A Piece of the Pi: mathematics explained

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Partitions and primes
The picture above shows all the partitions of the numbers 1 up to 8. For example, the partitions of 4 are the sequences of weakly decreasing positive…
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Richard Green
Arranging queens on a chessboard
What is the minimum number of chess pieces of a given type needed to attack every vacant square of the board? The answer turns out to be 9 for kings, 5…
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Richard Green
3
Friezes and Catalan numbers
A Conway–Coxeter frieze is an infinitely wide checkerboard of positive integers, such as the one in the picture above. Conway–Coxeter friezes were…
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Richard Green
The arithmetic derivative
Can you differentiate a number and get a nonzero answer? You can't with the usual kind of differentiation, but you can with the “arithmetic derivative…
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Richard Green
The Hadwiger–Nelson problem
What is the minimum number of colours needed to colour the plane in such a way that any two points that are at distance 1 from each other always have…
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Richard Green
2
Lozenges and Löschian numbers
These pictures come from the recent paper A lozenge triangulation of the plane with integers by Raghavendra N. Bhat, Cristian Cobeli, and Alexandru…
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Richard Green
PackIt! (a rectangle packing game)
The picture above comes from the recent paper PackIt! Gamified Rectangle Packing by Thomas Garrison, Marijn J.H. Heule, and Bernardo Subercaseaux…
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Richard Green
Picturing symmetry
The picture above by Matthew Macauley shows the multiplication table (or Cayley table) of the symmetries of a square. The symmetries themselves are…
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Richard Green
Edge colourings and Archimedean lattices
This picture of an edge colouring of an infinitely repeating graph comes from the recent paper Edge coloring lattice graphs by Joris Kattemölle. The…
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Richard Green
2
Pyritohedral symmetry
The mineral iron pyrite, also known as “fool’s gold”, usually forms cube-shaped crystals, but it can also form crystals like these. At first glance…
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Richard Green
The McNugget monoid
How many chicken nuggets can you buy if they are only sold in packs of 6, 9 and 20? The picture shows all the numbers from 0 to 44 that can be made from…
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Richard Green
9
Guillotine and diagonal rectangulations
The two pictures above are examples of rectangulations of size 7. Each one is a tiling of a rectangle by 7 smaller rectangles so that no four of the…
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Richard Green
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