mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 2016-12-23 
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

Mathsteroids
Optimal Pac-Man
Visualising MENACE's learning
Harriss and other spirals

Comments

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
                 Reply
HyperRogue is an example of a game that takes place on the hyperbolic plane. Rather than looping, the map is infinite.

See: http://zenorogue.blogspot.com.au/2012/...
maetl
                 Reply
Hyperrogue may be the only game in existence with a hyperbolic surface topology: http://www.roguetemple.com/z/hyper/
zaratustra
                 Reply
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.
Olaf
                 Reply
I don't know about video-games but there are puzzles by Jeff Weeks (http://www.geometrygames.org/) on torus.
gaurish
                 Reply
 Add a Comment 


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

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "h" then "e" then "x" then "a" then "g" then "o" then "n" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

braiding bubble bobble preconditioning people maths matrices final fantasy raspberry pi martin gardner chalkdust magazine wave scattering matrix of cofactors binary rhombicuboctahedron game show probability noughts and crosses logic games manchester finite element method hexapawn tmip sound london curvature a gamut of games probability stickers matrix of minors chess pythagoras european cup football hats triangles world cup javascript dates data map projections reddit tennis christmas card cambridge matt parker speed mathsteroids matrix multiplication pizza cutting graph theory statistics weak imposition approximation the aperiodical golden spiral ternary sport game of life countdown christmas palindromes python bodmas simultaneous equations golden ratio draughts fractals twitter harriss spiral light polynomials plastic ratio manchester science festival royal institution coins determinants craft books inverse matrices gerry anderson misleading statistics error bars programming pac-man ucl dragon curves folding paper news weather station frobel chebyshev accuracy electromagnetic field php hannah fry data visualisation folding tube maps phd estimation latex propositional calculus rugby bempp mathslogicbot nine men's morris computational complexity video games machine learning reuleaux polygons royal baby go talking maths in public sorting mathsjam puzzles interpolation numerical analysis radio 4 flexagons wool cross stitch realhats signorini conditions platonic solids inline code advent calendar asteroids arithmetic national lottery trigonometry london underground sobolev spaces dataset menace boundary element methods geometry gaussian elimination oeis captain scarlet big internet math-off

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020