and imo it shows why we use algebra. this video is super cool don't get me wrong, but it'd definitely easier to do this in algebra than through pure geometry.
@@officiallyaninja I feel like a lot of the use of this kind of stuff, is that while the algebra is very very useful for understanding that its true. The geometry gives you a very good look at why its true. And getting that kind of experience is really useful for actually solving problems. Being able to look at problems visually and see where the algebra needs to go.
As a person with super good visual understanding, this is just the way I think. I can prove some things to myself, without writing anything down. This one, however, was too far fetched for me to prove independently.
These are the types of things that made me realize how bad I am at math and how I still have no clue what is happening in the video no matter how many times he explains it
It's a lot easier though usually with coordinates and equations. For this example it could be done graphically more easily, but for more difficult examples, one would need to write down some variables
Not a generalisation but erecting equilateral triangles on the sides of a triangle and joining their centres yield another equilateral triangle whose centre coincides with the centroid of the original triangle. There are several more results, read about the Fermat point
Math can be both cool and beautiful. My issues with it comes when I don't understand things. A pedagocal and well made video is certainly a good way of making things more understandable. Good job!
Seems this is being recommended more, and I wanted to say this is very nice and intuitive. Haven’t seen this method used for many geometric proofs, but I hope to going forward.
VERY COOL. Everything was so good about this. The theorem itself, the method, the explanation, the animations, and the enthusiasm in your voice. I loved it
It appears he was baptised Henricus Hubertus van Aubel. His theorem was published in _Nouvelles Corresp. Mathematique 4 (1878), pp 40-44_ in French, and titled "Note concernant les centres des carrés construits sur les côtés d’un polygon quelconque". It probably makes the pronunciation of his first name somewhat moot.
Hi. Hair-splitting Devil's advocate nerd here: This doesn't work when the squares result in 2 pair of opposite-side squares with coinciding centers because then you don't have lines connecting the centers. In theory, it would not work with just one pair of opposite-side squares with coinciding centers, except that it is impossible. It is either both pairs or none. Prove it!
Ah, a hair-splitting nerd after my own heart! Well, if one pair shares a center, then both would, else the lines connecting them would not have equal lengths. But who's to say it "doesn't work" in that case? The lines exist if you accept the possibility of zero-length lines, and who's to say that two zero-length lines are not perpendicular? :-)
@@MathyJaphy Damn, beaten by another nerd using that ancient concept that 2 zero-length lines, which have no direction (or every), are always perpendicular to each other.
@@MansMan42069 I would argue so, yes. You can define such notions without decomposing a vector into a length and direction first. Two vectors are perpendicular if their inner product is zero, for instance. You can additionally require a nonzero exterior product, but that is only that: an additional requirement. I think it’s better to keep those separate by default, and maybe define _strictly perpendicular_ to incorporate both only when necessary.
@@EllipticGeometry Are you sure? Wouldn't a "zero length line" really be the same as a single point? For example, if LINE AB is defined with two POINTS like so: A(x, y) and B(x, y). Then, for line AB to be of "zero length", the (x, y) coordinate values of both points would have to the same ---- where, A = B, and (B - A) = 0. Because of this, any LINE of "zero length" would have to be "treated" like a 0-dimesnional (point-like) object... which makes it impossible for that LINE (or GLORIFIED POINT) to yield any 2-dimensional (or 3-dimensional) "vector-like qualities"... I would think.
This is incredible! A theory I didn't know existed shown and explained without any complicated lingo or prior knowledge, with graphics to make it easier to follow. Thank you!!
Totally cool! Who'd have thought of reaching for vectors for such a geometrical nicety? Wonder if it works for Ptolemy's Theorem (the intersecting chords with common ratio)
If maths was taught that cool when I was in school ,I wouldn't be struggling now. I always knew maths was fun ,but got no assistantce to it from any of my teachers . They were more into marks and stuff rather than learning the purpose of maths
Thanks. I'll keep doing videos with Desmos because it's my "thing", but I envy those who deftly use Manim as you have in your SOME1 video. One of these days, maybe I'll learn how to use it.
