 126, Issue 45 > ***************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

2 1

Re: [math-fun] More inhabitants of the "Goucher lego" zoo
by Adam P. Goucher 24 Aug '13

24 Aug '13

> Warren wrote: > > Now I remark it also is possible to make a "Goucher lego" not based on > a 1x1 square > tiling the plane, but rather based on a regular hexagon tiling plane > with unit center-separations. The legos will be hexagonal prisms with > knobbies on top and indentations on bottom. Then the lattice of > "knobbies" is based on > "Eisenstein integers" not Gaussian integers. > Gaussians: A+Bi, A,B plain-integers, i=squareroot(-1). > Eisensteins: A+Bw, A,B plain-integers, w=nontrivial cube root of unity. Nice. It has occurred to me that my Lego brick is not `primitive', and can be reduced to one where the upper knobbles are rotated by (2+i) and the lower indentations are rotated by (2-i). Similarly, for your hexagon, the upper knobbles can be rotated through (3+omega) and the lower indentations by (3-omega). This avoids the complication of having knobbles halfway between tiles. I shall send pictures shortly. Sincerely, Adam P. Goucher

1 0

Re: [math-fun] Lego aperiodic monotile (AP Goucher)
by Adam P. Goucher 24 Aug '13

24 Aug '13

Dan, Thanks for noticing that. I've amended it, together with the fact that the Taylor-Socolar tile isn't actually strongly aperiodic (there are tilings with translational symmetry), but rather distinctly more non-trivial than the Goucher lego or the SCD tile. Sincerely, Adam P. Goucher > ----- Original Message ----- > From: Dan Asimov > Sent: 08/24/13 04:32 PM > To: Adam P. Goucher > Subject: Re: [math-fun] Lego aperiodic monotile (AP Goucher) > > Dear Adam, > > In your very very nice CP^4 math blog, the August 23 entry includes this sentence: > > ----- > The first strongly aperiodic monotile (which admits no infinite cyclic group of symmetries) was found by Jean Taylor and Joshua Socolar. > ----- > > The link, however, leads to an arXiv paper with the author Taylor having the first name "Joan" rather than "Jean". > > (I wondered about this, since I know the mathematician Jean Taylor, emerita at Rutgers.) > > Regards from Berkeley, > > Dan

1 0

[math-fun] More inhabitants of the "Goucher lego" zoo
by Warren D Smith 24 Aug '13

24 Aug '13

Goucher's lego is based on a 1x1 square. We could equally well base it on a 2x1 rectangle (two glued together). Also, Goucher's lego as originally described actually will tile in more than one way. If you have 1x1 squares you can tile the plane in a non-unique way, make 1Xinfinity vertical strips and each strip can be slid vertically. Now for Goucher's lego, you can slide the strips certain discrete amounts within each layer, so there are an infinity of different ways to tile 3-space (or merely a finite number of layers) with it. However one could, if desired, slightly perturb the shape (e.g. the 1x1 square becomes a shape close to a 1x1 square, but with slightly S-curved edges) to force uniqueness of one of these ways. Now I remark it also is possible to make a "Goucher lego" not based on a 1x1 square tiling the plane, but rather based on a regular hexagon tiling plane with unit center-separations. The legos will be hexagonal prisms with knobbies on top and indentations on bottom. Then the lattice of "knobbies" is based on "Eisenstein integers" not Gaussian integers. Gaussians: A+Bi, A,B plain-integers, i=squareroot(-1). Eisensteins: A+Bw, A,B plain-integers, w=nontrivial cube root of unity. Both Gaussians and Eisensteins enjoy unique factorization. Now the magic choice Goucher made based on 3^2 + 4^2 = 5^2 [or any primitive Pythagorean triple (5,12,13), (8,15,17) etc], is replaced by 3^2 - 3*8 + 8^2 = 7^2 [or (5,8,7) or (7,15,13) or (16,21,19) etc; if (a,b,c) works so does (b-a,b,c)]. That is, the magic rotation of the knobbie lattice is the unit-norm complex number (3+8w)/7 instead of (3+4i)/5. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0

[math-fun] GITD Goscube (was: Lego aperiodic monotile (AP Goucher))
by Bill Gosper 23 Aug '13

