A Short History

Tilings: a short history of covering the plane

Look down at a tiled floor. Look at a brick wall, a honeycomb, a chessboard, the scales on a fish, a chain-link fence. Somebody — or something — has covered a whole surface with copies of a few shapes, with no gaps and no overlaps. That is a tiling, or a tessellation, and people have been making them for at least five thousand years.

What makes tilings interesting is that they are not a free-for-all. You cannot tile a floor with regular pentagons, no matter how clever you are. There are exactly three ways to tile a flat floor with one regular shape, exactly seventeen kinds of repeating flat pattern, and — if you are willing to leave the flat floor behind — infinitely many more worlds where the rules change. This essay is about how people worked all that out.

Why only three regular tilings

Start with the simplest question. Suppose you want to cover a floor with copies of one regular shape — all sides the same length, all corners the same angle. Which shapes work?

The answer comes from adding up angles at a corner. Wherever tiles meet, their corners have to fill up a complete turn, which is 360°. An equilateral triangle has 60° corners, and 6 × 60 = 360, so six triangles fit around a point. A square has 90° corners, and four of them make 360. A regular hexagon has 120° corners, and three make 360.

Now try a regular pentagon. Its corners are 108°. Three of them give 324° — a gap. Four give 432° — an overlap. Nothing fits, so regular pentagons cannot tile a flat floor. Go bigger and it gets worse: a regular heptagon has corners of about 128.6°, and no whole number of those makes 360 either.

Here is the part that makes the proof complete. A regular shape with n sides has an inside angle of

180° × (n − 2) ÷ n

For six sides, that gives 120°, and three fit. For more than six sides, each angle is greater than 120°, so even three overlap. Fewer than three polygons cannot surround a point. We have now ruled out every regular polygon after the hexagon, not just the pentagon and heptagon examples.

So there are exactly three regular tilings of the flat plane: triangles, squares, hexagons. Not "three that anyone has found" — three, provably, and the proof is the angle arithmetic above. This is the first taste of something that runs through the whole subject: tilings are governed by counting, and the counting is strict.

Bees, incidentally, use the hexagon. When a flat surface is divided into cells of equal area, the honeycomb arrangement needs the least total wall. That statement was conjectured for two thousand years and finally proved in 1999 by Thomas Hales.

A compact name for every regular tiling

Writing "regular heptagons, with three meeting at every corner" soon gets awkward, so mathematicians use a two-number name called a Schläfli symbol: {p,q}.

The order matters. {6,3} is the honeycomb: six-sided tiles, three at a corner. {3,6} uses triangles, six at a corner. These are different tilings, even though they use the same two numbers. They are partners called duals: put a dot at the centre of every tile in one pattern and connect dots in neighbouring tiles, and you draw the other pattern.

This notation lets one question cover every regular tiling. Does {p,q} live on a sphere, on a flat plane, or in the hyperbolic plane? Near the end of this essay, one short calculation will answer all three cases.

Regular tilings are only the beginning. If you allow more than one kind of regular polygon, the flat plane has eight semiregular tilings in which every corner has the same arrangement. The familiar pattern of octagons with small squares between them is one. If you allow irregular tiles, curved edges, or corners with different arrangements, the possibilities grow enormously.

Symmetry: the real subject

Here is the shift in thinking that turned tilings from decoration into mathematics. Instead of asking what shapes are in the pattern, ask what can you do to the pattern that leaves it looking exactly the same?

Slide a brick wall over by one brick and it looks unchanged. That slide is a symmetry of the wall. Spin a chessboard a quarter turn about the centre of any square and it looks unchanged: a rotation. Hold a mirror along the edge of a row of tiles and the reflection matches what was already there: a reflection. There is a fourth, sneakier one — slide and flip together, like the trail of footprints you leave walking in snow — called a glide reflection.

Collect every motion that leaves a pattern unchanged, and you have that pattern's symmetry group. Two patterns that look completely different can have the same group, and two patterns that look similar can have different groups. The group is the pattern's skeleton.

Groups compose: do one symmetry, then another, and the result is a symmetry too. That simple fact is what makes them mathematics rather than a list.

Seventeen wallpapers

If a flat pattern repeats in two different directions — like wallpaper, not like a stripe — then its symmetry group has to be one of exactly seventeen. Not sixteen, not eighteen. Any repeating flat pattern you have ever seen, or ever will see, belongs to one of seventeen families.

The Russian crystallographer Evgraf Fedorov proved the classification in 1891. George Pólya independently derived the seventeen in 1924, the same year Paul Niggli published another early account. Their work helped the result reach a wide audience. Crystallographers cared because the same reasoning in three dimensions tells you how atoms can stack, and there the answer is 230.

Long before any of that, artisans had found most of the seventeen by hand. Egyptian, Chinese, Roman and especially Islamic craftsmen filled walls with patterns of extraordinary sophistication; the tilework of the Alhambra in Granada, built in the 1300s, is the most famous example. It is often said that all seventeen groups appear there. Careful counts by later mathematicians find fewer — but many of the seventeen are unmistakably present, discovered by eye and by tradition, five centuries before anyone could prove the list was complete.

