Category Archives: math

Building a Homemade Trebuchet

“Warkitty” the Mischief Trebuchet at Fall PDF 2012

Summary:

Trebuchets are a really fun project to take on! There’s quite a bit written about them on the internet, but most of it tailored to specific projects. Here is the most generalized data I can give as to how to build a trebuchet of the size and scale of your choosing:

The swing arm must be constructed in such a way that the axle upon which it rests is located 1/5 of the distance between where the counterweight is anchored and the end of the swing arm itself. To achieve throws 10 or more times the length of the swing arm, you’ll want a pivoted counterweight that weighs at least 40 times as much as your payload. The length of the sling varies according to the trajectory you want your payload to have, but I’ve found that a length such that the sling when folded over is equal to the distance from the end of the swing arm to the axle is best for long shots. Shorter sling lengths may be used to achieve higher but shorter shots. Slings should consist of as little material as possible to avoid tangles. When looking at materials, focus your search for an axle on the most solid bar-shaped object you can obtain as the stresses on it will be quite high.

If you’d like to build a trebuchet according to the specifications of “Warkitty”, the trebuchet I spent the last four months building, you can download the blueprints and materials list here.

For a complete description of this project from beginning to end, read on below.

I. Introduction (What is a trebuchet? Why did I want to build one?)

Trebuchets are a type of medieval siege weapon that first appeared in both Europe and the Middle East in the 13th century and dominated siege warfare for the ensuing 300 years. They differed from other siege weapons of the day, including ballistas and mangonels, by using gravity to generate the force necessary to sling a heavy object at its target (usually a castle wall). I first became acquainted with trebuchets from a 2000 NOVA documentary on PBS in which a team of historians endeavored to build a replica of one such device on the shores of Loch Ness to test their destructive capabilities. I was instantly hooked on the device both for its historical significance and application of the physics of third-degree levers.

Four years ago, I encountered the device again at the mid-Atlantic regional burn, Playa Del Fuego. Two friends had brought a replica trebuchet that was approximately ten feet tall to the event, meaning for it to fire pillows over a small swath of trees into a field neighboring the event. We spent several hours working to increase the device’s range one afternoon, but never got a shot of more than 40 feet out of this device. Though this particular experiment was not a full success, it planted within me the desire to create a replica trebuchet that would be. In the spring of this year, I made it a goal to build a trebuchet of a similar scale with a range of no less than 100 feet.

In the course of researching how to build a trebuchet, I discovered that there is something of a cult community dedicated to their construction online and that there are more than a few successful designs to choose from. However, I had difficulties finding one specific to the capabilities I had in mind: easily transportable, able to break down, no component longer than ten feet, range of 100 feet. I thus set out through trial and error to create a trebuchet that would fulfill these requirements and having met this goal, I am setting down the lessons learned for other hobby engineers who wish to create their own trebuchets.

II. Research and Funding (Where did I get my design ideas? How did I pay for it?)

The first step in this process was to get the funds to put it together. It was also important to me that it be a project that would have community involvement and be a participatory work that others could engage in. As a Burning Man attendee, I knew that there was a local community of fellow Burners who’d found success in recent years hosting large events that involved just such projects and that they’d begun to develop a presence at the local regional burns. This group, called Mischief, seemed a perfect avenue through which to develop the project. I approached Mischief’s leadership in the spring with a tentative budget of $200-300 to create a modestly-sized trebuchet meant to hurl water balloons at event goers. After concerns with voiced about the potential for litter with such a project, it was instead suggested the intended payload be stuffed animals. With these requirements in place, I was given an art grant and the project was off and running!

I next did a bit of research into trebuchet design and construction. I was able to obtain a copy of the old NOVA broadcast to brush myself up on the basic principles of trebuchet design. The essential elements of a trebuchet are a long swing arm on a pivot that acts similar to a see-saw. When weight is dropped on one end the momentum is transferred to the opposite end, which is connected to a sling holding a payload. When the arm reaches a certain point along its arc, one end of the sling slides off the end of the arm and the sling releases the payload.

