Geometry 3/20/19

By Richard E. Bleil

Geometry plays a critical role in the world around us, although few of us ever think about it. I’ve written an article for pi day that touches on some geometric facts, but it’s far greater than that.

Here in America, most of the sewage covers are round. This might be something you never thought of before, but a circle is the only two-dimensional geometric shape that cannot fit through itself. If manhole covers were made, for example, as squares, the cover could be turned diagonally and fit through the hole. Because manhole covers are round, the cover can never be made to fit into the hole without physically damaging the cover, and, let’s be real about this…those things are made to resist damage!

Here’s some bad news for the flat-earthers. In three dimensions, the easiest shape is the sphere. Even easier than discs in three dimensions, although in two dimensions circles are easiest.

For fellow gamers, we have special dice for our role-playing. They are four-sided (d4), six sided (d6-the dice most familiar to people), eight sided (d8), twelve sided (d12) and twenty sided (d20). We also have two firms of ten-sided die, one labeled 0 through 9, the other 00 through 90 in tens. These two ten sided die are inferior to the other dice.

The first five dice are “regular polygons”. They are made of two-dimensional geometric shapes that are “regular”, meaning all sides are equal lengths. The d4 is made of equilateral triangles, meaning all three sides are the same. The d6 is made of squares (look at a six sided dice), d8 made of pentagons, d12 made of pentagons, and d20 also made of equilateral triangles. These are the only regular polygons known today. The two d10s are made of a form of trapezoid, with two long sides, and two short, but they are not all equal sided.

Triangles, especially equilateral triangles, are the strongest shapes known in two dimensions. These are seen frequently in bridge support structures, cross beams, scaffolding, even on the sides of some skyscrapers.

The “golden ratio” is something that frequently shows up as well. The golden ratio is 1:1.618. This ratio is naturally appealing to us, and tends to show up in nature from time to time. The ancient Greek Parthenon was built based on this ratio, as is the pentagram. There is also a relationship between the golden ratio and the Fibonacci sequence. If you’ve not heard of this, each new number in the sequence is the sum of the previous two (1 1 2 3 5 8 13 21 34 56 and so on). As the sequence continues, the ratio of the last two numbers gets closer and closer to the golden ration. So, when you divide it out the ratio of 34:56 is 1:1.647. This is not the exact value, but the further you go in the sequence, the closer the ratio becomes to the golden ratio. The conch shell is a naturally occurring shape that also follows the golden ratio. At any point, the ratio of the walls in parallel directions will be 1:1.618.

The “Rule of Thirds” is often confused with the “golden ratio”. The Rule of Thirds is frequently used in photography and art. You may have noticed that a lot of the best art and photography is “off center”, with the focus of the image appearing off center. If you look at the empty space in the image, you’ll see it occupies roughly 1/3 of the photo.

A friend of mine pointed out that this is often found in music as well. The “third refrain” was a common technique in rock and roll of the ’50’s and ’60’s, and you’ll sometimes still find songs that make use of it. Typically, music and lyrics will continue through the first two refrains, followed by a guitar solo or musical refrain without vocals for the third, hence the “rule of threes”.

Squares are interesting as well. From the source of point, the area quadruples. We see this frequently; sound decreases by squares, as does radiation, light intensity, and even farts. Yes, noxious gases. A unit of 1 away from the source (say 1 foot), you have a “base intensity”. If you double that, the intensity of the smell will be 1/4 as strong. At three times, it will be 1/9. At some distance x, the intensity will be 1/(x*x), or x squared. This is a weakness in chemical attacks; those close to the source are in great danger, but the danger decreases rapidly. This is also why smoke stacks are so high. From the height where the smoke is released into the atmosphere, the concentration of toxins from that height will be so low, thanks to the squares, that at ground level the toxins will be deemed “safe” by environmental laws.

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