One error that leapt out at me - it says:
“The symbol for showing that one set is a member of another set is ⊂”
That’s wrong. That’s the subset symbol, and I suspect it’s the symbol that’s right, and the English that’s wrong.
The example is right: P ⊂ N ⊂ Z ⊂ Q
But these are not members of each other, they are subsets.
Previously with the HN juice: https://lobste.rs/s/dzpi3p/spikey_spheres
Hah - I hadn’t seen that someone had cross-posted it. Sorry.
In truth, the actual math problem has nothing to do with pizza.
Dissect a square into congruent pieces that all touch the center point. This is a classical dissection, and yes, you can think of it as slicing up a square with scissors. In how many ways can you do this? Hint: probably more than you think. How many pieces can you use? There are probably fewer options than you think.
Next: Dissect a square into congruent pieces that do not all touch the center point. Again, i how many ways can you do this? The answer might be smaller than you think.
Next: Dissect a square so that the center point is in the interior of a piece. Again, not hard. Again, in how many ways can you do this?
Moving on - replace “square” with “triangle”. Then “pentagon”. All three options are still possible, and again counting families and characterising solutions is an interesting exercise.
And now we’re ready for the real problem.
Dissect a circle into congruent pieces that do not all touch the center point. I know of two infinite families, one consisting of infinitely many uncountable members. That’s a lot.
But this is an open question:
Can you dissect a circle into congruent pieces such that the center point is in the interior of one of the pieces?
This isn’t especially new - I wrote about it 5 years ago, and worked on it with Joel (one of the authors).
That’s why I get really annoyed when I’m getting maths news from general sites. I don’t know where else to get the news, though. It’s been a while since I’ve been in academia, so where is the news sites for mathematics people?
I don’t know of any single source - these days it seems unlikely that there can be. However, there are several sources that talk about math in general, and they usually pick up on major stories curiosities such as this. Included are:
They, in turn, list others. I write about this sort of stuff, but I’m not qualified to write about real advances in serious math.
In short, I don’t think there is a “math news” site that writes things “properly” without trying to make it “relevant” for ordinary people.
Excellent question. Upvoted in hope of an answer.
Actual error bound?
Someone else has taken the title literally and computed the error when you divide by 3000 instead of (54x53). However, that’s not the point of the article.
You can get a handle on the error in the article by identifying all the approximations and estimating from there. I’ll be doing that in a follow-up, but the main error comes from using e ~ 2.7. It’s closer to 2.72. That error is about 0.7%. When we raise to the power of 54 that gives an error of about 40% to 50%.
But that is offset by the approximation that 2^10 ~ 10^3, and raising that to the power of 5. It’s an interesting and useful exercise to refine the original by chasing down estimate of the errors. I might do that in a follow-up post.
By the definition of factorial as 1×2×…×n, you can get 52! exactly if you divide 54! by 53×54=2862. If you divide by the approximation 3000 instead of 2862, the result is exactly 95.4% of 52!.
There’s a great deal more approximation being done than just 3000 ~ 2862. The calculation of 54! is itself very approximate.
Anyway, now that I’m back at my laptop: (10^68 - 52!) / 52! ~ 0.24.
I really hope you understand that that’s not the point of the article.
I’m Rider of Giraffes, and my details are in my profile.
You can get access to the user list via the “About” link at the bottom of the page, and from there you can see the profiles people have chosen to put up. I haven’t time now, but it would be interesting to see if people gave details here, and yet have nothing in their profiles. I wonder if that would say anything about them?
Suggestion to the lobste.rs developers: when new users accept their invitations, lobste.rs should prompt them to fill in their profiles with the same link that they put in their invitation requests (or edit if they prefer).