Why are the formulas for the sphere so weird? (major upgrade of Archimedes' greatest discoveries)  Summary and Q&A
TL;DR
This video explores the mathematical history and discoveries surrounding the volume and surface area formulas of the sphere, using creative visualizations and demonstrations.
Key Insights
 💠 Turning a hemisphere inside out reveals a shape similar to a cylinder minus a cone, leading to a breakthrough in understanding the volume of a sphere.
 🥳 Archimedes discovered that the ratio of the volume and surface area of a sphere compared to a cylinder is 3:2, a relationship that fascinated him to the extent of engraving it on his tombstone.
 By unfolding the surface of a sphere using various techniques, mathematicians can demonstrate the areapreserving property of transformations, providing insights into the formulas and properties of the sphere.
Transcript
Welcome to an extraextraextraspecial Mathologer video. Why extraextraextraspecial, three extras? Well, first extra, it's the 100th Mathologer video. There, the thumbnails for the first 99 videos. It's been quite a journey :) Second extra: it's my first crossover episode. For the first time, I'm collaborating with another channel. Ma... Read More
Questions & Answers
Q: What is the significance of turning the hemisphere inside out into a shape resembling a cylinder minus a cone?
By turning the hemisphere inside out, mathematicians can derive the volume formula for a sphere by subtracting the volume of the cone from the volume of the cylinder, providing a new perspective on an ancient mathematical problem.
Q: How does the volume formula for a sphere relate to Archimedes' achievements?
Archimedes established the volume and area formulas of the sphere, but Cavalieri, a mathematician 400 years later, refined the understanding by comparing the shapes of a cylinder and a cone.
Q: How is the areapreserving property of transforming the sphere into a circle demonstrated?
The video showcases a folding technique using crescent moon shapes to unfold the surface of the sphere onto a circle, preserving the area. This property is visually demonstrated by the smooth transition of the moons without leaving gaps.
Q: How are the volume and area formulas of the sphere related to each other?
The video explains that the volume formula, 4/3πr^3, and the area formula, 4πr^2, share a common aesthetic appeal, with the area formula being derived from the derivative of the volume formula.
Summary & Key Takeaways

The video celebrates the 100th Mathologer episode and features a crossover collaboration with mathematician and 3D printing artist Henry Segerman.

The first discovery showcased involves turning a hemisphere inside out to reveal a new shape, which is approximately a cylinder minus a cone.

The video then explores how the volume formula of a sphere can be derived by subtracting the volume of the cone from the volume of the cylinder.

Additionally, the video delves into the surface area of the sphere, explaining its relationship to the area of its shadow circle and showcasing an areapreserving transformation.