11 ms·

Rational Trigonometry

95 points by KqAmJQ7 4y ago

dang 4y ago
Related: Show HN: Simple pendulum simulation with rational trigonometry - https://news.ycombinator.com/item?id=12525068 https://news.ycombinator.com/item?id=12525068 - Sept 2016 (28 comments)
eesmith 4y ago
Also, John Cook recently blogged about it at https://www.johndcook.com/blog/2023/01/06/rational-trigonometry/ https://www.johndcook.com/blog/2023/01/06/rational-trigonome... and wrote: > I became interested in rational trigonometry when I was working on a project to formally verify drone collision avoidance software. Because all calculations are purely algebraic, involving no transcendental functions, theorems are easier to prove. ... > Another advantage is accuracy. Because a computer can carry out rational arithmetic exactly, rational trigonometry applied to rational points yields exact results. No need to be concerned about the complications of floating point numbers. > A less obvious advantage is that the theory of rational trigonometry is independent of the underlying field
- prvc 4y ago
  >A less obvious advantage is that the theory of rational trigonometry is independent of the underlying field What sort of interesting results come out of using a non-real field here?
  - eesmith 4y ago
    You'll need to ask an actual mathematician. Cook recently wrote that blog post, so it's much better to ask him in the comment section of that post. Just because I can do a Google Scholar query using "rational trigonometry" and find papers saying things like: "Explicit tough Ramsey graphs" at https://arxiv.org/pdf/0807.2692.pdf https://arxiv.org/pdf/0807.2692.pdf . "We present here two new explicit constructions ... The first construction is based on a remarkable new approach of Wildberger to trigonometry and Euclidean geometry." does not mean I am qualified to judge what is "interesting", or figure out which use non-real fields.
- deleted 4y ago
  [deleted]
- bigbillheck 4y ago
  > Because all calculations are purely algebraic, involving no transcendental functions, theorems are easier to prove. ... Personally I would have said manipulation of trig identities and such is also purely algebraic.
  - koningrobot 4y ago
    They meant algebraic in the sense of https://en.wikipedia.org/wiki/Algebraic_operation https://en.wikipedia.org/wiki/Algebraic_operation.
    - bigbillheck 4y ago
      Sure, but to be honest I'm a little bit boggled at the idea of a phd mathematician finding trig manipulation hard enough to be worth commenting on.
      - eesmith 4y ago
        This review of Wildberger's book at https://philarchive.org/archive/FRADPR https://philarchive.org/archive/FRADPR says the point isn't that the PhD mathematician finds trig hard, but rather to beginners: > ... very becoming-numerate generation invests enormous effort in the painful calculation of the lengths and angles of complicated figures. Surveying, navigation and computer graphics are intensive users of the results. Much of that effort is wasted, Wildberger argues. The concentration on angles, especially, is a result of the historical accident that serious study of the subject began with spherical trigonometry for astronomy and long-range navigation, which meant there was altogether too much attention given to circles. ... > Having things done better is one major payoff, but equally important would be a removal of a substantial blockage to the education of young mathematicians, the waterless badlands of traditional trigonometry that youth eager to reach the delights of higher mathematics must spend painful years crossing Whether that's correct is different. https://handwiki.org/wiki/Rational_trigonometry https://handwiki.org/wiki/Rational_trigonometry seems to give a good description of opposing views.
        4y ago
munchler 4y ago
> I promise that rational trigonometry will pay off in a big way later. Sorry, I seem to have missed that part. The article just ends suddenly. It's an interesting idea, but what's the benefit?
- tgv 4y ago
  The article, and the other ones linked here, are too short, indeed. As if the idea gets dusted off, proudly presented, and then forgotten because it's just not practical enough. This wikipedia page also falls short of a conclusion: https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T... E.g., one thing that seems not very compatible is complex numbers: I can't see how e^(i*alpha) would benefit. So ... it's a specialized tool?
  - Chinjut 4y ago
    This works perfectly fine with complex numbers. Consider Gaussian integers or Gaussian rationals (complex numbers of the form a + b i, where a and b are integers or rationals). Like complex numbers in general, these amount to the combination of scaling by some magnitude and rotating by some angle. If we only care about the angle, we can consider two Gaussian rationals to be equivalent if their ratio is a positive real number. Any angle with a rational tangent or cotangent has some representation as a Gaussian rational (usually requiring a non-unit magnitude, though), and these angles are closed under addition and subtraction (corresponding to multiplying and dividing these Gaussian rationals). And given a Gaussian integer or rational, we can readily take its squared magnitude (a^2 + b^2) to get an integer or rational, and we can in the same way take the squared cosine or squared sine of its angle (a^2/(a^2 + b^2) or b^2/(a^2 + b^2)) to get a rational.
