Guest commentary: The mathematics of golf

By Mary Armstrong/For the Sun-News
Posted: 05/19/2011 09:33:19 PM MDT

The battle within oneself is at the heart of what makes golf so attractive to all of us. We study the game: we memorize the rules; we take lessons, we practice, we buy the most up to date equipment, and we read golf psychology articles and books. In short, we look for anything that will give us the slightest edge. It is often said that the game is played on a five-inch course - the distance between your ears. The game seems so inexact, so fleeting, and yet, it can be expressed quite well using the most precise field of study - mathematics.

Recently, while facing the most dreaded putt of all - a downhill, right breaking, three-footer. I wondered if realizing the margin of error for a 20 footer, which I normally stroke quite nicely, would relieve some of the stress over a three footer. When I returned home, I constructed a drawing in my AutoCAD application. The drawing indicated a ball location "A" and the hole location - a 4.25" diameter circle at point "B." I estimated that a ball traveling at the ideal speed with its equator no less than an inch inside the edge of the hole would fall into the cup. Therefore, I "shrunk" the cup (or target circle) by an inch in diameter and projected a line from point "A" tangent with the circle. Consequently, at three feet, there is a margin of error of only 3.875 inches. This translates into an angle of deviation of just under three degrees, which converts to a margin of error of 26 inches (13 inches left or right) on a 20 footer.

Whether this makes you feel better or worse about three footers is probably an individual thing. We often exaggerate facts in our minds and it sometimes helps us to objectively look at the problem. Math certainly takes speculation and indecision out of the picture, so I began exploring the math of golf.

Dr. Douglas N. Arnold, the McKnight Presidential Professor of Mathematics at the University of Minnesota, has devoted quite a lot of time to the mathematical study of golf. Dr. Arnold has a strong interest in in the public understanding of the role of mathematics in everyday life. About a year ago he made a presentation at Portland State University entitled "Mathematics that Swings: The Math Behind Golf." Although Dr. Arnold is not a golfer, he is apparently very interested in the sport, so much so that he indicated it would be possible to teach a full mathematics class on it.

In Dr. Arnold's presentation (which you can view on YouTube), he discusses many mathematical aspects of the game including ball flight, but my favorite subject was the swing. It is a commonly-held opinion that clubhead speed is the key to distance in a golf shot. While this is true, you may not be aware of the mathematical principles involved in achieving the greatest swing speed.

I have always understood that the wider the arc of the clubhead the faster it travels. This is mathematically proven, but that isn't the most important factor in achieving the fastest swing. Dr. Arnold explains that the golf swing is actually a double pendulum with fulcrums at the shoulders and the hands. To demonstrate the significant effect a pendulum can have think about a trebuchet or a whip. A trebuchet is a double pendulum similar to a golf swing. Also known as a siege engine, it was often used in medieval times to cast a projectile great distances. However, the most dramatic demonstration of the power of pendulums and one more familiar to us here in the southwest is the whip. The whip is a multiple pendulum, and its action is easily modeled mathematically. As we know, if you get the motion just right it is possible to make the whip "crack". This cracking noise is actually the sound of the whip tip breaking the sound barrier. So you can see that pendulums transfer and multiply energy very effectively.

So, when you hear Gary McCord talk about the tremendous "lag" in so and so pro's swing this is the energy engine for the swing. The "delayed hit" as it is also called, is the lower pendulum "whipping" from the upper pendulum and therefore multiplying the energy using our body and the golf club. This isn't the only factor in determining swing speed, but it does appear to me to be the most critical one. I, of course, had no clue that these mathematical factors were at work when I was a teenager. Back then, my father wanted my brother and I to cut the weeds in the ditch in front of our rural home. He would tell us that cutting the weeds with a manual "weed whip" would increase our tee shot distance. Weed Whips had about a 12 inch double sided cutting edge attached to a handle about 45 inches long. We quickly learned that the only way to get the blade through the lush, thick grass was with the fastest swing we could muster. Through trial and error, we developed a good "delayed hit" technique and probably maximized our swing speed and distance. I doubt my father knew the mathematics behind his prediction, but he certainly knew how to get the weeds in the ditch cut.

Note: Thanks to Mike Kirkpatrick, superintendent at Sonoma Ranch and Bruce Erhard, retired superintendent for assisting me with the article last week on warm and cool season grasses.

A golf architect in New Hampshire for over 20 years, Armstrong brought her craft to Las Cruces in January 2010. She is the founder of Armstrong Golf Architects, which provides planning, designing, permitting and construction monitoring services for golf course projects. You can comment on her writing and view past articles at her blog: