# Reuleaux Triangle

`warning`

Once you learn about the Reuleaux Triangle you may start to see it everwhere! In buildings, in coffee tables, in kids playground equipment, in beer logos... Reuleaux Rules Our World!

## Ever wanted to drill a square-hole?

Is it possible to build a drillbit that would produce such a shape? It seems impossible, but how impossible is it? Can we drill holes that are any shape other than circular?

A circle is a curve of constant width. As you rotate a circle its width doesn't change. Does any other shape have this property?

Why yes! The Reuleaux polygons all do. The simplest and best known of which is the Reuleaux Triangle.

## How do you *draw* a Reuleaux triangle?

Draw an equilateral triangle, and then inscribe 3 arcs, each centred on a point of the triangle and travelling between the other two points.

lt 30 repeat 3 [ fd 200 lt 120 lt 60 arc 200 60 rt 60 ]

Or, draw three circles, where the centre of each circle touches the outside of the other two:

## What can you *do* with this special shape?

You can make a drill bit that drills a square hole! (The Harry Watt square drill bit, where the drillbit needs to be mounted in a special chuck which allows for the bit not having a fixed centre of rotation)

You can build rotary engines! *

You can make a bike that has ##non-circular wheels## but still gives a smooth ride.

- Regarding rotary engines:

The Wankel rotary engine isn't quite Reuleaux shaped. It says on wikipedia:

The four-stroke cycle occurs in a moving combustion chamber between the inside of an oval-like epitrochoid-shaped housing, and a rotor that is similar in shape to a Reuleaux triangle with sides that are somewhat flatter.

Here's a program for drawing multiple triangles.

to shapo :size :i :j repeat :i [ rolly :size lt :j ] end to rolly :size lt 30 repeat 3 [ pu fd :size pd lt 120 lt 60 arc :size 60 rt 60 ] rt 30 end shapo 150 24 15

## Challenge

Can you write a program that draws a Reuleaux polygon of any number of sides?

Hint: here's a program that works only for odd-numbers of sides.

to rolly :size :sides repeat :sides [ pu fd :size pd lt 180 arc :size (180 / :sides) rt (180/ :sides) ] end rolly 150 7

## External Links

- Smooth-riding Bike with Reuleaux Triangle back-wheel and Pentagonal front-wheel
- Rolling with Reuleaux
- Mathworld: Reuleaux Triangle
- Wikipedia: Reuleaux Triangle