A Piece of the Pi: mathematics explained

A Piece of the Pi: mathematics explained

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Geometry 📐

Surfaces and sequence graphs
The golden ratio, φ=(1+√5)/2, is a famous example of an irrational number, which means that it cannot be expressed exactly as a ratio of two integers.
 • Richard Green
Colouring identical cuboids
How many colours do we need to colour a fixed stack of identical cuboids with integer coordinates in such a way that no two cuboids of the same colour…
 • Richard Green
Modular curve stitching
Consider the points of a clock face, numbered 1 up to 12, and for each point k, draw a straight line between k and 2k, taking remainders modulo 12.
 • Richard Green
How many cylinders can touch each other?
In 1968, J.E.
 • Richard Green
Transparent rectangle visibility graphs
A graph consists of a set of vertices, some of which are joined in pairs by edges.
 • Richard Green
How to make hyperbolic wallpaper
Given a repeating wallpaper pattern in the Euclidean plane, is there a good algorithm to turn it into a wallpaper pattern in the hyperbolic plane, as in…
 • Richard Green
The binary sphere packing problem
What is the densest way to fill space with spheres of the same size?
 • Richard Green
Tiling the plane with congruent polygons
It is impossible to tile the plane with regular pentagons of the same size, because the internal angle of a pentagon is 108°, and 360 is not a multiple…
 • Richard Green
Games in projective space
Projective geometry is a version of geometry in which there are points, lines, and planes, but in which there are no such things as distance or parallel…
 • Richard Green
Intersections of chords of a circle
The picture above shows a regular polygon with 15 vertices, including all the diagonals between the 15 vertices, and all the intersections between the…
 • Richard Green
Audio Note: Companion to “The geometry of musical keys”
The relationships between musical keys can be described either by means of the triangles in the “Tonnetz”, or by carefully designed geometric…
 • Richard Green
1:11
The geometry of musical keys
The Tonnetz (German for “tone network”) is a diagram representing musical tonal space, in which each vertex represents a musical note, and each small…
 • Richard Green
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