[HN Gopher] Pi from High School Maths
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Pi from High School Maths
Author : vonadz
Score : 47 points
Date : 2021-02-09 11:11 UTC (11 hours ago)
(HTM) web link (theartofmachinery.com)
(TXT) w3m dump (theartofmachinery.com)
| wizzwizz4 wrote:
| This is a very elegant approach. I have a project where I'm
| basically reconstructing mathematics from _many_ different sets
| of first principles in an attempt to give the computer
| "intuition" about mathematics (intuition a la expert systems, not
| machine learning), and this has opened my eyes to a _lot_ of
| elegant, useful sources of intuition (and not just about
| geometry).
| nayuki wrote:
| I haven't figured out how to compute p efficiently yet. But
| computing e with proper error bounds is feasible with high-
| school-level algebra: https://www.nayuki.io/page/approximating-
| eulers-number-corre...
| aesh2Xa1 wrote:
| > and this has opened my eyes to a lot of elegant, useful
| sources of intuition (and not just about geometry).
|
| Would you care to compile and share them?
| wizzwizz4 wrote:
| It's hard to compile intuition; seeing _that one diagram_
| unlocked an entire category of possibilities that I hadn 't
| spotted before, and I can _feel_ that they 're now accessible
| (and always were, but I hadn't looked). It's hard to
| enumerate them.
|
| Basically, a lot of things can be _thought of_ as something
| vaguely geometric, and then enumerated discretely using some
| method of enumeration (e.g. a grid) to form an approximation
| to the value, which is a known bound. It feels like the
| general idea can be used for things other than approximating
| real values (something to do with property membership and
| graphs?) but I can 't give any concrete examples of that.
|
| One of the ideas I got from this, which I _think_ was what
| spurred me to write that comment, was doing it in reverse.
| You can think of certain discrete systems as being, in some
| nebulous way stronger than a bound, inextricably linked to a
| continuous function. Kind of like Pisot-Vijayaraghavan
| numbers,[0] but more so. There was something about the
| resource requirements of an algorithm and applying this
| technique to compiler optimisation, but that 's an incomplete
| thought.
|
| Lots of things.
|
| [0]: https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghav
| an_n...
| tkgally wrote:
| This reminds me of a problem that was on the entrance examination
| to the University of Tokyo in 2003: "Prove that the ratio of a
| circle's circumference to its diameter is greater than 3.05." The
| problem was believed to have been posed as a protest against the
| dumbing down of the government-mandated curriculum, as elementary
| school children in Japan were being taught to use 3 as a
| substitute for pi in some calculations [1]. The elegance of the
| problem is that test-takers were unlikely ever to have
| encountered it before and that various creative solutions are, it
| seems, possible.
|
| [1] https://ja.wikipedia.org/wiki/Yuan Zhou Lu ha3
| [deleted]
| 908B64B197 wrote:
| > as elementary school children in Japan were being taught to
| use 3 as a substitute for pi in some calculations
|
| That's robbing them of getting an (intuitive) understanding of
| a whole class of numbers. And limits too!
| koolba wrote:
| What disgrace if a math program would use 3 as an estimate for
| pi? Not even 22/7?!
| wizzwizz4 wrote:
| 3 is no more of a disgrace than 22/7, in my book.
| angelgcuartero wrote:
| Wait... --> https://en.wikipedia.org/wiki/Indiana_Pi_Bill
| molticrystal wrote:
| Buffon's needle problem[0] remains my favorite way to calculate
| pi. You drop readily available sticks such as matches or
| toothpicks on top of parallel lines at least slightly wider than
| the sticks. In a Monte Carlo fashion, pi will emerge. An
| interactive example is available[1].
|
| pi is approximately: (2 x the stick length * the number dropped)
| divided by (the distance between lines * sticks crossing a line)
|
| [0] https://en.wikipedia.org/wiki/Buffon%27s_needle_problem
|
| [1] https://ogden.eu/pi/
| leephillips wrote:
| I made this for my students a while ago, approximating p with
| similar ideas:
|
| https://lee-phillips.org/pidaydarts/
|
| I think it was my first Svelte application.
|
| We did the paper airplane version in class, which was fun.
| [deleted]
| m4r35n357 wrote:
| I love playing "spot the programming language" . . .
| marcodiego wrote:
| I imagine that a good exercise for children who can already
| reason about proportionality is to calculate pi experimentally.
|
| You can basically make a few experiments to illustrate that the
| perimeter of a circle is proportional to its radius and then ask
| them to calculate the perimeter another circle knowing only its
| radius.
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