The trebuchet I’d encountered at Playa Del Fuego (PDF) had been constructed entirely out of wood 2x4s with a rope sling and a counterweight consisting of weights from an old barbell weight set with a chain threaded through them that hung from a pivot on one end of the swing arm. During our attempts to launch a pillow using this trebuchet, the counterweight had never exceeded 100 pounds. We’d done some adjustments to the length of the sling in an attempt to wrestle more range out of the device, but met with limited success.

In order to expand my knowledge of these devices, I also turned to numerous videos on YouTube of trebuchets created by other hobbyists. Many had impressive capabilities, but I found few that were to the scale that I envisioned. Most were either smaller (intended to hurl baseballs, etc) or much larger (meant to hurl pianos, people, etc), though a few of similar scale did appear in the noise. Few if any of the architects of these devices had bothered to include critical details as to their device’s construction. How much weight was necessary on the counterweight? How long did the sling need to be? What was the ideal proportion of the length of the swing arm to the distance between the axle and counterweight? How durable did the frame need to be?

Lacking easy answers to these questions, I endeavored to find the answers by constructing and testing small scale models and hoping the results of these experiments would scale up.

III. Scale Model Tests (Initial lessons learned from building a scale model of the trebuchet)

The scale models were constructed out of popsicle sticks acquired from a local hobby store and glued together with wood glue. I prepared multiple frames to test the ideal starting angle of the swing arm, multiple swing arms to test the ideal proportions of axle to counterweight, and crocheted a small sling to test the effect of the length of the sling on the distance traveled by the payload.

In an effort to accurately gauge the influence of the counterweight on distance traveled, I used a hoard of tiny fender washers with one set aside to be the payload. I reasoned that if I added weight to the counterweight by adding these washers, it would tell me the ideal proportion of weight between counterweight and payload (ie, if it took 40 washers on the counterweight to get a single washer to go the intended distance, that told me the weight ratio needed to be 40:1). Thus the trials began!

For each frame, I would try each swing arm with multiple weight amounts and two different sling lengths, measuring the distance traveled by the payload in 3 shots and working out the average. Frequently, the washers would bounce upon impact with the ground creating a small margin of error that could occasionally throw the average value off. Upon changing the variables of the experiment, however, there would be a noticeable difference in performance across the board, so the margin of error was deemed to be less than the margin of improvement with each subsequent test. My goal was to get the payload to travel at least ten times the length of the swing arm.

After tallying the results of these experiments, I came to the conclusion that the ideal swing arm would have the axle placed 1/4 the distance from the counterweight to the end of the swing arm (a measurement I’d later realize I made inaccurately) with a weight ratio of at least 40:1 (counterweight to payload) and that the angle of launch and length of sling weren’t nearly as important to performance as the counterweight was. You can see the spreadsheet of my numbers here.

With these numbers in hand, it was time to go to full-scale.

IV. Full-scale build (How was the trebuchet constructed? What problems did we encounter at full scale?)

Even with my numbers now in hand, there were quite a few other challenges to be solved at full scale. Among them: how to construct the frame? What would the sling be made of and how would it attach to the arm? In an effort to keep materials acquisition as simple as possible, I made a trip to the local Home Depot to see how many items could be acquired in a single trip there. The wood was the easy part–but it became clear on this trip that I could not get an angled cut made within the store and nobody that I knew had the equipment to perform such cuts, so my challenge with the frame was to find a way to construct it in such that it would not require any such cuts and still be strong enough to take the weight necessary to fling our intended projectiles. I designed the trebuchet to fling a payload of between 1 and 4 pounds, so the counterweight was to weigh up to 200 pounds.

