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Saturday, March 16, 2013

Five-Fold Symmetry

Nature is replete with five-fold symmetry, thanks to five-petaled flowers. While most of the works of humankind are designed around 4-fold symmetry and the cube, we have always been interested in five-fold symmetry as well. It's just that it doesn't always work as well as four-fold symmetry.

Our obsession with the number five clearly comes from our own hands, since we have five fingers on each hand. Our expression of the interest with the number five comes in several forms, though. My favorite is the five-pointed star.

Knowledge of the dodecahedron, a platonic solid bounded by regular pentagons, was actually kept secret in the school of Pythagoras. They believed that the general populace was simply not ready for it.

The US military uses five-pointed stars to denote the highest ranks: admirals, generals, and commandants. The five-star symbol was created during World War II for Eisenhower and Nimitz.

I read once where a draftsman invented the symbol for the US Government. But I haven't been able to find this information since. His designs were very interesting, to say the least.

The military loves the pentagon as well, as evidenced by their construction of one of the world's largest buildings: the Pentagon. It is remarkable that General Leslie Groves oversaw its construction. Earlier, he was put in charge of the Manhattan Project. So he was also in charge of the construction of another of the world's largest buildings, at Oak Ridge, Tennessee, used for the processing of Uranium.

Nature likes five-fold symmetry in part because five is a Fibonacci number and nature favors them. When I was a kid, I used to read the Fibonacci Quarterly religiously. I was like that.

It is notable that most music is rooted to four beats per measure. Sometimes, as in a waltz, we use three or six beats per measure. But five (or ten) beats per measure is exceptionally rare, except in Jazz. The Dave Brubeck Quartet made this famous in their rendition of Take Five. You can also hear this uncommon beat in the catchy original Mission Impossible theme, written by Lalo Schifrin (where ten beats is broken into two 3s and two 2s).

For me, it is fun to see the way that pentagons, stars, and the rhomboids interact. You can make a rhombus with 72 and 108 degree angles. This is the one used to create the rhombic dodecahedron.

Put stars and pentagons together and they make an irregular tiling of the plane. The designs at Alhambra in Spain are great examples of the attempt to make a regular tiling from five-fold symmetry. But, if you only use pentagons, it really can't be done.

Here is a basic chart of how they fit together. The rhomboids trim out the design. Roger Penrose has constructed aperiodic tilings out of rhombuses, including the one I show here and also a sharper one with 36 degree angles in it. The golden section (1.6180334...) figures in all the shapes in one way or another. Quasicrystals can be made of these designs.

Try to put pentagons together and you will have little luck in creating a seamless gapless tiling.

Here I have succeeded in putting some together using a star as a root and some crowns to glue them together!

The problem with the tilings stems from the angle of a pentagon: 108 degrees. The wide rhombus has an acute angle of 72 degrees, the complement of 108 degrees. The sharp point of the star is 36 degrees. So this means you get some other obtuse angles in there are well: 144 degrees for the thin rhombus.

A ten-sided regular polygon has outside angles of 144 degrees, the complement of 36 degrees. You can put a crown, three pentagons, and two thin rhombuses together and make a ten-sided polygon.

This construction is often used in Penrose and Kepler tilings. Two such decagons can join and use a common thin rhombus.

As I said before, the problem is that the primary angle, 108 degrees, does not evenly divide 360 degrees and so there must be some left over. After putting three pentagons together, exactly 36 degrees are left over.

So you must also have a star, a thin rhombus, or a crown to make pentagons tile the plane properly. If you distort the pentagon, however, you can solve the problem entirely.

Here is the common way of doing this. It is a novel pattern, for sure! Notice the squat hexagons that intersect each other in perpendicular patterns.

This pattern has been used to create concrete tilings in real life! It's cool that tiling patterns have real-world uses. Of course they occur in crystals all the time and so patterns are automatically embedded in the real world all around us.

It is worthwhile to peruse the images of Alexander Braun, an explorer of pentagonal tiling.

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