Hyperbolic Tiling Explorer

Lay three regular hexagons around a point and they close up with nothing to spare. Each corner of a hexagon is 120 degrees, three of them make 360, and the pattern can go on across a floor forever. Now try the same thing with heptagons. It fails at once.

The reason is a single subtraction. Every interior angle of a flat regular polygon with n sides measures (n - 2) times 180, divided by n. For seven sides that comes to a little over 128.5 degrees, so three heptagons need about 385.7 degrees around a point that only has 360 to give. Paper heptagons taped three to a corner cannot lie flat, because there is more paper at every corner than a tabletop has room for.

The corner that asks for too much

Geometers write a regular tiling as two numbers in braces: {p,q} means tiles with p sides, q of them meeting at every corner. The hexagon floor is {6,3}. The whole question of which pairs work, and where, comes down to comparing the corner angles with a full turn.

If the q angles add up to less than 360 degrees, the tiles fold upward and close into a solid; {5,3}, three pentagons at a corner, is the dodecahedron. If they add up to exactly 360, the tiling is flat, and only {3,6}, {4,4} and {6,3} manage it. If they add up to more, as with {7,3}, the tiling belongs to the hyperbolic plane. Written as fractions, the hyperbolic case is 1/p + 1/q less than 1/2. Multiply that out and it becomes a rule that can be checked on fingers: (p - 2)(q - 2) > 4.

That leaves infinitely many hyperbolic tilings against three flat ones and five solids. The flat plane is the rare, knife-edge case.

What makes the excess workable is a fact with no counterpart on a tabletop. In the hyperbolic plane the angles of a polygon depend on its size. A tiny heptagon is almost flat and has corners near 128.5 degrees. Make it bigger and every corner sharpens, without limit. Somewhere along the way the corners pass through exactly 120 degrees, and at that one size three heptagons fit around a point perfectly.

So a hyperbolic tiling has no choice about its scale. A square floor tile can be any size you like. A {7,3} heptagon comes in one size only.

One heptagon, worked through

That size can be computed. Join the centre of the heptagon to a corner and to the middle of an adjacent side, and you get a right triangle whose other two angles are 180/7 degrees at the centre and 60 degrees at the corner, half of the 120 that the tiling demands. Hyperbolic trigonometry has a rule for right triangles that flat trigonometry lacks: the hyperbolic cosine of the long side equals the product of the cotangents of the two acute angles.

For the heptagon that product is 2.0765 times 0.5774, or 1.1989. The distance from centre to corner is therefore 0.6207, in the natural unit of a plane with curvature minus one. The side of the heptagon comes out as 0.5663. A flat heptagon with that side would have its corners 0.6525 from the centre, noticeably farther; using the flat figure leaves gaps that widen with every ring. In the explorer, set p to 7 and q to 3 and press the measuring buttons: one edge reads 0.56626 and centre to corner reads 0.62067.

The angles tell the same story from another side. Seven corners of 120 degrees sum to 840 degrees, where a flat heptagon has 900. The missing 60 degrees is called the defect, and in the hyperbolic plane the defect of a polygon, taken in radians, is its area. This heptagon covers pi/3, about 1.0472 square units. Johann Heinrich Lambert first showed that the area of a hyperbolic triangle is proportional to its defect, and the modern statement, the Gauss-Bonnet theorem, covers every polygon.

Cut the heptagon into its fourteen right triangles and you reach the smallest piece of the pattern. Each has angles of 90, 60 and 180/7 degrees, and mirror images of that one triangle, reflected over and over in its own sides, fill the plane. Mathematicians call the group of those reflections the (2,3,7) triangle group, and its triangle is the smallest of all such hyperbolic triangles. Coloring the triangles alternately gives the pinwheel picture that the explorer shows first.

A map of an endless plane

None of this can be built at full size in the space we live in, so it has to be drawn as a map, and every flat map of a curved thing distorts something. The best known map squeezes the entire hyperbolic plane into the inside of a circle. It carries the name of Henri Poincaré, who used it in 1882 and described it for general readers in a book of 1905, although Bernhard Riemann had described it in a lecture of 1854 and Eugenio Beltrami published it in 1868.