The next challenge was to find appropriate materials to hang the weights from and to make the axle from. When it came to the counterweight, I decided to go by the design my friends Smokie and Jen had used at PDF–threading a heavy-duty chain through the center of barbell weights. The chain was easy enough to locate, but finding a reliable way to fasten it would be a challenge. I opted for a spring-driven quick snap link and went to plumbing to search for an axle. Here, my best option was cast iron pipes, which were hollow and that concerned me. I decided to hang the counterweight from a single 6 inch cast iron pipe and use a 48 inch iron bar for the axle. Two elements of this design immediately struck me as prone to failure: the axle and the bar around which the counterweight would hang. One interesting challenge I did not foresee until this stage was finding a way to keep the swing arm from sliding side to side as it turned on the axle. A random trebuchet video on YouTube provided an interesting solution: cut two pieces of PVC that had an inner diameter greater than the axle to act as spacers between the frame and the swing arm.

For the frame design, I consulted my friend Carlos Bustamante, a local performer with an expertise in theatrical construction and carpentry. He suggested to me that I utilize a pair of A-frames constructed as equilateral triangles  that could be bolted to a base piece and thus broken down relatively flat. The A-frames would sandwich three layers of 2x4s together and reinforce the side of the trebuchet facing the direction in which it would fire. With this design in hand, I laid out the blueprints for the device in Illustrator and put down the final materials list. A date was selected for the build and an invite sent out for help with the construction.

For the sling, I opted to cut out a square of fabric from an old gift bag I had laying around and grommet the corners, tying ropes to each of them that would then attach to the swing arm itself.

We had a number of problems to overcome on build day. The first and most severe was that we quickly realized no one on the build team had a drill bit longer than 4 inches, making the sandwiching of three pieces of 2×4 impossible to make precise. For the first part of the day, we compensated by instead using wood screws on each side of the frame, realizing they would not penetrate the entire assembly but hoping they would still keep the A-frames stable. Next, we discovered the bore drill bit we had acquired at 1 1/8 inch was still creating a hole too small for the iron pipes we had acquired (I had thought the measurement was of the external rather than internal diameter). After some trial and error, we sent a runner to the local hardware store to grab both a 1 3/8 inch bore bit as well as the 6 inch drill bit we’d so badly needed in the morning.

With these two last pieces of the puzzle in place, the trebuchet construction was completed by dusk and we were ready to have our first test firing. We put only 90 pounds on the counterweight to start with, not having any idea whether our design would work or be stable. Even at this point, it was clear that the method we’d employed for holding the counterweight was going to be problematic. To affix the counterweight to the end of the swing arm, one or more people would have to physically lift the weights while another would work to quickly thread the chain through the center of these weights and snap the quick snap link shut before the arms of the people lifting the weights gave out. At 90 pounds this wasn’t terribly difficult. At 135 plus pounds it started becoming very difficult. Our first launch wasn’t very successful at all–we found that the fabric I’d used for the sling wasn’t very durable and the grommeted corners kept being ripped out by the forces of the launch. I grommeted and regrommeted as the early evening wore on and finally we got a decent launch at 135 pounds, launching a volleyball 60 feet. After this test firing, we discovered to our great chagrin that the hollow iron pipe acquired from Home Depot had already begun to bend under the strain of the first few shots. We hastily deconstructed the trebuchet in the dark and I went to work on finding an alternative for the axle.

After receiving a number of helpful suggestions, I opted to visit a local steel works in Manassas, Virginia to see what they had in stock. They had an array of metal bars of an appropriate diameter, but full lengths were priced too high for our budget. I lucked out and found a cast-off solid steel bar of the perfect diameter that was 8 feet long. I had the steel works cut it in half so we would have both an axle and a backup axle for use in testing and firing.

A couple weeks after acquiring the new axle, I was able to find the time to reconstruct the trebuchet and test fire it a few more times. While the new axle was a major improvement, it was clear that the device wasn’t performing according to its intended specifications: the stuffed animals thrown from it rarely traveled more than 30 or 40 feet and the sling frequently would tangle and not release at all. I began to suspect that the stuffed animals were too light and had too much surface area to be effective as a payload. To improve upon the sling, our primary patron Josh had a supply of heavy-duty plastic paintin tarp and I opted to make a new sling by cutting a square out of it and grommeting the corners as I had the earlier fabric. During one of these shots my second concern became fully realized: the iron bar supporting the counterweight broke through the swing arm with the weights landing in a forceful pile upon the base piece. I hastily made for the local hardware store to replace the broken pieces and finished reinstalling them well after nightfall, making sure to locate the axle for the counterweight a few inches closer to the axle than before to avoid the issue in the future. Any further testing would have to wait till the trebuchet’s full debut at the North Carolina regional burn, TransformUs.

