People who can’t read Venn diagrams but want to

1. People who can’t read Venn diagrams but want to People who can read Venn diagrams People who can and can’t read Venn diagrams and want to People who can and can’t read Venn diagrams and want and don’t want to People who can’t read Venn diagrams and want and don’t want to People who can and can’t read Venn diagrams and don’t want to People who can’t read Venn diagrams and don’t want to by Sid Harris

2. Surfaces that can close up and have to Surfaces that can’t close up Surfaces that can and can’t close up and have to Surfaces that do better than standard double bubbles and can and can’t close up but have to Surfaces that do better than standard double bubbles but can’t close up Surfaces that can close up and have to, and that do better than standard double bubbles Surfaces that do better than standard double bubbles

3. If there is a double bubble in Rn that does better than a standard one, then the one that beats standard by the largest margin must consist of pieces with smaller area and smaller mean curvature, and thus smaller Gauss curvature (Jacobian of Gauss map), and thus smaller Gauss map image.

4. … and thus smaller Gauss map image. On the other hand, the image of the Gauss map must cover the whole sphere, with overlap. The overlap size is determined by the singular set, and is larger for competitors than for the standard. Smaller areas + smaller curvature + more doubling back  inability for exterior to close up.

5. Unification • Context: A whole family of conjectured minimizers • “Beating the spread” • Divide surface area by expected minimum • Similar to Lagrange multipliers in spirit and in effect.

6. Unification • Letting volumes vary gives control on individual mean curvatures • Letting weights vary gives control on individual surface areas • Letting slicing planes vary replicates the method of proving minimization by slicing • Combining all these can create a powerful tool.

7. Important details • Existence • Nonsingularity in the moduli space • Regularity

8. Gauss map overlap due to singular set • An annulus for each singular circle (sphere) • Width constant determined by weights • Isoperimetric solution on sphere implies result.

9. Triple bubble approach • Slicing • Equivalent problems (Caratheodory) • Paired calibration • Gauss’ divergence theorem • Metacalibration • Localized unification • Adaptive modeling • Weighted planar triple via unification

10. Triple bubble approach • Slicing • “Question me an answer” • Partition a proposed minimizer and ask what local problem each piece ought to solve. • Example: the brachistochrone • Contrasting example: a piece of equator • “Does it work on the margin?”

11. Triple bubble approach • Equivalent problems • Add a telescoping sum to local problems • Pieces borrow and lend to neighbors • How much to borrow or lend?

12. Triple bubble approach Bring in an investment counselor

13. Triple bubble approach