Lalbritton's "Ideal Harmonic Transformer" is a 3D printed mechanism for calculating sines and cosines: "
It is a thing to hold, enjoy turning the crank, and look at. If you can't find your calculator, and need to know the sine or cosine of an angle real quick, you can dial in the angle and read off of the Scotch Yokes. It also works in reverse."

Ideal Harmonic Transformer by lalbritton
(*via Make*)

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I don’t think the scope of sines and cosigns can be portrayed in physical form.

Sines and cosines vary from negative one to one, no matter what angle they relate to. It is impossible to construct a mechanical device to represent the tangent of any given angle, however.

You can’t do the whole thing, but you can easily construct a device which computes tan(x) for x in [-pi/2 + espilon, pi/2 – epsilon] for any epsilon >0.

That’s what we learn the half-angle formula for. Now range is no longer a problem, but precision is.

neat! I would add (if possible) some dark brown shoe polish to create a patina, making it look older… the shiny white just-out-of-the-oven look does not matches the antique looks of the tool.

This is genius. Every maths classroom should have one. If, as a 14 year old, I had known that sines and cosines actually meant something rather than a list of numbers on a chart I probably wouldn’t have thrown in the towel on trig, and by extension the rest of maths at high school.

But it’s much more efficient to teach you how to blindly slam numbers into an ugly calculator!

“list of numbers on a chart”

I’m guessing that he took trig before calculators made it to K12. They used to compute sines by linear interpolation of a lookup table of precomputed values. (For other transcendental functions too, I ran into a foot-thick book of values for the Bessel functions in my libraries basement once.) It had all the horrors of mindless plug-and-chug plus it was a big pain in the ass.

A lot of the math work on the manhatten project was done by young ladies sitting in a room doing math by hand, or aided by a mechanical calculator.

I am talking about 1990 here, so I had a calculator, but when I asked my maths teacher what the buttons meant, and what was happening when I pressed them, she just gave me a photocopy of the chart, so it might as well have been the 1950’s.

Don’t know if I can blame the teacher and curriculum completely, the girl I sat next to is now working at CERN and the one in front of me did her PhD in statistics.

Same here! The funny thing is, I really WANT to understand this stuff. Seeing it represented in meat-space might help…or maybe I just suck at math.

This is a fabulous tool! Kids and adults who are very visual-spatial, in particular, gravitate toward things that can be related to in physical form even if it is not a perfect representation. I am not unusual in that I “see” mathematical concepts, enabling me to perform mathematical operations on what I consider a mental playground. As a mother to two girls who tend to see things this way too, I am always looking for these kinds of tools.

When we talk about mathematics education, we need to revolutionize how it is taught from the earliest levels to accommodate those who are lost in computations and calculations and are crying out for a conceptual understanding–something that they don’t always get in elementary school. I’m not blaming the teachers; I’m blaming a system in which children tend to be discouraged from developing their own “conceptual” means to perform computations. Sure, there are visual aids but without an ability to explore them their use is often limited to the actual point of the lesson.

When I studied early mathematics education, I found most of the future teachers were less than excited about math, and rightfully so. They had inherited math anxiety or a stand-offish relationship with it from one of their own early teachers. Again, I cannot blame teachers. Many elementary teachers are fantastic at demonstrating the actual flexibility and conceptual nature of mathematics but the experiences of each student vary widely in this regard and it just takes one year where the light goes off to ruin math for a kid.

The solution: we need at least one generation of elementary Mathematics specialists in the schools to teach math with enthusiasm and openness–to teach computation but also the reasoning behind it and how there are many ways forward with a mathematical challenge or puzzle. Math is everywhere and computation-only systems are not the only way to approach it. I know, I know…nobody wants to pay more for kids’ education but it’s not necessarily about hiring new teachers, just re purposing the ones with the right skills and specialty. However it can be made to happen, it would be well-worth the effort to create the first generation of kids who will not be afraid of math.

I know I would have loved this thing if I’d had it in high school. Like you say, some of us need math to be concrete. My job is based on arithmetic, and one reason I’m good at it is that for me, arithmetic involves quantities I can “see”. I can add, subtract, multiply, and divide in my head because I can see how the numbers fit together.

In that way, I got far in geometry and algebra, but then I hit trig and it just didn’t make SENSE. Nobody showed me how the functions relate, and now I want to find someone who “sees” math to teach me how it works. I think a device like this would have been a huge help.

Wow – when I was in high school, the mechanical calculation systems we used to calculate trig functions and logarithms and exponents were much simpler than that – looked a lot like http://en.wikipedia.org/wiki/Slide_rule (or see Google Images for a lot more varieties of slide rules!)

This just makes me want a 3D printed Antikythera mechanism.

Very cool.

But talk to me when you can print one of these puppies:

I feel so relieved and heartened by some of the comments here from other people who “see” math or perceive of math within visual-spacial relationships. I, too, carried exemplary grades through my algebra and geometry class, yet hit a wall with trig and barely even passed the class. If I’d had a device like this when I was in school it would’ve changed the whole experience.

Anybody want to make a Curta?