# Strange Geometry

Perhaps we all remember our high school geometry class, or maybe we have all tried hard to forget it. Many of the topics in geometry seemed to have little relevance in ones life. And while not all of us took higher level geometry that deal with Non-Euclidean surfaces, exponential functions, hyperspace, and so on, many wonders of geometry had already been presented to us. All we had to do was dig a little deeper.

## A Truly Rigid Shape

Look at a pylon holding power lines. This structure has many interlacing beams of steel holding it together. When we look closely at the structure of it, we will find that many triangles are within the overall structure and that we can make many different sized triangles if we try hard enough. In fact if you see any quadrilaterals inside of the pylon that is really just two triangles put together (which is also a convenient way to find the number of degrees in a shape, for a triangle has 180 and if you can find the number of triangles in it then the number of degrees is a short multiplication problem away). So is the overall structure being made up of triangles play into its strength? The short answer is yes, but the long way is the math behind it (Barrow 21).

When we define a rigid shape we mean that it will not deform into another shape if bent. Squares are not rigid because if you push on it, the shape becomes a parallelogram. But triangles are rigid because if you push on it, the shape will not change. In the 19^{th} century Augustin-Louis Cauchy looked into the rigidness of triangle and eventually was able to show that in three dimensions that for a convex polyhedron with all rigid faces and hinges along the connecting edges, the overall polyhedron is also convex. Convex means that all of the faces are on the outside while a concave polyhedron has some or all of its faces on the inside (21-2).

Now, proving that concave polyhedrons are also rigid is a hard task. In fact, Robert Connelly in 1978 was able to find a concave polyhedron whose faces were not rigid. So not all polyhedrons are rigid, but we should have faith that those pylons are okay, right? Sadly, the convex polyhedron theory works so long as the polyhedron is constructed perfectly. No such structure exists, so it is better to say that all pylons are almost rigid (22-3).

## Halves and Doubles of the Universe

Taking half of something is an easy conceptualization. Just split it into two equal groups of the same size. So lets start with a piece of 8.5 by 11 paper and cut it in half. Take one of the pieces and cut it in half. Now we have ¼ of the paper. After 47 cuts we will be at 10^-13 centimeters, which is about the diameter of a proton. And after 114 cuts we will be at 10^-33 centimeters. This is about a Planck length, which is where quantum foam may exist. It would be the smallest thing in existence and could be the key to a theory of everything (71-2).

Now that we have seen the power of halves, what about doubling? Yes, this too takes us to great places. Take 2 sheets of those paper and put them together. Now double that amount for a total of 4 sheets of paper. After 90 total doubles we would actually reach the edge of the visible universe. Incredibly, it only takes a total of 204 halves and doubles to describe the total measurements of all we know around us (72).

## Mystery of the Cassini Identity

Area is the amount of squares that fit inside a two dimensional object. For any amount of squares we have initially, we can rearrange them and still have the same area. For example, a 4x4 square has 16 squares in it. This will be the same number as a 2x8 square. But now we will look at a situation where this equivalence seems to be in violation.

We will start with an 8x8 square, which will have a total of 64 squares in it. Now, let’s split up that huge square into several shapes. We will break it up into two triangles with legs of 3 units and 8 units and two trapezoids with bases of 5 units and legs of 5 and 3 units. We can rearrange this into a 13x5 rectangle. But notice something: this rectangle has an area of 65 square units. How did we get an extra square unit out of the same material? (167-8)

The answer lies in the Fibonacci Sequence. For those of us who are not familiar with it, it starts out as the following: 1,1,2,3,5,8,13,21,34,… and so on. What kind of pattern is that? Each term is the previous two terms added together. 2 = 1+1, 3 = 2+1, 5 = 3+2, 8= 5+3, etc. To put this in formal notation, the F_{n} term of the Fibonacci Sequence = F_{n-1} + F_{n-2}. So how does this play a role in our mystery? Well, we started with an 8x8 square and ended with a 13x5 rectangle. The area of the rectangle is one more than the square, or 65 = 64+1. 8 is the 6^{th} Fibonacci number, 13 is the 7^{th} Fibonacci number and 5 is the 5^{th} Fibonacci number. So 65 = 13x5 = F_{7 }x F_{5} and 64 = F_{6} x F_{6}. Notice how we have the next term and the previous term in these equations, so to phrase it more formally, F_{n} x F_{n} + 1 = F_{n-1 } x F_{n+1}. Amazingly, this is a result of an even greater truth known as the Cassini Identity. Formally, it states that F_{n} x F_{n} – (F_{n-1 } x F_{n+1}.) = (-1)^{n+1}. This identity shows that when n is an even number, then the right hand side will always equal 1 and the area of the corresponding rectangle will be greater than the area of the square. If n is an odd number then the right hand side is always 1 and the area of the square is greater than the rectangle (283). It must be emphasized that this only works with Fibonacci numbers and that the real-world reason for it is that the triangles and trapezoids do not actually have the same slopes even through it may seem so visually. That is, they do not actually line up perfectly. They slight imperfection though very small does in fact leave an extra square foot inside the shape.

## The Shape of the Eiffel Tower

The Eiffel Tower was designed by Alexandre-Gustave for a world fair in the late 1800’s. It was only meant to be a temporary structure yet here it stands to this day, withstanding the elements. Gustave designed this tower without any modern knowledge of science or engineering. So how does a supposedly temporary tower manage to hold up so well? Patrick Weidman of the University of Colorado decided to find the shape of the tower and was surprised when it fit the equation y = e^ln(x). This design allows for “zero wind load on the diagonal elements” and also “reduced weight” of the overall tower by having what would have been normally required for support no longer needed. As it turns out, many designs in nature also follow this curve (Stone 12).

## Works Cited

Barrow, John D. __100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World__. New York: W.W. Norton &, 2009. Print. 21-3, 71-2, 167-8, 283.

Stone, Alex. “Eiffel Equation.” __Discover__ Apr. 2005: 12. Print.

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