Penrose: 500 Years Late?

The master builder tradition was based on pattern, not fads. Pattern was largely associated to geometry and that geometry was also used to develop building proportions, down in some cases to the smallest of details. In this blog I’ve spent lots of time looking at such things as the Golden Ratio, the number 1.6180339887…, along with a classic medieval drawing method for producing various figures and patterns, the vesica piscis (here and here).

A related aspect to patterning involves tiling. It’s not uncommon for carpenters (who, unofficially at least, seem to also wear the hats of tile-setter, brick-layer, plasterer, etc.) to be engaged in placing floor and bathroom tiles, and parquet flooring, though now somewhat of a rarity, is an area allowing for much creative variation. The French in particular developed that area in depth. Laying bricks and cinder blocks, paving stones, acoustical ceiling tiles and so on are also examples of tiling. So, general carpentry practice often involves tiling in one form or another. In Japan, I might note, it is considered a disaster if a room is tiled so that tiles have to be cut to fit at the edges – they like to plan ahead and get a sublime result. Yep, that’s right, choose the tiles before you decide on the exact size of the room.

Tiles that fit together to one another without leaving gaps as they cover a given surface are said to ‘tessellate’ – the root of that word being tessella, which is Latin for a small cubical piece of clay, glass or stone (used to make mosaics). Tessellations may be composed of regular or semi-regular elements. ‘Regular’ means that the tiling elements are congruent regular polygons, and there are only three of those – equilateral triangles, squares, and hexagons. A bee hive’s honeycomb arrangement is a classic example of a natural tessellation using hexagons. Semi-regular tiling also involves polygons, however more than one type is mixed in, like octagons with equilateral triangles, or hexagons mixed with squares and equilateral triangles, like this:

That’s called a ‘rhombitrihexagonal’ tiling. Not a word you would typically bring out at the dinner table in polite conversation. For a fairly in-depth look at the possibilities of polygonal tessellation, take a look at Brian Galebach’s page (<– link) cataloging hundreds of variations.

Other irregular and curvilinear shapes can tessellate and Maurits C. Escher, a world-famous artist, produced quite a lot of work employing tessellations and morphing shapes, like this classic from the 1940’s, Reptiles:

I’ve found this matter of tessellation an interesting topic for a long time, and of course with the continual drawing, design and study I do, geometry and pattern are never far from my thoughts.

One of the curiosities of tessellation relates back to the above-mentioned golden ratio and the fact that the polygon which associates closely to that ratio, the pentagon, does not tessellate. Kepler noted this in his 1619 work Harmonices Mundi (meaning: The Harmony of the World), where he proposed a way of tiling which made use of pentagonal symmetry, a drawing he called ‘Aa’:

So, looking closely at this figure we can see that regular pentagons are mixed in with pentagrams, regular decagons, and a curious sort of fused-together pair of decagons. An excellent page discussing pentagon tilings can be found here.

So, the problem with pentagons and their apparent intractability as far as tessellation is concerned stymied mathematicians for many years. Until the 1970’s in fact, when British mathematician Roger Penrose discovered a pair of tiles, the ‘kite’ and ‘dart’, each having pentagonal geometry (i.e., 36˚ and 72˚ angles), would tessellate. He wrote about this discovery in a 1978 article entitled ‘Pentaplexity‘ (<– link).

Here are those two tiles, the ‘dart’ on the left and the ‘kite’ on the right:

With these two tile shapes, a surface can be covered leaving no gaps, and a pentagonal symmetry manifests. Here’s an example – note the floor Professor Penrose is standing upon, at Texas A&M University:

Possibly the kite and darts aren’t so obvious in the above photo? That’s because the shapes have been ‘welded’ together, as it were, and the abutment lines erased. Perhaps this example will be clearer:

Compare the two pictures to see how the form of the tiles can be made plain or obscured, as desired.

Yesterday I was looking at the news and saw that the Nobel Prize for Chemistry had been awarded to an Israeli scientist, Dan Shechtman, who discovered what are called ‘quasi-crystals’. Schectman, who made his discovery on April 8, 1982 while working in the US, “fundamentally altered how chemists conceive of solid matter” – this, the academy said in its citation for the 10 million kronor ($1.5 million US) award. In fact, Schectman’s discovery, like a lot of paradigm-breaking discoveries in Science, was met by reflexive skepticism and mockery by his peers, even prompting his expulsion from his research team, before it later won widespread acceptance as a fundamental breakthrough. First they ignore you, then they laugh at you, then they fight you, then you win, as Gandhi wrote. I like stories like that, and you have to admit, these icosahedral quasi-crystals of metal are pretty cool looking:

 I started doing more digging and reading about these unique forms, and that lead me to Peter Lu’s site. He’s a whiz-kid at Harvard, a physicist who has interests in ancient Chinese technology and Islamic art – tilings that is. If you have a spare 70 minutes and feel like geeking out, I strongly recommend taking in a  video on his site, a film of a lecture he gave entitled “Decagonal and Quasicrystalline Tilings in Medieval Islamic Architecture” (<– link). I found the talk so enthralling I watched it twice, but then I do have an obsessive side.  Anyway, if you check out the video, you will come to see why I chose the title for this post. I hope you enjoy it!


Thanks for coming by the Carpentry Way. Happy tiling.