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People
who can
read Venn
diagrams
People who
can’t read
Venn diagrams
but want to
People who can and
can’t read Venn
diagrams and want to
People who can
and can’t read
Venn diagrams and
don’t 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’t read
Venn diagrams and don’t
want to
by Sid
Harris
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 can
close up and have to,
and that do better than
standard double
bubbles
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 do better
than standard double
bubbles
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.
… 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.
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.
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.
Important details
• Existence
• Nonsingularity in the moduli space
• Regularity
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.
Triple bubble approach
• Slicing
• Equivalent problems (Caratheodory)
• Paired calibration
• Gauss’ divergence theorem
• Metacalibration
• Localized unification
• Adaptive modeling
• Weighted planar triple via unification
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?”
Triple bubble approach
• Equivalent problems
• Add a telescoping sum to local problems
• Pieces borrow and lend to neighbors
• How much to borrow or lend?
Triple bubble approach
Bring in an investment counselor
Triple bubble approach