Think not of what you see, but what it took to produce what you see
Randomness is all around you…or so you think. Consider the various shapes of the morning clouds, the jagged points of Colorado’s Rocky Mountains, the twists and turns of England’s coastline and the forks of a lightning bolt streaking through a dark, stormy sky. Such irregularity is commonplace throughout our natural world. One can also find similar irregular structures in biology. The branch-like structures in your lungs called Bronchi, for instance, fork out in irregular patterns that eerily mirror the way rivers bifurcate into smaller streams. It turns out that these irregular structures are not as irregular and random as one might think. They’re self-similar, meaning the overall structure remains the same as you zoom in or out.
The mathematics that describes these irregular shapes and patterns would not be fully understood until the 1970s with the advent of the computer. In 1982, a renegade mathematician by the name of Benoit Mandelbrot published a book entitled “The Fractal Geometry of Nature”. It was a revision of his previous work, “Fractals: Form, Chance and Dimension” which was published a few years before. Today, they are regarded as one of the ten most influential scientific essays of the 20th century.
Mandelbrot coined the term “Fractal,” which is derived from the Latin word fractus, which means irregular or broken. He called himself a “fractalist,” and often referred to his work as “the study of roughness.” In this article, we’re going to describe what fractals are and explore areas where fractals are used in modern technology, while saving the more technical aspects for a later article.
As we mentioned he starts off with a really succinct and well written tutorial on celestial coordinates that antiquity would have killed to have. If we were writing a bit of code to do our own positional astronomy system, this is the tab we’d have open. Incidentally, that’s exactly what he encourages those who have followed the tutorial to do.
The star pointer itself is a high powered green laser pointer (battery powered), 3D printed parts, and an amalgam of fourteen dollars of Chinese tech cruft. The project uses two Arduino clones to process serial commands and manage two 28byj-48 stepper motors. The 2nd Arduino clone was purely to supplement the digital pins of the first; we paused a bit at that, but then we realized that import arduinos have gotten so cheap they probably are more affordable than an I2C breakout board or stepper driver these days. The body was designed with a mixture of Tinkercad and something we’d not heard of, OpenJsCAD.
Once it’s all assembled and tested the only thing left to do is go outside with your contraption. After making sure that you’ve followed all the local regulations for not pointing lasers at airplanes, point the laser at the north star. After that you can plug in any star coordinate and the laser will swing towards it and track its location in the sky. Pretty cool.
Remember learning all about functions in algebra? Neither do we. Oh sure, most of us remember linear plots and the magic of understanding y=mx+b for the first time. But a lot of us managed to slide by with only a tenuous grasp of more complex functions like exponentials and conic sections. Luckily the functionally challenged among us can bolster their understanding with this demonstration using analog multipliers and op amps.
[devttys0]’s video tutorial is a great primer on analog multipliers and their many uses. Starting with a simple example that multiplies two input voltages together, he goes on to show circuits that output both the square and the cube of an input voltage. Seeing the output waveform of the cube of a ramped input voltage was what nailed the concept for us and transported us back to those seemingly wasted hours in algebra class many years ago. Further refinements by the addition of an op amp yield a circuit that outputs the square root of an input voltage, and eventually lead to a voltage controlled resistor that can attenuate an input signal depending on its voltage. Pretty powerful stuff for just a few chips.
[Mark Gibson] probably has nothing against silicon. He just knows that a lot that can be done with simple switches, relays, and solenoids and wants to share that knowledge with the world. This was made abundantly clear to me during repeat visits to his expansive booth at Denver Mini Maker Faire last weekend.
In the sunlight-filled atrium of the Museum of Nature and Science, [Mark] sat behind several long tables covered with his creations made from mid-century pinball machines. There are about two dozen pieces in his interactive exhibit, which made its debut at the first-ever Northern Colorado Maker Faire in 2013. [Mark] was motivated to build these boards because he wanted to get people interested in the way things work through interaction and discovery of pinball mechanisms.
