Life on a torus
The question
In some video games, walking off the right edge of the screen brings you back on the left, and walking off the top brings you back at the bottom. If that world were a real surface, what shape would it be?
Try the experiment before reading on. There’s no wrong way to explore.
Experiment
Surfaces from a square
Glue the sides of a square in pairs so that the arrows match. Walk a flatlander across a side and watch where — and how — it comes back.
Keys: ↑ forward, ↓ back, ← and → turn, Space walks or stops. You can also drag it.
- Orientable
- Yes
- Euler characteristic
- χ = 0
Gluing turns the 4 corners into V = 1, the 4 sides into E = 2 and the square into F = 1: V − E + F = 1 − 2 + 1 = 0
Same way round as at the start.
Sides crossed so far: 0
Opposite sides are glued with arrows pointing the same way. Walk off the right side and you come back on the left, the same way round. However it walks, the flatlander is never mirrored: the torus is orientable.
Hints, if you want them
What’s going on
What’s going on
On a wrap-around screen of width and height , a point has coordinates with and , and moving past an edge means counting on modulo or — clock arithmetic in two directions at once.
Gluing the left and right edges of a rectangle gives a cylinder. Gluing the two ends of the cylinder gives a torus: the surface of a doughnut. So the video-game world is a torus. A walker on it can go "around the hole" in two different ways — across the screen or up the screen — and neither loop can be shrunk to a point. Counting how many times a loop goes around each way is the start of the fundamental group of the torus, .
Now change the rule: walking off the top brings you back at the bottom mirrored, left and right swapped. The walker comes home as their own mirror image — their left hand has become a right hand. That surface is the Klein bottle, and a surface where this can happen is called non-orientable. The Klein bottle cannot be built in ordinary three-dimensional space without passing through itself, but the rule for its edges describes it perfectly well.
Mathematicians build many surfaces this way, by gluing the edges of a polygon, and the classification of surfaces says that every closed surface arises like this. Whether our own universe might be finite but wrap around in some such way is a question cosmologists have tested against observations; so far no evidence for it has been found.
Analogy, not a formal construction: The game screen is a flat rectangle with a rule for its edges; a doughnut in space is curved. They have the same topology but not the same geometry, so 'the screen is a doughnut' is a statement about shape up to stretching, not a picture of the game.
Where this leads
- Early undergraduate · Lesson availableSurfaces, polyhedra and the Euler characteristicA first course in the topologist's way of seeing: Euler's formula V − E + F = 2, the five Platonic solids, surfaces built by gluing the edges of polygons, orientability, the Euler characteristic as an invariant and the classification of compact surfaces.
- Advanced undergraduate · Lesson availableHomotopy and the fundamental groupHomotopy of maps and paths, the fundamental group, covering spaces, the computation π₁(S¹) ≅ ℤ, the Seifert–van Kampen theorem and applications such as the two-dimensional Brouwer fixed-point theorem.
- Research pathways · Lesson availableLow-dimensional topology: 3- and 4-manifoldsOptional research direction: 3-manifolds through Heegaard splittings, surgery and the eight Thurston geometries; the Poincaré conjecture and geometrisation (Perelman); hyperbolic 3-manifolds; and the strange world of 4-manifolds (Freedman, Donaldson, exotic ℝ⁴), separating established results from open questions.