Flatland, Sphereland, and Beyond: a Journey Through Two-Dimensional Universes

Flatland, Sphereland, and Beyond: A Journey Through Two-Dimensional Universes

Maxime Fortier Bourque

University of Glasgow

September 4, 2019 Flatland, Sphereland and Beyond We get a cube:

Gluing squares together

What happens if we take several identical squares and glue them along their edges so that 3 meet at each vertex? Gluing squares together

What happens if we take several identical squares and glue them along their edges so that 3 meet at each vertex?

We get a cube: We get an inﬁnite plane:

Gluing squares together

What happens if we take several identical squares and glue them along their edges so that 4 meet at each vertex? Gluing squares together

What happens if we take several identical squares and glue them along their edges so that 4 meet at each vertex?

We get an inﬁnite plane: We get a tetrahedron:

Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 3 meet at each vertex? Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 3 meet at each vertex?

We get a tetrahedron: We get an octahedron:

Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 4 meet at each vertex? Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 4 meet at each vertex?

We get an octahedron: We get an icosahedron:

Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 5 meet at each vertex? Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 5 meet at each vertex?

We get an icosahedron: Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 5 meet at each vertex? We get an inﬁnite plane:

Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 6 meet at each vertex? Gluing triangles together

What happens if we take several identical equilateral triangles and glue them along their edges so that 6 meet at each vertex?

We get an inﬁnite plane: We get a dodecahedron:

Gluing pentagons together

What happens if we take several identical regular pentagons and glue them along their edges so that 3 meet at each vertex? Gluing pentagons together

What happens if we take several identical regular pentagons and glue them along their edges so that 3 meet at each vertex?

We get a dodecahedron: Gluing regular n-gons together, k per vertex

What happens if we take several identical regular n-gons and glue them along their edges so that k of them meet at each vertex? H HH k H 3 4 5 6 7 n HH

3 bent

4 bent bent bent

5 bent bent bent bent

6 bent bent bent bent 7 bent bent bent bent bent The ﬁve Platonic solids

We have essentially shown that there are only ﬁve Platonic solids:

(convex polyhedrons whose faces are congruent regular n-gons, with a constant number k meeting at each vertex)

The reason is that if k is large enough compared to n, then the sum of angles around a vertex will be at least 360◦, so the surface will either be ﬂat or bend in diﬀerent directions around the vertex.

The interior angles of a regular n-gon are equal to (n − 2)180◦/n. The ﬁve Platonic solids

For example, in the pentagon (n = 5) the interior angles are 108◦. and 3 × 108◦ = 324◦ < 360◦, but 4 × 108◦ = 432◦ > 360◦.

This also explains why regular pentagons cannot tile the plane, as 108 does not divide 360. Tilings of the plane

We also obtained the three tilings of the plane by regular polygons: Tilings of the plane

If we do not insist that the tiles be regular, then there are many tilings by pentagons: Yes, with a little bending of the imagination.

Wrapping up the plane

Recall that the rules were to glue identical regular n-gons, with k of them meeting at each vertex.

{3, 6}{4, 4}{6, 3}

Can we do this with a ﬁnite number of polygons instead? Wrapping up the plane

Recall that the rules were to glue identical regular n-gons, with k of them meeting at each vertex.

{3, 6}{4, 4}{6, 3}

Can we do this with a ﬁnite number of polygons instead?

Yes, with a little bending of the imagination. The square torus The square torus The square torus The hexagonal torus The hexagonal torus Back to Platonic solids

Just as regular polygons are approximations of the circle:

The Platonic solids are in some sense approximations of the sphere: Spherical tilings

In fact, we can project the edges of each Platonic solic onto a sphere with the same center to get a tiling by “spherical polygons”:

On the sphere, great circles (like the equator or meridians) play the role of straight lines. One key diﬀerence between spherical geometry and ﬂat geometry is that the the sum of the interior angles of any triangle on the sphere is more than 180◦. Beyond... or Hyperland

Similarly, the bent surfaces we saw with too much angle at each corner were approximations of a surface called the hyperbolic plane which looks like a saddle around each point. The Poincar´emodel

It is not possible to visualize the whole hyperbolic plane in 3-dimensional space, but we can make (distorted) 2-dimensional models. The Poincar´emodel

In the hyperbolic plane, the interior angles of any triangle add up to less than 180◦. In fact, the angle sum can be arbitrarily close to zero!

For example, there are regular hyperbolic triangles with interior angles 360◦/7. We can put 7 of these around each vertex! H H k HH 3 4 5 6 7 n HH

7 M.C. Escher’s drawings Finite hyperbolic surfaces

Just like we could wrap up a square or hexagon into a ﬁnite torus, the hyperbolic plane can be wrapped up into surfaces with more than one handle. This example is called Klein’s quartic: Summary

The three simplest 2-dimensional universes are: Flatland (the Euclidean plane) Sphereland (the sphere) Hyperland (the hyperbolic plane)

Each of them admits many diﬀerent symmetric tilings.

The ﬁrst and last can be wrapped up into smaller universes.

Conclusion: One simple rule (equal number of polygons at each vertex) can lead to a rich variety of mathematics (Platonic solids, tilings of the plane, curved spaces).