I was lucky enough to buy two great books in the 1980s. They may or may not make anybody else’s list of enduring classics, and they certainly were not runaway best sellers in their time. Nevertheless, Peter Brancazio’s Sportscience: Physical Laws and Optimum Performance and The Bill James Baseball Abstract 1984 were inspirational. Without them, this book would not exist.
Brancazio’s Sportscience discusses a variety of physics concepts using sports. The extent to which it teaches physics through sports as opposed to explaining sports with physics is in the mind of the reader. It does both beautifully, and it opened up an exciting new world to me. Bill James is now well known to the sports public, often referred to as the father of the statistical analysis of baseball.1 The Baseball Abstract offers witty, outspoken commentary backed by a clearminded use of statistics to investigate interesting questions. In a talk at a recent mathematics conference,2 the speaker discussed his use of the Abstracts to teach students how to analyze and write about statistical experiments.
Despite the large differences in subject matter, both books gave me a much-needed spark of inspiration. I have always been a sports fan but had not considered that this could benefit my career as a mathematics professor. Both Brancazio and James transferred to paper the essence of intellectual exploration—a restless desire to fully understand important aspects of sports and a pure delight in their discoveries. I read their ideas and kept thinking, “Of course, that’s right. Why didn’t I think of that? That’s great!” The excitement of learning at the feet of two masters was invigorating.
By 1988, I had designed an interdisciplinary course on sports science, which I have taught in a variety of formats at Roanoke College. The popularity of these courses led me to incorporate sports problems into my calculus and other mathematics classes. The success of these problems was a factor in my becoming coauthor on a series of calculus books3 and led directly to the opportunity to write this book. I hope that, in some measure, this book creates pleasure for you in the way that Brancazio and James did for me.
That you are holding this book right now means that you have overcome a powerful cultural bias against mathematics. This bias is a regrettable and, to my mind, inconceivable attitude for a country like the United States.* If you get curious about something (anything!) and start to analyze it and understand it better, you are probably doing mathematics. Mathematics follows from trying to give unbiased answers to interesting questions. The precision required to remain unbiased can sometimes be daunting. In this book, I’ve tried to minimize the daunt and keep the focus on the interesting questions.
An unfortunate truth of modern mathematics is that, contrary to algebra class where everything is solved exactly, simple answers are rare. This is especially true about the mathematics of golf. If you give me a golfer’s swing speed, can I tell you how far the drive will go? In a word, no. It depends on what ball is being used, the properties of the shaft and clubhead, the altitude and latitude of the course, and so on. Most of what you will find here are sample calculations intended to give you a slice of golf truth,† and to improve your understanding and appreciation of the game (and, perhaps, mathematics).
If you have bought this book, you have personally disproved Stephen Hawking’s theory that every equation reduces the sales of a book by half.* Starting from the number of people for whom a book titled Golf by the Numbers might be purchased, there are enough equations in the book to reduce projected sales to one or less. My college library is required to buy a copy, so your purchase is an unexpected bonus. Thank you. More seriously, I have tried to segregate the mathematics to enhance the readability of the text. All overt calculus equations are in the endnotes.
The more technical arguments in each chapter are reserved for “The Back Tee” sections at the end of each chapter. On a golf course, playing from the back tees gives the greatest challenges and reveals the best the course has to offer. However, very enjoyable rounds can be had without venturing to the back tees. Similarly, the results in “The Back Tee” sections are interesting and accessible, but you can have an enjoyable read without facing every such challenge. My assumption is that you agree that equations are a convenient way to express precise relationships and are not a form of devil worship. I have used diagrams, graphs, and equations when I think that they help convey the information in a simpler and clearer form than the equivalent thousand words. If you so choose, you should be able to skip over the equations and find complete sentences that summarize the most important ideas to be gained from the equations.
Beyond standard high school mathematics, there is no specific mathematics course that I assume you have mastered. As in nearly every aspect of life, the more mathematics you know, the better. In my view, mathematics includes a variety of techniques for analyzing the world, so along with some equations you will see graphs, tables of numbers, logical arguments, and ideas from probability and statistics. It is all mathematics. If you are a person with a decent command of high school mathematics, a curiosity about how mathematics might be used to understand basic principles in golf, and some knowledge of golf as both player and fan, you are my ideal audience. However, even if you know little about golf, you will be able to follow along using the glossary to make sense of the arcane terms that are part of golf lingo.
As befits someone with Peter Brancazio and Bill James as muses, I have included a fair amount of physics and statistics in the book. However, I am neither a physicist nor a statistician. (My golfing buddies would be quick to jump in right now and tell you that I’m not much of a golfer, either.) Instead, I am a mathematics professor who has always done work in applied areas, incorporating physics and statistics (and biology and economics and so on) as needed. The material in the book is sometimes reproduced (with proper credit) from other sources, sometimes adapted from other sources to suit my goals (again, with proper credit), and sometimes developed specifically for this book. (In some cases, I should admit, being a mathematician makes something deeply interesting to me that might not be terribly compelling to someone else.)
