A Piece of the Pi: mathematics explained
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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…
Aug 6
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Richard Green
5
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A Piece of the Pi: mathematics explained
How to make hyperbolic wallpaper
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1
July 2025
Spiral Sudoku
Suppose we have a 5×5 grid with 15 circled positions in a spiral shape as shown above.
Jul 23
•
Richard Green
10
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A Piece of the Pi: mathematics explained
Spiral Sudoku
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5
The binary sphere packing problem
What is the densest way to fill space with spheres of the same size?
Jul 12
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Richard Green
9
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A Piece of the Pi: mathematics explained
The binary sphere packing problem
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June 2025
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…
Jun 29
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Richard Green
3
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A Piece of the Pi: mathematics explained
Tiling the plane with congruent polygons
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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…
Jun 16
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Richard Green
8
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A Piece of the Pi: mathematics explained
Games in projective space
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2
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…
Jun 4
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Richard Green
12
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A Piece of the Pi: mathematics explained
Intersections of chords of a circle
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2
May 2025
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…
May 23
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Richard Green
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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…
May 21
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Richard Green
23
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A Piece of the Pi: mathematics explained
The geometry of musical keys
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2
A robot that unknots knots
Imagine a robot walking a single lap of a knotted track in three dimensional space.
May 13
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Richard Green
2
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0:14
A robot that unknots knots
Imagine a robot walking one loop of a closed-circuit outdoor track.
May 12
•
Richard Green
4
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A Piece of the Pi: mathematics explained
A robot that unknots knots
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5
April 2025
Anti-inductive dice
David and Goliath each have a six-sided die.
Apr 28
•
Richard Green
13
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A Piece of the Pi: mathematics explained
Anti-inductive dice
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4
Geodesics and polyhedra
A geodesic on a convex surface is a curve on the surface that is “locally straight”.
Apr 20
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Richard Green
9
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A Piece of the Pi: mathematics explained
Geodesics and polyhedra
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1
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