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

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

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,

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

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, "

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 "

Thanks for coming by the Carpentry Way. Happy tiling.

Labels: designing with the golden mean, golden ratio, Roger Penrose, tiling