23 Aug '13

Jim Propp>Why does the Goscube fly apart the way it does? Is there some potential energy that got stored in the Goscube in the assembly process (did the prongs have to bend a bit)? No. I think you could summarize it as high stiction with low friction. Whereas the cube faces are smooth thanks to the glass build-plate, the slanted "3D dovetail" surfaces are finely stepped with physical jaggies, so they stridulate rather loudly as they slide apart. But the ABS is hard and slippery enough that friction doesn't much damp the separation velocity, giving the impression of a mild explosion. The hollow shape of the plates probably contributes to the noise, which is surprising even when you open it by merely spinning on a corner. And why is it called the "GITD" Goscube? Jim Propp It glows in the dark. http://gosper.org/kryptocube.jpg (Photo by Peter Bickford.) --rwg [...] I exported the object inhttp://www.youtube.com/watch?v=pvT5ySMnO8U as a (hugely "illegal") .STL, which took tedious wizardry on NeilB's part to actually print. --rwg

1 0

[math-fun] GITD Goscube (was: Lego aperiodic monotile (AP Goucher))
by James Propp 23 Aug '13

23 Aug '13

Why does the Goscube fly apart the way it does? Is there some potential energy that got stored in the Goscube in the assembly process (did the prongs have to bend a bit)? And why is it called the "GITD" Goscube? Jim Propp On Fri, Aug 23, 2013 at 7:13 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > APG> http://cp4space.wordpress.com/2013/08/23/new-aperiodic-monotile/ > > Warren Smith wrote: > > Brilliantly simple! Lovable! > > > apg>Thank you! > > > > Recently a new phase of matter was discovered. It supposedly appears amorphous > to Xray crystallography, but nevertheless exact chemical-composition > ratios and "nucleation+growth" behavior are seen. I suppose your sort > of lego is one possible explanation of that. > > > apg>X-ray crystallography would probably detect the `layered' aspect > of the tiling from certain angles, but appear amorphous from other > angles. Doesn't glass have an exact chemical composition (SiO2, > empirically) coupled with an amorphous structure (the crystalline > phase being quartz)? > > SiO₂ is quartz, quite crystalline. > "The most familiar type of glass, used for centuries in > windows<http://en.wikipedia.org/wiki/Window>and drinking > vessels <http://en.wikipedia.org/wiki/List_of_glassware>, is soda-lime > glass<http://en.wikipedia.org/wiki/Soda-lime_glass>, > composed of about 75% silica <http://en.wikipedia.org/wiki/Silicon_dioxide>(SiO > 2) plus sodium oxide <http://en.wikipedia.org/wiki/Sodium_oxide> (Na2O) > from soda ash, lime <http://en.wikipedia.org/wiki/Calcium_oxide> (CaO), and > several minor additives." > Oddity: Industrial lasers can cut quartz, but not glass. > > apg>Aperiodic tilings are observed in quasicrystals. I seem to recall that > certain alloys have local icosahedral symmetry, and generalise the Penrose > tiling to three dimensions. > > wds>(PS. How'd you make the cool graphics?) > > > apg>Wolfram Mathematica 8. I actually cheated by not including the > indentations on the bottom (since they wouldn't be visible from that > aspect): > > Module[{block = {Cuboid[{-5, -5, 0}, {5, 5, 3}], > GeometricTransformation[ > Table[If[Abs[i] + Abs[j] <= 3, > Cylinder[{{2 i, 2 j, 3}, {2 i, 2 j, 3.5}}, 0.6], {}], {i, -3, > 3}, {j, -3, 3}], {{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}]}}, > Graphics3D[{Red, block, Green, Translate[block, {10, 0, 0}], Purple, > Translate[block, {10, 10, 0}], Yellow, > GeometricTransformation[{block, Orange, > GeometricTransformation[ > block, {{{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}, {0, 0, > 3}}]}, {{{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}, {10, 0, > 3}}]}, Boxed -> False]] > > I think that Mathematica might be able to export 3D models, although > it would be far preferable to rebuild the block in a dedicated CAD > package. > > > Sincerely, > > > Adam P. Goucher > > True and true. I exported the object in > http://www.youtube.com/watch?v=pvT5ySMnO8U > > as a (hugely "illegal") .STL, which took tedious wizardry on NeilB's > part to actually print. > > --rwg > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