I have played golf most of my life. I learned to play at Hardy’s Driving Range and the numerous municipal courses in Dallas, especially Tenison Park. Unfortunately, I was born a couple of years too late to meet Hardy’s most famous employee, Lee Trevino. Where I sometimes accidentally slammed worm-burning drives into the 100-yard marker, Trevino had made considerable money hitting the marker intentionally with golf balls struck with Dr Pepper bottles.4 As well, I was too young to participate in or even know about the wild gambling that is one of Tenison Park’s main legacies.5 The high point of my golf career may have been a third-place finish in the Dallas Times-Herald city-wide tournament, 16-year-old division. I lost in the semifinals the day before my parents and I moved to Virginia. I played on the Virginia Commonwealth University golf team in college. (To be fair to John Rollins and the real VCU golfers, the team I played for had only club status, and all five of us who tried out made the team.) While researching material for this book, I discovered that my golf development was mathemagenic.6 That is a polysyllabic way of saying that I taught myself. I benefitted greatly from the one lesson I had and envy young golfers today who have access to fantastic golf instruction and technology.
This book is not about finding “the perfect swing.” Golfers my age who grew up watching Arnold Palmer, Gary Player, Lee Trevino, Doug Sanders, Miller Barber, and others have a visual library that proves there are many ways to swing a golf club well. A man named Homer Kelley7 distilled years of intense experimentation into a complete description of the golf swing. He allowed for personal variations in many stages of the swing: more flexibility here, greater height here, extra weight there, and so on. Including variations, the perfect swing comes in 446,512,500,000,000,000 versions. (Unfortunately, the evidence shows that my swing is not one of them.)
Instead, this book is about different aspects of golf which lend themselves nicely to mathematical analysis. How much do you slice if you leave the clubface open 5°? How much extra distance do you get if you are hitting downhill? Is the handicap system biased in favor of good golfers? Is “drive for show and putt for dough” an accurate assessment of the importance of driving and putting? Who is the best putter on the PGA Tour? How accurate are pros from 200 yards out? What are the strengths and weaknesses of Phil Mickelson’s game? Given that Tiger was the best on Tour in the 2000s, how much better was he than the second best? These are among the many questions that we will use mathematics to answer.
The timing is good for books like this. The computer revolution is inexorably turning the world into a more technical and quantitative place. Americans can no longer be technophobes without forfeiting a basic understanding of life. As Ian Ayres describes it in his book Super Crunchers8 aspects of life ranging from wine-tasting to online shopping to public policy decisions are all increasingly governed by mathematical analysis. Ayres says, “We are in a historic moment of horse-versus-locomotive competition, where intuitive and experiential expertise is losing out time and again to number crunching.” The wisdom of grizzled veterans is challenged by the results of numerical computer experiments. However, these do not have to be mutually exclusive; instead, they can enhance one another. This book in no way replaces Ben Hogan’s books on golf swing principles. I view it as complementary, filling in some of the gaps that Hogan could not have known about in the pre-computing age in which he lived. Hogan, in fact, was an enthusiastic user of scientific breakthroughs for his line of Ben Hogan clubs and balls.
Golf has benefited from modern technology more than most sports. Sophisticated high-speed cameras and testing procedures have increased our knowledge of what happens when club meets ball. The lucrative industries of golf club and ball manufacturing have become highly competitive and technical. Golf telecasts have become more detailed, but too often complex statistics are thrown at us with little or no context. So, Camilo Villegas has hit 50% of the fairways and made 82% of his putts inside of 10 feet. Are those good numbers? Are those important areas in which to excel? You will find answers here which can be determined only through statistical analysis.
The use of ShotLink by the PGA has significantly raised the ante on the accuracy of golf statistics. The classic statistics of scoring average and money earned have been supplemented with greens hit in regulation, fairways hit, and others. Now we can know that in 2007 Tiger Woods averaged 27 feet, 6 inches from the hole for all shots from the fairway9 (making him no. 1 on the Tour). Detailed knowledge of the location of each shot opens up the possibility of the development of new, meaningful statistics that can start to answer questions of course management and strategy. Perhaps you will become the Bill James of golf and fully develop the study of golf statistics.
Every year, our knowledge of the mechanics of a golf shot is stretched. New methods of analyzing a golf swing are developed. A barrage of numerical statistics is available for each tournament played. As a result, we are continuously revising our understanding of the game through the daily breakthroughs that occur in statistical analysis.
