To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour
~ from Auguries of Innocence by William Blake (1757 – 1827)
Have you ever wondered why nature is so extravagantly rich in shapes, forms, color and function? Why the world as we know it is so astonishing complex, so infinitely beautiful? The answer in part is that nature exploits the mathematics of combinatorial explosion.
Combinatorial explosion, in its simplest form, is the rapid growth in the number of overall possibilities as you make several independent choices. For example, if I can choose any one of ten digits to fill nine places, then I can make 109, or one billion different combinations – namely, the numbers 000000000, 000000001, 000000002, …. 999999999. Ten and nine are quite small numbers, but 109 is a huge one. This is the power of combinatorial explosion.
Nature is an expert in combinatorial explosions. With DNA, for example, nature gets to make four choices among the nucleotides guanine, adenine, thymine, cytosine or G,A,T,C to attach at spots along a lenghtly sugar-phosphate backbone, and there can be thousands upon thousands of spots to choose from. Similarly, the making of proteins involve choosing among twenty amino acids attached to stereotyped backbones of variable lengths. These architectures support combinatorial explosions of precisely the same type as the decimal expansion of numbers, but in base 4 or base 20 respectively. Thus, DNA sequences, which are used to store information, can record enormous amount of information. And proteins, which provide the structural and functional building blocks for life, form a huge inventory as well. Different proteins fold into an enormous variety of sizes and shapes, with diverse mechanical and electrical properties.
Molecules of other kinds, in both the organic and the inorganic worlds, can branch, form loops, agglomerate into membranes, stack regularly into crystals, and do many other tricks. This wealth of possibilities leads to a combinatorial explosion of combinatorial explosions! When we consider the fact that a single gram of matter contains billions of billions of atoms, it becomes clear that there’s no shortage of material to support life’s complexity on a grand scale. William Blake’s famous phrase – holding “an infinity in the palm of your hand” is not only poetically beautiful but has a sound scientific basis.