In addition to the other comments, what interested me about the findings mentioned in the 2016 quantamag article was the shared problem solving required to answer Joe Corbo's sense that there was a pattern to be found in the distribution of retinal cells.
To conclusively identify the pattern, the biologists needed to contact a theoretical chemist who studied optimal object packing density. Looking at the rest of that 2016 article shows the possible shared mechanisms and mathematics behind not just the hyperuniformity found in chicken retinas, but the same packing phenomena in other systems/objects. It is exciting to find organizational principles which can be seen in many different systems.
The other elements I like are that any study of receptors and neural organization, connection, and communication is that this outside-in approach ties in with study of the whole visual system and can connect other neurophysiological and cognitive findings. Not only is this relevant for some specialists, but many people are curious about the conscious experiences and capabilities of other species so learning about this may generate popular interest depending on the findings.
As a commentator (twic) mentions below, one of his colleagues did not care about the receptor packing patterns in her data. However, what one person ignores or deems a waste of time, another can pick up and use. Somewhere out there is a graduate student who is frantically looking for a finding to publish so I bet someone would be interested in looking at the control, visualization, quantification, or prediction of receptor spacing. I don't work at the cellular level, but this sounds fundamental. Mr. Twic - any reason not hand off your observation to an undergraduate or other interested party?
A simple proof of the Gaussian correlation
conjecture extended to multivariate gamma
distributions by Thomas Royen: https://arxiv.org/pdf/1408.1028.pdf
Mathematics has this funny habit of throwing around "it is easy to say that ...", or in this case, calling a proof that eluded mathematicians for decades "simple."
I had a professor who liked to talk about how "elementary" did not mean easy, it just means "uses only the foundations."
This proof is a perfect example of that statement. Formulating the problem the right way -- which is most often the hardest part -- was the real challenge here, not the mechanisms needed to do the formulation or the proof.
My favourite along those lines was "let epsilon be a small number which is not necessarily greater than zero". Everybody who spent the preceding year on epsilon-delta proofs did a double-take at that.
There's a wonderful book, by the way, Proofs from THE BOOK, of, well, simple beautiful proofs. The book is named after
> mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."
I had an undergraduate course my freshman year where we went through a circular proof of the equivalence of twelve or thirteen formulations of the axiom of choice. A hundred years ago, proving many of the steps of that proof might well have been non-trivial, perhaps even distinctly so.
Simple (easy to say) is not the same as being easy to deduce. A needle is a simple object, but finding one in a pile of hay takes a lot of work. Likewise, it is hard to pick out the features that make a problem simple when they are embedded in a sea of complexity. It becomes easy to say once you are no longer faced with the difficulty of knowing the right thing to say.
Well, no, the point isn't that simple propositions may have complex proofs, the point is that a proof may itself be simple, though hard to discover in the first place.
> A needle is a simple object, but finding one in a pile of hay takes a lot of work.
1. Gather all the hay in multiple trash bags (this assumes that the needle remains with the hay and doesn't fall out)
2. After gather the hay, use a roller-magnet pickup tool over the area the hay was at to find the needle if it fell out of the hay (assumes needle is made of ferrous magnetic material)
3. Place each bag, one by one, inside the torus of a CT scanner. Turn on CT scanner.
4. Remove bag of hay. Check inside of CT scanner torus for needle.
5. Repeat as needed with other bags.
Alternatively, one could set the hay pile on fire, then run over the ashes with the magnetic pickup tool.
That doesn't take away from the fact that it was very hard for it to be discovered.
And it's a good thing.
It's a good thing that things which were hard to discover can be communicated easily.
Otherwise we wouldn't be able to compress thousands of lifetimes worth of discoveries into a one-semester lecture.
The video mentions that physicists are looking for the "ripples" (gravitational waves) that would prove inflation, and while they haven't found them yet, they seem to be getting closer to detecting them.