[math-fun] Wanted/needed: video of morphing dissection of the plane
I just realized that, to illustrate Warren Smith's way of proving the Wall of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way. Can anyone dash off such a video? If I use it in my talk I will of course give credit. Thanks, Jim Propp
Hi James, There is an example of that about two minutes into this video, as a warmup to what happens when you slice the Menger Sponge: https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur... George http://georgehart.com On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving the Wall of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I get "Sorry. Because of its privacy settings, this video cannot be played here." On Wed, Jul 31, 2019 at 9:29 AM George Hart <george@georgehart.com> wrote:
Hi James,
There is an example of that about two minutes into this video, as a warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving the Wall of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
Mike, That's odd. (It works for me from the SF web server.) Are you in a distant country at the moment where videos might be blocked? There is also a YouTube copy here: https://www.youtube.com/watch?v=fWsmq9E4YC0 And a Scientific American copy here: https://www.scientificamerican.com/article/mathematical-impressions-the-surp... George http://georgehart.com On 7/31/2019 11:42 AM, Mike Stay wrote:
I get "Sorry. Because of its privacy settings, this video cannot be played here."
On Wed, Jul 31, 2019 at 9:29 AM George Hart <george@georgehart.com> wrote:
Hi James,
There is an example of that about two minutes into this video, as a warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving the Wall of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
George's video runs for me in Ireland. Mike, I'm afraid you'll just have to relocate to a more salubrious region! WFL On 7/31/19, George Hart <george@georgehart.com> wrote:
Mike,
That's odd. (It works for me from the SF web server.) Are you in a distant country at the moment where videos might be blocked?
There is also a YouTube copy here:
https://www.youtube.com/watch?v=fWsmq9E4YC0
And a Scientific American copy here:
https://www.scientificamerican.com/article/mathematical-impressions-the-surp...
George http://georgehart.com
On 7/31/2019 11:42 AM, Mike Stay wrote:
I get "Sorry. Because of its privacy settings, this video cannot be played here."
On Wed, Jul 31, 2019 at 9:29 AM George Hart <george@georgehart.com> wrote:
Hi James,
There is an example of that about two minutes into this video, as a warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving the Wall of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Mike No Move. Mike Stay. (Sorry) On Wed, Jul 31, 2019 at 12:19 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
George's video runs for me in Ireland.
Mike, I'm afraid you'll just have to relocate to a more salubrious region!
WFL
On 7/31/19, George Hart <george@georgehart.com> wrote:
Mike,
That's odd. (It works for me from the SF web server.) Are you in a distant country at the moment where videos might be blocked?
There is also a YouTube copy here:
https://www.youtube.com/watch?v=fWsmq9E4YC0
And a Scientific American copy here:
https://www.scientificamerican.com/article/mathematical-impressions-the-surp...
George http://georgehart.com
On 7/31/2019 11:42 AM, Mike Stay wrote:
I get "Sorry. Because of its privacy settings, this video cannot be played here."
On Wed, Jul 31, 2019 at 9:29 AM George Hart <george@georgehart.com>
wrote:
Hi James,
There is an example of that about two minutes into this video, as a warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving the Wall of Fire theorem at my August 7 talk, it'd be cool to have a video or
GIF
showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Wed, Jul 31, 2019 at 10:08 AM George Hart <george@georgehart.com> wrote:
Mike,
That's odd. (It works for me from the SF web server.) Are you in a distant country at the moment where videos might be blocked?
No, I'm in Utah near Rich.
There is also a YouTube copy here:
Thanks!
And a Scientific American copy here:
https://www.scientificamerican.com/article/mathematical-impressions-the-surp...
George http://georgehart.com
On 7/31/2019 11:42 AM, Mike Stay wrote:
I get "Sorry. Because of its privacy settings, this video cannot be played here."
On Wed, Jul 31, 2019 at 9:29 AM George Hart <george@georgehart.com> wrote:
Hi James,
There is an example of that about two minutes into this video, as a warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving the Wall of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
I think the video Jim is looking for would show three families of equally-spaced parallel lines, at 60-degree angles to each other, all moving at the same rate. The tilings shown would almost always consist of two families of equilateral triangles of two different sizes, and one family of hexagons whose edges alternate between the side lengths of the two triangular families. I have seen this animation -- I can't remember where -- and my recollection is that it is surprisingly boring. On Wed, Jul 31, 2019 at 1:16 PM Mike Stay <metaweta@gmail.com> wrote:
On Wed, Jul 31, 2019 at 10:08 AM George Hart <george@georgehart.com> wrote:
Mike,
That's odd. (It works for me from the SF web server.) Are you in a distant country at the moment where videos might be blocked?
