You may remember that I collect slide rules. If you don’t, it probably doesn’t surprise you. I have a large number of what I think of as normal slide rules. I also have the less common circular and cylindrical slide rules. But I recently picked up a real oddity that I had to share: the Smarty Cat. It isn’t exactly a slide rule but it sort of is if you stretch the definition a bit.
Real Slide Rules
A regular slide rule takes advantage of the fact that you can multiply and divide by adding logarithms. Imagine having two rulers marked in inches or centimeters — it doesn’t matter (see the adjoining image). Suppose you want to add 5 and 3. You count off 5 marks on one ruler and line it with up the zero inch mark on the other ruler. Now you count off 3 marks on the second ruler and that position on the first ruler will indicate the result. Here it lines up with the 8 mark, which is, of course, the correct answer.
That’s a simple addition. But if you can convert your numbers into logarithms, add the logarithms, and then back out to a regular number, you can multiply.
Continue reading “Hands-On: Smarty Cat Is Junior’s First Slide Rule”
We’ve often noted that whether had ancient man known binary, we could all count to 1023 on our fingers. We thought about that while watching [Numberphile’s] latest video about “Russian” multiplication (see below). Apparently, the method dates back quite a way, sometimes known as Ethiopian or peasant multiplication. Even the ancient Egyptians did a form of it.
If you’ve ever written long multiplication code for a microcontroller, you can probably tell how this works. Each halving of the number amounts to a right shift. Each doubling is a left shift. Throwing out the even numbers means you only take the values when the least-significant bit is zero. Booth’s algorithm is more efficient, but the “Russian” method is simple to do on paper.
Continue reading “Hacking Multiplication: Binary Multiply On Paper”
You can do a lot of electronics without ever touching a tensor, but there are some situations in which tensors are absolutely essential. The problem is that most math texts give you a very dry description that is difficult to internalize. That’s where [The Science Asylum] comes in. Their recent video (see below) starts with the dry definition and then shows you what it means and why.
According to the video, the textbook definition is:
A rank-n tensor in m-dimensions is a mathematical object that has n indices and mn components and obeys certain transformational rules.
That sounds a lot like an array but we are not sure what “certain transformational rules” really means to anyone.
Wikipedia does a little better:
[A]n algebraic object that describes a linear mapping from one set of algebraic objects to another.
These constructs are key to anything electromagnetic (including antennas) and show up a lot in stress calculations and quantum mechanics. Even Einstien’s theory of relativity uses tensors.
Continue reading “Tensors Explained”
Everyone learns (and some readers maybe still remember) the quadratic formula. It’s a pillar of algebra and allows you to solve equations like Ax2+Bx+C=0. But just because you’ve used it doesn’t mean you know how to come up with the formula itself. It’s a bear to derive so the vast majority of us simply memorize the formula. A Carnegie Mellon mathematician named Po-Shen Loh didn’t expect to find a new way to derive the solution when he was reviewing math materials for middle school use to make them easier to understand. After all, people have been solving that equation for about 4,000 years. But that’s exactly what he did.
Before we look at the new solution, let’s talk about why you want to solve quadratic equations. They are used in many contexts. In ancient times you might use them to determine how much more crop to grow to cover pay tax payments without eating in to the crop you needed to subsist. In physics, it can describe motion. There’s seemingly no end to how many things you can describe with a quadratic equation.
Babylonians, in particular, would solve simultaneous equations to find the roots of a quadratic. Egyptians, Grecians, Indians, and Chinese peoples used graphical methods to solve the equations. The entire history is a bit much to get into, but still a great read. For this article, let’s dig into how the new derivation was discovered.
Continue reading “The Quadratic Equation Solution A Few Thousand Years In The Making”
Many languages feature a random number generator library for help with tasks like rolling a die or flipping a coin. Why, you may ask, is this necessary when humans are perfectly capable of randomly coming up with values?
The data from gathered from running the script with 200 pseudo-random inputs 100,000 times resulted in a distribution of correct guess approximately normal (µ=50% and σ=3.5%). The probability of the script correctly guessing the user’s input is >57% from calculating µ+2σ. The result? Humans aren’t so good at being random after all.
It’s almost intuitive why this happens. Finger presses tend to repeat certain patterns. The script already has a database of all possible combinations of five presses, with a counter for each combination. Every time a key is pressed, the latest five presses is updated and the counter increases for whichever combination of five presses this falls under. Based on this data, the script is able to make a prediction about the user’s next press.
In a follow-up statistic analysis, [ex-punctis] notes that with more key presses, the accuracy of the script tended to increase, with the exception of 1000+ key presses. The latter was thought to be due to the use of a psuedo random number generator to achieve such high levels of engagement with the script.
Some additional tests were done to see if holding shorter or longer sequences in memory would account for more accurate predictions. While shorter sequences should theoretically work, the risk of players keeping a tally of their own presses made it more likely for the longer sequences to reduce bias.
There’s a lot of literature on behavioral models and framing effects for similar games if you’re interested in implementing your own experiments and tricking your friends into giving you some cash.
Anybody interested in building their own robot, sending spacecraft to the moon, or launching inter-continental ballistic missiles should have at least some basic filter options in their toolkit, otherwise the robot will likely wobble about erratically and the missile will miss it’s target.
What is a filter anyway? In practical terms, the filter should smooth out erratic sensor data with as little time lag, or ‘error lag’ as possible. In the case of the missile, it could travel nice and smoothly through the air, but miss it’s target because the positional data is getting processed ‘too late’. The simplest filter, that many of us will have already used, is to pause our code, take about 10 quick readings from our sensor and then calculate the mean by dividing by 10. Incredibly simple and effective as long as our machine or process is not time sensitive – perfect for a weather station temperature sensor, although wind direction is slightly more complicated. A wind vane is actually an example of a good sensor giving ‘noisy’ readings: not that the sensor itself is noisy, but that wind is inherently gusty and is constantly changing direction.
It’s a really good idea to try and model our data on some kind of computer running software that will print out graphs – I chose the Raspberry Pi and installed Jupyter Notebook running Python 3.
The photo on the left shows my test rig. There’s a PT100 probe with it’s MAX31865 break-out board, a Dallas DS18B20 and a DHT22. The shield on the Pi is a GPS shield which is currently not used. If you don’t want the hassle of setting up these probes there’s a Jupyter Notebook file that can also use the internal temp sensor in the Raspberry Pi. It’s incredibly quick and easy to get up and running.
It’s quite interesting to see the performance of the different sensors, but I quickly ended up completely mangling the data from the DS18B20 by artificially adding randomly generated noise and some very nasty data spikes to really punish the filters as much as possible. Getting the temperature data to change rapidly was effected by putting a small piece of frozen Bockwurst on top of the DS18B20 and then removing it again.
Continue reading “Sensor Filters For Coders”
If you ask most people to explain the Fourier series they will tell you how you can decompose any particular wave into a sum of sine waves. We’ve used that explanation before ourselves, and it is not incorrect. In fact, it is how Fourier first worked out his famous series. However, it is only part of the story and master video maker [3Blue1Brown] explains the story in his usual entertaining and informative way. You can see the video below.
Paradoxically, [3Blue1Brown] asserts that it is easier to understand the series by thinking of functions with complex number outputs producing rotating vectors in a two-dimensional space. If you watch the video, you’ll see it is an easier way to work it out and it also lets you draw very cool pictures.
Continue reading “Fourier Explained: [3Blue1Brown] Style!”