It is an unexplained fact that objects with icosahedral symmetry occur in nature only at microscopic scales. Examples include quasicrystals, many viruses, the carbon-60 molecule, and some beautiful protozoa in the radiolarian family.
Can you combine simple observations and mathematical thinking to show that atoms exist?
The mathematics of knot theory says that a simple loop and a trefoil are fundamentally different knots. But is that all there is to the question?
The Menger Sponge, a well-studied fractal, was first described in the 1920s. The fractal is cube-like, yet its cross section is quite surprising. What happens when it is sliced on a diagonal plane?
George Hart describes in this video how to create physical models of mathematical objects, surveying some examples of surfaces and polytopes.