Most of the pieces he has built are single units and simple systems from pinball machines—flippers, chime units, targets, bumpers, and so on—that he affixed to wooden boards so that people can explore them without breaking anything. All of the units are operated using large and inviting push buttons that have been screwed down tight. Each of the systems also has a display card with an engineering drawing of the mechanism and a short explanation of how it works.
[Mark] also brought some of the original games he has created by combining several systems from different machines, like a horse derby and a baseball game. Both of these were built with education in mind; all of the guts including the original fabric-wrapped wires are prominently displayed. The derby game wasn’t working, but I managed to load the bases and get a grand slam in the baseball game. Probably couldn’t do that again in a million summers.
If you want to sell a toy for the toddler crowd, it ought to be pretty close to indestructible. A lot of toys out there are just plain nonsense game-wise and therefore waste their beefy potential. [2dom]’s wife was close to throwing out such a toy—a Little Tikes Goofy Ball. The thing literally does nothing but let you push its big buttons in. After some time passes, it pops them back out again and giggles. Game over. [2dom] rescued it from the trash and turned it into a toy that plays math games.
[2dom] removed the existing board and replaced it with an Arduino Pro Mini and a Darlington array that drives the motor that pops the buttons back out, the speaker, and a Nokia 5110 screen. Upon startup, the user chooses between addition, subtraction, and multiplication questions using the appropriate button. Questions appear in the middle of the screen and multiple choice answers in the corners.
Choose the right answer and the ball cheers and shows one of a few faces. Choose the wrong answer and it makes a buzzing sound and shows an X. There is an adaptive level system for the questions that [2dom] doesn’t show in the demonstration video after the break. For every five correct answers, you level up. His 3- and 5-year-olds love it. For more advanced teachable moments, there’s this toy-turned-enigma-machine.
We all know and love OpenSCAD for its sweet sweet parametrical goodness. However, it’s possible to get some of that same goodness out of Fusion 360. To do this we will be making a mathematical model of our object and then we’ll change variables to get different geometry. It’s simpler than it sounds.
Even if you don’t use Fusion 360 it’s good to have an idea of how different design tools work. This is web-based 3D Modeling software produced by Autodesk. One of the nice features is that it lets me share my models with others. I’ll do that in just a minute as I walk you through modeling a simple object. Another way to describe what we’re going to learn is: How to think when modeling in Fusion 360.
If I asked you to find the area of a square, you would have no problem doing so. It would be the same if I asked you to find the volume of a cone or rectangle or any other regular shape. You might have to turn to Google to get the proper formula, but it would be a trivial process nonetheless. But what if I asked you to find the volume of some random vase sitting on a kitchen counter? How does one go about finding the volume of irregular shapes?
One way would be to fill the vase with much smaller objects of a known volume. Then you could add up the smaller volumes to get an estimate of the total volume of the vase. For instance, imagine we fill the vase with marbles. A marble is a sphere, and we can calculate the volume of each marble with the formula 4/3πr3. We count all of our marbles and multiply the total by the volume of a single marble and arrive at our answer. It is not perfect, however. There is a lot of empty space that exists between the marbles as they fill the vase. We are forced to conclude that our estimated volume will be lower that the actual volume.
It would be about this time when our good friend Isaac Newton would ask the question “What if you made the marbles smaller?” Reducing the size of each marble would reduce the empty space that exists between them as they pile up in the vase, giving us a more accurate total volume. But how small? Is there a limit to how small we can make them? “Do not trouble yourself with the limit.” says [Newton]. “You will find that as you make the marbles smaller and smaller, you will begin to converge on a single number – and that number will be the exact volume of your vase.”
Reducing the size of the marble to get a more exact volume demonstrates the idea of the integral – one of the two fundamental principles of The Calculus. The other principle is known as the derivative, which we explained in our previous article by taking a very careful and tedious examination of an arrow in flight. In this article, we shall take the same approach toward the integral. By the end, you will have a fundamental understanding of what the integral is, and more importantly, how it works. Our vase example gives you a good mental image of what the integral is all about, but it is hardly a fundamental understanding of it. Just how do you make those marbles smaller? To answer this question, let us look again at one of Zeno’s moving arrows.