Amid celebratory drinks after wrapping up the deconstruction, we christened the trebuchet “Warkitty” in tribute to “Warwolf”, the trebuchet Edward I had used in the siege of Sterling Castle.

V. TransformUs (The trebuchet’s first deployment–what went wrong and what went right)

One of the intended uses of the trebuchet when it was being pitched to Mischief was as a method for throwing stuffed animals into neighboring camps at TransformUs. In the time I had before setting the trebuchet up there, I set about looking at the two primary problems the device seemed to currently have: first, it wasn’t throwing nearly as far as it should have been, which could conceivably have been a problem with either the type of payload or something off in the design. Second, the sling was getting tangled in the swing arm as it fired so often that it was misfiring more often than it was firing. I looked back over my design materials and rewatched some of my source materials. Upon a second viewing of the PBS NOVA special, I realized there was a section where the process to determine the distance from the counterweight to the axle was demonstrated using computer graphics but not explained vocally. Curious, I repeated the steps myself in a notebook and discovered they indicated an axle that was 1/5 of the distance from the counterweight to the end of the swing arm. Curious how I could have missed such an obvious issue, I went back and remeasured my scale model. There I found I’d made a critical mistake in measuring the distance not from the location of the counterweight, but the very opposite end of the swing arm. The difference wound up putting the location of the axle at 1/5 the length of the swing arm as well and I chided myself for the oversight.

At TransformUs, with the trebuchet unpacked I discovered some of the weights were missing, so I’d be unable to put more than 145 pounds on the counterweight. With no other options, I dug in and worked to fine-tune all the other elements of the design. I started with the sling. Rather than having a loose rope that hung around the bare screw that functioned as the release finger on the end of the sling arm, I switched to a design wherein a single length of rope would go between neighboring grommets of the sling and then have a single length of rope going from the swing arm to tie to the middle of this loop between the grommets. There was an immediate improvement both in performance and in proportion between successful firings as opposed to misfirings. I quickly discovered that the arrangement of the sling had a huge effect on the quality of the shots wherein loading it sideways would almost inevitably lead to a misfire.

With this crucial piece in place, I remeasured the swing arm and sure enough found the axle to be in the wrong spot. The point 1/5 of the way between the counterweight and end of the swing arm was another four inches toward the counterweight. I bored a new hole and again saw an improvement in performance with smaller, denser stuff animals regularly flying 60 feet. Still below specifications, but at least more consistent. With little else that could be done to improve performance at this burn, I set aside a couple hours each day to fling stuffed animals into neighboring camps. One afternoon, a gentleman who’d constructed a similar trebuchet to fling bowling balls at the Georgia burn, Alchemy stopped by and we had a brief chat about the devices. He praised the work I’d already done, stating that the lessons I’d learned from my scale models had enabled me to skip work it had taken him a couple years to complete on a full-size model. On burn night, a crew of seven people helped to move the trebuchet out to the burn field where we launched stuffed animals into the gathered crowd, making one young lady very happy when a teddy bear miraculously dropped from the heavens into her lap and quickly getting shut down by the staff there after a misfire in the dark.

VI. Figment (The trebuchet’s second deployment–entertaining small children and improving on the counterweight)

The trebuchet sat unused at Josh’s house for all of August and much of September as Burning Man and other activities began eating up more of all of our time. Another chance to use it came up at the end of September when the Burning Man-inspired event Figment had its inaugural run in DC. It was suggested we once again bring out the trebuchet and I enthusiastically endorsed the idea. I resolved I wanted to fix two elements of its use: I wanted to ensure we had all the weights (205 pounds worth) to bring it up to full specifications and I wanted to make it easier to add the counterweights to the swing arm.