The coolest thing I saw today, and probably will be the coolest video I see this week, or month, or even year 😁 Math isn't always taught as such in schools, unfortunately!
Wow, very impressive. When I saw the thumbnail, I did the proof myself with vectors, although my proof was by far less insightful than your construction. Somehow it just all vanishes to zero at the end. The only thing that's really required is that two left rotations form a reflection, which is pretty obvious.
What if you deform the cube in which the attached squares are on the inside instead of the outside? If the starting shape were a perfect square and the attached squares are on the inside, wouldn’t it just make a single point?
What software do you use for the animations? Recently 3blue1brown did a sort of challenge/contest where lots of people used his software Manim to make their own educational math videos.
I use Desmos Graphing Calculater. There's a link you can follow in the video's description to see the graph that made the video. Yes, I would have entered that contest if I'd had a video ready to go by the deadline. Use of Manim wasn't required. I love what Manim can do and someday I'll learn Python and figure out how to use it.
@@nabeelsherazi8860 I am not from an English speaking country and we use secont, but I have watched maths and physics videos in English and they use secont too. Guess it comes down to choice.
I suppose, in this particular case, it is kosher to use vectors or complex numbers to prove a theorem in Euclidean geometry. But in general, I'm always suspicious of chimeric proofs. If you mix-and-match different branches, you have to be absolutely certain about the provenance of the results you're using. Otherwise you run the risk of going around in circles. To use a Graph Theory metaphor, your whole system of axioms, theorems, and proofs need to form a Directed Acyclic Graph (DAG); no loops allowed. True story: - My students have used the distance formula from Cartesian geometry to (trivially) prove Pythagoras Theorem. - They've then used Pythagoras Theorem to prove Euclid's Parallel Postulate / Playfair's Axiom! 😂 Notable exception: If you're solving a Putnam or a Math Olympiad problem, you may use any "well-known" result from any branch of math.
Nice proof and very nice graphics indeed. Here is another proof, no graphics. Think of A, B, C, D as complex numbers. Then the midpoint of the side AB of the quadrilateral ABCD is (A+B)/2 and your vector a is (B-A)/2 while a'=i(B-A)/2, i denoting the imaginary unit. The "Start 1" point, thought of as a complex number, equals (A+B+Bi-Ai)/2 and, similarly, the "End 1" is (C+D+Di-Ci)/2. The vector that joins "Start 1" and "End 1" is the difference: v=(C+D-A-B+Di-Ci-Bi+Ai)/2. In the same way the vector that joins "Start 2" and "End 2" is equal to w = (D+A-B-C+Ai-Di-Ci+Bi)/2 (one can get w by cyclic substitution A->B, B->C, C->D, D->A in v). Since iw = v the vectors v and w are equal length and perpendicular. a''=ia'=i^2a=-a so this is consistent with your proof.
Yes! Isn't it amazing how complex numbers can make it so much simpler? As I wrote in the description, I actually started with that proof and worked it into a graphical one.
You’re referring to Napoleon’s Theorem. It might be possible to prove it using vectors like this, but I suspect it wouldn’t be as visually obvious, and might require some non-trivial math. Definitely something to think about.
Not knowing the origin of the background music is killing me. It sounds like plants vs zombies but I don't think so, I think it's from a tv show. A sitcom. Seems like a jingle that plays between sections of the show. Will and Grace?
Very cool indeed. It's unique to see algebra presented without a single equal sign.
and imo it shows why we use algebra. this video is super cool don't get me wrong, but it'd definitely easier to do this in algebra than through pure geometry.
@@officiallyaninja I feel like a lot of the use of this kind of stuff, is that while the algebra is very very useful for understanding that its true. The geometry gives you a very good look at why its true. And getting that kind of experience is really useful for actually solving problems. Being able to look at problems visually and see where the algebra needs to go.
@@henryhowe769 Both? Both. Both is good.
There is a equal sign but it is not written, he just said it
As a person with super good visual understanding, this is just the way I think. I can prove some things to myself, without writing anything down. This one, however, was too far fetched for me to prove independently.
