Surfboard Calculus

By Mike Black

Surfing is a massive part of our lives. My ideal weekday
involves surfing in the morning, working through the afternoon, and ending with
quality family time. Regarding work, I get paid to solve math problems. My B.S.
in Math and my Master of Arts in Pure Math from the University of California
set that path for me decades ago.
Often as the morning colors go through orange, light blue and violet, I
find myself staring at the horizon waiting for a wave, thinking about math
since it’s what my day will be about.

It was one of these mornings that the seed for this article
was planted. I wanted to hear what contribution surfboard shapers and surfers from
around the planet thought mathematics has made to surfboard design. The
responses were varied.

In the late 70’s shaper and engineer Bill Barnfield was
keeping track of numbers like length, thickness, width, and rocker. Rusty
Preisendorfer of Rusty Surfboards said Bill was tracking ten or twelve numbers.
Bill taught Preisendorfer the importance of tracking numbers. Preisendorfer
said about his shaping before the use of machines: “I was tracking 30 or more
points per board. This ensured consistency. I started getting more consistent
positive feed back.“ The more points he had to hit, the more accurately he
could reproduce his curves. Preisendorfer continues: “One thing that still
needs some basic math skills is scaling the boards up and down. Volume is a
wonderful tool. The only way to get that number before software was with a displacement
tank. I don’t know of any shaper that went to the trouble to build one. Volume
is a number that everyone tracks now.”

Volume plays an important role in surfboard design. Australian
Andrew Kidman was telling me about a software program: “…a friend of mine that
uses computers to make his boards takes his customers’ weight and then somehow
figures out the volume they’ll need using some program he has.“ I responded: “I
wonder what that software is doing. How do other design elements contribute to
that volume number? For example, what would the difference in volume need to be
for a finless board designed for a 180-pound guy compared to a single fin or twin
fin. I'm interested in the input variables and the algorithm. I’m guessing it's
a regression deal. I suspect the algorithm was fed data based on surfboard
designs that were known to work but I’m not sure. So the programmers put in the
plan for the latest zip-zang model, then I go
to order it and the machine beefs it up to compensate for my plumpness. Does
this increase in volume represent a true proportional increase in every element
of the shape? Or does the volume just show up in some elements of the shape? If the increase in volume isn’t proportional throughout
every design element, won’t the board be different than what it was intended to
be? “

This exchange Andrew and I shared was insightful, but
off the mark somewhat. We are saying volume, but in this case we probably
should have been saying something else. A cube that is 10’ by 10’ by 10’ has
the same volume as a 3D rectangular solid that is 1000’ by 1’ by 1’. Clearly
they are different shapes.

Volume, like any design element, is practically impossible
to discuss in isolation. Two boards of equal volume can ride completely
differently. In an example of length, my 10’ pig rides WAY differently than my
10’ Simmons replica. That’s where
other dimensions of surfboards come into play.

I asked Richard Kenvin the project director of Hydrodynamica
what he thought about math and surfboard design, he wrote me a quote: ‘I went
over to Simmon’s place. He had all these equations written everywhere. It
looked like Chinese! –Dale Velzy’ A quote instead of an original thought surprised
me given the content on the Hydrodynamica website.

Manuel Caro of Mandala Surfboards loves the book

*The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture.*This book provides examples of objects that fit the golden ratio. The golden ratio (defined as a proportion) is sometimes called Phi, a math term that represents two fractions that are equal. These fractions are built using two positive numbers, a small one, and a large one. The first fraction is the large number over the small number. The second fraction is the sum of the numbers over the large number. It works out that the large number is roughly 1.61803 times greater than the smaller one. According to this book, if one is optimistic and keen to stretching their imagination, many naturally occurring (or man-made) objects can be thought of as fitting the golden ratio. “There’s always math underlying beautiful design, both natural or man-made.” Says Caro. “As far as mathematics is concerned, beyond the normal and obvious methods of measuring boards in terms of feet and inches, there are relatively few people using any ‘real math’ in design. I know I’m not.”
South African Donald Brink of Brink surfboards feels
similarly: “Sure there are some rudimentary measurements and
tasks of arithmetic involved for building a board or producing consistent
results.“

