Why Penrose Tiles Never Repeat

these incredibly pretty geometric patterns are Penrose tilings

and if you've heard anything about them it's probably

that they never repeat themselves I mean they look pretty similar all over

and there are patches

that are perfect matches

but if you slide the whole thing over

and around it will never completely line up with itself again patterns like this

that go on forever

and feel like they should repeat

but don't are called quasi-periodic

but I never really felt like I understood these patterns like how do you

make them how do we know they don't ever repeat I just had to

take somebody's word for it

that they worked the way people say they do until recently

when I learned there's a hidden pattern inside Penrose tilings a pentagrid

and it's quite possibly the best way to understand pandro's tilings at least it's

what finally helped me feel like I understood them here's how you find the

pentagrid start with a single tile

and highlight neighboring tiles whose edges are parallel

and their neighbors and you end up with a wobbly ribbon of tiles

that snakes around a bit

but overall follows a straight path

and if you pick another tile with the same orientation you can make a

ribbon that's parallel to the first

and you can keep going here's a whole set of parallel ribbons of course

we could have started with the other edges of our original tile

and ended up with a different ribbon of tiles

and there's a whole parallel set of these ribbons too in fact jumping ahead

a little bit if we make a slightly more complicated version of the Penrose

tiling and color the tiles based on how they're oriented you see a whole

mess of ribbons jump out at you these ribbons are the key to understanding

Penrose tilings because the ribbons form a pentagrid

and what exactly is a pentagrid

if you take a regular array of parallel lines you can copy

and rotate it so it forms a grid you're probably most familiar with a

square grid where two sets of lines have been evenly rotated from one another

and intersect at 90 degrees you might also have seen a triangular grid where

three sets of lines have been evenly rotated

and intersect at 60 degrees

and if you create a grid with 5 sets of lines evenly rotated from

each other and intersecting at either 36

or 72 degrees you get a pentagrid pentagrids are made up of five sets

of parallel lines and Penrose tilings are made up of five sets of parallel

ribbons of tiles because they're actually the same to make a Penrose tiling all

you have to do is start with a pentagrid

and then at every point where two lines intersect you draw a tile oriented

so the sides of the tiles are perpendicular to the two lines this way

at the next intersection along the line the sides of

that tile will be parallel to the sides of the first tile

and the same at the next intersection

and so on and you can slide them all together into a ribbon

And if you do the same for the next lineup in the pentagrid you

get another ribbon and another

and if you also do it for all the other lines in the other

directions all the ribbons combined together make a Penrose tiling you can also just

add a tile to every intersection

and slide them all together along the grid lines either way you get a

Penrose tiling every Penrose tiling is made out of five infinite sets of parallel

infinitely long ribbons because every Penrose tiling is a pentagrid in Disguise of course

you don't have to use this particular pentagrid we can also shift the various

different sets of lines by random amounts

and get a beautiful new tiling that's slightly different from Penrose tilings

and and were not limited to a pentagrid here's a heptagrid

and its corresponding penrose-like tiling

and here's an OCTA grid a nanogrid a Deca grid

and so on and

that beautiful ribbony pattern we showed before was from a grid with 17 different

sets of lines my friend atish made an interactive website where you can play

around with all of this

and make your own penrose-like patterns you can highlight the grid lines

and see their counterpart tiles

and vice versa you can change the colorings to bring out different aspects of

the patterns you can use it to generate a bunch of other famous patterns

you can save them for a phone

or computer background or to print on a shirt

or whatever it's really great

and as you've probably noticed it's where all the visuals in this video are

from but there's one more thing remember how I said

that pentagrids helped me see why these patterns never repeat themselves this isn't a

proof but it at least gives you a flavor of the non-repetition

so start with a single ribbon

if the ribbon ever did repeat itself

then after a certain point in time you'd have the same pattern of thin

and wide tiles over again

and again and again

so the ratio of thin to wide tiles would be a rational number the

number of thin tiles in a given chunk divided by the number of wide

tiles in this example there are six wide tiles for every four thin ones

in an actual Penrose tiling we can directly calculate the ratio of thin tiles

to wide tiles since the ribbons of tiles correspond to the intersections along the

line of the pentagrid the wide tiles are from the intersections with the 72

Degree lines and the thin tiles from the 36 degree lines some basic trigonometry

shows that the spacing between 36 degree lines is 1 over the sine of

36 degrees and the spacing between 72 Degree lines is 1 over the sine

of 72 degrees so the ratio of wide tiles to thin tiles is the

ratio of these which happens to be the golden ratio

which is irrational so there's no way the pattern could ever repeat

if it did the Golden Ratio would have to be rational remember

if the pattern did repeat the ratio of wide to thin tiles would have

to be rational which the golden ratio isn't of course this just proves the

tiling can't repeat in One Direction the hole proof is a little bit more

than we want to get into here the pentagrid allows us to directly calculate

that as you go out along any ribbon in a Penrose tiling for every

10 thin tiles you see there are on average 16.18 y tiles a golden

ratio worth and because the golden ratio is irrational sometimes there are slightly more

wide tiles for every 10 thin ones

and sometimes they're slightly fewer in a way

that is perfectly predicted by the value of the golden ratio

but never repeats and the more tiles you look at the more closely their

ratio matches the golden ratio of course there's nothing special about the golden ratio

here it happens to show up a lot

when you have five-sided things for the heptagrid

or Deca Grid or whatever the ribbons still don't repeat

because the ratio of the spacings with the grids

and the ratios of the numbers of types of tiles is some other irrational

number all these patterns are quasi-periodic they may never repeat

but they also aren't just a random jumble of tiles all right go play

with the beautiful Penrose tile patterns over at autishbead.com pattern collider

and send the prettiest ones to me on patreon at minutephysics

and speaking of beautiful geometric patterns head over to brilliant this video sponsor for

their interactive course on beautiful geometry you'll explore how to make tessellations fractals infinite

tilings and more brilliant has dozens of courses covering broad swaths of math

and science and there's something for everyone from entertaining puzzles to clever problem solving

strategies for high school math competitions to black holes actually all of those subjects

are for me you can choose your own by signing up for free at

brilliant.org minut physics the first 200 people get 20 off an annual premium subscription

with full access to all of Brilliance courses

and puzzles and more exclusive content added monthly

or you can give a brilliant subscription to somebody

as a gift again that's brilliant.org minutephysics

and thanks to brilliant for their support foreign