Quotes from Steven H. Strogatz

The solution to Schrödinger's equation shows that a small portion of the electron probability wave exists on the far side of an impenetrable barrier.

If it's right, it means the answer to the ultimate question of life, the universe, and everything is not 42, with apologies to fans of Douglas Adams and The Hitchhiker's Guide to the Galaxy. But Deep Thought was on the right track: the secret of the universe is indeed mathematical.

Alpha particles tunnel out of uranium nuclei at the predicted rate to produce the effect known as radioactivity. Tunneling also plays an important role in the nuclear-fusion processes that make the sun shine, so life on Earth depends partially on tunneling

We have no intuition for such events at the atomic scale, being the gargantuan creatures composed of trillions upon trillions of atoms that we are.

a cruel irony, both Newton and Leibniz, the pioneers of calculus, died in excruciating pain while suffering from calculi—a bladder stone for Newton, a kidney stone for Leibniz.

Hopfield's insight was that self-organized criticality might be intimately linked to synchronization in pulse-coupled oscillator systems. The tantalizing possibility of a relationship between those two areas spawned dozens of papers exploring the possible ties.

All of calculus, and hence all of theoretical physics, hinges on this assumption of continuous space and time. That assumption of continuity has been resoundingly successful so far.

At such small scales, space and time might seethe and roil at random. They might fluctuate like bubbling foam.

In 1899, the father of quantum theory, a German physicist named Max Planck, realized that there was one and only one way to combine these fundamental constants to produce a scale of length.

When you're trying to prove something, it helps to know it's true.

Most of everyday life is spectacularly nonlinear; if you listen to your two favorite songs at the same time, you won't get double the pleasure. The same goes for consuming alcohol and drugs, where the interaction effects can be deadly. By contrast, peanut butter and jelly are better together. They don't just add up—they synergize.

Space and time would no longer make sense below these scales. They're the end of the line.

Calculus, like other forms of mathematics, is much more than a language; it's also an incredibly powerful system of reasoning.

I want to stress that—only sixty digits. That's the most we would ever need to express one distance in terms of another.

In a nutshell, calculus wants to make hard problems simpler.

If real numbers are not real, why do mathematicians love them so much?

There are various ideas about the mechanism of synchronization, but the best guess is that it has something to do with pheromones: unidentified, odorless chemicals that somehow convey a synchronizing signal.

math always involves both invention and discovery: we invent the concepts but discover their consequences.

Our analysis revealed that whether the nodes in the network are neurons or computers, people or power plants, everyone is connected to everyone else by a short chain of intermediaries. In other words, the "small world" phenomenon is much more than a curiosity of human social life: It's a unifying feature of diverse networks found in nature and technology.

Creation is intuitive; reason comes later.

neutral stability normally occurs only at transitions, at critical settings of a system's parameters (the "knobs" that control its properties). But the Kuramoto model was breaking this rule. Its incoherent state was doggedly staying neutrally stable, even as we widened the bell curve to make the population more diverse. Turning that knob over a wide range of parameters made no difference.

Remarkably, the other 20 percent of blind people do manage to synchronize to the light-dark cycle. The likely explanation is that the circadian photoreceptors in their retinas are intact, even if their rods and cones are not. This allows light to work its resetting action on the clock, by striking the eyes and then traveling down the neural pathways to the pacemaker. In other words, although these people lack sight, they can still perceive light in a nonvisual, circadian sense.

The implication is that there are two pathways from the eyes to the brain: one for conscious vision and the other for circadian entrainment. This hypothesis is consistent with the known anatomy of the mammalian brain; the neural hotline to the pacemaker is separate from the brain's visual pathways.

that raises a profound mystery: Scientists have long been baffled by the existence of spontaneous order in the universe. The laws of thermodynamics seem to dictate the opposite, that nature should inexorably degenerate toward a state of greater disorder, greater entropy. Yet all around us we see magnificent structures—galaxies, cells, ecosystems, human beings—that have somehow managed to assemble themselves.