1 0

[math-fun] Five Math Experts split the check
by Victor Miller 23 Aug '13

23 Aug '13

http://mathwithbaddrawings.com/2013/08/21/five-math-experts-split-the-check/

1 0

Re: [math-fun] Lego aperiodic monotile (AP Goucher)
by Bill Gosper 23 Aug '13

23 Aug '13

APG> http://cp4space.wordpress.com/2013/08/23/new-aperiodic-monotile/ Warren Smith wrote: Brilliantly simple! Lovable! apg>Thank you! Recently a new phase of matter was discovered. It supposedly appears amorphous to Xray crystallography, but nevertheless exact chemical-composition ratios and "nucleation+growth" behavior are seen. I suppose your sort of lego is one possible explanation of that. apg>X-ray crystallography would probably detect the `layered' aspect of the tiling from certain angles, but appear amorphous from other angles. Doesn't glass have an exact chemical composition (SiO2, empirically) coupled with an amorphous structure (the crystalline phase being quartz)? SiO₂ is quartz, quite crystalline. "The most familiar type of glass, used for centuries in windows<http://en.wikipedia.org/wiki/Window>and drinking vessels <http://en.wikipedia.org/wiki/List_of_glassware>, is soda-lime glass<http://en.wikipedia.org/wiki/Soda-lime_glass>, composed of about 75% silica <http://en.wikipedia.org/wiki/Silicon_dioxide>(SiO 2) plus sodium oxide <http://en.wikipedia.org/wiki/Sodium_oxide> (Na2O) from soda ash, lime <http://en.wikipedia.org/wiki/Calcium_oxide> (CaO), and several minor additives." Oddity: Industrial lasers can cut quartz, but not glass. apg>Aperiodic tilings are observed in quasicrystals. I seem to recall that certain alloys have local icosahedral symmetry, and generalise the Penrose tiling to three dimensions. wds>(PS. How'd you make the cool graphics?) apg>Wolfram Mathematica 8. I actually cheated by not including the indentations on the bottom (since they wouldn't be visible from that aspect): Module[{block = {Cuboid[{-5, -5, 0}, {5, 5, 3}], GeometricTransformation[ Table[If[Abs[i] + Abs[j] <= 3, Cylinder[{{2 i, 2 j, 3}, {2 i, 2 j, 3.5}}, 0.6], {}], {i, -3, 3}, {j, -3, 3}], {{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}]}}, Graphics3D[{Red, block, Green, Translate[block, {10, 0, 0}], Purple, Translate[block, {10, 10, 0}], Yellow, GeometricTransformation[{block, Orange, GeometricTransformation[ block, {{{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}, {0, 0, 3}}]}, {{{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}, {10, 0, 3}}]}, Boxed -> False]] I think that Mathematica might be able to export 3D models, although it would be far preferable to rebuild the block in a dedicated CAD package. Sincerely, Adam P. Goucher True and true. I exported the object in http://www.youtube.com/watch?v=pvT5ySMnO8U as a (hugely "illegal") .STL, which took tedious wizardry on NeilB's part to actually print. --rwg

1 0

Re: [math-fun] Lego aperiodic monotile (AP Goucher)
by Adam P. Goucher 23 Aug '13

23 Aug '13

> George Hart wrote: > > But I must say that the SCD design seems to me to convey the same > ideas while being so much simpler and pleasantly convex. I agree that the convexity makes the Schmitt-Conway-Danzer biprism a superior monotile. However, as Warren Smith noted, my Lego aperiodic structure is somewhat different from the SCD tile, in that the union of any finite number of layers is periodic. Also, the tiles interlock (obviously impossible for convex tiles), and thus it's more obvious how to assemble these than SCD biprisms. > George also wrote: > > P.S. My latest Simons Foundation video is about math and music: > > https://www.simonsfoundation.org/multimedia/mathematical-impressions-making… Interesting. I presume this is a discrete analogue of the fact that the space of lines on an affine plane form a topological Möbius strip? Also, I suspect you can apply projective duality to interchange notes and chords. Sincerely, Adam P. Goucher > On 8/23/2013 4:39 PM, Warren D Smith wrote: > > Hey George Hart, that AP Goucher "lego" is talking to you! It is saying > > "print me on your 3D printer!" I can hear it! > > > > > > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

1 0

[math-fun] General Relativity in high school ??
by Henry Baker 23 Aug '13

23 Aug '13

I've been watching Prof. Suskind (Stanford) do his GR lectures available on iTunes U and YouTube. While these lectures have a lot of problems, they appear to be about the only (free) online GR lectures that I've been able to find on the Internet. It's a shame that the Khan Academy doesn't do GR; I think that Sal Khan's GR lectures would be immensely better than Suskind's. It occurred to me that a series of GR lectures, if properly done & properly motivated, could be accessible to the more advanced high school student -- particularly if the lectures avoided the incredibly intimidating tensor notation. Suskind himself uses some pretty basic geometric examples, involving no more than hyperbolic sines & cosines. It's now been 100 years, and GR as a geometric theory has had all of the kinks worked out. I think that it is high time that GR be taught in high school as an extension of geometry -- probably in conjunction with spherical geometry to set up an analogy. There's no need for GR to be so mysterious that it is accessible to only 1% of the population. Note that I'm _not_ talking about quantum mechanics or the integration of quantum mechanics with GR; I'm merely talking about a classical geometry of curved spaces & times.

2 2