Perspective On A Cycle


This picture was taken at sunrise 7 days before the 2015 winter solstice which occurred on 12/22/15. We are looking southeast towards the Oakland hills from above Eureka Valley in San Francisco. The lone, tall redwood tree marks the approximate location of the terminus of the sun’s journey south on the horizon, after which, the sunrise will move northwards again until it reaches its summer terminus in six months to come. That point will occur quite a bit to the left of the frame of this photo!

Earthquakes, Know Them, Love Them

I published a new app named Epicenter on the Apple app store yesterday. Epicenter plots the latest 30 days of worldwide earthquake data supplied by the USGS on a beautiful globe.  The globe provides super responsive panning and zooming capabilities and is a treat in of itself!  While great on an iPhone, Epicenter is superb on an iPad!

You can find Epicenter here:  Epicenter – 30 Days Of Worldwide Earthquakes. If you like it, please give it a review!

Update: Epicenter now supports Today Widgets and the Apple Watch!

Economics & Complexity Theory


The study of economics and the physical sciences has been an interest of mine for some time.  Over the past two years I have been seeking specific examples of the application of nonlinear dynamics and complexity theory in economic research.  It turns out it’s a hot new field!

Reflecting on my readings it occurred to me that a mapping of the books I came across might be illuminating.  The result is this Book Map that graphically presents the linkages between subject matter, author and book (the book titles are hyperlinked to Amazon for easy access).

The journey led me to unexpected discussions in areas such as political science, sociology, anthropology, language theory, cognitive science, computer science and finance.  For example, Albin’s search for a scale from which to measure degrees of complexity in his dynamical models led him to Chomsky’s early work on defining the complexity of languages.  That was a pleasant surprise to me!

Many of the books in the map are technical .  I do own all of these books.  Some I have read multiple times and many have become reference books.

Use this link “Book Map” to obtain a copy of a PDF document that possesses working hyperlinks to

A few of my very favorite titles are listed below:

  • Adam’s Fallacy – Duncan Foley
  • Barriers and Bounds to Rationality – Peter Albin
  • Currency Wars – James Rickards
  • Debt:  The first 5000 years – David Graeber
  • Nonlinear Dynamics and Chaos – Steven Strogratz
  • On Anarchism – Noam Chomsky
  • The Self Made Tapestry – Philip Ball

All of the books in this shortened list are approachable by anybody except Albin and Strogratz (Albin and Strogratz are very technical). Albin’s book contains an introduction by Duncan Foley that is the best overview description of complexity theory I have ever read (and I have read a few).  It’s a treat.  Strogratz is one of those rare teachers that can make the hard easy.

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:



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}}]



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

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],
(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]