Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. 35:45 Because the motion is smooth, we consider the local movement of one bug (being chased) relative to another (which I will refer to in the first person - I am the bug that chases), and we see that it's linear. Further, in this case we see that the movement is perpendicular (movement being it's change in position all happening relative to me) - this is because I am always facing it, and more importantly, the angle between me, it, and the bug it's chasing (the direction of it's movement) starts off at 90° and never changes - THIS is due to symmetry. It follows that (locally) relative to me, it's moving in a circle around me (and also I'm rotating in place to face it) - never getting either closer or farther from me. I'm the only one that has any affect on the distance between us, so the final distance covered is the initial distance I needed to cover.
    4
  2. I don't understand why mathematical theories attract me. I don't even know the multiplication table well
    4
  3. Very kind to put the solution in a reply to prevent accidental spoilers
    4
  4.  @manavgunnia  2^countable is uncountable afaik. PS: @tom13king has a wonderful argument above.
    4
  5.  @josephmathes  Definitely a great topic. We'll see .... :)
    4
  6. Thank you so much for your wonderful and inspiring videos!
    4
  7. Man.... i used to cry because of math.... I am now crying because of you... Amazing... and thank you
    4
  8. We need a Mathematica plug-in for algebra autopilot!!
    4
  9. Did you read The Archimedes Codex by Reviel Netz and William Noel? I would highly recommend, it includes both a very entertaining history of the remaining copies of his works, as well as insights into what he was working on :)
    4
  10. When I was at the Gymnasium in Germany, in 9th grade, approx. 1990, the formula was included in the problem section of the book we used (Lambacher Schweizer). If I remember correctly, the problem even consisted in proving the formula (given many hints on how to proceed).
    4
  11. Roger Penrose also presents the "cube-hegaxon" visual proof in his 1994 book " Shadows of the Mind: A Search for the Missing Science of Consciousness ". It is in chapter 2, section 2.4 (" How do we decide that some computations do not stop?")
    4
  12. Amazing....
    4
  13. I've never been more ready for a math video
    4
  14. Mathloger, one of the few channels that I'm excited to see a nearly hour long video posted. Love it! Thanks for the maths!
    4
  15.  @ffggddss  Wow nice argument!
    4
  16.  @gammaknife167  Thanks. But I think I may have simply remembered it from the Mathematical Games column. Not sure. After digesting enough beautiful arguments, they tend to become ingrained in your thinking, without your knowledge of their origin. But whoever came up with it, it is indeed a nice insight.
    4
  17. YES! Saturday morning Mathologer! 😎
    4
  18. :face-red-heart-shape:
    4
  19. I admit, I enjoyed this very much. The most fun with differing rates of convergence, and the distributive property, I've had in years. You have reawakened me. Thank you.
    4
  20. I first encountered the coin problem studying engineering over 60 years ago. The context was designing what we referred to as epicyclic gear trains. The rule of thumb is : if the small gear [ coin ] rolls round the outer of the centre gear [ coin ] its motion relative to an outside observer is the ratio of diameters [ in gearing parlance the pitch diameters ] PLUS 1 rotation. If the smaller rolls round the inner surface of the larger then the number of rotations of the smaller is the ratio of the pitch diameters MINUS 1 rotation. If you imagine the problem of an interconnected train of epicylic stages it's essential to have simple rules of thumb to work with. It's clear that the coin problem is actually only a simple system that one is almost never going to encounter in the real world of gearing. The basic real-world system would be THREE gears interconnected and referred to as a sun, planet and ring gear combination. Any one of the three elements of this system can be held stationary with respect to an outside observer. As a final glimpse into the world of gearing, it's important to remember that if the basic function of a gear train is as a speed reducer there is a concomitant increase in the torque [ and vice versa ]. It can happen then that if the train is driven in reverse this can lead to excessive torque or tooth forces which can cause failure of the gear teeth at the former input end.
    4
  21. Slide rules are *awesome*. I love them. They're like a physical manifestation of mathematics - the most abstract field of study there is. What would it take to make a slide-rule type instrument where the calculation performed is not one number multiplied by a second number, but instead, one number raised to the power of a second number? Bonus question, but maybe only for the most hardcore of mathematicians: what about a slide rule where the operation is addition?! The mind boggles.
    4
  22. What I disliked about calculus was seemingly endless questions asking me to integrate or differential nonsensical mashups of sins and cosines like wtf would be the actual utility of integrating sin^2(5x)cos^3(x) aside from exam questions so annoying
    4
  23. Great video. I missed the -1/12 in the end somehow 🙂
    4
  24. 0:04 I found it - it's in the bottom right corner.