The Dutch artist M. C. Escher visited the Alhambra in 1922 and again in 1936, copied the patterns obsessively, and spent the rest of his life turning them into interlocking fish, birds and lizards. He worked out much of the underlying theory for himself, in his own private notation, without the formal mathematics.

Reading a symmetry group's name

Mathematicians eventually needed short names for the seventeen. The one this app uses is orbifold notation, invented by John Conway, and it is worth learning because you can read it off a picture.

Two symbols do most of the work. A star * means "there are mirror lines". Digits after the star describe corners where two boundary mirrors cross. A digit n means that the angle between those mirrors is 180° ÷ n. Reflecting in both mirrors, one after the other, produces a rotation of 360° ÷ n, so the digit also tells you the order of rotation at that point.

For example, the 6 in *632 describes a 30° corner because 180 ÷ 6 = 30. Twelve copies of that narrow corner fit around a point, alternately reflected, while the finished pattern repeats after a 60° rotation. The notation is compact, but it records a complete recipe for rebuilding the pattern from one small mirrored region.

Conway also found a beautiful way to prove the list of seventeen using this notation. Each symbol is given a price, the total for any legal pattern must come to exactly 2, and enumerating the ways to spend exactly 2 gives seventeen possibilities and no more. He called it the magic theorem.

Patterns that never repeat

For a long time everyone assumed a tiling made from a few shapes would eventually have to repeat. In the 1960s and 70s that assumption fell apart.

Roger Penrose found, in 1974, a pair of simple four-sided tiles that cover the plane in a pattern that never repeats — you can slide it as far as you like and it never lines up with itself again. Yet it is not random: it is full of five-fold rosettes, and any patch you find somewhere appears infinitely often elsewhere. Order without repetition.

Then in 1982 Dan Shechtman found real crystals doing this, with the "impossible" five-fold symmetry showing up in his diffraction pictures. His result was rejected and ridiculed for years — he was asked to leave his research group over it. He received the Nobel Prize in Chemistry for it in 2011.

The question that stayed open was whether a single shape could do it. In March 2023, David Smith, an amateur working at his kitchen table, found one: a thirteen-sided shape now called the hat. With Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss he proved it works. Later the same year the team found the spectre, which does it without needing mirror-image copies. A problem open for sixty years was cracked, and it started with someone cutting out paper shapes.

Leaving the flat floor

Now back to the pentagon that would not fit, and the heptagon that would not either. They fail because the corner angles do not add to 360°. But why must they add to 360°?

Because the floor is flat. That is the only reason.

Take a ball instead. Draw a triangle on it with three right angles — start at the north pole, go down to the equator, along a quarter of the equator, and back up. Its angles add to 270°, not 180°. On a sphere, triangles are fat: their angles add to more than 180°, and shapes fit together differently. This is why a football can be stitched from pentagons and hexagons even though pentagons cannot tile a floor.

Now imagine bending the other way — a surface curved like a saddle or a Pringle at every point, in every direction, forever. On such a surface triangles are thin: their angles add up to less than 180°. There is more room around every point than a flat plane has. And with more room, the shapes that did not fit suddenly do.

The word "surface" can be misleading here. A saddle you can hold is only a small patch, and it sits inside ordinary three-dimensional space. The hyperbolic plane is the complete geometry obtained by continuing that negative curvature everywhere. Mathematicians do not need to bend it inside a larger room. People living in it could measure distances and angles entirely from within, just as we can measure the Earth's curved surface without looking at it from space.

This is the hyperbolic plane, and on it there are infinitely many regular tilings. Seven heptagons around a point? Fine. Five squares? Fine. Any pair of numbers works as long as there is enough room, which turns out to mean a single clean condition: writing {p,q} for "p-sided tiles, q of them around each corner", the tiling is hyperbolic exactly when

(p − 2) × (q − 2) > 4

The test comes from the flat angle formula above. On a flat plane, q copies of the polygon's inside angle must total 360°:

q × 180° × (p − 2) ÷ p = 360°

Cancel 180°, rearrange, and you get (p − 2) × (q − 2) = 4. Hyperbolic polygons have smaller angle sums than flat ones, so cases above 4 gain enough room to fit. Spherical polygons have larger angle sums, giving the cases below 4.

Check it: {6,3}, the honeycomb, gives 4 × 1 = 4 — exactly flat. {4,4}, the chessboard, gives 2 × 2 = 4 — flat again. {3,6}, the triangle tiling, also 4. Those three are the three regular flat tilings from the beginning of this essay, and they are precisely the cases where the product equals 4. Anything less than 4 lives on the sphere; anything more lives in hyperbolic space. One inequality sorts every regular tiling into one of three worlds.

Two hundred years of trouble

The hyperbolic plane was not discovered by someone hunting tilings. It came out of a two-thousand-year argument about a single line in Euclid's Elements.

