About 200BC, Euclid formalized the geometry that could be understood with a straight-edge and compass. Great job, good enough to build cathedrals, castles and mosques, and the basis for most progress in math and science for 1700 years afterwards. One of Euclid’s contemporaries in Greek times, Archimedes of Syracuse, almost got to calculus (lost to the Roman culture), that is the power of that line of thought.
This discusses a couple of things that CANNOT be done within Euclidean geometry, using straight-edge and compass:
So trisecting an angle resists human minds for a couple of thousands years until Galois invents a new mathematical tool in algebra that proves those tools can’t find cube roots and a school teacher started paying attention to origami which could. Straight-edge and compass to a flat sheet of paper, such a tiny change in the tools, yet the old problem disappears :
Paper reached Europe by the 1200s. So for 500+ years, European geometers failed to look past their straight-edges and compasses, stayed firmly in the intellectual box.
In fact, origami can be used to solve problems that require equations with 4th powers and roots.
Clearly, human minds use mental models to explore other mental models, and that process shows no kinds of stopping. What escapes from the box might come of :
All of that above should be really encouraging to us ordinary minds, we have such much better tools, now including Youtube and those videos, and so many intellectual barriers have been found in so many different areas. Yesterday’s thoughts on Zeno’s Paradox may be an example.