Pictures of Maths IV

Untitled by Gary Greenfield

This image is produced by visualising the foraging behaviour of the ant P. barbatus. 1000 ants stream out of the nest along 6 main trails and break off to individually search for seeds. The image only highlights 500 of their steps. The colour graduations distinguish the “search phase from the return phase and reinforce the dynamic aspects of the process”.

“Bunny” by Craig S. Kaplan and Henry Segerman

This sculpture is self-referential; it is a sculpture of a bunny whose surface is tiled by 72 copies of the word ‘bunny’. It is an example of an autologlyph, which is “a word written or represented in a way which is described by the word itself”. This type of autologlyph combines “Escher-like tessellation with typographical ideas related to ambigrams“.

Intrinsic Transformation I by Conan Chadbourne

This work shows the structure of the icosahedral group. This is the smallest non-abelian simple group and is the set of “orientation-preserving symmetries of the regular icosahedron and dodecahedron”. The groups elements are shown as yellow disks arranged at the vertices of a truncated icosahedron, whilst the group’s generators – orders 2 and 5, which are coloured red and blue respectively – are depicted by the regions between the disks.

Visualisation of the musical piece “Five Armies” by Kevin MacLeod

Musical Flocks is a project which produces animations by simulating the behaviour of agents that react to the sound of music. Slow musics results in the slow and gentle movement of the flock, whilst fast tempo music results in fast movement and abrupt changes. Sounds which have a high volumes and a “rich” frequency spectrum affect the majority of the boids, whereas low volume and less active frequencies gives “more subtle visual variations and slower graphic evolution”.

Touch-graph by Annie Verhoeven

Verhoeven used ‘thread art’ to create the sculpture shown above. It visualises a computer program that is used to show connections between people; the coloured lines symbolise the different types of connections between people. “All these different connections are used in our life on every level, be it scientific, religious, political or otherwise, and can help us determine where we stand in the world.”

Stars of the Mind’s Sky by Paul Salomon

In this image, 300 starts are positioned along concentric circles. The number of points on the star increases as the radius from the centre increases, and the stars with the same number of points are placed evenly along their circle according to the density (“jump number”) used to generate them. Mathematically, this can be seen as representing subgroups and cosets generated by elements of a cyclic group. The different colours highlight the number of cosets, with the red stars representing the generating element. Thus, “we may observe congruent stars with increasingly many cosets, shifting their way to blue along central rays through any red star.”

The Yang-Baxter scarf by Robin Endelman

The scarf above depicts the Yang-Baxter equation in statistical mechanics. A variation of this equation is used in braid theory and in the 3rd Reidemeister move in knot theory. In this scarf, the numbers 1, 2 and 3 are assigned to the colours blue, green and gold respectively. The Yang-Baxter equation states that R₁₂ R₁₃ R₂₃ = R₂₃ R₁₃ R₁₂, where R_ij denotes the point where strand i crosses strand j; the two sides of the equation are highlighted by the two ends of the scarf, and the middle section demonstrates the equality (=).

Hyperbolic Fractal Tiling 1 by Robert Fathauer

This image shows four different views of the same 3D object – a fractal tiling where every tile has a similar dart shape. This demonstrates how a complex organic structure can be created from the repeated application of a simple set of rules to a simple starting structure.

Image credits to Discover magazine.

Read my previous posts in this series here. M x