Colouring mathematical art

The n mice problem

Imagine n mice, each starting from a vertex of a regular n-gon. At the same time, each mouse tries to chase the mouse to its right. If we mark out their movement, we trace out n logarithm spirals, also known as curve of pursuit.

If instead we trace the path that each mouse wants to take at regular intervals, we end up with the interesting pictures illustrated below (for n being 3, 4, 5 and 6). While the drawings are made up only of straight lines, n logarithmic spirals naturally emerge.

While these shapes are interesting by themselves we can make more interesting drawings when we look at them with tiling of the Euclidean plane. Below are various tiling of the Euclidean plane with each tile consisting of a n mice problem. The notation [n^a.m^b] means that each vertex (where tiles meet), there is a regular n-gon, followed by b regular m-gon, etc.

While these drawings can be made using a pencil and ruler (see Tiling for Mindfulness, page 4, or Colouring for Mindfulness), all of the following were produced using the program Cinderella.2 or the TikZ package in LateX. Each pictures can be clicked on for a larger version that can be downloaded, some pictures also have a link to a pdf version.

Uniform tiling (k=1)

The first three are the three regular tilings in the Euclidean plane (i.e., those involving only one regular n-gon).
The next set are uniform tiling because all the vertex follow the same notation/set of polygon meeting.

2-uniform tiling [k=2]

These are pictures where there are two different set of polygons meeting at a vertex. The notation is the same as above, but with a semi-colon used to separate between vertex. For example, in the first picture at a vertex we could either have 6 triangles meeting, or 2 triangles and 2 hexagons meeting.

The hyperbolic plane

We can also turn our attention to the hyperbolic plane, in particular the Poincaré Disk Model. While the Euclidean plane had only three regular tiling (triangles, squares and hexagons), the hyperbolic plane has infinitely many. This is because in hyperbolic geometry the sum of the angles in a triangle add up to less than pi radiants (180 degrees). So there is a lot more freedom to the size of the angles in a regular polygon (for example you can have a pentagon whose internal angles are all right angles).

All the following drawings were done using Cinderella.2. The notation is the same as before, with a suffix of centred to say the centre of the unit disk is also the centre of a polygon, and the suffix vertex means that the centre of the unit disk is a vertex of our tiling.

Platonic Solids net

The following are nets of the 5 platonic solids, with each face having the n mice problem on them. Each of these were drawn using the TikZ package of LateX (hence why they are pdf format), you may need to scroll down to the second page of the pdf.

• Tetahedron (4 sided solid, i.e., Pyramid): Coloured; Uncoloured.

• Hexadron (6 sided solid, i.e., Cube): Coloured; Uncoloured.

• Octahedron (8 sided solid): Coloured; Uncoloured.

• Dodecahedron (12 sided solid): Coloured; Uncoloured.

• Icosahedron (20 sided solid): Coloured; Uncoloured.