The mathematics of "linkages" resembles classical geometry but incorporates motion.
While imagined movement is commonly used to aid reasoning in geometry, linkages are explicitly understood as jointed assemblies of rigid objects, and what's studied is their movement relative to each other.
Such theory, perfectly suited to application during the Industrial Revolution, began to occupy mathematicians in the 19th century, soon after the development of modern precision machining. See here: https://www.youtube.com/watch?v=djB9oK6pkbA
Proliferation of mechanical computing devices such as we already discussed was a direct consequence: https://www.youtube.com/user/MechanicalComputing https://en.m.wikipedia.org/wiki/Curta
Here are all the linkage articles: https://makezine.com/article/maker-news/math-monday-linkages-an-introduction/
(1) Study the first two articles above, and browse the remainder. Download the "kit" from the first article, paying attention to the general instructions/advice that follow the PDF link.
(2) Build a linkage of 4 or more bars. You can follow instructions from one of the articles, or invent your own. Explore its movements for a while, then make and submit a short video demonstrating its full range of motion.