Hi, Yoav! I guess at the time I didn’t know, or had forgotten, who conjectured that. I’m glad you checked small perturbations. And it’s great to let people try to come up with counterexamples!

]]>I’m 3 years late to this, but better late than never. I believe you’re talking about me (“whoever conjectured the heptagon was the worst possible convex shape for dense packing”). I did in fact consider small perturbations and I was able to prove that all sufficiently small perturbations allow for denser packing (see https://ykallus.github.io/gt2014.html). I also made a little javascript interactive where you can try to come up with a counterexample: https://ykallus.github.io/demo/shape.html

]]>I don’t think Hammersley ever wrote up a proof. By now a better solution is known. I believe this paper has a proof that this better solution is really a solution, but you’d have to read it to see what you think:

• Joseph L. Gerver, On moving a sofa around a corner,*Geometriae Dedicata* **42** (1992), 267–283.

With real plane curves we say \(y = 0\) intersects \(y = x\) in one point, \(y = x^2 – \epsilon\) intersects \(y = 0\) in two points for \( \epsilon \gt 0\), and \(y = x^2\) intersects \(x = 0\) in a double point: “two points that are infinitesimally far apart”.

With surfaces I guess ‘double point’ is used by analogy.

]]>The term “double points” seems like a misnomer! That cone looks as if it’s bunching together continuum many points at the origin, not just two.

]]>w is the hallway of width

r is inside hallway outer cycle ‘s radius

theory:

use the hallway outside create a inside cycle C1 and inside of hallway create outside cycle C2,

then use the area of C1 – C2 is the cycle can pass through the hallway, the area should be the max

issue:

did not figure the relation of w and r, is anyone can help me?

I don’t know the answer to these questions. They’re nice questions! I believe they should be easier than the double-lattice packing questions… but not easy enough for me answer them.

]]>the cross section of a red blood cell, if you cut it in half, looks similar to this.

]]>