We could find a speech topic just by looking down at the
floor of a room. Our bathrooms have a grid consisting of brown 20“ square
tiles. (Back when I was growing up we had much smaller white hexagonal tiles).
You also could use triangles. There are Wikipedia pages for a Triangular tiling,
a Square tiling, and a Hexagonal tiling. The mathematical term is Tessellation,
for which Wikipedia says:
“A tessellation or tiling is the covering of a surface,
often a plane, using one or more geometric shapes, called tiles, with no
overlaps and no gaps."
Tiling can have translational symmetry (be periodic) or not (be
aperiodic). There is a Wikipedia page for Penrose tiling that begins:
“A Penrose
tiling is an example of an aperiodic tiling. Here, a tiling is a
covering of the plane by non-overlapping polygons or other shapes, and a tiling
is aperiodic if it does not contain arbitrarily large periodic regions
or patches. However, despite their lack of translational symmetry, Penrose
tilings may have both reflection symmetry and fivefold rotational symmetry.
Penrose tilings are named after mathematician and physicist Roger Penrose, who
investigated them in the 1970s.
There
are several variants of Penrose tilings with different tile shapes. The
original form of Penrose tiling used tiles of four different shapes, but this
was later reduced to only two shapes: either two different rhombi, or two
different quadrilaterals called kites and darts. The Penrose tilings are
obtained by constraining the ways in which these shapes are allowed to fit
together in a way that avoids periodic tiling.”
More recently single aperiodic tiles (monotile or ein stein
for one stone) were found, like the hat shown above. The hat got mentioned at the
beginning of a monologue at Jimmy Kimmel Live on March 29, 2023 For more
detail, watch this 5-1/ 2 minute YouTube video by Aylien on June 2, 2023 titled
Finally, a true Aperiodic Monotile. There also is an 18 minute video from Up
and Atom on September 3, 2023 titled How a Hobbyist Solved a 50-Year-Old Math
Problem (Einstein Tile).
There also are a Turtle, and a Spectre.
If you want to see the math, there is one 91-page pdf article by David Smith et al. at Combinatorial Theory in 2024, (Volume 4,
Number 1) titled An Aperiodic Monotile. A second 25-page pdf article (also by David
Smith et al.) in Combinatorial Theory for 2024, Volume 4, Number 2 is titled A
Chiral Aperiodic Monotile.
How did I find this topic? I read about it in Matt Parker’s
2024 book Love Triangle: How trigonometry shapes the world. In his chapter
five, Well Fit, the final section is titled Don’t Make Me Repeat Myself:
“For a long time, the holy grail for mathematical tiling
patterns was a polygon that could perfectly cover a surface but in a way that
never repeats. One thing all the tiling patterns we’ve seen so far have in
common is that they repeat periodically. I only had to give the builders a
small diagram of snub-square tiling because once they got the pattern correct
it could be repeated forever. Easy.
Mathematicians dreamed of a shape that sat right on the cusp
between order and chaos. Some polygons are able to cover a surface with no gaps
in a neat pattern, and others cannot fit together without gaps. But imagine a
shape that brings both sides together; it cannot form a repeating pattern yet
it still can cover a surface.
This mystical tiling pattern is called ‘aperiodic.’ A lot of
tiles can form a ‘non periodic’ pattern: square tiles can be arranged with each
row offset a different, irrational amount from the previous. Technically, this
is a pattern that never repeats. But an aperiodic pattern involves the stronger
condition that it is impossible to arrange the tiles in a periodic fashion.
Square tiles could be knocked back into a periodic pattern and so they don’t
count.
The first set of aperiodic tiles was found in 1964, but it
involved combining 20,426 different shapes of tiles together. By 1974 this had
been reduced to a set of two shapes, called Penrose tiles, which were aperiodic
as a team, but the search was still on for a monotile that could be aperiodic all
on its own. This mysterious, hypothetical shape was often called the ‘einstein’
as a hilarious German-language pun on ‘one stone.’
Even though mathematicians had yet to find an Einstein tile,
they did know some things about what it must be like (if it did exist). Recall Rao’s
proof from 2017 showing that all the convex pentagons that could tile had been
found. This completed the search for all convex polygons, and every single one
that could cover a surface did so in a nice periodic fashion. If there was an
aperiodic monotile out there, it was not convex. It must have concave,
sticking-in bits.
In 2010 an Einstein was discovered! But it was a terrible
shape. The Socolar-Taylor tile, named after its discoverers, was an aperiodic
monotile but it wasn’t contiguous. Several little disjointed pieces all made up,
technically, ‘one tile.’ Having tiles each made from a collection of disparate
parts did feel unsatisfactory. In a follow-up publication the discoverers
described it as ‘an Einstein according to a reasonable definition’ Which is
absolutely true. But both mathematicians and builders agreed that each tile
being a solid piece was an even more reasonable definition.
Then in March 2023 it was found. The first ever aperiodic
monotile. See if you can guess if it was someone messing around on their
kitchen table at home or an advanced computer search! Answer to follow. I
remember the release vividly: the news broke on March 21 and on March 22 I was
due to give a public lecture at the Royal Society in London called ‘Every Interesting
Bit of Maths Ever.’ A swift rewrite ensued.
The excitement was instant. It swept through the maths world
very quickly, and the mainstream media was not far behind. Th mathematicians
who had found the shape had dubbed it ‘the Hat’ because thy thought it looked
like a hat. It has also been claimed to look a lot like a shirt. The point is,
it was a nice, tidy, public-friendly shape. Before long people were 3D printing
them, baking cookies shaped like them. My friend, Aliean MacDonald showed up
for my Royal Society lecture in a Hat-covered dress she had made herself.
