Show me a Random Blog Post


folding paper folding tube maps london underground platonic solids london rhombicuboctahedron raspberry pi weather station programming python php inline code news royal baby probability game show probability christmas flexagons frobel coins reuleaux polygons countdown football world cup sport stickers tennis braiding craft wool emf camp people maths trigonometry logic propositional calculus twitter mathslogicbot oeis pac-man graph theory video games games chalkdust magazine menace machine learning javascript martin gardner reddit national lottery rugby puzzles advent game of life dragon curves fractals pythagoras geometry triangles european cup dates palindromes chalkdust christmas card bubble bobble asteroids final fantasy curvature binary arithmetic bodmas statistics error bars estimation accuracy misleading statistics pizza cutting captain scarlet gerry anderson light sound speed


Show me a Random Blog Post
▼ show ▼

Video Game Surfaces

In many early arcade games, the size of the playable area was limited by the size of the screen. To make this area seem larger, or to make gameplay more interesting, many games used wraparound; allowing the player to leave one side of the screen and return on another. In Pac-Man, for example, the player could leave the left of the screen along the arrow shown and return on the right, or vice versa.
Pac-Man's apparent teleportation from one side of the screen to the other may seem like magic, but it is more easily explained by the shape of Pac-Man's world being a cylinder.
Rather than jumping or teleporting from one side to the other, Pac-Man simply travels round the cylinder.
Bubble Bobble was first released in 1986 and features two dragons, Bub and Bob, who are tasked with rescuing their girlfriends by trapping 100 levels worth of monsters inside bubbles. In these levels, the dragons and monsters may leave the bottom of the screen to return at the top. Just like in Pac-Man, Bub and Bob live on the surface of a cylinder, but this time it's horizontal not vertical.
A very large number of arcade games use left-right or top-bottom wrapping and have the same cylindrical shape as Pac-Man or Bubble Bobble. In Asteroids, both left-right and top-bottom wrapping are used.
The ships and asteroids in Asteroids live on the surface of a torus, or doughnut: a cylinder around to make its two ends meet up.
There is, however, a problem with the torus show here. In Asteroids, the ship will take amount of time to get from the left of the screen to the right however high or low on the screen it is. But the ship can get around the inside of the torus shown faster than it can around the outside, as the inside is shorter. This is because the screen of play is completely flat, while the inside and outside halves of the torus are curved.
It is impossible to make a flat torus in three-dimensional space, but it is possible to make one in four-dimensional space. Therefore, while Asteroids seems to be a simple two-dimensional game, it is actually taking place on a four-dimensional surface.
Wrapping doesn't only appear in arcade games. Many games in the excellent Final Fantasy series use wrapping on the world maps, as shown here on the Final Fantasy VIII map.
Just like in Asteroids, this wrapping means that Squall & co. carry out their adventure on the surface of a four-dimensional flat torus. The game designers, however, seem to not have realised this, as shown in this screenshot including a spherical (!) map.
Due to the curvature of a sphere, lines that start off parallel eventually meet. This makes it impossible to map nicely between a flat surface to a sphere (this is why so many different map projections exist), and heavily complicates the task of making a game with a truly spherical map. So I'll let the Final Fantasy VIII game designers off. Especially since the rest of the game is such incredible fun.
It is sad, however, that there are no games (at leat that I know of) that make use of the great variety of different wrapping rules available. By only slightly adjusting the wrapping rules used in the games in this post, it is possible to make games on a variety of other surfaces, such a Klein bottles or Möbius strips as shown below.

If you know of any games make use of these surfaces, let me know in the comments below!

Similar Posts

Optimal Pac-Man
Proving Pythagoras' Theorem
The Mathematical Games of Martin Gardner


Comments in green were written by me. Comments in blue were not written by me.
HyperRogue also has special modes which experiment with other geometries (spherical, and elliptic which is non-orientable). Hydra Slayer has Mobius strip and Klein bottle levels.
Zeno Rogue
HyperRogue is an example of a game that takes place on the hyperbolic plane. Rather than looping, the map is infinite.

Hyperrogue may be the only game in existence with a hyperbolic surface topology:
F-Zero GX had a track called Mobius Ring, that was... well, a Möbius ring.

F-Zero X had a more trivial track that was just the outward side of a regular ring, but it was rather weird too, because it meant that this was a looping track that had no turns.
I don't know about video-games but there are puzzles by Jeff Weeks ( on torus.
 Add a Comment 

I will only use your email address to reply to your comment (if a reply is needed).

To prove you are not a spam bot, please type "etogyx" backwards in the box below (case sensitive):
© Matthew Scroggs 2017