These are the types of things that made me realise how fascinating math is, but I was never intuitive enough to really understand the application.
These are the types of things that made me realize how bad I am at math and how I still have no clue what is happening in the video no matter how many times he explains it
@@dammit3048 SAAAMEEEE UGAA BUGGA UGH UGH
Pretty cool that you don’t need to do any math with coordinates, that’s all just encapsulated in the vector representation
It's a lot easier though usually with coordinates and equations. For this example it could be done graphically more easily, but for more difficult examples, one would need to write down some variables
what about generalizations, if we draw equilateral triangles on each sides or other shapes?
Not a generalisation but erecting equilateral triangles on the sides of a triangle and joining their centres yield another equilateral triangle whose centre coincides with the centroid of the original triangle. There are several more results, read about the Fermat point
Umm... yes. Math gud
Math can be both cool and beautiful. My issues with it comes when I don't understand things. A pedagocal and well made video is certainly a good way of making things more understandable. Good job!
Seems this is being recommended more, and I wanted to say this is very nice and intuitive. Haven’t seen this method used for many geometric proofs, but I hope to going forward.
Vector proofs are a huge portion of my current curriculum
VERY COOL. Everything was so good about this. The theorem itself, the method, the explanation, the animations, and the enthusiasm in your voice. I loved it
Very cool, very cool
The only sad part here is reading Henri as french would, when he is dutch😂
It pains me
This fills Henri with ennui
It kinda hurts xd
I know, it pains me too. A disclaimer in the description is all I can do for now.
It appears he was baptised Henricus Hubertus van Aubel. His theorem was published in _Nouvelles Corresp. Mathematique 4 (1878), pp 40-44_ in French, and titled "Note concernant les centres des carrés construits sur les côtés d’un polygon quelconque". It probably makes the pronunciation of his first name somewhat moot.
Very cool theorem! Thanks for your amazing animations!
Hi. Hair-splitting Devil's advocate nerd here: This doesn't work when the squares result in 2 pair of opposite-side squares with coinciding centers because then you don't have lines connecting the centers. In theory, it would not work with just one pair of opposite-side squares with coinciding centers, except that it is impossible. It is either both pairs or none. Prove it!
Ah, a hair-splitting nerd after my own heart! Well, if one pair shares a center, then both would, else the lines connecting them would not have equal lengths. But who's to say it "doesn't work" in that case? The lines exist if you accept the possibility of zero-length lines, and who's to say that two zero-length lines are not perpendicular? :-)
@@MathyJaphy Damn, beaten by another nerd using that ancient concept that 2 zero-length lines, which have no direction (or every), are always perpendicular to each other.
@@adb012 are zero length lines also parallel? And every intersect angle in between?
@@MansMan42069 I would argue so, yes. You can define such notions without decomposing a vector into a length and direction first. Two vectors are perpendicular if their inner product is zero, for instance. You can additionally require a nonzero exterior product, but that is only that: an additional requirement. I think it’s better to keep those separate by default, and maybe define _strictly perpendicular_ to incorporate both only when necessary.
@@EllipticGeometry Are you sure? Wouldn't a "zero length line" really be the same as a single point? For example, if LINE AB is defined with two POINTS like so: A(x, y) and B(x, y). Then, for line AB to be of "zero length", the (x, y) coordinate values of both points would have to the same ---- where, A = B, and (B - A) = 0. Because of this, any LINE of "zero length" would have to be "treated" like a 0-dimesnional (point-like) object... which makes it impossible for that LINE (or GLORIFIED POINT) to yield any 2-dimensional (or 3-dimensional) "vector-like qualities"... I would think.
This is incredible! A theory I didn't know existed shown and explained without any complicated lingo or prior knowledge, with graphics to make it easier to follow. Thank you!!
This was so well done! Your voice makes it feel like a comforting cartoon or something
Beautiful animations and explanations.
Love it. A light yet surprising theorem with a satisfying visual proof. Now I gotta go try out Desmos.
I don't know how I just found your channel, but I absolutely love stuff like this. Great explanations, great visuals, concise. Simply beautiful.
