Video: Glider discovered in Penrose tiles

Stable path

If someone asked you to walk in a straight line over a constantly shifting floor, you would probably declare it impossible after a few tries and a couple of grazed knees. Researchers studying a version of this task inside a particular mathematical universe felt the same – but now a nimble traveller capable of navigating a clear path has emerged.


The walk in question takes place inside a particular, irregular cellular automaton, a theoretical universe in which patterns of cells evolve, and appear to move, just by following simple rules. The traveller, meanwhile, is a highly repetitive object known as a glider.

Its creation is an achievement because gliders were previously thought to exist only in regular cellular automata, such as the most famous one, the Game of Life – and because these objects are building blocks for more complex creations.

Invented by mathematician John Conway in the 1970s, Life is composed of an infinite grid of identical square cells, which can either be “live” or “dead”. Play consists of simply choosing an initial pattern of cells. The pattern then evolves according to a short set of rules that state how cells should switch their state depending on the exact patterns of cells that surround them. The result can be a variety of patterns, some of which appear to morph or even move across the grid.

A dynamic demonstration of how complexity can arise from simple but deterministic beginnings, Life has spawned armies of hobbyists who push this idea to its limits. For example, players have managed to create “computers” that can theoretically run any software and, in a curious echo of real life, a self-replicating creature – from nothing but these simple evolving cells.

Death sentence

The glider is the most fundamental of these patterns. It runs on a four-step “cycle” that ends up moving the pattern across the grid in an infinite diagonal path. This capability has proved to be an invaluable building block for many of Life’s more advanced constructions – but was thought only to exist in cellular automatons with a highly regular structure.

Now Life enthusiast Adam Goucher has discovered a glider in an aperiodic cellular automaton. Unlike the regular-gridded surface of Life, Goucher’s world is a mish-mash of two types of rhombus that completely cover the two-dimensional plane without ever repeating their arrangement. This ever-changing surface is known as a Penrose tiling, after the mathematician Roger Penrose who first dreamed it up. It was seen as a death sentence for gliders: one irregularity could cause the pattern to disappear or veer off-course and loop back on itself.

Goucher’s discovery was born out of a recent online discussion sparked by Andrew Trevorrow and Tim Hutton, software developers behind cellular automata software Ready. They wondered whether gliders were possible in aperiodic cellular automata: Hutton said it seemed unlikely – Trevorrow countered by offering a $100 bounty to anyone who could find one.

A few days later, Goucher had it. The key was finding a roughly stable path through the constantly changing Penrose landscape. “I was convinced that we could design a glider in a cellular automaton on the Penrose tiling, although I was concerned about whether it would be stuck in a loop,” he says.

Rhombi ribbons

In Life, a glider has a specific shape and is made of just five tiles. Goucher’s Penrose glider looks quite different – “ribbons” of rhombi keep the glider on a straight line, while a “head” and “tail” give it a sense of direction. This allows it to move through the aperiodic Penrose landscape on an infinite, straight path – the defining feature of a glider.

Goucher’s Penrose universe is also vastly more complicated than Life and has several incarnations: in the version featured above, the cells take one of four possible states – shown as different colours, rather than just “live” or “dead”.

The Life community now plans to look for more advanced patterns in Penrose-like universes, using the gliders as building blocks just as they are in regular Life. Indeed, Goucher’s discovery demonstrates how a pattern or object that seemed impossible to create in a particular universe can be made to pop up, if given the right starting conditions. “It needs just the right amount of chaos to be habitable, similar to the Goldilocks zone of habitable exoplanets,” he says.

Susan Stepney a computer scientist at the University of York who has previously investigated Penrose-style Life agrees: “If the underlying cellular automaton supports rich behaviour, some interesting structures might well be constructed,” she says.