Ever since our first few experiments in firing on the first day the trebuchet was built, I knew we needed a better way to add weights. Even under 135 pounds, the counterweight was painfully difficult to set up and even more dangerous to take down. I’d envisioned a metal T-bar like the type used to store the very kind of weights we were using that we could easily slide the weights on and off of. I’d been racking my brains to find a friend with metal-working experience when it dawned on me the day before Figment that there was a much easier solution that I’d overlooked. Rather than fashioning a T-bar completely out of metal, I could improvise one out of a piece of wood and another iron pipe by taking a 30-inch 2×4 and drilling holes in either end of it. On one end we would thread through the iron pipe used to hang the counterweight on and through the other we would thread a 12 inch iron pipe upon which we would mount the weights.

“Warkitty” the Mischief Trebuchet at Figment DC 2012

The next day at Figment, the design came together without a hitch and resulted not only in us being able to load the trebuchet easier, but also re-aim it when we so chose by quickly removing the weights and dragging the frame to the new intended direction. Alas, only 180 pounds of the weights made it out, but not the final 25 pound plate. Nevertheless, we spent the day launching stuffed animals to the delight of the children who attended the event, many of whom opted to pick from a barrel full of them that we’d set out just for the occasion. Between high winds that frequently blew the stuffed animals back toward the trebuchet and the continued mysterious tendency of them to fire high and then fall straight back down to earth, no shot went terribly far that day, but the children were ecstatic to have the device there. We also learned that with the T-bar, the trebuchet was now so stable that it could take repeated and frequent firings over the course of many hours and no piece of it was likely to fail.

VII. Playa Del Fuego (The trebuchet finally performs as designed–what were the final pieces of the puzzle?)

For Fall Playa del Fuego (PDF) 2012, I was determined to bring out the trebuchet once again. In addition to wanting to get the trebuchet to perform as it had been originally designed, I had a strong desire to bring it to the place where it had originally been inspired more than two years ago. With some effort, I was able to find a friend who could transport it to Delaware so I could spend part of the long Columbus Day weekend perfecting the design.

Onsite, I had a friend named Emily Hanson helping me construct the trebuchet who had what turned out to be a critical piece of advice: use a heavier payload. After the first few shots with stuffed animals resulted in distances similar to those at Figment, despite now using the full weight available to us, Emily fetched a soccer ball that we then attempted to fire. The difference was night and day! Rather than 40 limping feet the ball was now easily coasting over 80. We spent some time working to find an ideal sling length before settling on a length equal to the distance from the end of the swing arm to the axle and found our shots easily besting our previous record of 60 feet. We measured shots easily crossing the 160 foot mark and wrote down each successive shot in sharpie on Warkitty’s frame. When an especially strong tailwind came up, we measured a 180 foot shot, tripling the trebuchet’s best performance up to PDF and repeatedly demonstrating it to delighted children and adults for two days during the burn. Here, our biggest problem was that the T-bar was just slightly crooked and sometimes the weights would veer dangerously close to popping off the ends as they fell during the firing of the trebuchet. We also found one of the C-Clamps we’d been using to hold the axle in place on the outsides of the A-frame had bent slightly out of shape and now was frequently falling off when we fired it, but it was replaced with another clamp that performed perfectly the rest of the weekend. This was the trebuchet we’d been waiting for!

