Mechanical Mathematics

[beero1000]

An interest of mine is in the physical manifestations of mathematical ideas. I really enjoy simple objects that implicitly utilize mathematics to perform a task. I've been toying with course ideas in that regard - maybe named something like "Mechanical Mathematics". I think that would be a cool class to teach (and take, I hope).

I came across this video on youtube recently. Contrary to it's title, there is math going on (a generalization of  Thales' theorem). Using a simple jig made of straight pieces of wood, you can trace curves that are arcs of large circles without finding the center or radius. Particularly useful for carpentry.



Another favorite of mine are the planimeters - tools that measure area (using  Green's theorem)



There are many more like that. Ingenious people solved problems well before the modern age allowed easy digital computation. There are mechanical governors, analog control systems, linkages, etc. Do you know any other good ones?

 

[untermensche]

In the top video he uses a mechanical device to do something that can be figured out with mathematics.

But mathematics has nothing to do with what he is doing.

 

[Malintent]

Metronome synchronization on a moving table:
"If you place 32 metronomes on a static object and set them rocking out of phase with one another, they will remain that way indefinitely. Place them on a moveable surface, however, and something very interesting (and very mesmerizing) happens."
https://www.youtube.com/watch?v=RjF1_eDEsqc

[YOUTUBE]https://www.youtube.com/watch?v=RjF1_eDEsqc[/YOUTUBE]

 

[Bomb#20]

There are many more like that. Ingenious people solved problems well before the modern age allowed easy digital computation. There are mechanical governors, analog control systems, linkages, etc. Do you know any other good ones?

The secret to Mt. Palomar's 45 years at the leading edge of astronomy:

"All large modern mirrors are also either composites of many small mirrors, like the Keck mirrors, or are built of thin glass blanks that sag dramatically under their own weight. In either case, the telescopes require complex systems of computer-controlled actuators to force the mirrors into the proper shape as they sag differently depending how high in the sky the mirror is pointed. The Hale mirror has the same problem, but the solution is strictly mechanical — the mirror supports have weights and springs that cause them to counterflex the mirror just the right amount. In effect, it's a distributed, mechanically based, analog computer — a startlingly effective solution."

(Source)

 

[beero1000]

The BBC had a documentary on mechanical automata, fantastic stuff. Some of them are beyond belief.

 

[Uwe]

A paper-and-pencil method of short-cutting systems of linear equations used in engineering comes to mind.
I was never taught it in school, what with CAD software eliminating pencils, but I re-discovered it using parametric 3D CAD later in life.

The idea is that forces travel along the lines of the structure under scrutiny, and when in equilibrium (that is: not moving or falling down) you can perform vector math with those forces by drawing them as groups of triangles. https://en.wikipedia.org/wiki/Cremona_diagram

 

[skepticalbip]

There is the old flyball governor that was used in early steam engines to control speed. It was a mechanical solution that bypasses a hell of a lot of math and physics.

 

[fromderinside]

Not only math but description too "balls out"

 

[beero1000]

There is the old flyball governor that was used in early steam engines to control speed. It was a mechanical solution that bypasses a hell of a lot of math and physics.

View attachment 5068

A super simple piece of mechanical equipment that can be easily tweaked to solve a large class of differential equations? Yeah, that one was Watt. Dude was a genius who almost single-handedly made the industrial revolution possible.

Speaking of Watt, I remember one of his quotes saying that the invention he was most proud of was his straight-line linkage. He figured out how, using bars and pivots, you can trace the pedal curve of a hyperbola, approximating a straight line. Nothing major, just the idea that made double-action engines possible.

 

[Alcoholic Actuary]

Mechanical Random Number Generator:



aa

 

[ronburgundy]

Mechanical Random Number Generator:

View attachment 5073

aa

Dork Alert!: The die in your pic, like most die, are actually non-random. Due to the numbers being represented by divots each side has different mass and different friction against the rolling surface. When combined with the slightly rounded and smoothed corners, this results in the die being more or less likely to lose momentum and stop depending on which side is up or down. Cheap die with air bubbles can also matter.

Apparently 1s are the most common result, occurring up to twice as often as they should by chance with typical die included with most games. Vegas uses die with no divots, perfect squares with 90 degree corners and clear so they can inspect for lack of uniform consistency and bubbles.

 

[skepticalbip]

You shouldn't forget toys like the Spirograph that lets five-year-olds create elaborate madalas.

 

[barbos]

There is the old flyball governor that was used in early steam engines to control speed. It was a mechanical solution that bypasses a hell of a lot of math and physics.

View attachment 5068

A super simple piece of mechanical equipment that can be easily tweaked to solve a large class of differential equations? Yeah, that one was Watt. Dude was a genius who almost single-handedly made the industrial revolution possible.

Speaking of Watt, I remember one of his quotes saying that the invention he was most proud of was his straight-line linkage. He figured out how, using bars and pivots, you can trace the pedal curve of a hyperbola, approximating a straight line. Nothing major, just the idea that made double-action engines possible.

View attachment 5072

double-action engines don't require this.

 

[connick]

Here's an example of a physical manifestation of mathematics that I found quite elegant when I first came across it: https://www.youtube.com/watch?v=CAkMUdeB06o

I'm sure there are others that just haven't come to mind yet. It is an interesting topic for sure.