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Monday, March 25, 2013

Seven

After my post on five-fold symmetry, I can hardly keep myself from writing about seven. It seems unlikely, but the number seven does have some surprising properties, which I will illustrate. For instance, despite being called an octave, the diatonic musical scale really consists of seven notes: C, D, E, F, G, A, and B. With a remarkable sense of synesthesia, some people like to think each note has a color to it. I have folded my concept of the colors of notes into a paper aperture for your amusement.

Musicians like Alexander Scriabin developed systems to assign colors to key signatures based on the circle of fifths. The famous Hungarian composer and piano virtuoso, Franz Liszt, had a famous quarrel with Russian composer Nikolai Rimsky-Korsakov about the colors of the various key signatures; they saw them quite differently.

Seven is an odd prime number. Because it divides evenly into 1001 (and 1001*999 is one less than one million) its reciprocal has a six-digit repeat block, and thus seven the first number to have a repeat block that has length equal to the number minus one. It is a noble prime.

999999 = 3*3*3*7*11*13*37

Note that 7 and 13 have six-digit reciprocals, but 7 is often associated with good luck and 13 is often associated with bad luck.

An odd, prime number like 7 would seem to be impossibly irregular until you try to lay out seven pennies upon the table, as I did when I was five or so. I was surprised that it made the most elegant, regular arrangement possible.

And the seven pennies introduced young me to hexagonal packing. You can see that seven hexagons can make a hexagonal cluster. This is because it is a hexagonal number. The numbers 1, 7, 19, 37, ... , expressed as

1 + 6*T(n)

(where T(n) is the nth triangular number), are called hexagonal numbers because they give the exact number of smaller hexagons that can be put together to form a larger hexagon.

The clusters themselves can be fitted together. into an elegant offset packing, here shown using my Tile Patterns application. And a little help from Painter.

When I constructed this tiling, I had to work it out by hand first before I could enter it properly into Tile Patterns.

Here is my sketch of this tiling, giving some indication of the way I wanted to see it. Perhaps if we had hexagonally-packed eyes like the honeybee, and saw everything in these patterns, we would make our homes like they make their honeycombs.

It is only because my eyes are not hexagonally packed, I know, that I couldn't quite get the proportions right.

The green dashed parallelogram shows the repeat block of the offset tile pattern. It is because I like to think in squares and cubes that I can see it.

Seven is an interesting number for cubes as well, because it is one less than the cube of two.

Here I have illustrated that concept for you. It's always easier to see it visually than to just read it, I think.

Put one cube in the missing corner and you can make a 2x2x2 block. Two cubed is eight. So this shows seven cubes. Plus, I like a good graphic!

When it comes to seven, we do spend a bit of time dancing around six and eight.

The first diagram I showed was a folded paper aperture with seven sides. Its outline is a seven-sided regular polygon, called a heptagon.

Connect the corners of a heptagon and you can make various forms of seven-pointed stars.

Many countries use five-, six-, seven-, and eight-pointed stars as their symbols. Normally there are the wide star and the thin star. The Sheriff's Badge symbol uses a seven-pointed star that's somewhere in-between the two.

Other than these I don't really know other ways that the seven-pointed star gets used. This illustration I have created is a mandala form. I have applied a little color so you can see the various shapes better.

Seven dots on a grid can be situated in several different ways. But if you look at seven as two times four minus one, then you can see how a corner of one square may be shared with the corner of another square.

Each number is unique and interesting. In music, there is more to seven than just the diatonic scale. There is also music that features seven beats per measure, like Money by Pink Floyd, Solsbury Hill by Peter Gabriel, and the final Precipitato from Prokofiev's Piano Sonata No. 7 in B-flat. When I get in a mood, I will use this time signature. Usually it is broken up into two-two-three.

Finally, did you know that graph theory is based upon Leonhard Euler's solution to the problem of the Seven Bridges of Königsberg? Walk through the city, crossing each of the seven bridges exactly once. Once again the number seven provokes thought. Euler abstracted the two sides of the river and the two islands into four nodes and the bridges were thus abstracted into the seven arcs between them. The number of arcs attached to each node is called the degree of the node. If a node has even degree, then any path can enter and leave the node in an equal pairing. But if a node has an odd degree, then either the path must start or end there. It is easy to see that if more then two nodes have odd degree it is impossible for a single path to traverse all nodes, using the arcs between them. This is because a path must have only two endpoints. Königsberg's graph has four nodes of odd degree. Thus no such walk can exist.

So the number seven was actually the doorway to graph theory in the eighteenth century!

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.