# Reuleaux Triangle

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 100 lt 120 lt 60 arc 100 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