Mathematical Miscellany

Miscellaneous mathematical stuff that I found interesting at one point or another.

Reuleaux triangles

A Reuleaux triangle is the shape you get, starting from an equilateral triangle, and adding three circular arcs, drawn from each corner. It’s not actually a triangle, but what’s in a name?

The interesting thing is that a Reuleaux triangle is a shape of constant width. That means it can be used as a roller (but not a wheel). The animation below shows this – we can see that although the centre of the triangle moves up and down, the top surface of the rotating triangle is always level.

Rolling Reuleaux Triangle

Follow the link below to see more, get links to 3D printable versions, and Matlab code to play with it.

Leslie Matrices

*Leslie Matrices* are a tool for modelling population demographic
 dynamics. [Wikipedia](https://en.wikipedia.org/wiki/Leslie_matrix)
 does a decent job of explanation of these. My megre contribution
 is a little online
 [Leslie Matrix Calculator](http://bandicoot.maths.adelaide.edu.au/Leslie_matrix/leslie.cgi).

Log-azimuthal maps

A Log-azimuthal map of the world (as seen from Adelaide, Australia, my home town) using log-distances, is a map that shows places close by are displayed in detail, and those further away, have less.

Sea Shells

There are a large variety of sea-shells (and land shells) that can be simply described by rotating a ellipse around a log-spiral that has been projected onto a cone. The image below was generated using this approach:



Roulettes

A Roulette is a curve derived by rolling one curve against another. In the image below, the blue point on the blue ellipse is rolled along a straight line (the x-axis) to generate the purple curve (which is called an undularly).

roulette

Included here is some Matlab code to generate roulette curves, with a fair degree of generality, but also showing how to generate specific instances such as cycloids, trochoids, the Cissoid of Dioclese, and the undularly and nodary.