No, I'm in Utah near Rich.
There is also a YouTube copy here:
Thanks!
And a Scientific American copy here:
https://www.scientificamerican.com/article/mathematical-impressions-the-surp...
George http://georgehart.com
On 7/31/2019 11:42 AM, Mike Stay wrote:
I get "Sorry. Because of its privacy settings, this video cannot be played here."
On Wed, Jul 31, 2019 at 9:29 AM George Hart <george@georgehart.com>
wrote:
Hi James,
There is an example of that about two minutes into this video, as
a
warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving
the Wall
of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
<< I have seen this animation -- I can't remember where -- and my recollection is that it is surprisingly boring. >> Quite so ... A modest amount of variety might come with the ability to vary the Miller index (plane angles) interactively, involving maybe a Javaview script rather than a simple GIF loop. I performed a half-hearted search for some relevant animation; however crystallographers seemingly to content themselves with static hand-drawn diagrams. WFL On 7/31/19, Allan Wechsler <acwacw@gmail.com> wrote:
I think the video Jim is looking for would show three families of equally-spaced parallel lines, at 60-degree angles to each other, all moving at the same rate. The tilings shown would almost always consist of two families of equilateral triangles of two different sizes, and one family of hexagons whose edges alternate between the side lengths of the two triangular families.
I have seen this animation -- I can't remember where -- and my recollection is that it is surprisingly boring.
On Wed, Jul 31, 2019 at 1:16 PM Mike Stay <metaweta@gmail.com> wrote:
On Wed, Jul 31, 2019 at 10:08 AM George Hart <george@georgehart.com> wrote:
Mike,
That's odd. (It works for me from the SF web server.) Are you in a distant country at the moment where videos might be blocked?
No, I'm in Utah near Rich.
There is also a YouTube copy here:
Thanks!
And a Scientific American copy here:
https://www.scientificamerican.com/article/mathematical-impressions-the-surp...
George http://georgehart.com
On 7/31/2019 11:42 AM, Mike Stay wrote:
I get "Sorry. Because of its privacy settings, this video cannot be played here."
On Wed, Jul 31, 2019 at 9:29 AM George Hart <george@georgehart.com>
wrote:
Hi James,
There is an example of that about two minutes into this video, as
a
warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving
the Wall
of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I don't quite follow exactly what it is that is supposed to be visualised, but I hacked together a quick edit of some WebGL code here : http://traxme.net/fractal/ccc.html If whatever is sought to be visualised can be expressed in the rather limited GL shader language in this function: var fragCode = 'precision mediump float;'+ 'varying vec3 vColor;'+ 'void main(void) {'+ 'float x = floor(vColor.x+100.5);'+ 'float y = floor(vColor.y+100.5);'+ 'float z = floor(vColor.z+100.5);'+ 'float sum = (x+y+z);' + 'float col = mod(sum, 2.0);' + 'gl_FragColor = vec4(col, col, col, 1.);'+ '}'; Then this could help an effort to visualize it in javascript (browser) - Controls etc can be added,, but I am to dizzy from watching this animation to make any further useful contributions now :-) /f On Wed, Jul 31, 2019 at 8:14 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< I have seen this animation -- I can't remember where -- and my recollection is that it is surprisingly boring. >> Quite so ...
A modest amount of variety might come with the ability to vary the Miller index (plane angles) interactively, involving maybe a Javaview script rather than a simple GIF loop.
I performed a half-hearted search for some relevant animation; however crystallographers seemingly to content themselves with static hand-drawn diagrams.
WFL
On 7/31/19, Allan Wechsler <acwacw@gmail.com> wrote:
I think the video Jim is looking for would show three families of equally-spaced parallel lines, at 60-degree angles to each other, all moving at the same rate. The tilings shown would almost always consist of two families of equilateral triangles of two different sizes, and one family of hexagons whose edges alternate between the side lengths of the two triangular families.