I can’t believe this whole video is animated in Desmos, some amazing dedication!
Totally cool! Who'd have thought of reaching for vectors for such a geometrical nicety? Wonder if it works for Ptolemy's Theorem (the intersecting chords with common ratio)
If maths was taught that cool when I was in school ,I wouldn't be struggling now. I always knew maths was fun ,but got no assistantce to it from any of my teachers . They were more into marks and stuff rather than learning the purpose of maths
Very pretty! Thanks for sharing.
I dont know why but i started to watch math problem solving as a way to entertain myself recently,keep it up the good work
just saw post of van aubel's theorem on my facebook feed. as a mathematician, had to see a proof. excellent presentation! thanks!
How cool was that? VERY cool. Your voice sounds enthusiastic and the video itself is edited greatly
"Well Rich, what did you think of Van Aubel's Theorem?"
Your presentation is gold!!!
You make it easy to see the problem!
Thank you, man!
i read about this in needham's complex analysis 3 days ago. Algorithm really did read me
This is the first video I watch on my new smartphone...
Super cool to see Desmos used for the visuals for a video!
Thanks. I'll keep doing videos with Desmos because it's my "thing", but I envy those who deftly use Manim as you have in your SOME1 video. One of these days, maybe I'll learn how to use it.
it is not only fascinating but there must be some incredible applications waiting to be discovered.
It's incredible that this didn't even involve multiplication (of vectors), nice
Wow, that’s a really neat way of presenting that.
The coolest thing I saw today, and probably will be the coolest video I see this week, or month, or even year 😁
Math isn't always taught as such in schools, unfortunately!
Thank you for telling about desmo calculator
That's really awesome. Subscribed. Hope to see more.
Wow, very impressive. When I saw the thumbnail, I did the proof myself with vectors, although my proof was by far less insightful than your construction. Somehow it just all vanishes to zero at the end. The only thing that's really required is that two left rotations form a reflection, which is pretty obvious.
That was very interesting, thanks for the video!
Very cool! Well explained and nicely animated.
Thank you! A fantastic way to demonstrate a proof!
Nice theorem and great presentation Very pretty
now this is a great math video, the visualizations are top-notch!
thanks bro for explaining this concept so easily.
Liked and subscribed juste for the prononciation of Henry, I love the fact that prononciation is important to you
Love from France 🇨🇵
to answer the last question: really cool
Damn that is brilliant.
I would have never thought of this.
nice, thank you for this illustration
How fun! An elegant proof.
Absolutely brilliant
Very nicely done.
New maths theorem added to my knowledge
You just contributed to the net Awesome in the universe. Thanks!
Some math dude discovered this and was like
"Nifty".
You pronounciation of henri van abel makes me cry
Alas, me too, now that I've been schooled by several commenters. I added an apology to the description a few days ago.
Awesome proof, quality video!
How cool? To speak with Balckadders' servant Baldrick: That was pretty cool, m'lord'
This was a very high quality video
It makes a change to hear properly edited audio. That was listenable.
What if you deform the cube in which the attached squares are on the inside instead of the outside? If the starting shape were a perfect square and the attached squares are on the inside, wouldn’t it just make a single point?
You were right, this was fun.
well done
Math is freaky. Love it! (And I'm now a new subscriber too!)
This was fantastic!
Very very cool.
Next: nine point circle
How you eliminated vectors c anad c' from the vector diagram, when they canceled out to zero, was cool.
Wow that's really cool.
From Morocco..genious..thank you
Oddly satisfying to one who has never used more than school math in his life. 😁
Well its actually fun thing to learn. Its a shame i never heard about it in school
What software do you use for the animations? Recently 3blue1brown did a sort of challenge/contest where lots of people used his software Manim to make their own educational math videos.
I use Desmos Graphing Calculater. There's a link you can follow in the video's description to see the graph that made the video. Yes, I would have entered that contest if I'd had a video ready to go by the deadline. Use of Manim wasn't required. I love what Manim can do and someday I'll learn Python and figure out how to use it.
No entendí cual es la finalidad de este teorema, que busca demostrar?
