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I'm reading Random Circles on a Sphere and the authors did the following to empirically check their results:

To make a partial test of the accuracy of the above approximations an experiment was carried out using table tennis balls. These had a mean diameter of 37.2 mm. with a standard deviation around this mean of 0.02mm. One hundred holes of diameter 29.9mm. were punched in an aluminium sheet forming one side of a flat box. The balls were held firmly against the holes by a foam rubber pad, and sprayed with a duco paint. After drying they were removed and replaced at random by hand. Forty sprayings were done in each of three sets of 100 balls. The number of balls not completely covered after N sprayings are shown in Table 2, and fit the theoretical curve rather more closely than the roughness of the approximations used would lead one to expect. The angle α was about 53.43° as used in the calculations.

What are other examples of mathematicians turning into carpenters to test theories in modern days (post 60s)?

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    $\begingroup$ Persi Diaconis springs to mind. $\endgroup$ – JP McCarthy Feb 26 at 14:23
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    $\begingroup$ By searching for "mathematicians turning into carpenters" I presume you wish to exclude computer experiments? Would mathematical physics count? (Testing a mathematical theory by comparing with a physics experiment.) $\endgroup$ – Carlo Beenakker Feb 26 at 14:57
  • $\begingroup$ @CarloBeenakker correct, computers should not be involved. Also, I would prefer things not related to physics experiments, since they are a more natural way for experimentation. My interest is in creative, non-digital ways of experimenting with mathematical theories, especially aiming for publication, like the authors I quoted. $\endgroup$ – guest Feb 26 at 15:05
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    $\begingroup$ I couldn't find a photo or even any details, so I don't want to add it as an answer, but I've read somewhere that in the early days of Conway's Game of Life, it was carried out by physical markers on a tiled floor. $\endgroup$ – John Coleman Feb 27 at 22:13
  • $\begingroup$ I thought it was Go stones on a 19x19 Go board. (It's convenient to have two colors of stones: you can use black for the current pattern and white for the cells born in the next generation before removing the black stones that don't have exactly 2 or 3 black neighbors.) But I don't know of a picture either. $\endgroup$ – Noam D. Elkies Feb 28 at 2:30
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I am encouraged to give this answer by the comment of the OP "My interest is in creative, non-digital ways of experimenting with mathematical theories, especially aiming for publication".

My colleague Hendrik Lenstra used his expertise with elliptic curves to fill in the empty hole at the center of a litho by Escher, see Artful Mathematics:

Original litho on the left, zoom into the completed hole at the right.

We shall see that the lithograph can be viewed as drawn on a certain elliptic curve over the field of complex numbers and deduce that an idealized version of the picture repeats itself in the middle. More precisely, it contains a copy of itself, rotated clockwise by $157.6255960832\ldots$ degrees and scaled down by a factor of $22.5836845286\ldots$

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I was amused by the work of Scott Aaronson on soap bubbles and Steiner trees. Given $n$ points $p_1$, $p_2$, ..., $p_n$ in $\mathbb{R}^2$, the Steiner tree through these points is the connected planar graph of shortest length containing these points. Computing the topology of the Steiner tree, given the coordinates of the points, is NP-hard.

On the other hand, if two parallel glass plates are separated by rods in positions $p_1$, $p_2$, ..., $p_n$ and dipped into a bucket of soapy water, the shape of the resulting soap film is a local minimum for this problem. Which lead to the question: How close is the local minimum, formed by whatever complicated PDE governs soap film, to the NP-hard global minimum?

Aaronson decided to try the experiment, and reported on his results in Section 3 of "NP-complete problems and physical reality". The answer was not very close: The stable shape of the soap film was often not even a tree!

enter image description here

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Possibly the OP might allow Lehmer's "bicycle chain sieves" https://en.wikipedia.org/wiki/Lehmer_sieve, which for several decades were the state of the art in factoring large numbers; according to http://ed-thelen.org/comp-hist/Mike-Williams-Lehmer.html the first was made in 1926 and "it was only some time in the early 1960s when computers were fast enough to match these speeds".

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Psychologist Frank Rosenblatt built the first neural networks in 1957/8. Today it is trivial to build a neural network using software, but Rosenblatt built a neural network using analogue hardware.

The perceptron could tell the difference between triangles, circles and and squares, or between different letters of the alphabet. Most critically, it could get better at doing so as it was given feedback on whether it was correct or incorrect about previous predictions. In this sense, it could 'learn'.

A hardware implementation of a neural network

The input to the perceptron was an array of 20 x 20 grid of light sensitive resistors

enter image description here

And the weights/bias values of the primitive neural network were stored in racks of cylindrical objects each consisting of an electrical motor and potentiometer (rotary resistor).

enter image description here

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I am not sure if this counts, because maybe it is physics more than mathematics, but the density of packing tetrahedra in space was experimentally investigated by Jaoshvili, Esakia, Porrati, and Chaikin, Experiments on the random packing of tetrahedral dice, Phys. Rev. Lett. 104 (2010), 185501. I believe that at the time they did their experiments, the empirically observed packing density was slightly higher than what mathematicians had been able to construct explicitly.

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(1) The history of the double bubble conjecture suggests that, in the 19th century, it was likely formulated from physical observation. It was only settled in 2000. (The image below would violate the conjecture.)


      DoubleBubble


(2) Closer to modern times, a 2006 MIT math Ph.D. thesis used physical experiments to formulate two conjectures.

"This thesis studies surfaces which minimize area, subject to a fixed boundary and to a free boundary with length constraint. Based on physical experiments, I make two conjectures." The physical experiments used a "heart curve" and a "bent-paperclip curve."

Stephens, Benjamin Keith. "Thread-wire surfaces." PhD diss., Dept. Math., Massachusetts Institute of Technology, 2006.

The closest I could get to one his physical examples is in a poor-quality photo in a 2006 talk Thread-wire surfaces: "I’ll begin my talk by showing videos of experiments I did with wire, thread and soap surfaces."

ThreadWire_20-Figure2-1-1

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    $\begingroup$ Not sure if this is related to the second point, but William Irvine has done some amazing works on knotty vortex loops. Here is a nice video demonstration. $\endgroup$ – polfosol Feb 27 at 7:13
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Berlekamp, I believe in the late 60s at Bell Labs, built a 10x10 version of a lightbulb switching game, which was studied earlier by Gleason and is related to covering radius in coding theory.

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This is more of a physics example, but there are a few articles in American Journal of Physics where the authors come to conclusions about different types of unrestrained brachistochrone (ie. the problem of finding the path of a frictionless track between two horizontally separated points along which a block with an initial velocity will travel in the shortest time) and then build tracks out of acrylic to roll or slide objects down and test the conclusions.

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