I have seen this animation -- I can't remember where -- and my recollection is that it is surprisingly boring.
On Wed, Jul 31, 2019 at 1:16 PM Mike Stay <metaweta@gmail.com> wrote:
On Wed, Jul 31, 2019 at 10:08 AM George Hart <george@georgehart.com> wrote:
Mike,
That's odd. (It works for me from the SF web server.) Are you in
a
distant country at the moment where videos might be blocked?
No, I'm in Utah near Rich.
There is also a YouTube copy here:
Thanks!
And a Scientific American copy here:
https://www.scientificamerican.com/article/mathematical-impressions-the-surp...
George http://georgehart.com
On 7/31/2019 11:42 AM, Mike Stay wrote:
I get "Sorry. Because of its privacy settings, this video cannot be played here."
On Wed, Jul 31, 2019 at 9:29 AM George Hart <george@georgehart.com>
wrote:
Hi James,
There is an example of that about two minutes into this video,
as a
warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote: > I just realized that, to illustrate Warren Smith's way of proving
the Wall
> of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF > showing how the intersection between the 2-skeleton of a moving cubical > network and a fixed plane evolves in time. For instance, say the plane is > {(x,y,z): x+y+z=0} and the cubical network is the standard one in > Z^3 > moving at constant speed in the (1,1,1) direction, which one can write as > {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic > dissection of the plane in which equilateral triangles grow and turn into > hexagons and then turn into shrinking triangles pointing the other way. > > Can anyone dash off such a video? If I use it in my talk I will of course > give credit. > > Thanks, > > Jim Propp > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I don't understand enough WebGL to modify this to do the right thing. But, this is certainly warm. In the animation (I think) Jim wants, the families of lines don't rotate -- they only move perpendicularly to the direction of the lines, at a constant speed. The three families all have the same spacing, and move at the same speed (but in directions spaced by 120 degrees). On Wed, Jul 31, 2019 at 2:33 PM Frank Stevenson < frankstevensonmobile@gmail.com> wrote:
I don't quite follow exactly what it is that is supposed to be visualised, but I hacked together a quick edit of some WebGL code here : http://traxme.net/fractal/ccc.html
If whatever is sought to be visualised can be expressed in the rather limited GL shader language in this function:
var fragCode = 'precision mediump float;'+ 'varying vec3 vColor;'+ 'void main(void) {'+ 'float x = floor(vColor.x+100.5);'+ 'float y = floor(vColor.y+100.5);'+ 'float z = floor(vColor.z+100.5);'+ 'float sum = (x+y+z);' + 'float col = mod(sum, 2.0);' + 'gl_FragColor = vec4(col, col, col, 1.);'+ '}';
Then this could help an effort to visualize it in javascript (browser) - Controls etc can be added,, but I am to dizzy from watching this animation to make any further useful contributions now :-)
/f
On Wed, Jul 31, 2019 at 8:14 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< I have seen this animation -- I can't remember where -- and my recollection is that it is surprisingly boring. >> Quite so ...
A modest amount of variety might come with the ability to vary the Miller index (plane angles) interactively, involving maybe a Javaview script rather than a simple GIF loop.
I performed a half-hearted search for some relevant animation; however crystallographers seemingly to content themselves with static hand-drawn diagrams.
WFL
On 7/31/19, Allan Wechsler <acwacw@gmail.com> wrote:
I think the video Jim is looking for would show three families of equally-spaced parallel lines, at 60-degree angles to each other, all moving at the same rate. The tilings shown would almost always consist of two families of equilateral triangles of two different sizes, and one family of hexagons whose edges alternate between the side lengths of the two triangular families.
I have seen this animation -- I can't remember where -- and my recollection is that it is surprisingly boring.
On Wed, Jul 31, 2019 at 1:16 PM Mike Stay <metaweta@gmail.com> wrote:
On Wed, Jul 31, 2019 at 10:08 AM George Hart <george@georgehart.com> wrote:
Mike,
That's odd. (It works for me from the SF web server.) Are you
in a
distant country at the moment where videos might be blocked?
No, I'm in Utah near Rich.
There is also a YouTube copy here:
Thanks!
And a Scientific American copy here:
https://www.scientificamerican.com/article/mathematical-impressions-the-surp...
