The Mysterious World of Numbers

    Math isn't always about number crunching. Step into the rabbit hole and explore some of the stranger areas in the world of numbers by playing my latest quiz, The Mysterious World of Numbers.

Fun facts from the quiz:

1. A curious pattern emerges when you plot all non-negative numbers in a spiral formation. Circling all the prime numbers on this spiral creates the Ulam spiral. The interesting thing about the Ulam Spiral is that all the prime numbers tend to line up on diagonal lines. The number you start with doesn't even have to be 1; it can be as large as you want and the primes will still line up in diagonal clusters. Nobody knows why prime numbers do this.

2.  Polar coordinates can be used to graph a number of objects found in nature, such as hearts, flowers, and leaves.  Instead of x and y, radius and angle are the two variables used to plot them. Everyone who's taken algebra is familiar with Cartesian coordinates (x, y), but polar coordinates are different. The radius is measured as the distance of a point from the origin, while the angle is the number of degrees from the x-axis. Using trigonometric and periodical functions, polar coordinates are able to graph a wide range of figures. 

3.  If you've ever wondered how two radio signals can be heard at the same time, the answer can be found in a function that decomposes a wave-form into its constituent parts, called a Fourier Transform. The Fourier Transform is arguably the most important function in all of math. It essentially uses calculus to break up a single wave into all its sinusoidal parts. At any given time there are a multitude of electromagnetic signals zooming around us through space. The Fourier Transform allows receivers and antennae to isolate them so they're easier to identify.

4.  Say you place a random point anywhere inside an equilateral triangle. Viviani's theorem states that the height of the triangle is equal to the distance to its perpendicular intersections with each side. When you draw three perpendicular lines from the triangle's edges to that point, they always add up to its height. This even holds true in higher dimensions.

5.  In any statistical analysis, Benford's Law can be used to predict how often the number "1" will appear first in a string of numbers. In tables that list populations, death rates, stock prices, the area of lakes, etc, the number "1" appears first about 30 percent of the time. The number "9" appears least often, at just under 5 percent. Several theories explaining how this happens have been offered, but it still boggles the mind of many people. Benford's Law has been used to detect fraudulent activities in a number of areas, including accounting and elections.

6. The Fundamental Theorem of Calculus tells us the area under a curve between two points can be computed by using its antiderivative. An easy way to think about this is to imagine the area under a curve as an infinite sum of rectangular widths between two points on a function. By using the curve's antiderivative, or the derivative of the function that created it, you can find the answer to whatever depends on its rate of change (for example: the acceleration of an object depends on its velocity). Many other applications in science and engineering can be used with it. It's remarkable that the area under a curve could yield such results.

7. Bombelli's solution for the square root of negative numbers was rejected by many mathematicians of his time. Chief among them was Descartes, who came up with a disparaging term that we still use today, called imaginary numbers. In the 16th century it was hard to believe that imaginary numbers could have any practical use, but key advances in modern physics wouldn't have been possible without them. Signal processing, fluid dynamics, and quantum mechanics depend on them. Many theoretical concepts use them as well, including Einstein's Relativity, Schroedinger's wave equation, and String Theory. Maybe Einstein was onto something when he said, "Imagination is more important than knowledge."

8. A polyhedron is any three dimensional shape that has multiple faces. With any convex polyhedron (tetrahedron, cube, octahedron, etc.), when the number of its edges are subtracted from its vertices and added to its faces (V - E + F), the result of the equation always equals 2. This mysterious result is called Euler's polyhedron formula. The formula is so powerful that it serves as a cornerstone of topology.

9.  By iterating a shape or function over and over again, you can find the same patterns emerging on smaller scales inside the main one. The Sierpinski Gasket, Koch Snowflake, and Mandlebrot Set all demonstrate this kind of pattern, called a fractal. There are many shapes in nature that are similar to fractals. Leaves, coastlines, wind currents, blood vessels, and snowflakes all appear as fractals from a distance. But once you zoom in, disorder emerges. Something known as Chaos Theory (tiny variations that distort orderly systems) is thought to be the culprit behind nature's inability to generate predictable systems.

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