What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented
Bit of a mystery Mathologer today with the title of the video not giving away much. Anyway it all starts with the quest for equilateral triangles in square grids and by the end of it we find ourselves once more in the realms of irrationality. This video contains some extra gorgeous visual proofs that hardly anybody seems to know about.
0:00 Intro
0:47 First puzzle
2:24 Second puzzle
3:50 Edward Lucas
4:41 Equilateral triangles
13:15 3d & 3rd puzzle
19:52 30 45 60
29:31 Credits
Here are links to/references of some of the things I mention in the video:
Joel Hamkin's blog posts that inspired this video:
http://jdh.hamkins.org/no-regular-polygons-in-the-integer-lattice/
http://jdh.hamkins.org/no-regular-polygons-in-the-hexagonal-lattice/
There is also a whole chapter about all this and much more related maths in his new book
https://www.amazon.com/Proof-Mathematics-Joel-David-Hamkins/dp/0262539799
Here is another really good article which includes a nice characterisation of the triangles that can be found in square grids plus a very good survey of relevant results:
Michael J. Beeson, Triangles with Vertices on Lattice Points, The American Mathematical Monthly 99 (1992), 243-252, https://www.jstor.org/stable/2325060?seq=1
Scherrer's and Hadwinger's articles:
Scherrer, Willy, Die Einlagerung eines regulären Vielecks in ein Gitter, Elemente der Mathematik 1 (1946), 97-98.
https://tinyurl.com/y45p64t7
https://gdz.sub.uni-goettingen.de/id/PPN378850199_0001?tify={%22pages%22:[101]}
Hadwiger, Hugo Über die rationalen Hauptwinkel der Goniometrie, Elemente der Mathematik 1 (1946), 98-100.
https://tinyurl.com/yx98kkqt
https://gdz.sub.uni-goettingen.de/id/PPN378850199_0001?tify={%22pages%22:[102],%22view%22:%22info%22}
Another, nice paper on rational (and algebraic) cosines
https://arxiv.org/pdf/1006.2938.pdf
Here is a solution to the first puzzle (one way to find the general formula):
https://nrich.maths.org/657/solution
The music in this video is by Chris Haugen, Fresh Fallen Snow (playing in the video) and Morning Mandolin (for the credits)
A couple of remarks:
1. Probably the simplest way to deduce the sin and tan parts of the rational trig ratio theorem is to realise that they follow from the cos part via the trigonometric identities: sin(x)=cos(90-x) and tan^2(x) = (1-cos(2x))/(1+cos(2x)). Note that the second identity implies that if tan(x) is rational, then cos(2x) is rational (if tan(x)=c/d, then tan^2(x)=c^2/d^2=C/D and cos(2x)=(D-C)/(D+C)).
2. Bug report.
a) Here I redefine cos(120◦) = 1.
https://youtu.be/sDfzCIWpS7Q?t=1362
Remarkable :(
b) This transition to the good stuff I clearly did not think through properly.
https://youtu.be/sDfzCIWpS7Q?t=1018
It's possible to make this work for all regular n-gons. There is only one complication that occurs for n's that are of the form 2 * odd. For the corresponding regular n-gons, if you pick up the edges in the order that they appear around the n-gon and assemble them into a star, things close up into (n/2)-stars. For all other n, things work exactly as I showed in the video. Having said that you can also assemble the edges of one of the exceptions into stars. Have a look at this https://imgur.com/68A3fEe and you'll get the idea. Anyway lots more nice side puzzles to be explored here if you are interested :)
Enjoy!