George http://georgehart.com
On 7/31/2019 11:42 AM, Mike Stay wrote:
I get "Sorry. Because of its privacy settings, this video cannot
be
played here."
On Wed, Jul 31, 2019 at 9:29 AM George Hart < george@georgehart.com> wrote: > > Hi James, > > There is an example of that about two minutes into this video, as a > warmup to what happens when you slice the Menger Sponge: > > >
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
> > > George > http://georgehart.com > > > On 7/31/2019 8:10 AM, James Propp wrote: >> I just realized that, to illustrate Warren Smith's way of proving the Wall >> of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF >> showing how the intersection between the 2-skeleton of a moving cubical >> network and a fixed plane evolves in time. For instance, say the plane is >> {(x,y,z): x+y+z=0} and the cubical network is the standard one in >> Z^3 >> moving at constant speed in the (1,1,1) direction, which one can write as >> {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic >> dissection of the plane in which equilateral triangles grow and turn into >> hexagons and then turn into shrinking triangles pointing the other way. >> >> Can anyone dash off such a video? If I use it in my talk I will of course >> give credit. >> >> Thanks, >> >> Jim Propp >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I was able to view the video, and it's definitely cool, but it doesn't give the morphing Kagome picture I'm looking for. The part of the video that shows a single cube being sliced is a good warmup, for my purposes as it was for George's. But whereas George is gearing up for what happens to a COMPACT fractal assembly of UNEQUAL cubes, I'm gearing up for what happens to an UNBOUNDED assembly of EQUAL cubes. In both scenarios, hexagons and triangles are starring players, but the arrangement of them is quite different. Jim On Wed, Jul 31, 2019 at 11:29 AM George Hart <george@georgehart.com> wrote:
Hi James,
There is an example of that about two minutes into this video, as a warmup to what happens when you slice the Menger Sponge:
https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-sur...
George http://georgehart.com
On 7/31/2019 8:10 AM, James Propp wrote:
I just realized that, to illustrate Warren Smith's way of proving the Wall of Fire theorem at my August 7 talk, it'd be cool to have a video or GIF showing how the intersection between the 2-skeleton of a moving cubical network and a fixed plane evolves in time. For instance, say the plane is {(x,y,z): x+y+z=0} and the cubical network is the standard one in Z^3 moving at constant speed in the (1,1,1) direction, which one can write as {(x,y,z); x≡t (mod 1) or y≡t (mod 1) or z≡t (mod 1)}. We see a dynamic dissection of the plane in which equilateral triangles grow and turn into hexagons and then turn into shrinking triangles pointing the other way.
Can anyone dash off such a video? If I use it in my talk I will of course give credit.
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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2D Intersection of (1,1,1) plane with unit cell and nearest neighbors: TriVerts = Sqrt[3] {Sin[2 Pi #/3], Cos[2 Pi #/3]} & /@ Range[3]; poly[time_, or_] := Switch[{time >= 1/3, time <= -1/3}, {True, _}, Polygon[or + # & /@ (TriVerts (1 - time))], {_, True}, Polygon[or + # {1, -1} & /@ (TriVerts (time + 1))], {_, _}, Polygon[{}]] G[t_] := Graphics[{EdgeForm[Thick], {ColorData["TemperatureMap"][t/2], poly[1/3 + (2/3) t, {0, 0}], poly[1/3 + (2/3) t, {2, 0}], poly[1/3 + (2/3) t, {1, Sqrt[3]}]}, {ColorData["TemperatureMap"][1/2 + t/2], poly[-1 + (2/3) t, {1, -Sqrt[3]/3}], poly[-1 + (2/3) t, {2, 2 Sqrt[3]/3}], poly[-1 + (2/3) t, {0, 2 Sqrt[3]/3}]}, ColorData["TemperatureMap"][1 - t], Polygon[{ poly[1/3 + (2/3) t, {0, 0}][[1, 1]], poly[1/3 + (2/3) t, {0, 0}][[1, 3]], poly[1/3 + (2/3) t, {1, Sqrt[3]}][[1, 2]], poly[1/3 + (2/3) t, {1, Sqrt[3]}][[1, 1]], poly[1/3 + (2/3) t, {2, 0}][[1, 3]], poly[1/3 + (2/3) t, {2, 0}][[1, 2]]}]}, PlotRange -> {{-2, 4}, {-2, 3}}] ListAnimate[G[#/100] & /@ Range[0, 100]] On Wed, Jul 31, 2019 at 7:11 AM James Propp <jamespropp@gmail.com> wrote:
Can anyone dash off such a video?