As a mathematics professor at a liberal arts college, I am sometimes asked to explain the relevance of mathematics and liberal arts in the modern world. One partial answer is this. A solid mathematical background lets you revel in the wonders of new possibilities instead of being overwhelmed by rapid technological change. A liberal arts education opens up new worlds,10 some of which require mathematics to fully appreciate the nuances and intricacies of their residents’ interactions. This book chronicles my brief excursion into the brave new world of mathematics and golf. I hope it opens up new worlds of insight for you.
I want to thank Trevor Lipscombe and the staff at the Johns Hopkins University Press. This book was Trevor’s idea, and his encouragement and sound advice were crucial in helping me overcome the inevitable bumps and roadblocks that arise in getting a book to publication. Many thanks to the man “wearing the British accent.” Jen Malat and Vince Burke oversaw the details of getting the text in shape to print. Kim Johnson is my fantastic copy editor. She corrected mistakes, improved explanations, reduced my use of jargon, and streamlined the structure of the text. She measured each aspect of the book against an exacting standard and helped me bring the text closer to the book I dreamed about than I could ever have achieved on my own. Thanks, Kim. My calculus coauthor Bob Smith and then-editor Liz Covello were very supportive of my desire to see this through, even though it represented a significant energy drain from our ongoing calculus work.
The general ideas that form the foundation of this book were developed in 2008, when much of the programming and analysis were done. I am indebted to Roanoke College for funding a sabbatical in the spring of 2008, which enabled me to tackle this enjoyable project.
Numerous friends were kind enough to read portions of the manuscript and provide valuable feedback. I especially thank Geoff Boyer, Brian Gray, Reid Garst, and Rich Grant for their time and helpful suggestions. Several colleagues at Roanoke College served as sounding boards for both good and bad ideas and provided much-needed expertise. I especially thank Dave Taylor, Frank Munley, and Adam Childers for friendship and advice. Thanks to Scott Berry for his generosity in sending me several of his excellent papers and for helping with my description of his work. My brother George has always been a friend and inspiration, even if he never quite made it to the green of that par 6.
The friends who have played golf with me while I worked on the book have heard more about it than anyone would ever want to. John Selby, Jeff Sandborg, and Garry Fleming are, I assume, still working on the “Sounds of Golf” CD that was to accompany this book. Thanks also to Lee Hipp, Joe Austin, Bob Stauffer, Jim Stevens, Rob Almond, and Paul Reed.
When you work at a small institution like Roanoke College, students, colleagues, and staff often become friends and supporters. Students Amanda Coughlin, Danielle Shiley, Hannah Green, and Richard Goeres worked with me on various aspects of this project. I appreciate their hard work and friendship. Linda Davis and Laura Bair both helped with numerous administrative details, and department members Chris Lee, Jeff Spielman, and Karin Saoub complete the “Trexler family” that makes work enjoyable. Ed Parker of James Madison University and Dan Kalman of American University gave helpful feedback. I’m hoping that Ed’s business venture pays off. Bruce Torrence of Randolph-Macon College helped me produce a nice article for Math Horizons. The late Howard Penn organized the Mathematics Awareness Month–April 2010 (mathaware.org/mam/2010/) website; the articles were later collected in a book edited by Joe Gallian.
When I started this project, I was hoping to do some statistical analysis of ShotLink data and started by typing online stats into a spreadsheet. This is not the way to go! The PGA Tour was amazingly open and helpful, and this part of the project expanded beyond my highest expectations. Thanks to Steve Evans, Senior VP of Information Systems at PGA Tour, and his executive administrative assistant, Stephanie Chvala, for their extraordinary cooperation. My PGA Tour liaison, Mike Vitti, was a pleasure to talk with and a helpful guide as I started exploring the data set.
An unexpected joy was a round at Oakhurst Links, the first golf course in the United States. Thanks to Nancy Midkiff for arranging the visit, for an excellent tour of the museum, and for helping John Selby and me get around the course in good shape. Oakhurst’s owner, Lewis Keller, took time from a busy day to chat with us, giving us a sense of the history of the site and its largely unknown role in golf history. Both John and I would love to grow up to be like Mr. Keller. My “home course,” Hanging Rock Golf Club, appears in a couple of places in the book. Many thanks to pros Billy McBride (who, sadly, passed away during the writing of the book) and Chip Sullivan.
Finally, and foremost, I thank my wife, Jan, and children, Kelly and Greg. After more than 30 years of marriage, Jan still encourages me to play golf. Granted, it could be to get me out of the house, but it is a true blessing to have a talented mate and best friend who genuinely wants good things to happen. As just one example, Jan researched and set up the round at Oakhurst as a surprise Christmas gift. Thank you. Kelly and Greg are not kids anymore. They are incredibly talented and good people. I’m lucky to have had a part in their upbringing and to get to follow their exploits. They will make the world a better place.