Isaac Newton developed calculus back in the mid-1600s as a tool to explain physical phenomena, in particular the motions of planets. As it turned out, it was also a way to encode how things move and change more generally, from economies to populations of rabbits and foxes. When Newton did it, his reasoning was mostly geometrical, not so much based on the large amounts of algebra and symbols we associate with calculus today. Those were mostly the brainchild of his contemporary (and competitor) Gottfried Wilhelm von Leibniz.
There is a lot of scary-sounding terminology that puts people off calculus, like derivatives and integrals. Too often, it is thought of as something to survive, be tested on, and never address again. However, these concepts are not all that hard, and can give insight into just about anything that changes or moves. A few basic relationships underlie everything else in calculus, one of which is so central that it is usually called the Fundamental Theorem.
As it turns out, Lego bricks are a perfect medium to do a very basic geometry-first demonstration of this theorem. We think Newton would have approved. If you want to follow along at home, you’ll need a base plate and some 2×2 square bricks.
After that, we’ll try a more accurate way of looking at a smooth curve, instead of a blocky one made of columns of Lego bricks. We’ll give you some downloadable models to 3D print or to print on paper to play along there, too.
Our new Make: Calculus book explores calculus concepts with hands-on explorations and a minimum of calculation. It also links key topics with traditional approaches so that it can be used either stand-alone to learn intuitively, or as a companion to a traditional textbook to give more insights.
Let’s make a wall of red Lego bricks using columns. For the first column, leave a space with “zero bricks.” Then, place 1 brick in the next column, then 4, 9, 16, and finally 25, as shown in Figure A. The height of each column in the red wall contains bricks corresponding to the square of its position. We alternated colors in the tallest one, just to help you count them.
Now, let’s take some blue bricks, and look at the differences from one red column to the next (Figure B). As we go along this curve, the differences get bigger, too.
Next, let’s move those differences to make a wall of their own, putting each blue brick in-between the columns of red ones (Figure C). That signifies that the blue brick is the difference between those two red columns (including the red “column” of zero bricks). There are 1, 3, 5, 7 and 9 bricks in the blue wall.
This blue wall made up of all the differences of the original wall has some interesting properties. It is climbing up by two bricks per column. In algebra you may have learned about the slope of a wall (sometimes expressed as rise/run). A straight line (like the one made by the blue wall) will have a slope that is some constant number. In this case, the blue wall has a slope of 2; it rises two bricks for every column.
Remember that the blue wall represents how the red curve is changing. In calculus, we call this more-general version of a slope the derivative of the original curve. From here on out we will call the blue wall of differences the “derivative wall.” (There are technicalities we are passing over here, because our wall is stair-stepping rather than being a smooth curve, but we’ll ignore all that for now.)
Now, about those interesting properties of the blue derivative wall. Suppose we added up the bricks in that wall. The number of bricks in the first blue column is 1. That’s the same as the number of bricks in the first nonzero column of the original (the difference between 0 and 1). If we now add up the first two columns of the derivative wall, we get a column the same height as the second (nonzero) column in the original wall. This is true all the way to the end (Figure D) where we get a blue totaled-up column equal to the tallest red column.
You can prove to yourself that this will always be true. If you start off with the first value in the original curve and keep adding the differences between subsequent points as you go along, the running total of blue bricks is the same as the number of red bricks at that point in the original curve. It is also proportional to the area under the blue derivative curve, since each Lego brick contributes one rectangle of area we can add up to get the whole.
In calculus, adding up a quantity represented by a curve is called taking its integral. Generally speaking, an integral is what you get when you add up what is underneath a curve: the area under a curve, or the volume under a surface.
Let’s recap what we did.
• We started with a red wall that was 0 bricks high, then 1, 4, 9, 16, up to 25.
• Then we created a new blue wall made up of the differences from column to column in the original wall. This new wall, we discovered, could be called the derivative of the first one.
• Next, we added up the number of bricks in our derivative wall. We discovered that the running total (integral) of the derivative is just the number of bricks in the original curve at that point!
To put it another way, finding derivatives and integrals are what mathematicians would call inverses of each other, or operations that cancel each other out. This is similar to the relationship between multiplication and division, or squaring and taking a square root. The fact that the derivative of an integral gives you back the original curve you started with, and likewise the integral of the derivative, is called the Fundamental Theorem of Calculus. And no equations required (so far)! In more complicated cases, the integral may have a constant offset, but we constructed our examples to avoid that.
As we just saw, the brick model works for curves that can be easily chopped up into a small number of whole bricks. But what if we had a smooth curve, that was changing in big swoops in some places and barely changing in others? Isaac Newton’s big “aha” moment (one of them, anyway) was that you could create ways of figuring out the slope of a curve instantaneously, between two “columns” that are incredibly close together. We could imagine that we create a zillion columns of teeny bricks and do the same exercise.
Since that’s not very practical, we have created a 3D-printed model to think about this (Figure E) and put it in front of our red wall to show that it, too, is a curve that rises as the square of its position. (It’s a little shorter because the size of our print bed did not allow us to go all the way to 5².) We will give you a template to make a paper one instead, but it is a little easier to see what is going on with this translucent model. We can see there that the smooth blue model is crossing through the center of the top brick in each stack.
There is a part of the smooth blue model sticking out toward us in Figure E. Let’s rotate the model and see how that part fits our blue wall (Figure F). Yes, it lines up perfectly.
So, we can see that one part of this 3D print (Figure G) follows the original (red) curve, and if we rotate it away from us, we follow the derivative (blue) curve (Figure H). Thus, we take this model (all of which is visible in Figure E) and rotate it away from us to get a derivative.
If instead we started with Figure H and wanted to know how the area of the shape accumulates as we move from left to right, we could rotate it the other way, to get Figure G, the integral. In this case, we can see that the integral grows faster and faster, because as the line slopes upward, more and more area is added for each tiny step to the right. So, the exact same pair of curves can show us a curve and its derivative, or an integral and the curve it integrates — another hands-on view of the Fundamental Theorem. We rotate one way to find a derivative, and rotate back the other way to get an integral.
Purists will note some details here that we are ignoring. As we mentioned earlier, there is a constant floating around we will need to deal with to get numerical answers in general, but let’s enjoy our victory for a moment!
Our new book, Make: Calculus (available soon in the Maker Shed and other retailers) details how to make a 3D printable version of this model and many others. This model was created in OpenSCAD, and will be available in the book’s open-source GitHub repository.
Meanwhile, if you’d like to play with it, we have provided both a 3D-printable STL and a paper-printable PDF file (Figure I)that is scaled to match models made from 2×2 Lego bricks, just like the ones in the photos. You can download these. For the paper version, be sure to print it out at 100% size. Cut it out and fold it along the fold line. Then think about how one side represents how the other side is changing or accumulating, respectively. (If you use different construction toys, you’ll need to scale these accordingly.)
People often plot a curve, its derivative, and the equation for its integral on the same 2D graph, since all of them are just functions of some variable. Often, we think of a variable as “dimensionless” and not having any units. But, in real life, variables are measured in feet or meters or bricks or dollars. Their derivatives will have units like bricks (high) per brick (long) in the case of the wall.
Integrating quantities of bricks with respect to other bricks will result in units of (bricks)². We feel that plotting derivatives, curves, and integrals together on the same 2D graph is misleading at best. That’s why we came up with this right-angle way of looking at curves. In these models, the independent axis (usually labeled x, or sometimes t for time), is shared by the two graphs.
We hope these models have given you some intuition about some of the fundamental principles of calculus. There are a lot more cool things to learn about calculus. We hope you are inspired to learn more!
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