The speaker is asked to choose a math proof to demonstrate the beauty of mathematics. They select the proof that the definitions of an ellipse as a stretched circle, a set of points with a constant sum of distances to two foci (thumbtacks and string), and a conic section (slicing a cone) are equivalent. The proof involves introducing two spheres to the cone, one above and one below the plane, which are tangent to the cone and the plane. The speaker guides the viewer through the proof, showing that the sum of the distances from any point on the ellipse to the two foci is constant, and that this is equivalent to the constant-sum property of the thumbtack definition. The proof is attributed to Germinal Pierre Dandelin in 1822, and the spheres are known as "Dandelin spheres." The speaker argues that this proof is a good representation of mathematics because it is both beautiful and substantial, and it reflects the common feature of mathematics that there is no single "most fundamental" way of defining something, but rather showing equivalences. The speaker also discusses the creative construction involved in the proof, and how it is an example of mathematical discovery. They quote Paul Lockhart, who says that the origin of such ingenious arguments is unknown, but the speaker suggests that it can be attributed to experience and practice in geometry.
Here are the key facts extracted from the text:
1. There are at least three main ways to define an ellipse.
2. One way to define an ellipse is by stretching a circle in one dimension.
3. Another way to define an ellipse is using the classic two-thumbtacks-and-a-piece-of-string construction.
4. The two thumbtack points in the construction are called the foci of the ellipse.
5. The constant sum of distances from each point on the ellipse to the two foci is a defining property of an ellipse.
6. Slicing a cone with a plane at an angle produces an ellipse.
7. The curve of points where the plane and cone intersect forms an ellipse.
8. An ellipse is a family of curves, ranging from a perfect circle to something infinitely stretched.
9. The specific shape of an ellipse is quantified by its eccentricity.
10. The eccentricity of an ellipse is determined by the distance between the foci divided by the length of the longest axis of the ellipse.
11. The eccentricity of an ellipse can also be determined by the slope of the plane that slices the cone.
12. The proof that slicing a cone produces an ellipse was first found by Germinal Pierre Dandelin in 1822.
13. The two spheres used in the proof are sometimes called "Dandelin spheres".
14. The same trick can be used to show why slicing a cylinder at an angle will give an ellipse.
15. Projecting a shape from one plane onto another tilted plane has the effect of simply stretching that shape.