mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

The end of coins of constant width

 2017-03-27 
Tomorrow, the new 12-sided one pound coin is released.
Although I'm excited about meeting this new coin, I am also a little sad, as its release ends the era in which all British coins are shapes of constant width.

Shapes of constant width

A shape of constant width is a shape that is the same width in every direction, so these shapes can roll without changing height. The most obvious such shape is a circle. But there are others, including the shape of the seven-sided 50p coin.
As shown below, each side of a 50p is part of a circle centred around the opposite corner. As a 50p rolls, its height is always the distance between one of the corners and the side opposite, or in other words the radius of this circle. As these circles are all the same size, the 50p is a shape of constant width.
Shapes of constant width can be created from any regular polygon with an odd number of sides, by replacing the sides by parts of circles centred at the opposite corner. The first few are shown below.
It's also possible to create shapes of constant width from irregular polygons with an odd number, but it's not possible to create them from polygons with an even number of sides. Therefore, the new 12-sided pound coin will be the first non-constant width British coin since the (also 12-sided) threepenny bit was phased out in 1971.
Back in 2014, I wrote to my MP in an attempt to find out why the new coin was not of a constant width. He forwarded my letter to the Treasury, but I never heard back from them.

Pizza cutting

When cutting a pizza into equal shaped pieces, the usual approach is to cut along a few diameters to make triangles. There are other ways to fairly share pizza, including the following (that has appeared here before as an answer to this puzzle):
The slices in this solution are closely related to a triangle of constant width. Solutions can be made using other shapes of constant width, including the following, made using a constant width pentagon and heptagon (50p):
There are many more ways to cut a pizza into equal pieces. You can find them in Infinite families of monohedral disk tilings by Joel Haddley and Stephen Worsley [1].
You can't use the shape of a new pound coin to cut a pizza though.
Edit: Speaking of new £1 coins, I made this stupid video with Adam "Frownsend" Townsend about them earlier today:

Infinite families of monohedral disk tilings by Joel Haddley and Stephen Worsley. December 2015. [link]
                        
(Click on one of these icons to react to this blog post)

You might also enjoy...

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li> <logo>
To prove you are not a spam bot, please type "n" then "u" then "m" then "b" then "e" then "r" in the box below (case sensitive):

Archive

Show me a random blog post
 2024 

Feb 2024

Zines, pt. 2

Jan 2024

Christmas (2023) is over
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

python frobel errors phd stirling numbers matt parker matrix multiplication preconditioning the aperiodical sport triangles countdown bubble bobble wave scattering martin gardner video games programming standard deviation matrix of minors golden spiral finite element method finite group estimation mathslogicbot chebyshev accuracy interpolation reuleaux polygons chalkdust magazine sobolev spaces flexagons oeis puzzles national lottery inverse matrices dates gaussian elimination platonic solids misleading statistics squares curvature probability statistics manchester science festival electromagnetic field rugby recursion football london underground royal baby newcastle pythagoras menace folding tube maps numerical analysis sorting dragon curves inline code weather station turtles pascal's triangle determinants hexapawn craft logo light nine men's morris draughts royal institution dataset numbers fence posts palindromes books approximation live stream dinosaurs anscombe's quartet captain scarlet plastic ratio game show probability graphs pi approximation day bempp pi noughts and crosses arithmetic zines tennis signorini conditions fractals fonts map projections european cup matrix of cofactors pizza cutting mathsteroids quadrilaterals boundary element methods convergence gather town asteroids a gamut of games games hyperbolic surfaces weak imposition logs big internet math-off braiding propositional calculus data visualisation christmas tmip rhombicuboctahedron guest posts speed 24 hour maths reddit radio 4 geometry game of life realhats harriss spiral hats raspberry pi geogebra computational complexity correlation machine learning youtube ternary cambridge error bars pac-man datasaurus dozen crossnumber polynomials coins cross stitch latex news people maths edinburgh bodmas php mean christmas card javascript matrices stickers talking maths in public london hannah fry sound databet logic data graph theory go mathsjam final fantasy simultaneous equations manchester golden ratio advent calendar binary world cup chess folding paper trigonometry ucl gerry anderson wool crochet exponential growth runge's phenomenon

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2024