Ornaments and icons, symbols of complexity or evil, aesthetically appealing and endlessly useful in everyday ways, knots are also the object of mathematical theory, used to unravel ideas about the topological nature of space. In recent years knot theory has been brought to bear on the study of equations describing weather systems, mathematical models used in physics, and even, with the realization that DNA sometimes is knotted, molecular biology.

This book, written by a mathematician known for his own work on knot theory, is a clear, concise, and engaging introduction to this complicated subject. A guide to the basic ideas and applications of knot theory, Knots takes us from Lord Kelvin’s early—and mistaken—idea of using the knot to model the atom, almost a century and a half ago, to the central problem confronting knot theorists today: distinguishing among various knots, classifying them, and finding a straightforward and general way of determining whether two knots—treated as mathematical objects—are equal.

Communicating the excitement of recent ferment in the field, as well as the joys and frustrations of his own work, Alexei Sossinsky reveals how analogy, speculation, coincidence, mistakes, hard work, aesthetics, and intuition figure far more than plain logic or magical inspiration in the process of discovery. His spirited, timely, and lavishly illustrated work shows us the pleasure of mathematics for its own sake as well as the surprising usefulness of its connections to real-world problems in the sciences. It will instruct and delight the expert, the amateur, and the curious alike.

A revolutionary notebook that challenges us to play outside (and with) the lines.
 
A standard notebook displays page after page of horizontal lines. But what if we break the pattern? What if the ruled pages grew unruly? In this Nonstandard Notebook, lines twist, fragment, curve, and crisscross in beautiful formations. Each sheet is a distinctive work of imagination, asking us to draw, doodle, and journal in the same spirit.
 
Page after page, as we journey from lines to parabolas to waves, deep questions arise—about form, art, and mathematics. How do we harness the infinite? Why do patterns permeate nature? What are the limitations and possibilities of human vision? Nonstandard Notebook explores these questions and more through its provocative and inspirational images, each displayed with the mathematics that generated it. We see how straight lines can form fractal crenellations, how circles can disrupt and unify, and how waves and scaling can form complex landscapes (or even famous faces). Created by mathematicians, educators, and math popularizers Tim Chartier and Amy Langville, and with a foreword from Ben Orlin (bestselling author of Math with Bad Drawings), Nonstandard Notebook shows that rules—both the rules of mathematics and the rules of a notebook—do not mark the end of creativity, but the beginning.

An essential guide to the ways data can improve decision making.
 
Statistics are everywhere: in news reports, at the doctor’s office, and in every sort of forecast, from the stock market to the weather. Blogger, teacher, and computer scientist Allen B. Downey knows well that people have an innate ability both to understand statistics and to be fooled by them. As he makes clear in this accessible introduction to statistical thinking, the stakes are big. Simple misunderstandings have led to incorrect medical prognoses, underestimated the likelihood of large earthquakes, hindered social justice efforts, and resulted in dubious policy decisions. There are right and wrong ways to look at numbers, and Downey will help you see which are which.
 
Probably Overthinking It uses real data to delve into real examples with real consequences, drawing on cases from health campaigns, political movements, chess rankings, and more. He lays out common pitfalls—like the base rate fallacy, length-biased sampling, and Simpson’s paradox—and shines a light on what we learn when we interpret data correctly, and what goes wrong when we don’t. Using data visualizations instead of equations, he builds understanding from the basics to help you recognize errors, whether in your own thinking or in media reports. Even if you have never studied statistics—or if you have and forgot everything you learned—this book will offer new insight into the methods and measurements that help us understand the world.

From the ancients’ first readings of the innards of birds to your neighbor’s last bout with the state lottery, humankind has put itself into the hands of chance. Today life itself may be at stake when probability comes into play—in the chance of a false negative in a medical test, in the reliability of DNA findings as legal evidence, or in the likelihood of passing on a deadly congenital disease—yet as few people as ever understand the odds. This book is aimed at the trouble with trying to learn about probability. A story of the misconceptions and difficulties civilization overcame in progressing toward probabilistic thinking, Randomness is also a skillful account of what makes the science of probability so daunting in our own day.

To acquire a (correct) intuition of chance is not easy to begin with, and moving from an intuitive sense to a formal notion of probability presents further problems. Author Deborah Bennett traces the path this process takes in an individual trying to come to grips with concepts of uncertainty and fairness, and also charts the parallel path by which societies have developed ideas about chance. Why, from ancient to modern times, have people resorted to chance in making decisions? Is a decision made by random choice “fair”? What role has gambling played in our understanding of chance? Why do some individuals and societies refuse to accept randomness at all? If understanding randomness is so important to probabilistic thinking, why do the experts disagree about what it really is? And why are our intuitions about chance almost always dead wrong?

Anyone who has puzzled over a probability conundrum is struck by the paradoxes and counterintuitive results that occur at a relatively simple level. Why this should be, and how it has been the case through the ages, for bumblers and brilliant mathematicians alike, is the entertaining and enlightening lesson of Randomness.