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Forum Index : Microcontroller and PC projects : Aperiodic Monotiles

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matherp
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Joined: 11/12/2012
Location: United Kingdom
Posts: 9131
Posted: 05:51pm 05 May 2023
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If you have been reading the press  you will know about the discovery of aperiodic monotiles.

This little program plots the two forms that have been discovered using the coordinates given here




Dim r(1)=(1,0)
Dim d(1)=(Cos(Rad(60)),-Sin(Rad(60)))
Dim u(1)=(Cos(Rad(60)),Sin(Rad(60)))
Dim c(1),e(1),f(1)
Dim xy(1,12)
CLS
hat
Math scale xy(),MM.HRes/9,xy()
For i=0 To 11
 Line xy(0,i)+MM.HRes*.6,MM.VRes\2-xy(1,i),xy(0,i+1)+MM.HRes*.6,MM.VRes\2-xy(1,i+1)
 Circle xy(0,i)+MM.HRes*.6,MM.VRes\2-xy(1,i),3,,,RGB(white),RGB(white)
 Pause 1000
Next
 Circle xy(0,12)+MM.HRes*.6,MM.VRes\2-xy(1,12),3,,,RGB(white),RGB(white)
 Line xy(0,12)+MM.HRes*.6,MM.VRes\2-xy(1,12),xy(0,0)+MM.HRes*.6,  MM.VRes\2-xy(1,0)
'
Text MM.HRes*0.7,MM.VRes/6,"HAT",C,,MM.HRes/200
Pause 1000
turtle
Math scale xy(),MM.HRes/16,xy()
For i=0 To 11
 Line xy(0,i)+MM.HRes*.1,MM.VRes*.3-xy(1,i),xy(0,i+1)+MM.HRes*.1,MM.VRes*.3-xy(1,i+1)
 Circle xy(0,i)+MM.HRes*.1,MM.VRes*.3-xy(1,i),3,,,RGB(white),RGB(white)
 Pause 1000
Next
 Circle xy(0,12)+MM.HRes*.1,MM.VRes*.3-xy(1,12),3,,,RGB(white),RGB(white)
 Line xy(0,12)+MM.HRes*.1,MM.VRes*.3-xy(1,12),xy(0,0)+MM.HRes*.1,MM.VRes*.3-xy(1,0)
'
Math scale xy(),40,xy()
Text MM.HRes*0.2,MM.VRes/6,"TURTLE",C,,MM.HRes/200


Do :Loop
'
Sub hat
 xy(0,0)=0
 xy(1,0)=0
'
 scale r(),0.5,c()
 xy(0,1)=c(0)
 xy(1,1)=c(1)
'
 xy(0,2)=u(0)
 xy(1,2)=u(1)
'
 scale u(),2,c()
 scale d(),0.5,e()
 add c(),e(),f()
 xy(0,3)=f(0)
 xy(1,3)=f(1)
'
 add r(),u(),c()
 xy(0,4)=c(0)
 xy(1,4)=c(1)
'
 scale u(),1,c()
 scale r(),1.5,e()
 add c(),e(),f()
 xy(0,5)=f(0)
 xy(1,5)=f(1)
'
 scale r(),2,c()
 xy(0,6)=c(0)
 xy(1,6)=c(1)
'
 scale u(),-0.5,c()
 scale r(),3,e()
 add c(),e(),f()
 xy(0,7)=f(0)
 xy(1,7)=f(1)
'
 scale r(),2,e()
 add d(),e(),f()
 xy(0,8)=f(0)
 xy(1,8)=f(1)
'
 add r(),d(),f()
 xy(0,9)=f(0)
 xy(1,9)=f(1)
'
 scale d(),0.5,e()
 add r(),e(),f()
 xy(0,10)=f(0)
 xy(1,10)=f(1)
'
 xy(0,11)=d(0)
 xy(1,11)=d(1)
'
 scale u(),-0.5,c()
 xy(0,12)=c(0)
 xy(1,12)=c(1)
'

End Sub

Sub turtle
 xy(0,0)=0
 xy(1,0)=0
'
 xy(0,1)=r(0)
 xy(1,1)=r(1)
'
 add r(),d(),c()
 xy(0,2)=c(0)
 xy(1,2)=c(1)
'
 scale r(),3,c()
 xy(0,3)=c(0)
 xy(1,3)=c(1)
'
 scale r(),5,c()
 scale d(),2,e()
 add c(),e(),f()
 xy(0,4)=f(0)
 xy(1,4)=f(1)
'
 scale r(),4,c()
 scale d(),4,e()
 add c(),e(),f()
 xy(0,5)=f(0)
 xy(1,5)=f(1)
'
 scale r(),3,c()
 scale d(),4,e()
 add c(),e(),f()
 xy(0,6)=f(0)
 xy(1,6)=f(1)
'
 scale r(),2,c()
 scale d(),5,e()
 add c(),e(),f()
 xy(0,7)=f(0)
 xy(1,7)=f(1)
'
 scale d(),4,e()
 add r(),e(),f()
 xy(0,8)=f(0)
 xy(1,8)=f(1)
'
 scale r(),-1,c()
 scale d(),5,e()
 add c(),e(),f()
 xy(0,9)=f(0)
 xy(1,9)=f(1)
'
 scale r(),-1,c()
 scale d(),4,e()
 add c(),e(),f()
 xy(0,10)=f(0)
 xy(1,10)=f(1)
'
 scale d(),3,e()
 xy(0,11)=e(0)
 xy(1,11)=e(1)
'
 scale r(),-1,c()
 scale d(),2,e()
 add c(),e(),f()
 xy(0,12)=f(0)
 xy(1,12)=f(1)
End Sub

Sub mul a(),b(),c()
 c(0)=a(0)*b(0)
 a(1)=a(1)*b(1)
End Sub

Sub add a(),b(),c()
 c(0)=a(0)+b(0)
 c(1)=a(1)+b(1)
End Sub

Sub scale a(),b,c()
 c(0)=a(0)*b
 c(1)=a(1)*b
End Sub

Edited 2023-05-06 04:09 by matherp
 
stanleyella

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Joined: 25/06/2022
Location: United Kingdom
Posts: 2129
Posted: 07:06pm 06 May 2023
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Hi sir, not many replies. Er... don't squares do a better job? Obviously I'm missing your point but psychic abilities fade with age.
You must have a point to make .. but what is it?
 
