Quotes from Steven H. Strogatz

When subjects go to sleep later in their body temperature cycles, they actually sleep less, even though they have been awake longer.

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

Mindless, lifeless things can sync spontaneously. The sympathy of clocks taught us that the capacity for sync does not depend on intelligence, or life, or natural selection. It springs from the deepest source of all: the laws of mathematics and physics.

No matter how erratically something moves, the area accumulated under its speed curve up to time t always equals the total distance it has traveled up to that time.

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Superfluid helium is a realization of the hypothetical quantum liquid that we imagined when performing the thought experiment with the buckets on the staircase. Its behavior is almost surreal.

This weird behavior is a manifestation of quantum sync. All liquids become highly ordered when cooled to very low temperatures. Normally they freeze into a crystal. But the two isotopes of helium, helium-3 and helium-4, never solidify, at least not at ordinary pressures. They remain liquids all the way down to absolute zero.

Pecora started the transmitter and receiver in different states, and then asked the computer to predict their behavior far into the future. As the numbers poured out, they bobbled erratically—the aperiodicity expected of chaos—but amazingly, their values converged toward each other. They were synchronizing. By driving the receiver with a chaotic signal transmitted from a duplicate of itself, Pecora had coaxed them to fluctuate in lockstep.

But my best hunch (and, full disclosure, I personally love geometry) is that people enjoy it because it marries logic and intuition. It feels good to use both halves of the brain.

I BET I CAN guess your favorite math subject in high school. It was geometry.

indeed, Johannes Kepler fell into a state of self-described "sacred frenzy" when he found his laws of planetary motion—because those patterns seemed to be signs of God's handiwork.

How does incoherence give birth to synchrony? It dawned on me one day that there was a straightforward way to frame the question as an exercise in differential equations: I needed to view incoherence as an equilibrium state and then calculate its stability.

Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.

To shed light on any continuous shape, object, motion, process, or phenomenon—no matter how wild and complicated it may appear—reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.

Circles are the simplest curves in geometry. Yet, surprisingly, measuring them—quantifying their properties with numbers—transcends geometry.

The equations showed that the answer depends on how similar the oscillators are. If they're identical, or nearly so, I found that the disturbances grow exponentially fast as oscillators clump together in phase, in an embryonic form of sync. Then out popped a formula for the exponential growth rate (analogous to the interest rate for how fast your money compounds in the bank). No one had ever found such a formula before.

determinism does not imply predictability.

In 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.

Proofs can cause dizziness or excessive drowsiness. Side effects of prolonged exposure may include night sweats, panic attacks, and, in rare cases, euphoria. Ask your doctor if proofs are right for you.

Whereas when you listen to her on digital, every aspect of her music is minced into tiny, discrete steps and converted into strings of 0s and 1s. Although conceptually the differences are gigantic, our ears can't hear them.

Anything that's continuous can be sliced exactly (not just approximately) into infinitely many infinitesimal pieces.

Do such infinitesimally small things exist in the real world? Quantum mechanics has something to say about that.

Along with numbers, shapes mattered too. In ancient Egypt, the measurement of lines and angles was of paramount importance. Each year surveyors had to redraw the boundaries of farmers' fields after the summer flooding of the Nile washed the borderlines away. That activity later gave its name to the study of shape in general: geometry, from the Greek ge, "earth," and metres, "measurer.

Feynman asked Wouk if he knew calculus. No, Wouk admitted, he didn't. "You had better learn it," said Feynman. "It's the language God talks.