 very different spatial scales—one familiar example is the vasculature of a leaf, whose branching geometry is replicated in miniature across the leaf’s surface. They can be found in a dizzying variety of systems, including our nervous system, blood vessels, capillaries in plant leaves, lung bronchioles, the calyxes of kidneys, neurons, root systems, and mitochondrial cristae.
Why would nature use fractal geometry so regularly? Mathematically, fractals are interpreted as having a fractional dimension higher than the space they reside in: A fractal drawn on a two-dimensional sheet of paper, for example, has a higher dimension—say, 2.1. This is a useful feature, allowing nature to pack some part of a fourth dimension into three-dimensional space.
To see why it might be useful, we can turn to a well-known result in the semiconductor industry, called Pollack’s Rule. It states that the speedup achievable by adding computing elements to a processor is proportional to the number of added elements, raised to the power of (1-1/D), where D is the dimension in which the processors are arranged. A three-dimensional arrangement of, say, 100 processors will give a greater boost than a two-dimensional arrangement of those same processors, due to reduced signal latency. The same kind of speedup can be achieved in our nervous and circulatory systems with a D of greater than three. A rule of thumb common in the halls of Intel was programmed into our bodies eons ago; the fractal arrangement of our biological sensors and the nerves that connect them are designed for speed.
The brain represents the height of biological information processing and transmission. But for years, it’s been considered a noisy instrument. Neurons code information with binary electrical events called spikes, shocking one another like tiny batteries. Once a neuron receives enough input that its voltage passes a certain threshold, it will spike in turn, sending its signal to downstream neurons. Fluctuations in the summed output of the brain’s 100 billion neurons