The rule of the map is simple to state. A point at true distance d from the centre is drawn at tanh(d/2) of the way to the rim. Going the other way, a mark halfway out is 1.10 units from the centre, a mark nine tenths of the way out is 2.94 units away, and a mark at 0.99 is 5.29 units away. The rim itself is never reached. It stands for infinity.

That is why the tiles appear to shrink. They do not. Every heptagon in the picture has the same side, the same corners and the same area of pi/3; the map simply spends less and less paper on each unit of distance as it nears the rim. The corners of the central heptagon sit at 0.3007 of the radius, its neighbors are already visibly smaller, and a few rings out they are specks.

There are a great many of those specks. Count the heptagons ring by ring around one tile and the numbers run 1, 7, 21, 56, 147, then 385, then 1,008: before long each ring holds about 2.6 times as many as the one before. In the plane of pentagons meeting four at a corner the rings run 1, 5, 15, 40, 105. A flat floor grows by a fixed number of tiles per ring. A hyperbolic one multiplies, which is the plainest way to feel how much room this plane has.

Straight lines suffer on this map as well. A hyperbolic straight line appears as an arc of a circle that meets the rim at right angles, or as a diameter. What the map keeps exactly is angle: two lines that cross at 120 degrees in the plane cross at 120 degrees on the paper, which is why the heptagon corners look right even when the heptagons look tiny.

Other maps, other honesty

A second disk map makes the opposite bargain. In the Klein disk, which Beltrami also published in 1868, every straight line is a straight chord of the circle, and the tiles become ordinary straight-sided polygons. The price is that angles are bent everywhere except at the centre, and circles flatten into ellipses toward the edge. The two disks agree on where each line meets the rim.

A third map unrolls the rim into a straight boundary line and draws the plane in the half of the page above it. Straight lines become half circles standing on that boundary, or upright rays. Angles are true here as in the Poincaré disk, and scale is easy to read: a figure twice as high above the boundary is drawn twice as large.

Behind all three sits a model with an awkward home: one sheet of a hyperboloid, a bowl-shaped surface in a space with its own rule for measuring distance, named after Hermann Minkowski. Put a flat disk one unit below the lowest point of the bowl and an eye one unit below the disk. Draw a line from the eye to each point of the bowl, mark where it pierces the disk, and the marks form the Poincaré map. Sight lines through the origin, caught on a plane, give the Klein disk instead.

Coxeter, Escher and the limits of the picture

These pictures were mathematicians’ diagrams until an artist saw one. Maurits Escher met the geometer Donald Coxeter in 1954. In 1957 Coxeter asked permission to use two of Escher’s drawings in a paper and sent him a copy. One of its figures showed the hyperbolic plane divided into triangles with angles of 30, 45 and 90 degrees, shrinking toward a circular rim. Escher wrote to Coxeter in 1958 that this figure had set him on the path to his Circle Limit prints.

According to his MacTutor biography, Escher could not follow the text and worked out the rules from the diagrams alone. The woodcut Circle Limit III followed in 1959. Coxeter later studied it closely and found something its maker had not known: the white curves that run through the fish are not hyperbolic straight lines at all. They meet the rim at about 80 degrees instead of 90, which makes them curves that keep a fixed distance from a straight line. Coxeter spent sixty years at the University of Toronto and died in 2003; Escher had died in 1972.

A computed picture has limits that are worth stating. It shows only regular tilings, one polygon repeated, and none of the figures an artist fits inside them. It stops where tiles fall below a pixel or two, since nothing smaller can be told apart, so the rim is always a haze and never a boundary. Efficient ways to generate such tilings without visiting any tile twice are still a topic of research papers.

The exponential room has a cost in arithmetic too. Travel far from the start and back along the same path, and tiny rounding errors are magnified by the same growth that multiplies the tiles. The explorer keeps its numbers small by always working from the tile under the centre, and it still measures a drift of under a millionth of the disk radius after going 20 units out and back, and about a pixel after 30. The tiling drawn at each moment remains exact. Only the memory of where you started fades.