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Month: July 2014

The Battle of Platea

We are looking approximately north across the Greek plains from the ancient city of Platea (above).  In 479 BC the Greeks defeated Xerxes and the Persian army in their third and final clash thereby setting the stage for the Golden Age and all that came afterward (In the first battle, the Greeks famously lost at Thermopylae and in the second decisively won a surprising yet painful sea victory at Salamis). The Persians set up camp on the far ridge with the Greeks residing in the foreground.  The river Asopus ran east to west in the now agricultural valley that lies between.  The final confrontation saw the Athenians, Spartans...

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Phyllotaxis With Mathematica

Expanding on the Fermat’s spiral example, these pretty depictions look like “daisy” patterns. Mathematically the two plots are the same, the only difference being the left plot uses circles and the right one diamonds. The use of the irrational number φ (Phi = 1.618…), known as the Golden Ratio, is integral to the formation of this pattern, which can readily be observed in nature, for example, in daisies, sunflowers, pineapples and pinecones. As it turns out, if the expansion of the opposing spirals from the center outwards grow proportionally to φ, then an even packing of the circles or diamonds is...

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Fermat’s Spiral With Mathematica

With their opposing pattern, Fermat spirals are surprisingly beautiful.  They look like something you might encounter in nature and yet result from a very basic mathematical relationship. Here’s the Mathematica code I used to generate them. (*Fermat’s Spiral*) width = 0.005; graphColor = RGBColor[70/255, 137/255, 102/255]; k = GoldenRatio; g[t_] := k*Sqrt[t] PolarPlot[{-g[t], g[t]}, {t, 0, 15*Pi}, PlotStyle -> {{graphColor, Thickness[width]}, {graphColor, Thickness[width]}}, PlotRange -> {{-12, 12}, {-12, 12}}, Axes ->...

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