Maxwell's Demon

The Klein Quartic is a absolutely fascinating object and worthy of a post in its own right, or even a book. It is clear evidence of the explosion of imagination and creativity in geometry that was taking place in the nineteenth century, as it cut its ties to the “real world”. Since then it has turned up all over mathematics.

One way to consider the Klein quartic is as a generalisation of a regular polyhedron. The tetrahedron has three equilateral triangles meeting at each corner, the cube has three squares and the dodecahedron three pentagons. Three hexagons gives a tiling of the plane. Why stop there? What about three regular heptagons? There are important reasons why this does not work in a simple manner. By playing fast and loose with what we mean by “regular heptagon” however we can do something. One object we can make is the Klein quartic. It does not produce something like a sphere, as the tetrahedron, cube and dodecahedron do, instead it is more like a pretzel with three holes.

Combining these ideas with little spherical magnets, we can make a model of the Klein Quartic. To do this we obviously have to start by making a heptagon

You start with a ring of seven balls, then put another ring of 14 balls around it. Note as this happens the heptagon buckles into a saddle shape. This is because the balls naturally create angles of 120˚ at the corners. As we move round the shape therefore we turn through a total of 7*120 = 480˚, this is greater than 360˚. We say the resulting surface has negative Gaussian curvature. We may also consider the length of the second loop. It is roughly distance 2 from the centre of our shape, yet it has length 14. If it were a circle of radius 2 the circumference would be 2*2π, which is less than 14.

Two of these heptagons can fit together on an edge:

For fans of Indra’s pearls and sphere reflections the balls make a pretty pattern.

As the angle at the corner is 120˚ three will fit round a corner:

We could now continue this, bringing three heptagons together at each corner, but we want to create the finite object. Next attach an additional heptagon to each of the outer three:

Now connect the three outside heptagons together. to make a surface with three holes:

You need to repeat this four times, using a total of 24 heptagons. As you make them, be careful of one thing, the magnets line up so that you get  all N poles on one side of the surface and all S on the other. As you connect each surface, therefore, make sure that it agrees with the others:

When you have all four, put one at the center and then connect the others to each of its four holes

To finish, technically we should connect up the remaining six holes so each branch is connected to both the others. The resulting shape has three heptagons meeting at every corner, and a wonderful collection of symmetries many of which cannot be easily seen in this model, or any model in 3d!

Just for kicks, lets finish with the work of one of Klein’s contemporaries a Möbius strip: