because there are rational numbers arbitrarily close to pi, and many such constructions have been found – sure, but I think more to the point, there are constructible numbers arbitrarily close to pi. The more-interesting constructions, like Ramanujan's, use irrational numbers.
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
The irrational constructible numbers do not provide more-accurate approximations than the rational ones; how could they? But I think the explanation for why good approximations exist is misplaced in the lead, so I removed the "because" clause. —
David Eppstein (
talk)
07:07, 25 May 2022 (UTC)reply
Cool. And all I meant was that a reader might think that only constructions of rational approximants are particularly good, or even possible, instead of the many approximations which construct irrational lengths. I like how it is now.
Ovinus (
talk)
07:43, 25 May 2022 (UTC)reply
Do we have a neusis construction for squaring the circle? I don't think the use of the quadratrix or Archimedes spiral are quite the same thing? My impression is that neusis, at least in its more strict definitions, can solve some higher-order algebraic problems like trisection or cube-doubling but isn't powerful enough for circle-squaring. Knorr's section on Archimedes' neuses discusses only trisection and heptagon construction (with also the tangent to a spiral early in the Archimedes chapter), for instance. —
David Eppstein (
talk) 07:28, 25 May 2022 (UTC
Ah I should have been more specific. Indeed neusis can't solve this problem. I meant in the sentence "Therefore, more powerful methods than compass and straightedge constructions..." but perhaps there are other constructions you didn't think fit to mention. Neusis was just the one that came to mind, since it can solve those problems. (Didn't know paper folding could though!)
Ovinus (
talk)
07:43, 25 May 2022 (UTC)reply
I'd like one sentence in History that says something about later professional work being mostly about finding clever approximations
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
Any info on how Ramanujan found his approximation? Or was this one of his "determined empirically" pieces of divine inspiration
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
Ramanujan writes "This value was obtained empirically, and it has no connection with the preceding theory." It's easy to obtain it from the continued fraction representation of by stopping just before the huge term 16539. Why has a huge term in its continued fraction is beyond me, though. —
David Eppstein (
talk)
19:16, 25 May 2022 (UTC)reply
I named Milü in History, but if you don't want it there, I'd suggest putting it in Constructions using 355/133.
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
Sources look great, citations are formatted nicely, prose is clear and engaging (and appropriately fun in places).
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
because there are rational numbers arbitrarily close to pi, and many such constructions have been found – sure, but I think more to the point, there are constructible numbers arbitrarily close to pi. The more-interesting constructions, like Ramanujan's, use irrational numbers.
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
The irrational constructible numbers do not provide more-accurate approximations than the rational ones; how could they? But I think the explanation for why good approximations exist is misplaced in the lead, so I removed the "because" clause. —
David Eppstein (
talk)
07:07, 25 May 2022 (UTC)reply
Cool. And all I meant was that a reader might think that only constructions of rational approximants are particularly good, or even possible, instead of the many approximations which construct irrational lengths. I like how it is now.
Ovinus (
talk)
07:43, 25 May 2022 (UTC)reply
Do we have a neusis construction for squaring the circle? I don't think the use of the quadratrix or Archimedes spiral are quite the same thing? My impression is that neusis, at least in its more strict definitions, can solve some higher-order algebraic problems like trisection or cube-doubling but isn't powerful enough for circle-squaring. Knorr's section on Archimedes' neuses discusses only trisection and heptagon construction (with also the tangent to a spiral early in the Archimedes chapter), for instance. —
David Eppstein (
talk) 07:28, 25 May 2022 (UTC
Ah I should have been more specific. Indeed neusis can't solve this problem. I meant in the sentence "Therefore, more powerful methods than compass and straightedge constructions..." but perhaps there are other constructions you didn't think fit to mention. Neusis was just the one that came to mind, since it can solve those problems. (Didn't know paper folding could though!)
Ovinus (
talk)
07:43, 25 May 2022 (UTC)reply
I'd like one sentence in History that says something about later professional work being mostly about finding clever approximations
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
Any info on how Ramanujan found his approximation? Or was this one of his "determined empirically" pieces of divine inspiration
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
Ramanujan writes "This value was obtained empirically, and it has no connection with the preceding theory." It's easy to obtain it from the continued fraction representation of by stopping just before the huge term 16539. Why has a huge term in its continued fraction is beyond me, though. —
David Eppstein (
talk)
19:16, 25 May 2022 (UTC)reply
I named Milü in History, but if you don't want it there, I'd suggest putting it in Constructions using 355/133.
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply
Sources look great, citations are formatted nicely, prose is clear and engaging (and appropriately fun in places).
Ovinus (
talk)
00:58, 23 May 2022 (UTC)reply