    4
  25. For the 3D version, the fourth color d we put on top of three colors a, b and c is a⊕b⊕c where ⊕ is the nim-addition https://en.wikipedia.org/wiki/Nimber#Addition_and_multiplication_tables . We also have a⊕b⊕c⊕d = 0, which explains why rotating any such pyramid gives another pyramid of the same type.
    4
  26. So this is why sudoku named one of their optional rule as magic square.
    4
  27. I asked ChatGPT to flesh out your idea Title: Pentagon’s Shadow Synopsis: In the heart of a bustling city, a series of seemingly random crimes baffle the local police. Each crime scene forms a perfect pentagon, its vertices marked by the locations of heinous acts: arson, robbery, assault, kidnapping, and murder. The pattern is unmistakable, but its origin and purpose remain elusive. Main Characters: 1. Detective Laura Mathis: A seasoned detective with a sharp mind for patterns and an unconventional approach to solving crimes. She has a background in mathematics, which becomes pivotal in cracking the case. 2. Dr. Petr Novák: A reclusive mathematician known for his work on complex geometric theorems. His breakthrough, known as “Petr’s Miracle,” describes the transformation and center-shifting of pentagons, a theorem that holds the key to the mystery. 3. Lucas Grey: A brilliant but troubled mathematician turned criminal mastermind. Obsessed with proving his intellectual superiority, he uses Petr’s Miracle to orchestrate his crimes. 4. Captain James O’Neil: Laura’s superior, a pragmatic and seasoned officer who struggles to understand Laura’s mathematical approach but trusts her instincts. Plot: Act 1: The Initial Crime Scenes • Scene 1: Detective Mathis is called to a grisly murder scene, the fifth in a series of crimes that seem unrelated except for one peculiar detail: each crime scene location, when connected, forms a perfect pentagon. • Scene 2: Laura revisits each crime scene, mapping out the pentagon. She notices subtle clues suggesting the involvement of higher mathematics. • Scene 3: Captain O’Neil is skeptical but authorizes Laura to pursue her hunch. Laura reaches out to her old professor, Dr. Petr Novák, who explains his theorem and how it could apply to the crimes. Act 2: Uncovering the Pattern • Scene 4: Laura discovers that each pentagon transforms according to Petr’s Miracle, with the center point and vertices shifting in a predictable manner. • Scene 5: The team realizes that each new pentagon is formed based on the previous crime’s locations, creating a series of nested pentagons. • Scene 6: Laura and Dr. Novák work together to predict the next vertices. They identify a pattern and pinpoint the next potential crime scene. Act 3: The Chase • Scene 7: As they rush to the predicted location, they narrowly miss catching Lucas Grey, who leaves behind a taunting message: “Catch me if you can.” • Scene 8: With the stakes higher, Laura delves into Lucas’s background and discovers his motivation: a twisted game to prove his genius. • Scene 9: A cat-and-mouse game ensues, with Lucas always one step ahead, using Petr’s theorem to stay elusive. Act 4: The Final Showdown • Scene 10: Laura finally deciphers the ultimate goal of Lucas’s geometric game – the final pentagon points to a significant, symbolic location in the city. • Scene 11: A high-stakes confrontation at the final predicted crime scene. Laura uses her knowledge of the theorem to outmaneuver Lucas, predicting his moves. • Scene 12: Lucas is captured, but not before a tense standoff where he reveals his admiration for Laura’s intellect, acknowledging her as a worthy opponent. Epilogue: Resolution • Scene 13: The city breathes a sigh of relief as Lucas is put behind bars. Laura reflects on the case, realizing that her unique skills in mathematics have not only caught a criminal but also saved lives. • Scene 14: Dr. Novák publishes a paper on the practical application of his theorem, crediting Laura for her intuitive leaps and deductive prowess. Themes: • Intellectual Duel: The battle of wits between Laura and Lucas showcases the power of intellect and strategy. • Math in the Real World: The plot highlights how abstract mathematical concepts can have real-world applications, even in crime-solving. • Moral Ambiguity: Lucas’s genius is both his strength and downfall, raising questions about the ethics of using intellect for harmful purposes. Style: The narrative is a blend of crime drama and intellectual thriller, with detailed explanations of the mathematical concepts woven seamlessly into the plot. The tension escalates with each new crime, keeping readers on the edge of their seats, while the unique use of geometry adds a fresh twist to the genre. Potential for Adaptation: “Pentagon’s Shadow” is ripe for adaptation into a TV series or film, with its intricate plot, compelling characters, and the unique blend of crime-solving and mathematics. Each episode could explore a new facet of the theorem and its application, culminating in a thrilling final showdown.
