Wednesday, 14 June 2006

Creation of Logo: Synapse Logo

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The Synapse logo is designed to convey a sense of lightning (we’re fast!), neurons (we’re smart!), and electricity (that’s what we do!). Zap.

The lightning/neuron/branching image is a fractal-like object that repeats its structure at different scales. It is a “Julia set” generated at fifteen iterations. Here’s how we create the image.

There is a famous recursion in mathematics:

Depending on the value of c and where you start ( z(0) ), the magnitude of z may converge to a finite value, oscillate between a finite set of values, diverge without bound, or dance chaotically through an infinite set of values.

For example, if c = -1, this equation is z(i+1) := z(i)^2-1.

For z(0)=0, z(1) = 0^2-1 = -1. For z(0)=-1, z(1)=-1^2-1=0, so z will oscillate back and forth between 0 and 1.

For z(0) = 2, z(1) = 2^2-1=3, z(2) = 3^2-1=8, and z will diverge without bound.

For z(0) = (1/2) + sqrt(5)/2, z(1) = z(0) so z will stay at a fixed point.

The Julia set, for a given parameter c is the set of starting points that do not diverge as i goes to infinity.* The initial value z(0) and parameter c are usually given the complex domain, so the numbers have a real part and an imaginary part.

Depending on the value of c, the Julia set could be anything from the null set, to a disc, to an infinite dust of point. The Synapse logo, depicts a Julia set with c=sqrt(-1).
The real part of z is plotted on the horizontal axis from [-1.5,1.5] while the imaginary part of z is plotted on a vertical axis from [-1.5,1.5]. Points inside the Julia set are colored white.

* The traditional definition is actually the points that move chaotically through an infinite set of points, but this is not the definition that the Synapse logo uses.


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