Thanks! I won’t have access to Mathematica till tomorrow, but I look forward to trying it out. Jim On Fri, Aug 2, 2019 at 5:16 PM Brad Klee <bradklee@gmail.com> wrote:
2D Intersection of (1,1,1) plane with unit cell and nearest neighbors:
TriVerts = Sqrt[3] {Sin[2 Pi #/3], Cos[2 Pi #/3]} & /@ Range[3];
poly[time_, or_] := Switch[{time >= 1/3, time <= -1/3}, {True, _}, Polygon[or + # & /@ (TriVerts (1 - time))], {_, True}, Polygon[or + # {1, -1} & /@ (TriVerts (time + 1))], {_, _}, Polygon[{}]]
G[t_] := Graphics[{EdgeForm[Thick], {ColorData["TemperatureMap"][t/2], poly[1/3 + (2/3) t, {0, 0}], poly[1/3 + (2/3) t, {2, 0}], poly[1/3 + (2/3) t, {1, Sqrt[3]}]}, {ColorData["TemperatureMap"][1/2 + t/2], poly[-1 + (2/3) t, {1, -Sqrt[3]/3}], poly[-1 + (2/3) t, {2, 2 Sqrt[3]/3}], poly[-1 + (2/3) t, {0, 2 Sqrt[3]/3}]}, ColorData["TemperatureMap"][1 - t], Polygon[{ poly[1/3 + (2/3) t, {0, 0}][[1, 1]], poly[1/3 + (2/3) t, {0, 0}][[1, 3]], poly[1/3 + (2/3) t, {1, Sqrt[3]}][[1, 2]], poly[1/3 + (2/3) t, {1, Sqrt[3]}][[1, 1]], poly[1/3 + (2/3) t, {2, 0}][[1, 3]], poly[1/3 + (2/3) t, {2, 0}][[1, 2]]}]}, PlotRange -> {{-2, 4}, {-2, 3}}]
ListAnimate[G[#/100] & /@ Range[0, 100]]
On Wed, Jul 31, 2019 at 7:11 AM James Propp <jamespropp@gmail.com> wrote:
Can anyone dash off such a video?
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I was able to run it just now and it looks good. Thanks, Brad! Unfortunately I'm not enough of a Mathematica-wizard to modify Brad's code to give me what I want, namely, a morphing tessellation. I need the visuals to convey the sense that the pattern repeats throughout the whole plane. Jim On Fri, Aug 2, 2019 at 6:57 PM James Propp <jamespropp@gmail.com> wrote:
Thanks! I won’t have access to Mathematica till tomorrow, but I look forward to trying it out.
Jim
On Fri, Aug 2, 2019 at 5:16 PM Brad Klee <bradklee@gmail.com> wrote:
2D Intersection of (1,1,1) plane with unit cell and nearest neighbors:
TriVerts = Sqrt[3] {Sin[2 Pi #/3], Cos[2 Pi #/3]} & /@ Range[3];
poly[time_, or_] := Switch[{time >= 1/3, time <= -1/3}, {True, _}, Polygon[or + # & /@ (TriVerts (1 - time))], {_, True}, Polygon[or + # {1, -1} & /@ (TriVerts (time + 1))], {_, _}, Polygon[{}]]
G[t_] := Graphics[{EdgeForm[Thick], {ColorData["TemperatureMap"][t/2], poly[1/3 + (2/3) t, {0, 0}], poly[1/3 + (2/3) t, {2, 0}], poly[1/3 + (2/3) t, {1, Sqrt[3]}]}, {ColorData["TemperatureMap"][1/2 + t/2], poly[-1 + (2/3) t, {1, -Sqrt[3]/3}], poly[-1 + (2/3) t, {2, 2 Sqrt[3]/3}], poly[-1 + (2/3) t, {0, 2 Sqrt[3]/3}]}, ColorData["TemperatureMap"][1 - t], Polygon[{ poly[1/3 + (2/3) t, {0, 0}][[1, 1]], poly[1/3 + (2/3) t, {0, 0}][[1, 3]], poly[1/3 + (2/3) t, {1, Sqrt[3]}][[1, 2]], poly[1/3 + (2/3) t, {1, Sqrt[3]}][[1, 1]], poly[1/3 + (2/3) t, {2, 0}][[1, 3]], poly[1/3 + (2/3) t, {2, 0}][[1, 2]]}]}, PlotRange -> {{-2, 4}, {-2, 3}}]
ListAnimate[G[#/100] & /@ Range[0, 100]]
On Wed, Jul 31, 2019 at 7:11 AM James Propp <jamespropp@gmail.com> wrote:
Can anyone dash off such a video?