VIII. Conclusion

This has been one of the more enjoyable projects I’ve taken up in the past few years and seeing it perform so beautifully at PDF was the culmination of months of hard work and design. I highly recommend this project to anyone who wants a fun physics or construction based project that will leave you in awe. I hope you’ve found this outline of the process helpful as to how many of the problems encountered in design and construction were solved–profit from the lessons I’ve learned the hard way! If you’ve built your own trebuchet or are building one based either upon your design or the one I’ve shared here, please leave me a comment or email me at ben (dot) drexler (at) gmail (dot) com and share with me whatever photos, videos, or written materials you may have. You may also share suggestions with me on how to improve my design if you like, but please note that after months of working on this project I’ve come into the habit of taking the advice of those who have not built their own trebuchets with a grain of salt. It’s very difficult to know how to diagnose problems with these devices if you have not had the physical experience of building one yourself. Thanks for taking the time to read all this and enjoy 🙂

Thanks to:

Josh and the Mischief crew for funding this project–and Josh especially for going above and beyond the call to make this project a success
Ethan Sapperstein for helping me build and take the trebuchet down more times than I can count
Emily Hanson for being such a trooper at PDF and for your excellent suggestions
Devin, Aaron, Josh, Ethan, Jessica, Debbi, Kate, Shamal, Will, JoAnna, and all the rest of the build crew for the epic day of construction
Debbi Arseneaux for putting up with my endless scale tests in the living room and showing me how best to glue the popsicle sticks together
The older gentleman who helped me with the trebuchet at TransformUs and whose name I’m ashamed to admit I cannot now remember–your generosity of time and effort are hugely appreciated 🙂
Matthew Blakey and Patrick Oberman for the incomparable service of transporting the trebuchet to and from Figment
Alan Foran for transporting the trebuchet to PDF
Scott Crum for seriously bailing us out and transporting the trebuchet back from PDF–and also for the badass castle that matched it sheets to drapes 🙂

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Filed under Engineering, math, Random fun, science, trebuchet

Math hacking

I had a hell of a breakthrough in learning some math this morning…and I wish it had happened ten years ago. But first a little background:

For most of my life I’ve thought I was bad at math, which really sucked because I was otherwise a relatively bright kid. When I was really young I had a talent for memorizing things and it helped me coast through tests to the delight of my parents and teachers. It was also a huge help at the easy stuff such as basic arithmetic and multiplication, but it when the time came to hit up algebra I hit a brick wall. I could not for the life of me make the uber-abstract world of algebra work inside my head and it made me feel stupid for the first time in my life. In middle school and high school, algebra study sessions at the dinner table frequently left me in tears after spending hours and hours not understanding what was going on.

It’s not like math didn’t work for me at all…I just didn’t realize the things I was also interested in counted as math. Geometry was easy for me and the proofs for it made complete sense to me. In one middle school math class I lost focus in, I spent one long afternoon deducing what a 4-dimensional hypercube must look like. My teacher upon seeing the work was out of her depth and tried to redirect my attention to my class work. I found out five years later I’d gotten it right. Likewise, by high school I’d immersed myself in the world of comic book art and fancied a career for myself as a penciler. In learning this art, I’d assimilated the canon proportions of the human body and face, learned rules of perspective and how it can make the sizes of things shift to an observer. I’ve also had a lifelong fascination with patterns–finding arrangements of things like the tiles on the ground in the subway, arrangements of rivets on merry-go-rounds, etc, and working to find symmetrical arrangements of these patterns that can be infinitely repeatable. During phone conversations in my parents’ home I took to finding an algorithm for which of the kitchen tiles to step on such that I could navigate around an island countertop and always perfectly arrive back at the original tile (it took a couple years and countless phone calls, but I finally got it). But I never, ever thought of any of these things as math.

Math in my mind was abstract, it was something minds more nimble than myself did. For the life of me, I could see no beauty in the arrangements of numbers and variables. They were just things…staid, boring facts that were presented as sacrosanct rules which I must repeat the usage of in problem sets over and over again to prove I’d learned them (I frankly never did learn most of them). Calculus in particular was a word that filled my mind with dread. Calculus was something the smartest of the smart did…it was math akin to doing magic that only the sharpest of kids got to do in high school and which I never had a desire to work toward or do myself. It was something that you had to get through algebra to get to, after all, and if it required more of that unholy work, I wanted no part of it. Quite frankly, I didn’t even know what it was until my late 20s.