Ryan Burch of

*bobbersandsinkers;*who is mentioned on Hydrodynamica’s website had this to say regarding mathematics’ contribution to surfboard design: “I don't use math when I'm shaping except for when I'm trying to scale boards down just to get accurate proportions. I never even got to calculus in school. As far as math contributing to design, I'm not too sure what others do, but it doesn't do shit for me. “
Australian mathematical thinker and surfboard shaper Pieter
Stockert, figures the less tools one has in their shop the more they have in
their head. He talks to a person and figures out what that person needs. He has
different rail templates measured to the millimeter that have been proven
through the years to be right for most people. These rail templates define the
thickness each board will end up with. There are exceptions. Rarely, someone’s
skill level might cause deviance from the rail template to board thickness
relationship. Pieter has a continuous curve profile template that he is able to
use on boards from five to eight feet. This curve was built from Pieter keeping
an eye on the boards that guys (pros) brought back in the shop and said
performed well. He found that these boards had an eight and a half meter radius
curve on the bottom surface of the board by the tail. According to feedback
from Pieter’s team riders, a surfboard with this curve won’t loose speed
through the cutback.

Pieter is defining mathematics as “human beings measuring something like when we figured out
the calendar, that there is repetition and so forth.” Pieter believes the shaping machines with all their liter
calculations are rubbish. Pieter feels the machine can’t do one millionth of
what a human brain can do. The shaper talks to the client and figures stuff
out, the computer is just a zeros machine, a bloody copier that spits outs
copies for a couple of cents each. Contrast this to a proper surfboard designer
and shaper: he is making all those calculations in his head to make sure the
surfer gets the best board they want.

Adrian of Fluid Juice Surfboards from Cornwall said the
following regarding math and surfboard design: “An interesting little
anecdote that gets people thinking is the fact that the very first board that
an ancient Polynesian islander made was probably all guesswork but you could
pretty much guarantee that the second one was compared to the first in which
case the first was effectively used as a unit of measurement so the second
board was the product of feedback and measurement, in other words, science and
maths.”

Rob Wright of Slide 65 out of the United Kingdom says
about mathematics and surfboard design: “For me when I am hand-shaping,
mathematics allows me to replicate a shape accurately. I have figures for the
outline shapes and more importantly, the rail banding which is critical to the
board’s foil and functional volume. Thickness measurements at the nose, tail
and centre give me accurate reference points for replication. Once I have
roughed the blank out using figures I then shelve the mathematics and take on
the role of sculptor. This is where the figures are blended and the board comes
alive. I see my hand shaping as a combination of figures (mathematics) and
sculpting. Without the figures replication would be impossible.”

Ricardo Muniz shaping RickyMuniz
surfboards out of Puerto Rico said: “I only involve math when I get specific
about measurements regarding templates or rocker of a specific shape or model. There
is no equation or technical mathematical analysis that helps me define those
curves. Most of the process is more of a creative and artistic nature; like
sculpting.”

Ryan Lovelace out of Santa Barbara, Ca. mentioned the boat
designers when asked about math and surfboard design: “…the main users of the number
game are guys who look after boat design generally, as boat hull design has
drawn heavily from math over its history.”

Ricardo Muniz echoes Ryan Lovelace’s statement: “After reading a lot about physics of sailing and
hull design I can say that in the boat design industry mathematics has
contributed a lot to the design process, development and maximizing performance
of sailing and motorboats with technical mathematical analysis. In contrary to that, surfboard design I think has
come more from the soul than from mathematical analysis. Backyard shapers and
professionals alike work mainly on the power of feeling and observation, and by
keeping notes and template reference of what has worked and what didn't.”

Dr. Lindsay Lord author of the famous book regarding
math and boat hull design:

*Naval Architecture of Planing hulls*, demonstrated how a boat’s performance can be modeled in a laboratory and how you can modify isolated elements and observe the results. You can apply physical principles and feel confident the theory matches the end product. Boats have a throttle; Kelly Slater and I push a throttle forward similarly. Surfboards don’t quite fit this idea of modeling. There are general design principles that are true: flat is fast, wide is stable. However, a magic board for me might or might not be a magic board for you. Consider a surfers stance. Each surfer has a different stance, height, and each carry their weight differently. Along with other variables this affects a riders’ center of gravity.
Dr. Lord assumes a non-variable static load with force
coming from

**one**direction (other than the force of gravity, and the opposing force of buoyancy). How many directions does force come from when we are surfing? A surfboard is built around the function it serves. Customers seek out shapers that build boards that have a reputation for functioning a certain way. If Kelly Slater and I are the same height and weight, I am not guaranteed to surf his board the same way he does. Skill is yet another random variable that throws us off Dr. Lord’s trip.
Dr. Lord’s entire book is not off the mark regarding
surfboard design. Dr. Lord states: “It was axiomatic that there could be no
‘best’ boat, only best compromise”. This is something that seems to be directly
applicable to the surfboard design paradigm.

