Volume, Surface, and the Infinite

Why could giants never actually roam the Earth? Joel David Hamkins begins with a question from Galileo: how do volume, surface area, and strength change as creatures scale up or down? The square-cube law reveals that giants would collapse under their own weight, while tiny humans would be bizarrely strong for their size. This insight about dimensional scaling opens the door to a series of mathematical surprises. Gabriel's Horn holds finite volume but has infinite surface area, raising the question of whether you can paint a surface that stretches on forever. The Koch Snowflake challenges our intuitions about area and perimeter. Hypersphere volumes peak unexpectedly at dimension five and then shrink toward zero. Hypercubes become "all corners" in high dimensions, and a blue sphere somehow escapes its bounding box. Each paradox deepens our understanding of how geometry behaves in ways that defy everyday experience, revealing the strange and beautiful consequences of infinity and dimension.