In college, I’d already decided math wasn’t for me and had figured out a way to game the requirements of my major to avoid taking any math classes. At CU Boulder, the college of arts and sciences would accept any engineering course as a math course, assuming that math was all that engineers actually did. I found an obscure freshman level engineering class called Telecommunications I to take. The course was on how to write HTML–something I’d already been doing for 4 years. It was the easiest “A” I got in college and with it I managed to skip Statistics, Differential Equations, and the host of other crazy-sounding classes my peers constantly complained about. Success! Or at least I thought…

Then in 2006 I went to Burning Man for the first time and it introduced me to fire dancing. More specifically, poi spinning. Poi is a really fascinating art because it inspires such incredibly different reactions from the people who practice it. At one end of the spectrum are people to treat it as ornamentation for movement and dance and at the other end of the spectrum are people for whom the analysis of how the tool moves is a seriously and decidedly complex pursuit. This latter end of the spectrum tends to attract people with backgrounds or interests in mathematics, physics, and programming. I picked up the tool myself in the spring of 2007 and was immediately hooked. My inclination toward pattern recognition and reorganization had found a very comfortable hook to hang its hat on and I found the practice of the art taking up more and more of my time.

As I assimilated the tool more and more, I began thinking of its movement not in terms of the arrangements of disparate movements into recognizable patterns, more colloquially known as “tricks,” but as complex combinations of simpler basic movement elements and it was these that fascinated me the most. By controlling and rearranging these elements in different fashions, I found I could for the first time create movements and tricks that no one else had yet performed or recorded. It didn’t happen overnight and to be honest I couldn’t tell you when I crossed over the line into doing something original, but what it did do was get me hooked on understanding the mechanics of the tool. At one point in the pursuit of this knowledge, I became obsessed with finding the distance the poi head would travel in all the different shapes I could make.

A good deal of poi movement bears significant similarity to a type of geometry called roulettes or trochoids, which are complex curves that tend to overlap with themselves and cycle in such a way that they produce flower-like patterns much like a spirograph. Mathematically, these patterns can be described using parametric or polar equations. To find the distance traveled by the poi head inside these shapes, it meant that I had to not only learn this type of math, but also the dreaded calculus. For, as I eventually learned, calculus was the mathematical study of curves and now everything I was doing relied on them.

And this is where things got really hard…math is traditionally taught in such a way that properties of it–say, the distributive property of multiplication, the quadratic formula, etc, are presented almost as canonical law. Students are then given problems that require these properties to solve and when the produce enough correct answers, they are thought to have learned the property successfully. This was always my issue with math: I understood the properties as presented, but I never understood WHY they worked. This is something that no math class I’ve ever taken has bothered to do–there seems to be an assumption that, this property having been proven, there is no need to understand how it was proven or why it works. Ironically, given that students are always required to show their work on a problem, we were never shown the work of how the elements we were working with worked themselves.

This became a problem in tackling my poi problem: I could find many, many references online as to what the parametric equations I needed to know to draw out poi patterns looked like, but none on why those equations worked and how they could be altered to produce different patterns entirely. I got very, very lucky in that at a fire festival two years ago I had a chance encounter with a juggler named Adam Dipert who took me through the process of creating the equations I needed. I do not think it’s a stretch to say that the ten minutes I spent with Adam that day taught me more about math than the entirety of high school did, nor that Adam is easily the best math teacher I’ve ever had. Far from just giving me the equation, Adam started by showing me how an equation could describe the movement of the poi head around then hand, and how the properties of that equation could be used to extrapolate a way of describing the hand’s movement as well. I won’t claim that instantly all was revealed, but I did now have the tools to figure out everything I wanted to know. It was as close to eureka as I could hope to get. I began creating problems for myself, not just of the flower-like roulette patterns, but also of three-dimensional poi moves that knotted and bent their way through space like corkscrews and doughnuts. And this was when math switched over from being a set of staid, lifeless facts and became a living, breathing thing.