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Hi Jim, Just create a parallelogram and use parameter "or" to change the origin: (* slightly revised, v0.2 *) TriVerts = Sqrt[3] {Sin[2 Pi #/3], Cos[2 Pi #/3]} & /@ Range[3]; poly[time_, or_] := Switch[{time >= 1/3, time <= -1/3}, {True, _}, Polygon[or + # & /@ (TriVerts (1 - time))], {_, True}, Polygon[or + # {1, -1} & /@ (TriVerts (time + 1))], {_, _}, Polygon[{ poly[1/3 + 2 time, or - {1, Sqrt[3]/3}][[1, 1]], poly[1/3 + 2 time, or - {1, Sqrt[3]/3}][[1, 3]], poly[1/3 + 2 time, or + {0, 2/Sqrt[3]}][[1, 2]], poly[1/3 + 2 time, or + {0, 2/Sqrt[3]}][[1, 1]], poly[1/3 + 2 time, or + {1, -1/Sqrt[3]}][[1, 3]], poly[1/3 + 2 time, or + {1, -1/Sqrt[3]}][[1, 2]]}]] G[time_, or_] := Graphics[{EdgeForm[Thick], { ColorData["TemperatureMap"][time/2], poly[1/3 + (2/3) time, {0, 0} + or]}, { ColorData["TemperatureMap"][1/2 + time/2], poly[-1 + (2/3) time, {2, 2 Sqrt[3]/3} + or]}, Blend[{ColorData["TemperatureMap"][1 - time], Green}, 2 (1/4 - (1/2 - time)^2)], poly[1/3 time, {1, Sqrt[3]/3} + or] }, PlotRange -> {{-2, 4}, {-2, 3}}] ListAnimate[G[#/100, {0, 0}] & /@ Range[0, 100]] ListAnimate[Show[Table[ G[#/100, {2*i + j, Sqrt[3] j}], {i, 0, 5}, {j, 0, 5}], PlotRange -> {{-4, 20}, {-2, 13}}, ImageSize -> 800] & /@ Range[0, 100]] To explain relation to Z^3, you can also draw a cube and put the RGB color gradient directly onto the faces of the cube. Cheers --Brad On Sat, Aug 3, 2019 at 6:43 AM James Propp <jamespropp@gmail.com> wrote:
I was able to run it just now and it looks good. Thanks, Brad! Unfortunately I'm not enough of a Mathematica-wizard to modify Brad's code to give me what I want, namely, a morphing tessellation. I need the visuals to convey the sense that the pattern repeats throughout the whole plane.
