1. People who cant read Venn diagrams but want to People who can read Venn diagrams People who cant read Venn diagrams and dont want to People who can and cant read Venn diagrams and want to People who cant read Venn diagrams and want and dont want to People who can and cant read Venn diagrams and want and dont want to People who can and cant read Venn diagrams and dont want to by Sid Harris

2. Surfaces that can close up and have to Surfaces that do better than standard double bubbles Surfaces that can and cant close up and have to Surfaces that do better than standard double bubbles but cant close up Surfaces that do better than standard double bubbles and can and cant close up but have to Surfaces that can close up and have to, and that do better than standard double bubbles Surfaces that cant close up

3. If there is a double bubble in R n 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. On the other hand, the image of the Gauss map must cover the whole sphere, with overlap. … and thus smaller Gauss map image. 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. 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. Unification

6. 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. Unification

7. Existence Nonsingularity in the moduli space Regularity Important details

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

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

10. 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? Triple bubble approach

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

12. Bring in an investment counselor Triple bubble approach