#symmetrymatters — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #symmetrymatters, aggregated by home.social.
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When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.
That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.
You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754
#calculus #math #symmetrymatters #halfangleapproach #euler #integration
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Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html
#math #geometry #trigonometry #halfangleapproach #symmetrymatters
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USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.
Article (draft): https://arxiv.org/abs/2505.03754
#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters
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The first and most complete unification of classical integration techniques ever! Say hello to USM.
Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
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The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.
Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters -
The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.
Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters -
The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.
Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters -
The USM is superior to traditional methods!
Using the USM’s IV transformation formula, from the summary at the link below, you convert integrals of this type into polynomial-type integrals. The traditional method reduces it to the integral of csc³, which would require complicated reduction formulas or integration by parts. The third Euler substitution reduces it to the integral of √(t² + a²), which also requires memorizing a standard formula or integrating it from scratch. None of this is as simple as integrating a three-term polynomial-type function, which is taught in the first few weeks of any integral calculus course.
Check out the technique here: http://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integration #USM #newmethod #halfangleapproach #symmetrymatters -
I gave this integral as an answer at MathSE.
\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]
The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.
#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus
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An 'impossible' integral to solve (Mathematica fails):
\[\int_{2}^{3} \frac{{1 - \tan\frac{{\sec^{-1}x}}{2}}}{{1 + \tan\frac{{\sec^{-1}x}}{2}}}\sqrt{\tan\frac{\csc^{-1}x}{2}} \,dx\]
Unless you know my new technique:
https://mathoverflow.net/questions/463459/identities-that-simplify-tedious-integrals?noredirect=1#comment1203657_463459
#math #impossibleintegral #calculus #halfangleapproach #symmetrymatters -
In my latest blog post, I have derived what appears to be a new formula for solving quadratic equations. Were you aware of it?
Link: https://geometriadominicana.blogspot.com/2024/01/trigonometric-formula-for-solving.html
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Do you know what I mean, right? Half-angle formulas are central! Their utility for you will depend on how ingenious you are. Look at Euler, learn from him, he is the master of all of us, someone once said.
A proof : https://m.youtube.com/watch?v=eYow30t9aQg&si=S8J4Kk43VgbybtXx&fbclid=IwAR1jf1tM-rgZMapC07vH0W6v7QMQDfM_EXDy_-7p4358XZFGASyZ8gnH1-Q -
Another theorem falls under the grasp of the half-angle approach: Burlet's theorem.
I give a generalization here: https://geometriadominicana.blogspot.com/2024/01/a-generalization-of-burlets-theorem-to.html?m=1&fbclid=IwAR1oQEwlKoQl2Ajv5BDoXdGHtUUew46WiB9iHqWALL4p1o58I8m8JdMzuT8
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I GOT PUBLISHED!!
The MATINF magazine has officially published my generalization of Newton's formula.
I am immensely grateful to Masters Leo Giugiuc and Costel Balcau for being instrumental in making this publication possible.
You can download and view the article on page 19 at the following link: http://matinf.upit.ro/MATINF9_10/mobile/index.html
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After discovering the central role that half-angle formulas play in relation to classical metric geometry, the natural question that arises is, why on earth are these formulas so useful? And my answer is symmetry, the same underlying concept in Galois theory or Noether's theorems.
#math #halfangleapproach #symmetrymatters -
The half-angle formulas are central! And as a picture is worth a thousand words...
If you don't know what I'm talking about, read my essay 'The Theoretical Importance of Half-Angle Formulas' and discover the most powerful formulas of all time.
Essay: https://drive.google.com/file/d/1jPrWxUhB1Tnsd5RH5UiMhwmkp2jCv11v/view?usp=sharing
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High school teachers in Singapur are already using the half-angle approach.
