Believe it or not, counting is not special. Quite a few animals have figured it out over the years. Tiny honeybees compare what is less and what is more, and their brains are smaller than a pinky nail. They even understand the concept of zero, which — as anyone who has had to teach a toddler knows — is rather difficult to grasp. No, counting is not special, but how we count is.

I don’t mean to toot our own horn, but humans are remarkable for having created numerous numeral systems, each specialized in their own ways. Ask almost anyone and they will at least have heard of binary. Hackaday readers are deeper into counting systems and most of us have used binary, octal, and hexadecimal, often in conjunction, but those are just the perfectly standard positional systems.

If you want to start getting weird, there’s balanced ternary and negabinary, and we still haven’t even left the positional systems. There’s a whole host of systems out there, each with their own strengths and weaknesses. I happen to think seximal is the best. To see why, we have to explore the different creations that arose throughout the ages. As long as we’ve had sheep, humans have been trying to count them, and the systems that resulted have been quite creative, if inefficient.

One of the things hard about engineering — electrical engineering, in particular — is that you can’t really visualize what’s important. Sure, you can see a resistor and an LED in your hands, but the real stuff that we care about — electron flow, space charge, and all that — is totally abstract. If you just tinker, you might avoid a lot of the inherent math (or maths for our UK friends), but if you decide to get serious, you’ll quickly find yourself in a numerical quicksand. The problem is, there’s mechanically understanding math, and intuitively understanding math. We recently came across a simple site that tries to help with the latter that deserves a look.

If you don’t know what we mean by that, consider a simple example. You can teach a kid that 5×3 is 15. But, hopefully, a teacher at some point in your academic career pointed out to you what the meaning of it was. That if you had five packages of three items, you have 15 items total. Or that if you have a room that is five feet on one side and three feet on the other, the square footage is 15 square feet.

Electronics can be seen as really just an application of physics, and you could in turn argue that physics is the application of math to the real world. Unfortunately, the way most of us were taught math was far from intuitive. Luckily, the Internet is full of amazing texts and videos that can help you get a better understanding for the “why” behind complex math topics. Case in point? [3Blue1Brown] has a video showing how to solve 2D equations using colors. If you watch enough, you’ll realize that the colors are just a clever way to represent vectors and, in fact, the method would apply to complex numbers.

Honestly, we don’t think you’d ever solve equations like this by hand — at least not with the colors. But the intuitive feel this video can give you for how things work is very valuable. In addition, if you were trying to implement an algorithm in software this would be tailor-made for it, although you wouldn’t really use colors there either we suppose.

When a 13-year old Marie-Sophie Germain was stuck in the house because of the chaotic revolution on the streets of Paris in 1789, she found a refuge for her active mind: her father’s mathematics books. These inspired her to embark on pioneering a new branch of mathematics that focussed on modeling the real world: applied mathematics.

Post-revolutionary France was not an easy place for a woman to study mathematics, though. She taught herself higher maths from her father’s books, eventually persuading her parents to support her unusual career choice and getting her a tutor. After she had learned all she could, she looked at studying at the new École Polytechnique. Founded after the revolution as a military and engineering school to focus on practical science, this school did not admit women.

Anyone could ask for copies of the lecture notes, however, and students submitted their observations in writing. Germain got the notes and submitted her coursework under the pseudonym Monsieur Antoine-August Le Blanc. One of the lecturers that she impressed was Joseph Louis Lagrange, the mathematician famous for defining the mathematics of orbital motion that explained why the moon kept the same face to the earth. Lagrange arranged to meet this promising student and was surprised when Germain turned out to be a woman.

Gauss and Germain

‘Le Blanc’ also corresponded with German mathematician Carl Friedrich Gauss on number theory. When Napoleon’s armies occupied the town the famous mathematician lived in, Germain enlisted a family friend in the army to check that Gauss had not been harmed. Gauss didn’t realize who had helped him out, until he discovered that ‘Le Blanc’ was Sophie Germain, he wrote to her thanking her for her concern and praising her mathematical prowess given the hurdles set before her.

“How sweet is the acquisition of a friendship so flattering and precious to my heart. The lively interest you took during this terrible war deserves the most sincere recognition….But when a person of this sex, who, for our mores and prejudices, must recognize infinitely more obstacles and difficulties than men to become acquainted with these thorny searches, knows how to get rid of these obstacles and to penetrate what they have, most hidden, must undoubtedly, she has the most noble courage, talents quite extraordinary, genius superior.”

As well as working on the thorny and theoretical problems of number theory, Germain worked on applying mathematics to real world problems. One of these was a challenge set by the Paris Academy of Science to mathematically describe the elasticity of metal plates. An experimenter called Ernest Chladni had demonstrated that a metal plate would resonate in odd ways when vibrated at certain frequencies. If you put sand on the plate, it would collect in different patterns created by the resonance of the plate, called Chladni figures. To win the prize, the solution had to predict these figures.