Mixtel90

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Joined: 05/10/2019
Location: United Kingdom
Posts: 6800
Posted: 07:43pm 06 May 2023
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It's clever stuff, Stan.
https://cs.uwaterloo.ca/~csk/hat/
https://aperiodical.com/2023/03/an-aperiodic-monotile-exists/
Edited 2023-05-07 05:46 by Mixtel90
Mick

Zilog Inside! nascom.info for Nascom & Gemini
Preliminary MMBasic docs & my PCB designs
 
stanleyella

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Joined: 25/06/2022
Location: United Kingdom
Posts: 2129
Posted: 09:39pm 06 May 2023
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It may be clever but what is the point for say pico users? Is it a polygon fill?
The links are interesting but that's all. I'm still missing the point.
Your code works pasted into mmedit and ili9341 but what to do??

 
matherp
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Joined: 11/12/2012
Location: United Kingdom
Posts: 9131
Posted: 09:56pm 06 May 2023
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Have you ever heard of intellectual curiosity? Everything doesn't need to have a point
 
Mixtel90

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Location: United Kingdom
Posts: 6800
Posted: 09:59pm 06 May 2023
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It's amazing that you can get a shape to do that. I wonder how many possibilities there are?
Mick

Zilog Inside! nascom.info for Nascom & Gemini
Preliminary MMBasic docs & my PCB designs
 
phil99

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Joined: 11/02/2018
Location: Australia
Posts: 2140
Posted: 10:06pm 06 May 2023
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The point is almost all tiles produce a repeating pattern. A pair of Penrose tile shapes can cover a surface, without gaps without ever repeating. With just one of the new tile shapes that can be done. The mathematics of proving that you can tile to infinity without ever producing a pattern is impressive.
 
stanleyella

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Joined: 25/06/2022
Location: United Kingdom
Posts: 2129
Posted: 10:11pm 06 May 2023
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does mmbasic have filled polygons or how to make shapes from filled triangles is more relevant. What can it do a square can't? Sorry, I'm an old git.
 
phil99

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Joined: 11/02/2018
Location: Australia
Posts: 2140
Posted: 02:15am 07 May 2023
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Squares, rectangles, triangles, hexagons and almost every other shape produce patterns. If you don't want a pattern squares are useless. So far out of countless possible shapes those two shapes are the only ones that will work using just one type of tile.
 
stanleyella

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Joined: 25/06/2022
Location: United Kingdom
Posts: 2129
Posted: 01:32pm 07 May 2023
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If you don't want a pattern squares are useless.
chess boards? :)
Is there any practical use please for the code?
Please explain what the point is about these shapes. The links say little to me.
I looked up Tessellation, that's inspiring but still can't see the point of hat shape.
I found mmbasic does have filled polygons.
 
Mixtel90

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Joined: 05/10/2019
Location: United Kingdom
Posts: 6800
Posted: 01:43pm 07 May 2023
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The shapes are a mathematical and topographical curiosity. Whether they will have any practical use is yet to be seen.

A chess board is full of patterns. They are everywhere - literally if you expand it to infinity in both directions. An aperiodic monotile is a shape that can be put side by side without leaving gaps, very much like a square, but without forming any patterns, not even when covering  an infinite surface.

There may be some use in the future, possibly in the arrangement of atoms in new materials, but at the moment they remain a recently discovered curiosity.

Making patterns comes naturally to us. We do it all the time. Finding things that refuse to form patterns when we want them to is very interesting. :)
Mick

Zilog Inside! nascom.info for Nascom & Gemini
Preliminary MMBasic docs & my PCB designs
 
Plasmamac

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Joined: 31/01/2019
Location: Germany
Posts: 554
Posted: 03:26pm 07 May 2023
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Its great Matherp.
Plasma
 
stanleyella

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Joined: 25/06/2022
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Posted: 03:35pm 07 May 2023
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It's interesting and worth the google time. over and out on this but thanks for all the fish :)
 
JohnS
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Joined: 18/11/2011
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Posts: 3805
Posted: 06:38pm 07 May 2023
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Various things in mathematics had no practical/useful applications - until they did.

E.g. complex numbers.

E.g. the realisation that some equations for electrical (R C etc) circuits were like those for air frames.

E.g. one-way (trapdoor) functions - yes, they turned out to be useful... in the s of https for example.

Maybe these shapes will be useful... maybe not... maths is often about the curiosity and the roads travelled.  If it turns out one day to be useful, so much the better.

I seem to recall a parallel regarding electricity (what use is it? was the question, basically).

Anyway, thanks Peter.

John
 
Andrew_G
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Joined: 18/10/2016
Location: Australia
Posts: 847
Posted: 11:20pm 07 May 2023
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Peter et al,
Thank you for this. Fascinating!
It seems to me that there are four 'variants' of the shape i.e. two rotations and two mirrored images.
Is there an algorithm for locating and rotating them on screen to progressively fill the screen?
(Only an academic interest, as apart from MB4W, I don't have VGA capability.)

Cheers,

Andrew
 
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