    - tgv 4y ago
      I don't see how that would help: e^(i*√s) is only further analyzable when s is the square of a rational number, but even then it won't help you go further: e^i*k is not rational (k≠0). In something like DFT or FIR analysis, it wouldn't help to work with spread instead of angle, I think. Integrating over s^2 only adds more steps, and I also don't see how it could help numerical approximation.
threatofrain 4y ago
N. J. Wildberger is a real mathematician but beware that he's somewhat idiosyncratic and a bit of a ultra-finitist. https://www.youtube.com/@njwildberger https://www.youtube.com/@njwildberger His Trig series: https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47 https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C5849871...
- KrugerDunnings 4y ago
  I like is idiosyncratic take on math, it is good to know there is someone with a slightly different opinion than everyone else. His videos about multisets released in the last weeks have been super interesting. They seem like a very promising alternative to classical set theory that maybe could be used for some better type system.
chriswarbo 4y ago
It slightly obscures the simplicity of spread to define it in terms of angles and transcendental functions. It can instead be defined directly, as opposite/hypotenuse on a right-angled triangle. Likewise, we can take quadrance as primary; and, if we're so inclined, we can define distance as its square root.
contravariant 4y ago
I'm not sure this is any more useful than just linear algebra with an inner product.
- chriswarbo 4y ago
  It works well with linear algebra, if you want to do that. If you don't, then there's no need to mess around with matrices, etc.
  - contravariant 4y ago
    There isn't really any need to deal with matrices for just geometry. The ability to use matrices however is a great tool. For one I don't know of a more elegant expression for the area of a triangle than just |P Q R| |1 1 1| / 2 Rational trigonometry seems to me like it's just linear algebra, except it uses the squared sine instead of the cosine just to get a notion of angle that is 0 for parallel line. Lot's of linear algebra features are removed, but I'm not seeing much benefit from doing so.
    - chriswarbo 4y ago
      The same formula appears in rational trig, e.g. in section 9.3 of Divine Proportions http://www.ms.lt/derlius/WildbergerDivineProportions.pdf#page=137 http://www.ms.lt/derlius/WildbergerDivineProportions.pdf#pag...
hgibbs 4y ago
I made a public bet with Norman in early 2020 (late Feb) about whether the university we were at would hold classes until the end of the term (he believed they would not due to covid). Norman was the only one with the guts to publicly point out the obvious (anybody with the Wikipedia page for China case numbers and a spreadsheet could have figured out what was going to happen in late Feb). Among other things he was a great lecturer and a talented go player, although I never really thought that rational trig was worth all the effort - regular math seems to have plenty of utility even if you think it is built on shaky philosophical foundations.
- deleted 4y ago
  [deleted]
- chriswarbo 4y ago
  > I never really thought that rational trig was worth all the effort - regular math seems to have plenty of utility even if you think it is built on shaky philosophical foundations For those familiar with "normal" (irrational?) trig, it's certainly an extra effort to learn, and may hinder communication, which gives a poor cost/benefit ratio. For those unfamiliar with "normal" trig (e.g. not reaching that stage in school, or having later forgot it all), then rational trig certainly seems easier to learn and understand. I also like the way it generalises, e.g. to hyperbolic space (relativity), by simply changing the dot-product.
- roenxi 4y ago
  > Norman was the only one with the guts to publicly point out... Sorry; what context was this in? UNSW maths department or some other social circle? I know I was anticipating lockdowns some time around then for exactly the same reasons as you list, but I assumed that all the mathematicians would have figured it out around the same moment.
  - lynndotpy 4y ago
    In my CS department this was anticipated by a few of us, pretty universally so by the end of February.
  - 4y ago
deleted 4y ago
[deleted]
bigbillheck 4y ago
I'm reminded of Chebyshev polynomials, which do a similar kind of thing, and I'd bet that polynomials in Wildberger's 's' are related to even-powered Chebyshev polynomials of the first kind.
- chriswarbo 4y ago
  That seems to be the case, based on e.g. https://mathoverflow.net/questions/47520/spread-polynomials https://mathoverflow.net/questions/47520/spread-polynomials
garbagecoder 4y ago
Just like non-standard analysis it's mostly a solution in search of a problem. There's no evidence this improves pedagogy. So, it's just an interesting exploration of a different way of doing math, which is very much what mathematicians do. But as with many things, the cost of transitioning have to justify the switch even if you had a blank slate it might be better (something I don't buy necessarily).

