# 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.

en wiki user LEMeZza, CC BY-SA 3.0

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:

#### References

[1]

**by***Infinite families of monohedral disk tilings***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.

## Archive

Show me a random blog post**2019**

### Sep 2019

A non-converging LaTeX documentTMiP 2019 treasure punt

### Jul 2019

Big Internet Math-Off stickers 2019### Jun 2019

Proving a conjecture### Apr 2019

Harriss and other spirals### Mar 2019

realhats### Jan 2019

Christmas (2018) is over**2018**

**2017**

**2016**

**2015**

**2014**

**2013**

**2012**

## Tags

cross stitch final fantasy coins reuleaux polygons european cup rugby folding paper gerry anderson geometry approximation chess mathsjam statistics hexapawn python javascript oeis data london underground games football programming go london frobel chalkdust magazine platonic solids ternary radio 4 sorting matt parker latex flexagons craft curvature nine men's morris big internet math-off light asteroids game of life machine learning game show probability binary countdown wool books manchester science festival php pizza cutting chebyshev stickers people maths map projections plastic ratio a gamut of games christmas realhats menace accuracy error bars reddit folding tube maps polynomials golden ratio talking maths in public raspberry pi manchester video games twitter dataset tmip royal baby national lottery hats palindromes logic sport bodmas pythagoras puzzles weather station harriss spiral martin gardner electromagnetic field graph theory world cup dates cambridge speed noughts and crosses rhombicuboctahedron the aperiodical captain scarlet pac-man triangles interpolation fractals tennis golden spiral trigonometry christmas card dragon curves mathslogicbot braiding bubble bobble mathsteroids misleading statistics arithmetic draughts probability inline code sound news propositional calculus estimation**© Matthew Scroggs 2019**

Add a Comment