Agreed, not least because:
- area-based visualizations make the effect hard to distinguish; bar charts or data clouds with numbers and confidence intervals would have been way more immediate.
- the colors make the negative group (usually) more visually prominent, since it has higher contrast with the background, exacerbating the area-estimation problem. (e.g. me wondering, "are there more overweight pink people as a fraction of pink people?")
While I agree the LSD is somewhat meaningless, I personally find it fun to test my color matching/mixing. Binary search circumvents that (at least for the most significant digit); though I agree it really only applies for the first and maybe second guess.
For instance, my initial guess was off by (+1,-1,-1,+3,-1-2) and my first impulse was to look at the target and see that I had too much red and not enough green.
That intuition makes a lot of sense to me, especially if I picture the person's frame of reference.
Kind of reminds me (a layman) of winding numbers. I suppose there are topologically inspired variations of this problem that might be even more "paradoxical" (or perhaps just silly). If you moonwalk the second half, you undo your rotation? Or if you follow specially designed subterranean tunnels, you can end up doing 0 or negative rotations!
"Travois" seemed the most interesting one. Based purely on the first page of Google images, it struck me as somewhere between a horse and a carriage, but ever more on the vehicle side.
it has no wheels so it is just dragged, maybe behind a horse but people can drag them too. horses walk and carriages roll, so it's not really between them. it's the asymptotic limit as wheels shrink to zero.
What a great example! As a recent astronomy enthusiast, I found myself doubting this comment initially ("well, eclipses ARE related"), and this despite the fact that I have a toy tellurion right by my desk.
But hearing a particular phrase in the below video helped correct my model. One sanity check is that you can see non-full moons during the day (although I definitely would have just assumed it was still a matter of angles).
Side comment: This is the first time I've seen my preferred username used for its actually meaning. And I've been using it for almost 3 decades. Neat.
I was going to use tellurian, which means an inhabitant of the earth, then saw that tellurion was sometimes used as an alternate spelling in an old Webster's dictionary I was looking through, and preferred it.
And yes, I did search through a dictionary to find a username.
I never did math past linear algebra/real analysis, so the only concept of sizes I have are countable/uncountable infinities.
Apparently the crux of this proof was showing that "the space of all modular forms with bounded denominators" and "the space of all congruence modular forms" were the same size.
I wonder what kind of expression "size" is here. Presumably not some finite integer, nor one of the simple infinities, since their first step was showing one is "a bit bigger" than the other. I wish this article went into more detail on that.
I definitely remember nerding out about modular forms via Andrew Wiles as a younger self.
If I understand the intro correctly, the "size" they're referring to is the growth rate of a sequence, where the sequence is counting the dimensions of certain subsets of bounded denominator modular forms.
Let BDMF = bounded denominator modular form.
They show congruence BDMFs grow at least N^3, but all BDMFs grow at most N^3*log(N). (The latter bound is the hard part of the proof.) To get the contradiction, they show a hypothetical noncongruence BDMF example would imply additional counterexamples that (just barely) get over the N^3*log(N) bound.
So is this a kind of result akin to the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem) where you have things that are asymptotically guaranteed to be/remain very close to one another?
Of particular note are some reinforcements from Old Norse (circa 850s onward):
- are (displacing bēoþ, sind, and sindon)
- their (displacing heora)
- they (displacing hīe)
As a linguistic game, sounds fun. Hopefully its pupils don't take it too seriously, hem hem, e.g. William Strunk the Lesser: "Saxon is a livelier tongue than Latin, so use Anglo-Saxon words"