95 points by KqAmJQ7 4y ago

dang 4y ago

Related: Show HN: Simple pendulum simulation with rational trigonometry - https://news.ycombinator.com/item?id=12525068 https://news.ycombinator.com/item?id=12525068 - Sept 2016 (28 comments)

eesmith 4y ago

Also, John Cook recently blogged about it at https://www.johndcook.com/blog/2023/01/06/rational-trigonometry/ https://www.johndcook.com/blog/2023/01/06/rational-trigonome... and wrote: > I became interested in rational trigonometry when I was working on a project to formally verify drone collision avoidance software. Because all calculations are purely algebraic, involving no transcendental functions, theorems are easier to prove. ... > Another advantage is accuracy. Because a computer can carry out rational arithmetic exactly, rational trigonometry applied to rational points yields exact results. No need to be concerned about the complications of floating point numbers. > A less obvious advantage is that the theory of rational trigonometry is independent of the underlying field

prvc 4y ago
>A less obvious advantage is that the theory of rational trigonometry is independent of the underlying field What sort of interesting results come out of using a non-real field here?
- eesmith 4y ago
  You'll need to ask an actual mathematician. Cook recently wrote that blog post, so it's much better to ask him in the comment section of that post. Just because I can do a Google Scholar query using "rational trigonometry" and find papers saying things like: "Explicit tough Ramsey graphs" at https://arxiv.org/pdf/0807.2692.pdf https://arxiv.org/pdf/0807.2692.pdf . "We present here two new explicit constructions ... The first construction is based on a remarkable new approach of Wildberger to trigonometry and Euclidean geometry." does not mean I am qualified to judge what is "interesting", or figure out which use non-real fields.
deleted 4y ago
[deleted]
bigbillheck 4y ago
> Because all calculations are purely algebraic, involving no transcendental functions, theorems are easier to prove. ... Personally I would have said manipulation of trig identities and such is also purely algebraic.
- koningrobot 4y ago
  They meant algebraic in the sense of https://en.wikipedia.org/wiki/Algebraic_operation https://en.wikipedia.org/wiki/Algebraic_operation.
  - bigbillheck 4y ago
    Sure, but to be honest I'm a little bit boggled at the idea of a phd mathematician finding trig manipulation hard enough to be worth commenting on.
    - eesmith 4y ago
      This review of Wildberger's book at https://philarchive.org/archive/FRADPR https://philarchive.org/archive/FRADPR says the point isn't that the PhD mathematician finds trig hard, but rather to beginners: > ... very becoming-numerate generation invests enormous effort in the painful calculation of the lengths and angles of complicated figures. Surveying, navigation and computer graphics are intensive users of the results. Much of that effort is wasted, Wildberger argues. The concentration on angles, especially, is a result of the historical accident that serious study of the subject began with spherical trigonometry for astronomy and long-range navigation, which meant there was altogether too much attention given to circles. ... > Having things done better is one major payoff, but equally important would be a removal of a substantial blockage to the education of young mathematicians, the waterless badlands of traditional trigonometry that youth eager to reach the delights of higher mathematics must spend painful years crossing Whether that's correct is different. https://handwiki.org/wiki/Rational_trigonometry https://handwiki.org/wiki/Rational_trigonometry seems to give a good description of opposing views.
      - 4y ago

munchler 4y ago

> I promise that rational trigonometry will pay off in a big way later. Sorry, I seem to have missed that part. The article just ends suddenly. It's an interesting idea, but what's the benefit?

tgv 4y ago
The article, and the other ones linked here, are too short, indeed. As if the idea gets dusted off, proudly presented, and then forgotten because it's just not practical enough. This wikipedia page also falls short of a conclusion: https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T... E.g., one thing that seems not very compatible is complex numbers: I can't see how e^(i*alpha) would benefit. So ... it's a specialized tool?
- Chinjut 4y ago
  This works perfectly fine with complex numbers. Consider Gaussian integers or Gaussian rationals (complex numbers of the form a + b i, where a and b are integers or rationals). Like complex numbers in general, these amount to the combination of scaling by some magnitude and rotating by some angle. If we only care about the angle, we can consider two Gaussian rationals to be equivalent if their ratio is a positive real number. Any angle with a rational tangent or cotangent has some representation as a Gaussian rational (usually requiring a non-unit magnitude, though), and these angles are closed under addition and subtraction (corresponding to multiplying and dividing these Gaussian rationals). And given a Gaussian integer or rational, we can readily take its squared magnitude (a^2 + b^2) to get an integer or rational, and we can in the same way take the squared cosine or squared sine of its angle (a^2/(a^2 + b^2) or b^2/(a^2 + b^2)) to get a rational.
  - tgv 4y ago
    I don't see how that would help: e^(i*√s) is only further analyzable when s is the square of a rational number, but even then it won't help you go further: e^i*k is not rational (k≠0). In something like DFT or FIR analysis, it wouldn't help to work with spread instead of angle, I think. Integrating over s^2 only adds more steps, and I also don't see how it could help numerical approximation.