Two boards identical in every aspect except fin
placement or fin shape will ride completely differently. I can test a fin in a
water tank all day long. I can publish results claiming some advanced design
leading to improved functioning. What meaning do those results have outside the
laboratory’s specific conditions? Each surfer has a different stance; every
wave is different outside a lab. This idea is relevant beyond fins. Recall my
statement regarding volume and length. How many variables would it take to map
surfboard design? How many differentials are there between these variables? A
differential is a method of recording the change in a mathematical function
with respect to one variable. The entropy that occurs when the concert of
design elements play is the challenge a computer savvy surfboard designer has
with mathematical modeling.

Consider a surfer’s speed on a wave. My Simmons feels
way faster than my pig, and my Simmons has more extreme rocker. This seems to
contradict the basic fact: flat is fast. The key is the idea of “feels”. Once
someone starts telling me numbers corresponding to their speed on a wave, all I
hear is white noise. Distance divided by time equals speed. According to this
definition of speed, when I’m surfing a standing wave, my board has no speed.
When I surf a point break, my board goes fast. Both experiences left me
“feeling” like I was going fast. How could we possibly model something with
mathematics if we can’t even accurately measure results?

Most shapers asked mention the idea of math being a tool to
ensure consistent results. There is no doubt that math is used in surfboard
design. Anyone that orders a board has undoubtedly said a number in the
description of the board to the shaper they were ordering from.

It is bizarre when one looks at board and “knows” how it
will ride. If you know how a board is going to ride by looking at it, then it
must also be possible to predict how that board will ride by looking at some
spreadsheet of numbers. The question remains, is there some mathematical function
(Riemann, Euler, trigonometric, partial differential or other) that can fuel an
algorithm to fill the spreadsheet with numbers that define the perfect
mathematical surfboard? Can an equation dictate design? If there were some
mathematical equation that gave birth to some shape, what would the shape be?
Would the board have fins, would it be flexible, would it be symmetrical? How
much would it weigh? What would it be made of? Would it take you different
places on the wave than our current boards? What could cause us to believe that
this equation is the beginning, middle, and end of all “proper” design
elements? How does one quantify proper surfboard design?

Dr. Lord addresses the issue regarding a mathematical
equation defining design quite eloquently: “There have been serious attempts to
generate hull lines by mathematical formula, but for practical purposes it is
evident that the merit of such procedures lies very largely in the resulting
similitude for duplication in other sizes according to whatever shape was first
worked out, rather than in achieving any possible virtue inherent in a
mathematical formula.”

Greg Noll said it best when he said: “Here is how you build
a good board. You build the damn thing. If you like it, it’s a good one, if you
don’t, it isn’t”. Brian Hilbers updated that quote in the following way: “Here
is how you build a good board. You build the damn thing. If you like, it’s a
good one, if you don’t, it isn’t a good one for you.”

It is beautiful when an experienced surfboard designer
navigates through what seems like a mathematical nightmare so effortlessly. All
the chalkboard scribbling in the world won’t top time in the water. Matt Calvani
has designed many surfboard models that work in a variety of conditions.
Although mathematically modeling surfboard design might be impossible,
functional designs continue to be developed. Does this make surfboard design
more of an art than a science?

Mathematics, the science seemingly shrouded in mystery, is
definitely a trendy way to advertise a surfboard. Mathematics is not some
wizard science reserved for mousy geeks studying under fluorescent lights. It’s
clear logic. A surfboard isn’t some 3D mapping of a mathematical function. It
is from the hand and mind of a human. Sure a surfboard can be drawn on a
computer, but that drawing is not the result of any one mathematical function
(equation).