Impossible Cookware and Other Triumphs of the Penrose Tile - Issue 13: Symmetry - Nautilus

Impossible Cookware and Other Triumphs of the Penrose Tile - Issue 13: Symmetry - Nautilus: One of the curious aspects of aperiodic division of the plane is that information about positioning is somehow communicated across great distances—a Penrose tile placed in one position prevents the placement of other pieces hundreds (and thousands and millions) of tiles away. “Somehow a local constraint imposes a global constraint,” says Harriss. “You impose that at no scale will these tiles give you something that is periodic..."

It turns out crystals don’t always form atom-by-atom. “In very complex intermetallic compounds, the units are huge. It’s not local,” says Shechtman. When large chunks of crystal form at once, rather than through gradual atom accretion, atoms that are far apart can affect one another’s position, exactly as do Penrose tiles.

Turing machine gives order to chaotic Penrose universe

Turing machine gives order to chaotic Penrose universe: The scientists constructed logic gates for their universal Turing machine by assigning one of eight different states to each Penrose tile, with the states changing over time according to a few simple rules.

Tiles in the first state act as wires that transmit signals between the logic gates, with the signal itself consisting of either a "front" or "back" state. Four other states manage the redirecting of the signal within the logic gates, while the final state is simply an unused background to keep the various states separate.

At first it wasn't clear whether Imai's team would be able to keep their logic gates wired together, as the gates can only appear in certain places where the tiles come together in the right way.

However, the team found that a long enough wire would always make the connection, proving that a universal Turing machine is possible in the Penrose universe...

 Imai didn't know about the Penrose glider at the time, so he was forced to take an alternative approach.

First gliders navigate ever-changing Penrose universe

First gliders navigate ever-changing Penrose universe: 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 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".