Measure order
When mathematicians were sorting geometric figures, they found a connection to an entirely different realm. This could finally bring them closer to the goal of arranging algebraic equations according to their basic building blocks.
Imagine that there are two paper polygons in front of you. Is it possible to cut and reassemble the first one to get the second shape? What sounds like a typical brain teaser for puzzle lovers has occupied mathematicians for thousands of years now.
Because no matter how simple the question may seem, researchers are not satisfied with just picking up a pair of scissors and trying things out. Instead, they look for features that unequivocally determine in advance whether one object is "scissors congruent" to another.
In fact, there is a surprisingly simple criterion for the above example of two-dimensional polygons: such objects are scissor-congruent if they have the same area. This realization immediately opened up new questions. What about higher-dimensional figures, such as a tetrahedron? And what happens when you look at the two-dimensional polygons, three-dimensional polyhedrons, or higher-dimensional polytopes in curved geometries, in which their sides no longer correspond to straight lines, but geodesics, similar to the longitudes of the globe? …
