A Piece of the Pi: mathematics explained

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Geometry 📐

Penny graphs
A penny graph can be created from a non-overlapping arrangement of unit circles on a flat surface.
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Richard Green
2
How many triangles are there?
A well-known type of brain teaser invites the reader to count the number of triangles formed by dividing up a larger triangle using straight lines.
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Richard Green
3
Facially complete graphs
The Four Colour Theorem proves that no more than four colours are required to colour the regions of any map in such a way that no two adjacent regions…
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Richard Green
8
Knots and the Menger sponge
The Menger sponge is a fractal formed by iteratively subdividing a cube into 27 equal cubes, and then removing the central cube of each face and the…
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Richard Green
3
Tilings and metallic means
Two of the most famous two-dimensional models of quasicrystals are the Penrose rhomb tiling and the Ammann–Beenker tiling. Each of these types of tiling…
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Richard Green
2
Wallpaper groups and interlocking assemblies
The patterns in the picture above illustrate an example of a wallpaper group, which is a two-dimensional repeating pattern with a particular kind of…
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Richard Green
2
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.
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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.
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Richard Green
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
Generalizations of Euler’s Tonnetz
The picture above by Konstanze Rietsch shows a Tonnetz of type G2. This is an infinite tessellation of 30°–60°–90° triangles whose edges are labelled by…
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Richard Green
Grids and folding curves
This picture of a grid in the plane comes from the recent paper Edge-covering plane-filling curves on grid colorings: a pedestrian approach by Jörg…
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Richard Green
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