Structure:
00:00 Introduction
00:27 "Asteroids" and topology
03:12 Mathematics in music
05:59 Castlevania and sine waves
10:05 Extraordinary maths in game engines
12:53 Mathematics in Islamic art
17:24 Portal's expressive level design
27:50 The Doppler effect brought out in level design
37:47 Closing comments

A lecture delivered at Game Connection Europe. I argue that in the pursuit of enjoyable, meaningful gameplay, we should look to mathematics for sublime and practical source material. I start with examples from Asteroids (Atari, 1979) and Castlevania (Konami, 1986), and I go on to look closely at the expression of the "fling" in Portal (Valve, 2007) and the Doppler effect in Music of the Spheres (me, 2013).

I point out that our industry is fortunately already full of mathematically informed people, and many cutting-edge game engines already contain fascinating maths. All we need to do is get into the habit of bringing out that maths in the mechanics, and to take a few hints from the expressive level design of games like Portal.

There's a tradition of expressing maths artistically that goes back thousands of years. In the past it has defined cultures, and I believe it can again.

Buy Music of the Spheres on Desura: http://www.desura.com/games/music-of-the-spheres
More about level design surrounding the Medusa head: http://www.kotaku.com.au/2012/02/finding-the-fun-medusa-heads-in-castlevania/
Here's a much-expanded version of the Portal analysis, including animations:
http://www.rockpapershotgun.com/2013/09/20/untold-riches-an-analysis-of-portals-expressive-level-design/

*A small clarification: I am aware that the things in Portal are not really wormholes, that real wormholes have many different properties. However, there are some very unique properties of wormholes that Portal does introduce, and that is still quite inspiring to me.

I'm indebted to Game Connection Europe for hosting the lecture, to Vi Hart for the "Doodle Music" video, and to Steven Malinowski (smalin) for the Contrapuntus 7 video.

You can visit the game connection website here http://www.game-connection.com/gameconn/