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@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
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Message: 3 Date: Fri, 23 Aug 2013 16:54:18 -0400 From: Mike Speciner <ms@alum.mit.edu> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] absolute value (branchless) Message-ID: <5217CBFA.6020206@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@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@gmail.com>wrote:
suppose my computer wants |x| but it refuses to perform an "if" statement.
What to do?
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Message: 4 Date: Fri, 23 Aug 2013 22:58:13 +0200 From: "Adam P. Goucher" <apgoucher@gmx.com> To: ms@alum.mit.edu,"math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] absolute value (branchless) Message-ID: <20130823205813.29990@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
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@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@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@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Message: 5 Date: Fri, 23 Aug 2013 16:59:45 -0400 From: Mike Speciner <ms@alum.mit.edu> To: ms@alum.mit.edu, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] absolute value (branchless) Message-ID: <5217CD41.9060505@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@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@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@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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------------------------------
Message: 6 Date: Fri, 23 Aug 2013 14:03:58 -0700 From: Tom Karzes <karzes@sonic.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] absolute value (branchless) Message-ID: <21015.52798.19613.694138@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@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@gmail.com>wrote:
suppose my computer wants |x| but it refuses to perform an "if" statement.
What to do?
------------------------------
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