#aperiodic

Penrose Chainmail

Dr. T's World of Chainmail

He does other tessellations, too!

But it is actually created using a single hexagonal tile.

Aperiodic Voxels is the name of a Minecraft-like that someone is making.

It's called Aperiodic Voxels because it uses an aperiodic voxel grid.

that's the video of devlog 1

Much excitement last week with the publication by David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss of their discovery of a single tile which tiles aperiodically. Not only that but it does not need any supplementary rules, as Penrose's dart/kite pair do, to ensure aperiodicity. Christian Lawson-Perfect wrote about the story in aPeriodical. Later: There is now an excellent online talk by Craig and Chaim on the discovery and its proof hosted by the National Museum of Mathematics in New York.

The shape was discovered by David Smith, a hobbyist geometer. There is a wealth of research in David's blog, which starts with the declaration "I am not a mathematician but I do like shapes, interesting symmetry, polyhedra, tessellations and geometric patterns. " Skimming though it, one can see the deep exporation of shapes which makes his discovery seem almost inevitable. I plan to read more of his blog for inspiration. Well done ye, David!

The mathematical work of proving that the tile is aperiodic is a masterful exercise, clearly explained. A truly inspiring collaboration between a tinkerer, acedemic mathematicians and computer scientists.

OpenSCAD

Naturally I had a look at this with my OpenSCAD tiling tools.

tl:dr: A customizer on Thingiverse.allows both kinds of tiles in normal or mirror forms and all intermediate tiles to be printed or laser-cut.

I began by defining the perimeter as a sequence of sides each defined by side length and interior angle taken from the diagram, where we see up of 16 right triangular segments ( 8 kites) of adjacent hexagons. In this formulation, r is the long side of the triangle, a and b the other sides.

function hat_peri(r) = // r is the radius of the base hexagon // returns perimeter as a sequence of [side length,internal angle]    let (a = r * sin(60), b = r*cos(60) )    [[b,180],[b,120],[b,270],[a,120],[a,90],[b,120],[b,270],[a,120],     [a,90],[b,240],[b,90],[a,240],[a,90],[b,120]];

The first two sides form a straight line but for the purposes of tiling are distinct edges.

The perimeter is converted to a sequence of points using turtle-like geometry:

function peri_to_points(peri,pos=[0,0],dir=0,i=0) =     i == len(peri)       ? [pos]       : let(side = peri[i])         let (distance = side[0])         let (newpos = pos + distance* [cos(dir), sin(dir)])         let (angle = side[1])         let (newdir = dir + (180 - angle))         concat([pos],peri_to_points(peri,newpos,newdir,i+1))      ;

Tiling with this shape requires the use of the mirror image of the base tile:

function reverse(l) = [for (i=[1:len(l)]) l[len(l)-i]];

function mirror_peri(q) = let(p=reverse(q)) [for (i=[0:len(p)-1]) [p[ (i - 1 + len(p) ) %len(p)].x,p[i].y] ];

[This is a bit of a pity. Penrose's tiles are symmetric so the question doesnt arise but the search continues for a monotiling which does not require the use of reflections.]

My library of OpenSCAD functions supports the creation of tilings by defining a sequence of edge-to-edge placements. ATM this list is created manually. Each line specifies the alignment of one of the base tiles and edge to a tile and edge in the assembly.

hat_assembly_6 = [ [[0,0]] ,[[0,0],[0,1]] ,[[0,5],[0,2]] ,[[0,3],[0,4]] ,[[0,13],[0,6]] ,[[1,1],[0,9]] ,[[0,7],[0,12]] ];

module hat_tile_6(d) { p = hat_peri (10,d); t=peri_to_points(p); m=peri_to_points(mirror(p)); unit=group_tiles([t,m],hat_assembly_6); fill_tiles(unit,["red","green","blue","yellow","pink","black","coral"]); }

Parametric tiles

More amazingly still, this tile turns out to be one of a family of aperiodic tiles, which in the limit are simple periodic tiles. They are beautifully animated here. The parametric perimeter description enables all members of this family to be constructed:

function hat_peri_family(r,d) = // r is the radius of the base hexagon

// d is the angle which varies from 0 to 90 // returns perimeter as a sequence of [side,internal angle] let (a = r * sin(d), b = r * cos(d) ) [[b,180],[b,120],[b,270],[a,120],[a,90],[b,120],[b,270],[a,120], [a,90],[b,240],[b,90],[a,240],[a,90],[b,120]]);

At the limits , the tiles tesselate:

d=0 (nicknamed the comet)

d=45

d=90 (nicknamed the chevron)

All the tiles inbetween are aperiodic:

The Turtle

David also discovered a second tile he called the turtle. Defined as a perimeter, this also has 14 sides and is a family of tiles:

function turtle_peri(r,d) = // r is the radius of the base hexagon // returns perimeter as a sequence of [side,internal angle] let (a = r * sin(d), b = r * cos(d) ) [[a,240],[a,90],[b,240],[b,90],[a,120],[a,180],[a,120],[a,270],[b,120],[b,90],[a,120],[a,270],[b,120],[b,90]];

d=60

The periodic endpoints d=0 and d=90 are the same as for the hat.

This is a small section of a tiling in David's blog:

No mirroed tiles in this example.

turtle_assembly_6 = [ [[0,0]] ,[[0,12],[0,13]] ,[[0,13],[0,2]] ,[[0,5],[0,6]] ,[[0,4],[0,7]] ,[[0,5],[0,10]] ,[[0,10],[0,11]] ];

To do

I'd like to add a couple of the suggested decorations.

A big task is to automate the tile placement to construct an assembly. Craig's software does this so I dont expect it to be easy! The approach I'd like to explore (perhaps it's the obvious appraoch?) is to start with the perimeter description of the base tile and extend this as each tile is added. The next tile can be placed by matching a subsequence of the tile perimeter to a subsequence of the expanding perimeter. Bit complicated to ensure that such a match will result in a fit especially with concave shapes.

Further

Veritasium has a nice youtube on the background to aperiodic tiling

A pedestrian street in central Helsinki, the Keskuskatu, is paved with a Penrose tiling—the most famous example of an aperiodic tiling. This means that the same basic building blocks repeat everywhere throughout the street, but the way they fit together does not: the tiling lacks translational symmetry.

The street paving was finished in 2014, and yes, you can pay a virtual visit on Google Maps!

An Aperiodic Tessellation Featuring Regular Decagons and Regular Pentagons, Along With Concave Pentagons and a Few Rhombi

It is not possible to tessellate a plane using only regular pentagons and decagons, but that doesn’t stop me from trying. To get it to work, I had to add rhombi and concave pentagons — but even then, it’s an aperiodic tessellation.