#math #halfangleapproach #symmetrymatters -
On every platform where I've shared my work on half-angle formulas, except for my personal blog and my social media, there's always been someone trying to sabotage my post. Recently, I posted my essay on Reddit. Although my post was initially accepted and garnered many positive votes, the supportive comment from RussianChattus was strangely pushed to the bottom and then deleted for several days until I complained in the comments. After my protest, RussianChattus's comment reappeared, but oddly enough, it was surpassed by a comment criticizing the lack of punctuation within parentheses, becoming the most upvoted comment. A hilarious situation!
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The Pythagorean theorem, merely a specific instance of the half-angle formulas, derives its remarkable utility from these very formulas.
#math #halfangleapproach #symmetrymatters -
Akin to the ascent of a tall tree, the identification of fundamental theorems within a field is a critical task. This analogy draws parallels between understanding the core principles of a subject and recognizing the trunk of a towering tree when one endeavors to reach its pinnacle. While it is acknowledged that not everyone, particularly the more adept individuals in this metaphorical climb, may need to traverse the trunk directly, the importance of this identification becomes apparent, especially for the average person.
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In terms of understanding the body of metric relationships in classical geometry, I have far surpassed my predecessors, constructing a unified framework that will aid students in better comprehending what is happening here.
#math #halfangleapproach #symmetrymatters -
The fact that I have been able to derive the most important theorems of classical geometry and generalize a good number of them without ANYONE being able to find a reference in 3 years is compelling evidence that the half-angle approach is more profound.
#math #halfangleapproach #symmetrymatters -
You can be trained to solve the most challenging problems in the most prestigious mathematics Olympiads in the world. But there are no training programs to create alternative approaches from which you can derive all the most important theorems in a discipline. No one invented inversion in the middle of a math competition.
#math #halfanglesmatter #symmetrymatters #halfangleapproach -
I shared my essay "The Theoretical Importance of Half-Angle Formulas" on Reddit: https://www.reddit.com/r/math/comments/16ojbnd/the_theoretical_importance_of_halfangle_formulas/
#math #geometry #trigonometry #halfanglesmatter #symmetrymatters
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If the half-angle formulas aren't considered fundamental, then how did Viète manage to derive his formula for π using them? And why is it that the Weierstrass substitution, hailed as the most ingenious substitution in the world by Spivak, is actually based on those very formulas? If these half-angle formulas lack significance, then how on earth have I successfully derived the formulas of Heron, Brahmagupta, and Bretschneider, as well as the laws of cosines, sines, and tangents, the Mollweide formula (or perhaps we should credit Newton), the angle bisector formula, the inradius formula for mixtillinear incircle, and the sum and difference of angle identities, not to mention Euler's remarkable triangular inequality, all stemming from these supposedly non-fundamental formulas? And if that's not enough, how have I managed to extend the Pythagorean trigonometric identity (with over 2000 years of history), the Mollweide formula (or Newton's, if you prefer) (with over 300 years of existence), and even the half-angle formulas themselves (likely over 2000 years old) into broader generalizations? Still skeptical? Well, I invite you to click on the link below and see for yourself.
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Method vs trick according to George Polya
ChatGPT agrees that my half-angle approach is a method rather than a trick.
#newmethod #halfangleapproach #math #geometry #trigonometry #georgepolya #halfanglesmatter #symmetrymatters #ChatGPT
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Earlier I had said that the half-angle formulas are an essential ingredient in the derivation of Viète's \(\pi\) formula. According to Eli Maor, Viète's formula marks the beginning of mathematical analysis, and Jonathan Borwein calls its appearance "the dawn of modern mathematics".
The work described in the link below is another example of the centrality and versatility of the half-angle formulas (although the authors mistakenly call them double-angle formulas), a centrality that transcends the field of geometry and trigonometry in the plane.
https://www.sciencedirect.com/science/article/pii/S0021904513001159?via%3Dihub#br000045
#math #geometry #trigonometry #halfanglesmatter #symmetrymatters
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My wake-up call for the extraordinary explanatory power of half-angle formulas has reached 3,000 views on MathSE. Are the half angle formulas more fundamental than the Pythagorean theorem or the law of cosines?
#math #trigonometry #geometry #halfanglesmatter #symmetrymatters