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The Worlds of our Solar System

The cosmic calendar - from the very beginning of time itself (January the 1st) till now. Scaled down to a year. Dinosaurs would have died one day ago on this scale - check it out.

1. Take a circle and draw some points on the boundary. For every point you draw, you must also draw its antipode (point on the opposite side of the circle).

2. Draw some points in the interior wherever you want.  3. Label the points either +1, -1, +2, or -2 as you wish. The only stipulation is that antipodes must have opposite sign.

4. Draw triangles however you want without crossing lines.

Tucker’s Lemma says that you will ALWAYS end up with at least one line that has endpoints of either +1 and -1 or +2 and -2. Try it! More info and proof here.

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Mathematical Spirals

According to Wikipedia, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point (similar to helices [plural for helix!] which are three-dimensional). Pictured above are some of the most important spirals of mathematics.

Logarithmic Spiral: Equation: r=ae^bθ. I must admit that these are my favorite! Logarithmic spirals are self-similar, basically meaning that the spiral maintains the same shape even as it grows. There are many examples of approximate logarithmic spirals in nature: the spiral arms of galaxies, the shape of nautilus shells, the approach of an insect to a light source, and more. Additionally, the awesome Mandelbrot set features some logarithmic spirals. Fun fact: the Fibonacci spiral is an approximation of the Golden spiral which is only a special case of the Logarithmic spiral. 

Fermat’s Spiral: Equation: r= ±θ^(½). This is a type of Archimedean spiral and is also known as the parabolic spiral. Fermat’s spiral plays a role in disk phyllotaxis (the arrangement of leaves in a plant system). 

Archimedean Spiral: Equation: r=a+bθ. The Archimedean spiral has the property that the distance between each successive turning of the spiral remains constant. This kind of spiral can have two arms (like in the Fermat’s spiral image), but pictured above is the one-armed version. 

Hyperbolic Spiral: Equation: r=a/θ. It is also know as the reciprocal spiral and is the opposite of an Archimedian spiral. It begins at an infinite distance from the pole in the center (for θ starting from zero r = a/θ starts from infinity), and it winds faster and faster around as it approaches the pole; the distance from any point to the pole, following the curve, is infinite.