Euclid's fifth postulate says, in effect, that through a point beside a line there is exactly one parallel. It is wordier and less obvious than his other four, and for twenty centuries mathematicians tried to prove it from them. Everyone failed. Some published proofs that turned out to assume what they were proving.

In the 1820s and 30s, Nikolai Lobachevsky in Russia and János Bolyai in Hungary independently did the audacious thing: they assumed the postulate was false — many parallels through the point — and worked out the consequences, expecting a contradiction. None came. Instead a whole new geometry unfolded, consistent and strange. Carl Friedrich Gauss had reached the same place years earlier but published nothing, fearing the controversy.

Bolyai's father, himself a mathematician, had begged him to leave the problem alone: "I entreat you, leave the science of parallels alone... it may take all your time, and deprive you of your health, your peace of mind and your happiness in life." His son took no notice, and wrote to him instead: "Out of nothing I have created a strange new world."

That world is where {7,3} lives.

Drawing an infinite world on a finite screen

The hyperbolic plane has a practical problem: it does not fit. It has too much room. You cannot lay it flat on a page without distorting it, in the same way you cannot flatten an orange peel — which is why every world map lies about something.

So mathematicians built models: honest, complete pictures of the hyperbolic plane that each tell one particular kind of lie. Eugenio Beltrami built the first ones in 1868, and Felix Klein and Henri Poincaré developed them further; their names are on the two you meet first.

The Poincaré disk squeezes the entire infinite plane inside a circle. The rim is infinitely far away — it is not an edge you can reach, it is the horizon. Tiles look smaller and smaller as they approach it, but that is the picture lying about size, not the tiles shrinking. Every one of them is the same size as every other. What the disk gets right is angles: every angle in the picture is the true angle, which is why the patterns look so convincing.

This trade-off is unavoidable. A map of Earth can preserve directions, areas, distances, or straight routes, but not all of them at once. Pictures of the hyperbolic plane face the same problem. The Poincaré disk preserves angles but curves most straight lines. The Klein disk makes straight lines look straight but changes angles. Neither picture is wrong; each is designed to answer a different question.

Escher met this picture in 1958, when the geometer H. S. M. Coxeter sent him a paper containing a hyperbolic tiling diagram. Escher had been struggling for years with how to show infinity in a finite frame, and here was the answer, handed to him in a journal offprint. He made the four Circle Limit prints from it. The two men corresponded for years afterwards; Escher once complained, affectionately, that Coxeter's mathematics was "hocus-pocus" to him, and Coxeter later wrote papers analysing exactly how mathematically precise Escher's construction had been.

What this app is doing

Everything above is what Kaleido lets you handle directly.

The flat scenes are three of the seventeen wallpaper groups. The curved scenes are hyperbolic {p,q} tilings, shown through six different models — six honest pictures of the same world, each lying about something different, and you can switch between them and watch the same tiling re-flow.

The rendering does not build the tiling tile by tile. There is no list of polygons anywhere in this app. Instead, for every pixel on your screen, it asks a question: if I follow the pattern's mirrors backwards from here, which part of the single master tile do I land on? Then it draws whatever is at that spot. Every pixel answers independently, which is why the app does not need a stored patch of tiles that eventually runs out. This is also how it keeps producing new tiles as they crowd toward the rim.

The mathematical rule describes an unbounded pattern rather than one large saved picture. The computer still uses finite-precision numbers, so at extreme views it has ordinary numerical limits; infinity belongs to the rule, not to the machine.

Four experiments to try

Reading gives you the vocabulary, but the quickest way to understand these ideas is to test them.

  1. Open the flat hexagonal scene and choose a line-only pattern. Find the 6-fold, 3-fold and 2-fold points named by *632. Turn the pattern and watch how far it moves before looking unchanged.
  2. Compare {7,3} with {3,7}. In the first, count the sides of one tile and the tiles at one corner. In the second, count again. The numbers exchange places because the two tilings are duals.
  3. Keep {7,3} and switch between the Poincaré and Klein disks. Follow the same tile edge. In one view it bows but keeps its angles; in the other it stays straight while the tile changes shape.
  4. Draw one short stroke across a guide line. Its reflected copies should join it. Move the stroke toward a corner and see how the order of rotation changes the flower it creates.

Each experiment asks you to compare something that changes with something that does not. That habit — looking for what a transformation preserves — is the central habit of geometry.

Why any of this matters

Symmetry groups turned out to be the language of crystals, and crystallography built modern chemistry, materials science and molecular biology — the structure of DNA was worked out with crucial evidence from diffraction photographs. Hyperbolic geometry, invented by people trying to settle an argument about parallel lines, now appears in relativity, network science, and the study of curved surfaces. Lettuce leaves and some corals even grow with similar negative-curvature patterns, crinkling because they cannot lie flat.

And none of that is why people make tilings. The craftsmen of the Alhambra were not doing crystallography. They were covering a wall beautifully, and they found their way to deep mathematics by paying close attention to what looked right. That is still a reasonable way to explore this app.