The Battle of Platea

Platea

We are looking approximately north across the Greek plains from the ancient city of Platea.  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 and Tegeans attacking from the right, sweeping down into the valley towards the Asopus while the Persians crossed the Asopus to engage.

In 2012 Rachel and I celebrated the battle victory over a fine Greek lunch in a small village off the beaten track near Platea and on our way to Delphi.

Lunch!

Phyllotaxis With Mathematica

 click to enlarge

Expanding on Fermat’s spiral, these pretty figures represent a “daisy” pattern.  Mathematically the two plots are the same.  In one case I plot circles (left) and in the other I plot diamonds (right).  The use of the irrational number φ (Phi = 1.618…), known as the Golden Ratio, is integral to the formation of these specific patterns, which can readily be observed in nature (daisies, sunflowers, pineapples, pinecones, etc.).

As it turns out, if the expansion of the spiral from the center outwards grows proportionally to φ, then an even packing of the circles or diamonds is achieved.  Nature has clearly discovered this space saving optimization.

For me, exercises like these provide a wonderfully accessible window into the relationship between math, numbers and nature.  And anybody can play!  No worries though, there is still plenty of mystery left in the end!

For those interested, here is the corresponding Mathematica code:

Circles:

(*Spiral*)

k = 1.;
d = 2*Pi/N[GoldenRatio];
f[t_] := k*Sqrt[t];
x[t_] := f[t]*Cos[t];
y[t_] := f[t]*Sin[t];
plotTo = 55.;
data = Table[
Graphics[{Hue[.17 – 0.00002*t],
Disk[{x[t], y[t]}, 0.8 + 0.0005*t]}], {t, 1, 2500, d}];
Show[data, PlotRange -> {{-plotTo, plotTo}, {-plotTo, plotTo}}]

Diamonds:

(*Spiral*)

k = 1.;
d = 2*Pi/N[GoldenRatio];
f[t_] := k*Sqrt[t];
x[t_] := f[t]*Cos[t];
y[t_] := f[t]*Sin[t];

(*Diamond*)
c = 5.;
d = 3;
diamond[xx_, yy_, rot_, s_, h_] := Graphics[Rotate[{Hue[h], Polygon[
{{xx, yy + s*c},
{xx + s*d, yy},
{xx, yy – s*c},
{xx – s*d, yy}}]}, -1*(Pi/2 – rot)]];

(*Table of Graphics objects mapped as spirals*)

data = Table[Graphics[diamond[x[t], y[t],
ArcTan[y[t]/x[t]],
(1 + 0.00025*t),
(0.15 – 0.00002*t)]],
{t, 1, 2500, d}];

(*Plot all Graphic objects together*)
plotTo = 60.;
Show[data, PlotRange -> {{-plotTo, plotTo}, {-plotTo, plotTo}}]

 

 

 

Fermat’s Spiral With Mathematica

With its 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 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 -> False]