Burkard
Two ways to support Mathologer
Mathologer Patreon: https://www.patreon.com/mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)
Видео What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented канала Mathologer
0:00 Intro
0:47 First puzzle
2:24 Second puzzle
3:50 Edward Lucas
4:41 Equilateral triangles
13:15 3d & 3rd puzzle
19:52 30 45 60
29:31 Credits
Here are links to/references of some of the things I mention in the video:
Joel Hamkin's blog posts that inspired this video:
http://jdh.hamkins.org/no-regular-polygons-in-the-integer-lattice/
http://jdh.hamkins.org/no-regular-polygons-in-the-hexagonal-lattice/
There is also a whole chapter about all this and much more related maths in his new book
https://www.amazon.com/Proof-Mathematics-Joel-David-Hamkins/dp/0262539799
Here is another really good article which includes a nice characterisation of the triangles that can be found in square grids plus a very good survey of relevant results:
Michael J. Beeson, Triangles with Vertices on Lattice Points, The American Mathematical Monthly 99 (1992), 243-252, https://www.jstor.org/stable/2325060?seq=1
Scherrer's and Hadwinger's articles:
Scherrer, Willy, Die Einlagerung eines regulären Vielecks in ein Gitter, Elemente der Mathematik 1 (1946), 97-98.
https://tinyurl.com/y45p64t7
https://gdz.sub.uni-goettingen.de/id/PPN378850199_0001?tify={%22pages%22:[101]}
Hadwiger, Hugo Über die rationalen Hauptwinkel der Goniometrie, Elemente der Mathematik 1 (1946), 98-100.
https://tinyurl.com/yx98kkqt
https://gdz.sub.uni-goettingen.de/id/PPN378850199_0001?tify={%22pages%22:[102],%22view%22:%22info%22}
Another, nice paper on rational (and algebraic) cosines
https://arxiv.org/pdf/1006.2938.pdf
Here is a solution to the first puzzle (one way to find the general formula):
https://nrich.maths.org/657/solution
The music in this video is by Chris Haugen, Fresh Fallen Snow (playing in the video) and Morning Mandolin (for the credits)
A couple of remarks:
1. Probably the simplest way to deduce the sin and tan parts of the rational trig ratio theorem is to realise that they follow from the cos part via the trigonometric identities: sin(x)=cos(90-x) and tan^2(x) = (1-cos(2x))/(1+cos(2x)). Note that the second identity implies that if tan(x) is rational, then cos(2x) is rational (if tan(x)=c/d, then tan^2(x)=c^2/d^2=C/D and cos(2x)=(D-C)/(D+C)).
2. Bug report.
a) Here I redefine cos(120◦) = 1.
https://youtu.be/sDfzCIWpS7Q?t=1362
Remarkable :(
b) This transition to the good stuff I clearly did not think through properly.
https://youtu.be/sDfzCIWpS7Q?t=1018
It's possible to make this work for all regular n-gons. There is only one complication that occurs for n's that are of the form 2 * odd. For the corresponding regular n-gons, if you pick up the edges in the order that they appear around the n-gon and assemble them into a star, things close up into (n/2)-stars. For all other n, things work exactly as I showed in the video. Having said that you can also assemble the edges of one of the exceptions into stars. Have a look at this https://imgur.com/68A3fEe and you'll get the idea. Anyway lots more nice side puzzles to be explored here if you are interested :)
Enjoy!
Burkard
Two ways to support Mathologer
Mathologer Patreon: https://www.patreon.com/mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)
Видео What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented канала Mathologer
Показать
Комментарии отсутствуют
Информация о видео
Другие видео канала
What is the best way to lace your shoes? Dream proof.How did Ramanujan solve the STRAND puzzle?Secret of row 10: a new visual key to ancient Pascalian puzzlesVisualising irrationality with triangular squaresWin a SMALL fortune with counting cards-the math of blackjack & Co.Epicycles, complex Fourier series and Homer Simpson's orbitEllipsoids and The Bizarre Behaviour of Rotating BodiesWhy did they prove this amazing theorem in 200 different ways? Quadratic Reciprocity MASTERCLASS700 years of secrets of the Sum of Sums (paradoxical harmonic series)Why is there no equation for the perimeter of an ellipse‽Visualising Pythagoras: ultimate proofs and crazy contortionsNew Reuleaux Triangle MagicIrrational Rootse to the pi i for dummies2000 years unsolved: Why is doubling cubes and squaring circles impossible?500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can't handle?This completely changed the way I see numbers | Modular Arithmetic Visually ExplainedEuler's and Fermat's last theorems, the Simpsons and CDC6600Fermat’s HUGE little theorem, pseudoprimes and Futurama