Jim
Thanks, Brad. That's closer to what I want, but Lucas V. B. has gone ahead and done something that does the job perfectly. So I think I'll use his video in my talk, provided I can figure out how to display a .webm file from within PowerPoint (or convert it to an .mp4). Thanks, Jim On Sat, Aug 3, 2019 at 12:50 PM Brad Klee <bradklee@gmail.com> wrote:
Hi Jim,
Just create a parallelogram and use parameter "or" to change the origin:
(* slightly revised, v0.2 *)
TriVerts = Sqrt[3] {Sin[2 Pi #/3], Cos[2 Pi #/3]} & /@ Range[3];
poly[time_, or_] := Switch[{time >= 1/3, time <= -1/3}, {True, _}, Polygon[or + # & /@ (TriVerts (1 - time))], {_, True}, Polygon[or + # {1, -1} & /@ (TriVerts (time + 1))], {_, _}, Polygon[{ poly[1/3 + 2 time, or - {1, Sqrt[3]/3}][[1, 1]], poly[1/3 + 2 time, or - {1, Sqrt[3]/3}][[1, 3]], poly[1/3 + 2 time, or + {0, 2/Sqrt[3]}][[1, 2]], poly[1/3 + 2 time, or + {0, 2/Sqrt[3]}][[1, 1]], poly[1/3 + 2 time, or + {1, -1/Sqrt[3]}][[1, 3]], poly[1/3 + 2 time, or + {1, -1/Sqrt[3]}][[1, 2]]}]]
G[time_, or_] := Graphics[{EdgeForm[Thick], { ColorData["TemperatureMap"][time/2], poly[1/3 + (2/3) time, {0, 0} + or]}, { ColorData["TemperatureMap"][1/2 + time/2], poly[-1 + (2/3) time, {2, 2 Sqrt[3]/3} + or]}, Blend[{ColorData["TemperatureMap"][1 - time], Green}, 2 (1/4 - (1/2 - time)^2)], poly[1/3 time, {1, Sqrt[3]/3} + or] }, PlotRange -> {{-2, 4}, {-2, 3}}]
ListAnimate[G[#/100, {0, 0}] & /@ Range[0, 100]]
ListAnimate[Show[Table[ G[#/100, {2*i + j, Sqrt[3] j}], {i, 0, 5}, {j, 0, 5}], PlotRange -> {{-4, 20}, {-2, 13}}, ImageSize -> 800] & /@ Range[0, 100]]
To explain relation to Z^3, you can also draw a cube and put the RGB color gradient directly onto the faces of the cube.
Cheers --Brad
On Sat, Aug 3, 2019 at 6:43 AM James Propp <jamespropp@gmail.com> wrote:
I was able to run it just now and it looks good. Thanks, Brad! Unfortunately I'm not enough of a Mathematica-wizard to modify Brad's
code
to give me what I want, namely, a morphing tessellation. I need the visuals to convey the sense that the pattern repeats throughout the whole plane.
Jim
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Jim, good, glad you have a perfectionist working for you. If my geometry was going into a talk at MoMA, of course I’d demand to be the speaker. Not likely this year or ever, haha... Did you think of taking a plane intersection with a tiling of rhombic dodecahedra (RD) along the same symmetry plane? Start from triangular tiling through three vertices of each RD. Call the set of all vertices L1. As the plane moves up or down, choose up or down triangles and shrink them linearly to a set of centroids L2u or L2d. The sets L2u + L1 and L2d + L1 are the vertices of two translation-Eq. regular hexagonal tilings. The two intersections are similar, but with noticeable differences. Your preferred cubic intersection geometry has three different polygons, while the RD intersection has only two. Also, the cubic intersection continually changes, while the RD intersection freezes the hexagonal tiling for a finite interval. I also liked George’s idea about intersecting fractals. Has anyone tried taking plane sections of the Icosahedral Danzer tetrahedra tiling? (I’m guessing no.) —Brad
On Aug 3, 2019, at 10:38 PM, James Propp <jamespropp@gmail.com> wrote:
Lucas V. B. has gone ahead and done something that does the job perfectly.
Brad, Jim, good, glad you have a perfectionist working for you. If my
geometry was going into a talk at MoMA, of course I’d demand to be the speaker. Not likely this year or ever, haha...
Did you think of taking a plane intersection with a tiling of rhombic dodecahedra (RD) along the same symmetry plane?
Hadn’t thought about the specifics of the symmetrical case, though I did think about the average case. Start from triangular tiling through three vertices of each RD.
Call the set of all vertices L1. As the plane moves up or down, choose up or down triangles and shrink them linearly to a set of centroids L2u or L2d. The sets L2u + L1 and L2d + L1 are the vertices of two translation-Eq. regular hexagonal tilings.
I can’t really envision this; I’d need to see a picture. (Or pass a RD through the Wall of Fire.) The two intersections are similar, but with noticeable differences.
Your preferred cubic intersection geometry has three different polygons, while the RD intersection has only two. Also, the cubic intersection continually changes, while the RD intersection freezes the hexagonal tiling for a finite interval.
I also liked George’s idea about intersecting fractals. Has anyone tried taking plane sections of the Icosahedral Danzer tetrahedra tiling? (I’m guessing no.)
Not that I know of. Jim
participants (8)
-
Allan Wechsler -
Brad Klee -
bradklee＠gmail.com
-
Frank Stevenson -
Fred Lunnon -
George Hart -
James Propp -
Mike Stay