A collection of maths based activites and articles.
Mostly with a geometrical flavour.
Cogs and gear ratios - simulation
Create your own gear trains. Drag different sized gears/cogs onto the board, connect a motor and choose the speed. How do the speeds of the cogs (in rpm) depend on the number of teeth they have?
In the first example we plot the angle of a pendulum against its speed. In the second example we plot the number of predators against the number of prey. In both cases we end up with rather unusual graphs - more usual would be to plot the position of the pendulum against time, and the number of each animal against time. But by choosing these alternative coordinates a picture of the system emerges called the 'phase plane'. In these activities you can experiment with different parameters and observe the phase portrait changing.
Image rescaling apparatus - London Science Museum
I visited the Maths section of the Science Museum in London recently. I've always found the interplay of engineering and maths interesting. Here is a Flash recreation of one exhibit - a simple but clever apparatus which uses the idea of similar triangles to accurately scale an image.
Requires Macromedia Flash Player
Two tile-based mathematical games. See if you can solve them.
Flash recreation of the fun pattern-making game Spirograph (TM). Choose what colour to use and then drag a circular disc around the inside or outside of another disc. Mathematically these patterns are called hypocycloids and hypercycloids.
Requires Macromedia Flash Player
L systems are a formal grammar (whatever one of those is) invented by Lindenmayer to study the structure of plant growth. Loosely speaking an L system is specified by a re-writing rule, which gives a string of letters to replace a given letter by. Starting with a single letter F (meaning one step forwards) and replacing it by the string gives a new longer string of letters, which encodes the shape of Plant 1 (1st generation). Feeding this string back and doing the replacement on that gives a more complicated longer string, encoding the information for Plant 2. Repeating this procedure can give a realistic looking fractal plant structure. Enter your re-writing rule and click 'draw', or use the presets. Rotate the planet using the mouse.
Requires Macromedia Flash Player
Create planets, moons and space craft.
Choose the initial velocities, click animate and let gravity take over.
In particular simulate elliptical orbital motion of planets/moons, and n-body motion.
Select the shape/outline of your surface and then create a rotatable surface of revolution. The faces are z-ordered by depth and coloured according to inclination to the observer.
Requires Macromedia Flash Player
This is a model of three balls in space that collide into each other and the walls.
Uses conservation of momentum to calculate paths, features gravity and damping.
This is the Icosahedron. Try rotating it - isn't it cool?
The Platonic solids are the convex solids all of whose faces are identical regular polygons, with the same number of polygons meeting at each vertex
It is written in flash and is part of a program I have been writing covering many different sorts of polyhedra, Platonic, Archimedean and Catalan.
I have recently rewritten them all in Actionscript 2.0 and now they run slower than before. Wahey.
Copyright Me, Requires Macromedia Flash
In July 2005 I visited a children's home near Chennai (formerly Madras) in South India.
Myself and colleagues organised Math-based activities for the children. My group made mathematical patterns with coloured string and a circle of nails.
Simple equipment you may think, but the patterns you get are quite interesting and artistic and can also lead to a good deal of higher level maths such as cyclic groups and equations for envelopes of straight lines.
Copyright Me, Requires Macromedia Flash Player
Choose the number, position and rotating speed of linked struts.
You can choose up to four linkages and vary their initial positions, rotating speed and length. Generate interesting patterns: ellipses, cycloidal curves, even straight lines if you're very clever!
Copyright Me, Requires Macromedia Flash Player
Wallpaper Groups
Create your own patterns based on the 17 plane symmetry (wallpaper) groups
These are the finite groups of symmetries of two-dimensional space, and there are exactly 17 of them. Draw on the red fundamental region and your drawing will be transformed according to the symmetry group, to cover the plane.
Copyright Me, Requires Macromedia Flash Player