Part of the problem, at least in my case, is that I am a hacker by nature. I see systems and patterns and upon figuring out the rules that govern these systems and patterns I want to find ways to recombine those patterns into new ones. It’s little wonder poi appealed to me, then, but this is a type of thinking that seems altogether alien to the way math is taught in most public schools. As students we are presented with the dry properties and problem sets, but rarely if ever presented with real-world problems for which the math we are being presented is the answer. We quickly forget all the knowledge our parents and the other tax payers of our school districts have paid handsomely for because it seems unrelated to the things we spend our time doing. When math had an application for something I was interested in: drawing human bodies in a way that seemed proportionally correct, finding the correct number of tiles to skip to navigate the kitchen in an easy-to-repeat pattern, I was fascinated. When math was presented as an isolated dalliance with inscrutably abstract numbers and figures I was lost and convinced I was stupid.

I’m still working my way toward an answer to my problem on poi distances. A friend wrote a computer program for me a couple years ago that took parametric equations and measured the distances traveled by the curve, but it’s only accurate to four decimal places. As repeated patterns have emerged from the data, I need to know how to write the equations that produce these distances themselves so I can isolate mathematically what these proportions are instead of just knowing what integer they are. To do this I’ve finally had to step up and teach myself calculus. I’ve had some wonderful help: Khan Academy has been hugely helpful for some of the basic step-by-step knowledge. Some of the a ha moments have come from a wonderful book that was a Christmas present from my girlfriend’s family: “The Calculus Diaries”, a wonderful book that is short on explanation of how calculus works but wonderfully detailed in all the problems it can solve. But by far the best tool I’ve had to work with is the brain of a hacker.

Many of the tools I need to simplify the equations I work have long since been lost to the annals of time. I’d love to claim I held onto all the quadratic, binomial, and trinomial properties I was supposed to have learned in high school, but it would be a lie. Now, however, I’m able to look at the numbers as yet one more thing I can hack. After tackling basic derivatives in Khan Academy’s hugely helpful video, I decided I wanted to take some of the solutions presented at the end of the last video, showing the derivatives for f(x)=x^whatever and rather than just take them at face value, create my own problems to test them and find out why they are what they are. It took a long time…I’m positive that if I’d remembered more of my high school algebra there are many steps I could have skipped or simplified, but in all honesty the extra work made it that much more rewarding when I finally got the answers myself. I tried here and there to find guides online for some of the work I was doing, but I still lack the vocabulary to describe most of this work. When it came right down to it, the solution I found was satisfying not just for reaching the answer, but knowing WHY the answer worked and thus how it can be obtained in situations that veer wildly off the grid.

As a quick note, I will say that there is another disincentive to learning math in this fashion and it has more to do with the attitudes of people who don’t consider themselves math people. People who are fascinated by math growing up may be marginalized as nerds or geeks at an age when children are known to be cruel by nature, but it’s quite another to encounter it adulthood. More times than I’d like to count, I’ve tried to shared some of the breakthroughs I’ve had with close friends only to find them recoiling at the idea of having to comprehend any complex math as an adult. The marginalization that kids who excel at math find isn’t just limited to childhood and it’s very disheartening to find it rearing it’s head even among my 30-something friends. It’s hard to take seriously the claim of friends who consider themselves open minded and whole-heartedly support their friends exploring art, music, and other means of personal expression but somehow find expression through math abhorrent.

At 31, I’m a decade past when most people who are going to learn calculus have learned it (and probably a few years after they’ve already forgotten it). There are countless lessons on it I’ve missed and I can’t help but feeling a great sense of regret that an education of this type wasn’t available to me as a child when my self-image was being ravaged by the onset of algebra. But now, being a math hacker, I know that the knowledge of it is going to be far more ingrained than anything I would have learned in high school or college. The value of the thing, after all, is the use we accrue from it. So, parents with kids who are having math problems, see if you can’t get your child more engaged by finding the real world problems that they solve. See if you can’t find how to make math applicable to the things they care about. Math isn’t hard…it’s just very good at convincing people it is.

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