The Mathematics of Stress and Strain

Mathematically predicting the behavior of metal plates could make it easier to design metal objects and predict how they would behave under stress. The prize was set in 1808 but was so difficult that Germain was the only one who decided to try to solve it, as it required coming up with a whole new way to analyze and describe how materials bend and change under stress.

The first two solutions that she submitted were rejected due to mathematical errors, but the third version won her the prize in 1816. However, due to the Academy policy of not allowing women to join (and to only attend events if they were wives of members), Germain was not able to attend the ceremony where the prize was granted. She was also not allowed to attend meetings of the Academy. After the Academy failed to publish her prize-winning work, Germain had to pay to publish the work herself in 1821.

Later, her friend Joseph Fourier allowed her to attend meetings and presentations, but the mathematical establishment never really accepted her, or her work. In a letter to a colleague in 1826 she complained about the way they rebuffed her:

“These facts are my domain and it is to me alone that they remain hidden. That’s the privilege of the ladies: they get compliments and no real benefits.”

In the same letter, Germain complained of suffering fatigue and she was diagnosed with breast cancer shortly afterwards. She died in 1831. Her final years were spent working on a solution to Fermat’s Last Theorem, and just before her death she published a partial solution that was the basis for much research into the theorem, which was finally solved only with computer help in the late 1990s.

Although Sophie Germain never earned a degree in her lifetime, she was given an honorary degree in 1837 from the University of Göttingen at the suggestion of Gauss, who noted that

“she proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree.”

The Academy that snubbed her also now offers an annual prize for mathematics in her name. Perhaps more importantly, her work formed the basis of the study of elasticity and stress in metals that allowed engineers to build larger objects and buildings. Creations such as the Eiffel tower in 1887 were directly influenced by her work, and it laid some of the groundwork for Einstein’s theory of General Relativity.

Modern scholars argue that Germain could have been more than she was: her work, they argue, was hamstrung by a limited understanding of some of the fundamental concepts that Gauss and others had described. Although her work was fundamental and important, if she had been given free access to the education that she wanted and deserved, it’s easy to imagine that she would have gone farther.

In 1962, John Glenn sat in his capsule waiting for his rocket engines to light-up and lift him to space. But first, he insisted that Katherine Johnson double-check the electronic computer’s trajectory calculations. While that’s the dramatic version of events given in the recent movie, Hidden Figures, the reality isn’t very far off. Glenn wasn’t sitting on the launchpad at the time, but during the weeks prior to launch, he did insist that Johnson double-check the computer’s calculations.

So who is this woman who played an important but largely unknown part of such a well-known historical event? During her long life, she was a wife, a mother, an African-American, a teacher, and a human computer, a term rarely used these days. Her calculations played a part in much of early spaceflight and in 2015, she was awarded the Presidential Medal of Freedom by President Obama. She also has a building named after her at the Langley Research Center in Hampton, Virginia.

The best rummage sale purchase I ever made was a piece of hardware that used Reverse Polish Notation. I know what you’re thinking… RPN sounds like a sales gimmick and I got taken for a fool. But I assure you it’s not only real, but a true gem in the evolution of computing.

Sometime in the 1980s when I was a spotty teen, I picked up a calculator at a rummage sale. Protected by a smart plastic case, it was a pretty good condition Sinclair Scientific that turned out when I got it home to have 1975 date codes on its chips, and since anything with a Sinclair badge was worth having it became mine for a trifling amount of money. It had a set of corroded batteries that had damaged one of its terminals, but with the application of a bit of copper strip I had a working calculator.

And what a calculator! It didn’t have many buttons at a time when you judged how cool a scientific calculator was by the prolific nature of its keyboard. This one looked more akin to a run-of-the-mill arithmetic calculator, but had button modes for trigonometric functions and oddly an enter key rather than an equals sign. The handy sticker inside the case explained the mystery, this machine used so-called Reverse Polish Notation, or RPN. It spent several years on my bench before being reverently placed in a storage box of Sinclair curios which I’ve spent half a day turning the house over to find as I write this article.

[Scott Cramer] is a retired professional woodworker who specializes in geometric art made from beautifully joined wood. In this project he’s carving four interlocked cloverleaf rings from a block of basswood. First he made a series of cuts to turn the block into a cuboctahedron, a geometric solid comprising six squares and eight triangles. Then he drew on the basic lines of the rings on the wood and went to work with a chisel, smoothing and separating the rings and carving out the interior. You can see more shots of the project on his Facebook post, which is included after the break.