"A double prime" as you called it is actually called a secont (second in latin, prime is first). Nice video
In mathematics the term double prime is the standard
@@nabeelsherazi8860 I am not from an English speaking country and we use secont, but I have watched maths and physics videos in English and they use secont too. Guess it comes down to choice.
this was definately cool
Great video!
Loved it man🔥🔥🔥
awesome video man
Pretty cool! I like
I suppose, in this particular case, it is kosher to use vectors or complex numbers to prove a theorem in Euclidean geometry.
But in general, I'm always suspicious of chimeric proofs.
If you mix-and-match different branches, you have to be absolutely certain about the provenance of the results you're using. Otherwise you run the risk of going around in circles. To use a Graph Theory metaphor, your whole system of axioms, theorems, and proofs need to form a Directed Acyclic Graph (DAG); no loops allowed.
True story:
- My students have used the distance formula from Cartesian geometry to (trivially) prove Pythagoras Theorem.
- They've then used Pythagoras Theorem to prove Euclid's Parallel Postulate / Playfair's Axiom! 😂
Notable exception: If you're solving a Putnam or a Math Olympiad problem, you may use any "well-known" result from any branch of math.
This tool would have been nice to have in the 90s.
Here, take my sub.
Could you make a tutorial on how you made the desmos demo? Seems very complex and interesting.
Interesting idea. I will give it some thought.
Nice proof and very nice graphics indeed. Here is another proof, no graphics. Think of A, B, C, D as complex numbers. Then the midpoint of the side AB of the quadrilateral ABCD is (A+B)/2 and your vector a is (B-A)/2 while a'=i(B-A)/2, i denoting the imaginary unit. The "Start 1" point, thought of as a complex number, equals (A+B+Bi-Ai)/2 and, similarly, the "End 1" is (C+D+Di-Ci)/2. The vector that joins "Start 1" and "End 1" is the difference: v=(C+D-A-B+Di-Ci-Bi+Ai)/2. In the same way the vector that joins "Start 2" and "End 2" is equal to w = (D+A-B-C+Ai-Di-Ci+Bi)/2 (one can get w by cyclic substitution A->B, B->C, C->D, D->A in v). Since iw = v the vectors v and w are equal length and perpendicular. a''=ia'=i^2a=-a so this is consistent with your proof.
Yes! Isn't it amazing how complex numbers can make it so much simpler? As I wrote in the description, I actually started with that proof and worked it into a graphical one.
@@MathyJaphy Dear me, I should have read the description. Thanks!
it was pretty cool dude
Are there any applications for this though?
Beautiful!!
Fantastic!
There a lot of shortcuts in this proof. Ex: why a'+b'+c'+d'=0?
This is so cool
Vety good question
Excellent and very clear, but absolutely not in the category of FUN PROOF.
Very cool!
This looks like the quadrilateral version of the pythogor-- whatever theorem
This is trippy
Super cute; thanks!
Super cool!!!
Pretty cool.
Does it work with triangles variant where you need to prove equilateral?
You’re referring to Napoleon’s Theorem. It might be possible to prove it using vectors like this, but I suspect it wouldn’t be as visually obvious, and might require some non-trivial math. Definitely something to think about.
Thanks for the inspiration! th-cam.com/video/_DpH_JUP0eY/w-d-xo.html
Does it generalize somehow when the four lines aren't in a plane?
what if all four points are the same point
The way he said henri van aubel is hilarious for me as a Dutch person
Apologies! I should have done more research. :-(
@@MathyJaphy no worries! I’ve never heard a non dutch person say dutch things correctly. It’s just funny not offensive or anything
A very cool theorem .Could not understand the explanation tho.
Loved it
Not knowing the origin of the background music is killing me. It sounds like plants vs zombies but I don't think so, I think it's from a tv show. A sitcom. Seems like a jingle that plays between sections of the show. Will and Grace?
None of the above. th-cam.com/video/6Y4Y0RzbaNY/w-d-xo.html
@@MathyJaphy Ah that makes sense. Thank you!
Figured he was using desmo, lol