in the morning and a steaming cup of cocoa in the evening, they offer the comfort of regularity.

Within the framework of equations, scientists like to distinguish two types of elements: variables and constants. Traditionally, constants aren’t supposed to alter over time. For instance, when Einstein composed his energy-mass relationship, he fully expected that the speed of light, a constant, would be the one permanent factor. So for any type of material under any kind of circumstance, he posited that this value would never change.

Another well-known equation is Newton’s inverse-square law of gravity. It too contains a seemingly enduring fixture of nature, the gravitational constant. When Einstein proposed general relativity as a theory of greater scope than that of Newton, he kept the same constant. Both cases represent invariant relationships—first formulated theoretically and later confirmed by observation as being correct descriptions of our world.

However, there is another kind of law, logically distinct from the type to which Newton’s and Einstein’s theories belong. When we examine nature’s vast array of phenomena, we sometimes observe patterns of a wholly different sort. Rather than the products of predictable equations, they constitute much subtler relationships that sometimes only the remarkable organizational capacities of our minds can perceive.

Consider, for example, the intricate designs of seashells and the elaborate lacework of snowflakes. Neither of these is governed by immutable equations. Instead, these spectacles emerge through self-organization—wonderful instances of order stemming from chaos. In the first case, the Fibonacci sequence of numbers, formed by adding each pair to produce the next (1, 1, 2, 3, 5, etc.), serves to characterize the length of successive turns in a spiral. In the second, the molecular geometry of water delimits the six-pronged symmetries of icy shapes. In each case, mathematical features manifest themselves in surprising ways.

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