threatofrain 4y ago

N. J. Wildberger is a real mathematician but beware that he's somewhat idiosyncratic and a bit of a ultra-finitist. https://www.youtube.com/@njwildberger https://www.youtube.com/@njwildberger His Trig series: https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47 https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C5849871...

KrugerDunnings 4y ago
I like is idiosyncratic take on math, it is good to know there is someone with a slightly different opinion than everyone else. His videos about multisets released in the last weeks have been super interesting. They seem like a very promising alternative to classical set theory that maybe could be used for some better type system.

chriswarbo 4y ago

It slightly obscures the simplicity of spread to define it in terms of angles and transcendental functions. It can instead be defined directly, as opposite/hypotenuse on a right-angled triangle. Likewise, we can take quadrance as primary; and, if we're so inclined, we can define distance as its square root.

contravariant 4y ago

I'm not sure this is any more useful than just linear algebra with an inner product.

chriswarbo 4y ago
It works well with linear algebra, if you want to do that. If you don't, then there's no need to mess around with matrices, etc.
- contravariant 4y ago
  There isn't really any need to deal with matrices for just geometry. The ability to use matrices however is a great tool. For one I don't know of a more elegant expression for the area of a triangle than just |P Q R| |1 1 1| / 2 Rational trigonometry seems to me like it's just linear algebra, except it uses the squared sine instead of the cosine just to get a notion of angle that is 0 for parallel line. Lot's of linear algebra features are removed, but I'm not seeing much benefit from doing so.
  - chriswarbo 4y ago
    The same formula appears in rational trig, e.g. in section 9.3 of Divine Proportions http://www.ms.lt/derlius/WildbergerDivineProportions.pdf#page=137 http://www.ms.lt/derlius/WildbergerDivineProportions.pdf#pag...

hgibbs 4y ago

I made a public bet with Norman in early 2020 (late Feb) about whether the university we were at would hold classes until the end of the term (he believed they would not due to covid). Norman was the only one with the guts to publicly point out the obvious (anybody with the Wikipedia page for China case numbers and a spreadsheet could have figured out what was going to happen in late Feb). Among other things he was a great lecturer and a talented go player, although I never really thought that rational trig was worth all the effort - regular math seems to have plenty of utility even if you think it is built on shaky philosophical foundations.

deleted 4y ago
[deleted]
chriswarbo 4y ago
> I never really thought that rational trig was worth all the effort - regular math seems to have plenty of utility even if you think it is built on shaky philosophical foundations For those familiar with "normal" (irrational?) trig, it's certainly an extra effort to learn, and may hinder communication, which gives a poor cost/benefit ratio. For those unfamiliar with "normal" trig (e.g. not reaching that stage in school, or having later forgot it all), then rational trig certainly seems easier to learn and understand. I also like the way it generalises, e.g. to hyperbolic space (relativity), by simply changing the dot-product.
roenxi 4y ago
> Norman was the only one with the guts to publicly point out... Sorry; what context was this in? UNSW maths department or some other social circle? I know I was anticipating lockdowns some time around then for exactly the same reasons as you list, but I assumed that all the mathematicians would have figured it out around the same moment.
- lynndotpy 4y ago
  In my CS department this was anticipated by a few of us, pretty universally so by the end of February.
- 4y ago

deleted 4y ago

[deleted]

bigbillheck 4y ago

I'm reminded of Chebyshev polynomials, which do a similar kind of thing, and I'd bet that polynomials in Wildberger's 's' are related to even-powered Chebyshev polynomials of the first kind.

chriswarbo 4y ago
That seems to be the case, based on e.g. https://mathoverflow.net/questions/47520/spread-polynomials https://mathoverflow.net/questions/47520/spread-polynomials

garbagecoder 4y ago

Just like non-standard analysis it's mostly a solution in search of a problem. There's no evidence this improves pedagogy. So, it's just an interesting exploration of a different way of doing math, which is very much what mathematicians do. But as with many things, the cost of transitioning have to justify the switch even if you had a blank slate it might be better (something I don't buy necessarily).