Hexagons and Geospatial Encoding

Original article was published by Mark Cleverley on Artificial Intelligence on Medium


If you’re like me, you’ve always had a strange feeling that bees (much like dolphins) know more than they’re letting on.

They know architecture, and communicate navigational information through dance. Their elaborate caste-based hierarchy and collective survival impetus is striking. There is something mathematical about them, and that is deeply unsettling.

I’ve recently been looking into encoding data to binary representations for use in neuromorphic AI systems. I’ve always been fascinated with mathematical and artistic representations of space, distance and volume.

Searching for the intersection of these ideas has led me to the same destination as bees: the hexagon.

Let me make this quite clear: we are only starting to learn the extent to which honey-gathering insects know more than us. Bees build their hives with tessellating hexagons. What do they know?

A Hex Upon Us All

Evolution drives efficiency like a function approaching a limit. Consider bees building a honeycomb: they use wax (derived from honey) to build a structure to store honey. What’s efficient in this scenario?

  1. Maximizing honey storage capacity
  2. Minimizing wax use
  3. Ensuring stability

If you had to stack equally sized square boxes to take up the least amount of space, you’d line them up nicely to minimize wasted space in between — this is tessellation. But it wouldn’t work with octagonal boxes.

Only three shapes can tessellate without space in between: equilateral triangles, squares and hexagons. Why have bees chosen the latter?

I’m far from the first to eye our small yellow-striped neighbors with suspicion. 2,000 years ago, the great Roman scholar Marcus Terentius Varro proposed a “Honeycomb Conjecture”.

source

He reasoned that hexagonal hives are more compact, generally speaking, than other assortments, and had overall smaller perimeters. Thomas Hales formally proved this in 1999.

Bees that made inefficiently shaped hives stored less honey, which also had to be used to make more wax (for the same relative size). Colonies with the most efficient architecture outcompeted their rivals across the globe.

The image above demonstrates a this as an intuitive hypothesis: incremental improvements over time lead from circular comb-arrangement to hexagons. This ensures structural stability by connecting neighboring cells in a unified arrangement.

Krulwich at NPR also provides an interesting analysis of labor efficiency: with tessellating cell shapes, many workers can assemble and connect cells concurrently without need for organizational bottlenecks wasting valuable time.

Welcome to the Grid

This efficiency has created a world where entire ecosystems hinge upon their pollination. We are reliant on these insects for our very sustenance. I don’t trust these buzzing busybodies, and after you hear about grid cells, neither will you.

Back in 2010 some scientists hooked up an electrode to a single neuron in a rat’s entorhinal cortex, and let it run around a small enclosure: