“Forbidden Geometry” — Penrose Tiling in HYBYCOZO

There are two sculptures in HYBYCOZO family that contain a geometry that deemed so impossible that researching it would have caused to you be deemed a laughing stock in the community.

Known as the Penrose Tiling, discovered by mathemagician Roger Penrose in the 1970s, the pattern carries a deep significance for our artwork. For us, it represents the eternal and relentless search for a connecting thread in the universe; one that seems to speak to the existence of a meaning or reason for life, the universe and everything. It is this quest which ties all of humanity together, whether it be through math, science, nature, religion, or spirituality.

The two sculptures below.. Rhombi and inSpire

‘Rhombi’ by HYBYCOZO

Rhombi, the rhombic dodecahedron that won the People’s Choice Award for favorite 3D shape at the 2016 Burning Man festival.

The 800 pound Gorilla

The inSpire, most recently re-named ‘The 800 Pound Gorilla” by the Downtown Las Vegas Consortium on the Arts.

Tiling in 5-fold symmetry was thought to be impossible!

To explain the discovery of this hidden area of mathematics:

Planes can be filled completely and symmetrically with tiles of 3, 4 and 6 sided shapes, but it was long believed that it was impossible to fill an area with 5-fold symmetry, as shown to below:

In the early 1970’s, however, Roger Penrose discovered that a surface can be completely tiled in an asymmetrical, non-repeating manner in five-fold symmetry with just two 5 sided shapes based on the golden ratio or phi.

This is accomplished by creating a set of two symmetrical tiles, each of which is the combination of the two triangles found in the geometry of the pentagon.

Five fold symmetry creating an infinite tiling on plane made of 5 sided based shapes.

Phi (or the Golden Ration) plays a pivotal role in these constructions.

The relationship of the sides of the pentagon, and also the tiles, is Phi, 1 and 1/Phi. The curious thing about these tilings is they use only two kinds of tiles, and will tile a plane without repeating the pattern. ever.

Making a Penrose Tiling

A Penrose tiling is made of two kinds of tiles, called kites and darts. A kite is made from two acute golden triangles and a dart from two obtuse golden triangles, as shown above.

As you expand the tiling to cover greater areas, the ratio of the quantity of the one type of tile to the other always approaches phi, or 1.6180339…, the Golden Ratio.

Why is this exciting?

Well, a couple of reasons. Pentagons not tiling in 2D space was the reason why ancient Greeks thought the dodecahedron was ‘mystical’ or related to the ether. The impossibility of 5 fold tiling was long held that this was an immutable rule of mathematics. But as we all know, rules are meant to be BROKEN.

It was also another unexpected manifestation of the golden ratio, appearing to us in purely mathematical form. But wait. There’s more… read below to find some more Penrose related surprises.

In the inside of the Rhombi sculpture, detail of the Penrose tiling

But who really discovered the Penrose tiling?

When Peter J. Lu, a student at Harvard, traveled to Uzbekistan, he had no idea of the mathematical journey that he was about to embark on as well. He began to be fascinated by how the builders were able to produce such exact patterns, with no small errors developing and increasing over the size of the tiling. So he began to study the patterns on different buildings across the Middle East and Central Asia.

Archway from the Darb-i Imam shrine in Isfahan, Iran, which was built in 1453 C.E.

The Harvard graduate student in physics was fascinated by the beautiful and intricate geometric “girih” patterns on the 800-year-old buildings there, and he wanted to know how ancient artisans had created them.

What he discovered, though was much more than just a clever construction method.

An archway in the Sultan’s Lodge in the Green Mosque in Bursa, Turkey from 1424.

He found an entirely unexpected level of mathematical sophistication in the designs, pointing at mathematical ideas that weren’t formally developed until hundreds of years later.

Lu figured out that the girih tiles could be broken up into the kites and darts of Penrose tiles. When he divided the tiles in this way, one building, the Darb-i Imam shrine, had a near-perfect Penrose tiling. The shrine was built in 1453, and it would be another 500 years before the mathematics behind Penrose tiles was developed.

The Darb-i Imam shrine was particularly remarkable because it showed girih tile patterns at two different scales, so that large scale sets of girih tiles were broken up into smaller girih tiles, creating a fractal. In principle, by repeatedly scaling up the tiling in this way, they could have covered an arbitrarily large wall with a Penrose tiling.

So the question remains — when an new idea appears, how do we gather around it? Do we denounce it and call it forbidden? Or do we look around ourselves and see clues to where it may have been present all along? How do we create communities that share in this search for truth and beauty?

Aperiodic tiling in the next dimension

The Penrose 2D tile discovery in the 1970s is also intimately related to the discovery that won the 2011 Nobel prize, quasicrystal 3D formations which you can read about here.