    4
  28. The same you would do with a half baby
    4
  29. 5:00 Isn't there a Pascal Triangle also? I have just started the video and its already intriguing
    4
  30. This entire process of an “adding ears algorithm,” along with the 4 “basis pentagons” for every pentagon, along with the importance of directional adding of ears, along with the points being described by complex numbers, etc etc all seems like any polygon can be described using some sort of complex polynomial where each power term refers to a different “basis polygon” that when rotated and scaled in some way then added with all the others can form any polynomial. This has some interesting implications as it would mean that an n-sided polygon can be described using a polynomial of nth degree where the roots of the polynomial correspond to the vertices of the polygon, and would also be another demonstration as to why one of the “basis pentagons/polygons” goes to 0 + 0i when the ear adding algorithm is applied as it is just taking the derivative of some complex polynomial where that power goes to zero, and also showing that when enough “derivatives” are taken of the polygon it becomes regular then to a point like going to some constant then to zero. This could also imply the existence of polygons described by power series, like a circle or any continuous loop being described as an infinite degree polynomial which could also indicate the power series forms for sine and cosine and e^(ix) = cos(x) + i sin(x) describes more than just a cool identity but also a polynomial describing a circle as a polygon in the complex plane. I wonder how this could be applied in terms of sequences to differential equations like the fibioci sequence and sequences being described by polynomials through taking “discrete derivatives” of the sequence the have the first numbers be coefficients into a formula which forms a polynomial describing the sequence, could there be a way to term a sequence into a polynomial into a complex polygon, or find complex sequences using complex polynomials and then complex polygons which shape could describe some property and maybe get into topology? Sorry I’m rambling lol Has this been studied at all or does anyone know if this is a valid way to understand polygons? I noticed it again and again throughout the video and thought it was very interesting but I don’t think I have the skill to explore it on my own quite yet
    4
  31. This might be the most mindblowing piece of math I have ever seen. I had problems focusing on the rest of the video because my mind was reeling from the extreme and simplistic beauty of this structure!
    4
  32. Amazing video (I am assuming) EDIT: It was indeed a good one
    4
  33. I really like the design of your shirt :)
    4
  34. lost at 50:00 will watch again
    4
  35. Math(s)ologer!!
    4
  36. Wow, I am early haha :)
    4
  37. As someone who's just now getting to grips with non-integer applications of the binomial theorem (I have a weird idea of fun) I'm now wondering if there's any way to extend Moessner's Miracle to the reals, or beyond...
    4
  38. And indeed if you look at the formula for rectangular boards, it always spits out an even number.
    4
  39. Kudos to the discoverer of this. A very organized mind.
    4
  40. Brought my jacket and a tie for the grand premiere :) Thank You for the great format of Your videos. They help me stay sharp long time after university studies.
    4
  41. I made it all the way to the very end, but I'm still not sure how to split the pastrami sandwich. These are real-world problems Professor! 😁 (jk - I see the scenario where this works for that purpose too ... unless someone either does or doesn't like the bread crust and so each cut isn't valued equally.)
    4
  42. I think the proof that shows that you can make a dual triangle by composing the three medians is a more rigorous and easy to understand proof. It also immediately shows why isoceles triangles duals are isoceles: because two of the medians are the same!
    4
  43. Maybe 2000 years from now, we solve quantum gravity.
    4
  44. At 5:20: the original marble you consider cannot be the final marble simply because it’s movement is always in steps of two. If you label all possible places it can move on the board, you see that the center is not one of those spots. Using the same argument for the other marbles, only marble ‘e’ can survive.
    4
  45. This man is straight up a beast.
    4
  46. I'm so sad humans are going extinct :(
    4
  47. the polynomial you observe this pattern from could be manipulated in other ways, i.e. (x+3)^n instead of (x+2)^n , I wonder what those patterns correspond to.
    4
  48. I loved it as always. Didn't want it to end.
    4
  49. Reminds me of the matrix exponential. The power expansion of { e^(A) * e^(B) } for commutative matrices A & B can be rearranged to fit the binomial theorem. Fun!
    4
  50. This is a really great presentation, but at every point I can think of an architectural example from history built and still existing today which demonstrates the points of the presentation. For example think of Bernini’s Baldacchino at St. Peter’s. The twisted ‘rope’ columns have rational and calculable areas and volumes. This was built a few years before the mathematics presented by Cavalieri. Imagine what the ‘3D printer’ used to build it was like, - 400 years ago. 😊
    4