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> 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 interes

> 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.
    - chriswarbo 4y ago
      Wildberger's take (e.g. https://www.youtube.com/watch?v=by0Gy1ZJ_hU https://www.youtube.com/watch?v=by0Gy1ZJ_hU ) is that rational trigonometry is well-suited to 'triangle geometry' (i.e. anywhere we might use Pythagoras); whilst angles are well-suited to uniform circular motion (although even there he prefers turns instead of radians ;) https://www.youtube.com/watch?v=CnQXRdgN_7I https://www.youtube.com/watch?v=CnQXRdgN_7I ).
- ogogmad 4y ago
  e^{it} would get replaced with \frac{1+it}{1-it}. t=n/d then implies \frac{1+it}{1-it}=\frac{d+in}{d-in}. RT boils down to doing trigonometry with dot products and without square roots. There's a temptation to replace dot products with ordinary trigonometry, but RT says don't do it because it's a step back.
- eesmith 4y ago
  I am not a mathematician but I follow one on the internet. Quoting https://www.johndcook.com/blog/2023/01/06/rational-trigonometry/ https://www.johndcook.com/blog/2023/01/06/rational-trigonome... > You can work over the rational numbers, but you don’t need to. You could work over real or complex numbers, or even finite fields. Because you don’t take square roots, you can work over fields that don’t necessarily have square roots. If you’re working with integers modulo a prime, half of your numbers have no square root and the other half have two square roots. ... > Why would you want to do geometry over finite fields? Finite fields are important in applications: error-correcting codes, cryptography, signal processing, combinatorics, etc. And by thinking of problems involving finite fields as geometry problems, you can carry your highly developed intuition for plane geometry into a less familiar setting. "Orientation Modeling Using Quaternions and Rational Trigonometry" at https://www.mdpi.com/2075-1702/10/9/749 https://www.mdpi.com/2075-1702/10/9/749 has an example of using it with complex numbers: We can define the quadrance of z by: Q(z) = z · z̅ = a² + b². Spread: The spread can be defined in several ways ... in a more general way, the spread can be specified for any complex number by the next expression: s(w) ≡ b² / a² The paper includes the section "Rotations on the Plane" which may address your e^(i*alpha).
stijnoomes 4y ago
I am developing a novel theory on 3D vision. Rational trigonometry allows me to derive rather simple expressions. Here is an example: https://stijnoomes.com/height-of-the-center-of-projection/ https://stijnoomes.com/height-of-the-center-of-projection/

> 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.
  - chriswarbo 4y ago
    Wildberger's take (e.g. https://www.youtube.com/watch?v=by0Gy1ZJ_hU https://www.youtube.com/watch?v=by0Gy1ZJ_hU ) is that rational trigonometry is well-suited to 'triangle geometry' (i.e. anywhere we might use Pythagoras); whilst angles are well-suited to uniform circular motion (although even there he prefers turns instead of radians ;) https://www.youtube.com/watch?v=CnQXRdgN_7I https://www.youtube.com/watch?v=CnQXRdgN_7I ).
ogogmad 4y ago
e^{it} would get replaced with \frac{1+it}{1-it}. t=n/d then implies \frac{1+it}{1-it}=\frac{d+in}{d-in}. RT boils down to doing trigonometry with dot products and without square roots. There's a temptation to replace dot products with ordinary trigonometry, but RT says don't do it because it's a step back.
eesmith 4y ago
I am not a mathematician but I follow one on the internet. Quoting https://www.johndcook.com/blog/2023/01/06/rational-trigonometry/ https://www.johndcook.com/blog/2023/01/06/rational-trigonome... > You can work over the rational numbers, but you don’t need to. You could work over real or complex numbers, or even finite fields. Because you don’t take square roots, you can work over fields that don’t necessarily have square roots. If you’re working with integers modulo a prime, half of your numbers have no square root and the other half have two square roots. ... > Why would you want to do geometry over finite fields? Finite fields are important in applications: error-correcting codes, cryptography, signal processing, combinatorics, etc. And by thinking of problems involving finite fields as geometry problems, you can carry your highly developed intuition for plane geometry into a less familiar setting. "Orientation Modeling Using Quaternions and Rational Trigonometry" at https://www.mdpi.com/2075-1702/10/9/749 https://www.mdpi.com/2075-1702/10/9/749 has an example of using it with complex numbers: We can define the quadrance of z by: Q(z) = z · z̅ = a² + b². Spread: The spread can be defined in several ways ... in a more general way, the spread can be specified for any complex number by the next expression: s(w) ≡ b² / a² The paper includes the section "Rotations on the Plane" which may address your e^(i*alpha).

stijnoomes 4y ago

I am developing a novel theory on 3D vision. Rational trigonometry allows me to derive rather simple expressions. Here is an example: https://stijnoomes.com/height-of-the-center-of-projection/ https://stijnoomes.com/height-of-the-center-of-projection/