# Abdalla G. M. Ahmed

Originally I am a telecommunications engineer. Accidentally
discovering AA-patterns 15 years ago eventually lead me to know
Bridges and its community. These patterns are so simple yet so rich,
and they do not cease to inspire me. So far I found applications for
them in pixel art (still and animated), weaving design, and most
recently sampling (in computer graphics).

My artworks this year are all inspired by AA patterns.

"Are these cubes projecting out or engraved in" is a common
optical illusion.

Here we present a probably richer 3D illusion.

As I showed in a previous Bridges paper¹, interesting patterns are
obtained by connecting every seconds pair of a regular grid of
points (in rows and columns).

In this artwork we also connect every second pair of grid points,
but we use a triangular gird. In each row we start randomly with a
dash or gap. The resulting pattern looks complex, but a careful
inspection reveals that every vertex is trivalent. Since edges
meet at 120°, the angles of isometric projection, it gives the
illusion of cubic corners, sometimes in, sometimes out, sometimes
solid, and sometimes hollow.

This pattern is produced from AA(sqrt(3)) by connecting each point
to vertices of its Voronoi cell. This seemingly simple layout in
fact embodies a lot of math from different areas; e.g:

- Geometry: tiles are quadrilaterals with perpendicular
diagonals

- Tessellation: the few distinct tiles are an example set which
can tile the plane periodically or aperiodically

- Topology: pattern points can be distinguished by valence ( >
3)

- Symmetry: (partial) translational symmetries (high
auto-correlation) are evident at steps reflecting best rational
approximations of the pattern parameter

12-fold rotational symmetries are also visible thanks to the
parameter connection to angle π/6

Finally, it resembles Islamic arts, but is aperiodic!

This is similar to previous one, but instead of changing the grid
layout of AA outlines, we use a regular grid but make the dashes 2
steps long. The result is a puzzling layout of boxes. An artistic
application which comes to mind is carpets and wallpapers.

Besides its aesthetic appeal this structure can be the subject for
challenging problems in probability. For example, (easy) what is
the probability of having a block of 4 boxes (which are always
isolated), or (hard) what's the probability density function of
contiguous areas.

A possibly challenging problem in topology is to prove (or
disprove) that 4-boxes are the only possible disjoint blocks.