There was something about the Hat that made it popular with
the public and mathematicians alike: it was surprisingly simple. Given this
shape had been eluding the entire mathematics community for over half a
century, nobody expected it to be so straightforward. It’s a 13-sided polygon,
far fewer sides than I would have predicted. It’s concave, as expected, but it
doesn’t have any detached, fragmented bits or any holes. When I look at it, I
see a slightly modified equilateral triangle. Even in the research paper
announcing its discovery says, ‘The shape is almost mundane in its simplicity.
None of this is to devalue the incredible feat of finding
the Hat. It was discovered by a retired print technician, David Smith, doing some
recreational maths at home on his kitchen table. He had been designing shapes
in a tiling software package when he outlined the Hat and realized there was no
obvious way to arrange it in a tiling pattern But it looked like it should fit
together nicely. David cut 30 of them out of cardboard and found they did fit
together but with no obvious pattern Another 30 copies were cut out and added
to the tiling; still no pattern.
He contacted mathematician Craig Kaplan, who used some
adopted software to explore how far the Hat could tile. It tiled further than
any other known non-tiling shape, which strongly suggested the Hat could indeed
cover any infinite surface. Yet the patterns it formed were not periodic. More
mathematicians were recruited, and soon they managed to prove that the Hat was
indeed an aperiodic monotile. For completeness they even proved it two
different ways. The first proof was done using a computer, which worked but
didn’t offer any insight into why the shape was aperiodic. As they said in the
paper, ‘These calculations are necessarily ad hoc, and are essentially
unenlightening.’ So they proved it again in a much more satisfactory way. There
was now no doubt this was the Einstein shape everyone had been looking for.
Then David found another one.
Dubbed ‘the Turtle’, it was a second example of an Einstein.
It felt wildly unlikely that two unrelated einsteins would be found so close
together by the same person. And, after a bit of digging, the tile team found
that the Hat and the Turtle were in fact two members of the same ‘family’ of
tiles. This is the same as how we consider all rectangles part of the same
family of shapes, each member of the family having a different ratio between
the two edge lengths. Actually, since the ratio in a rectangle can be anything,
the family of rectangles is infinite. The same is true of the Hat family, but
it’s a less straightforward ratio. The original Hat is made from two different
side-lengths ( 1 and Ö3)
and those lengths can be varied to produce other einsteins.
When the side-lengths are the other way around, Ö3 and 1, the resulting
shape is the Turtle. All the other ratios work as well, with three exceptions.
If the entire infinite family of Hat tiles was put in a line, and labeled with
their distinguishing two side-lengths, it would start with tile 0, 1 and end
with tile 1, 0. Both of these end tiles are not technically aperiodic. They can
be arranged to be nonperiodic but also have alternate periodic arrangements.
Strangely, the very middle 1, 1 tile is also not aperiodic.
For a shape to be aperiodic it needs to walk a very fine line between order and
chaos; too much order and it becomes periodic; too much chaos and it ceases to
completely cover a surface. Having the two edge lengths the same tips this
middle class into having just enough order to be periodic. But, on the plus
side, we are still left with infinitely many other shapes that do work.
Classic maths. You wait half a century for one aperiodic
monotile and then infinitely many of them show up at once. The only slight
disappointment was that all of these tilings use the reflection of the tile
within the tiling. Which is something the mathematicians are OK with, but
actual bathroom tiles and paving blocks come with a front and a back. So,
annoyingly, the Hat would not make a good bathroom tile. For that a new Einstein
will need to be found that tiles without using its reflection. We can only
hope.
And that hope has already paid off! In May 2023 the same
team came back with a chiral aperiodic monotile – one that tiles without using
reflections – just over two months after the first Einstein had been announced.
I will add that two months was the perfect amount of time for the
mathematics-communication community to have just finished work on all manner of
podcasts, videos, blog posts, and magazine articles telling the ‘definitive’
story of the Einstein before bam: all obsolete. (Goodness knows what will be
announced the second this book gets published.)
This new shape was named ‘the Specter’ [sic], and it was
also hiding in plain sight. David found it right in the middle of the Hat
family: it’s the shape with 1, 1 edges that we had previously discounted! All
of the Hat tilings that were aperiodic needed to use their reflections, but it
was the reflection of the 1, 1 tile that stopped it from being aperiodic. If
reflections were banned then it would become aperiodic. David and the team
realized that by curving the edges in a special way they could remove the
ability for the reflected version to fit at all, turning the Specter [sic] into
a ‘strictly chiral aperiodic monotile.’ Mission accomplished!
I feel like, over time, the general public gradually builds
up the capacity to pay attention to a breaking maths story (like a video-game
power bar), and the Hat came out at just the right time, depleting the
reservoir of excitement. When the Specter [sic] was announced two months later
it didn’t even register as a blip on the mainstream media or in the public
consciousness. Sure, maths people were super excited – this was arguably the
more amazing result – but the general populace had no need for another new
shape so soon. Even though this one is ideal for tiling a bathroom.
At the time of writing I am wondering what the next startling
new shape will be. It could come from anywhere. I have contacted the Hat team
to double-check they don’t have some other new tilings to be announced the
moment I finalize this manuscript. Because after they found the whole Hat
family and saw how nonexotic the shapes were, they wrote, ‘We might therefor
hope that a zoo of interesting new monotiles will emerge in its wake,’
I also hope it does. But not until the next edition of this
book.”
