mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 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]

Similar posts

New machine unfriendly £1 coin, pt. 2
New machine unfriendly £1 coin
World Cup stickers 2018, pt. 3
World Cup stickers 2018, pt. 2

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>
To prove you are not a spam bot, please type "r" then "a" then "t" then "i" then "o" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Mar 2020

Log-scaled axes

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

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

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020