Pythagoras twisted squares: Why did they not teach you any of this in school?
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- Опубліковано 30 кві 2024
- A video on the iconic twisted squares diagram that many math(s) lovers have been familiar with since primary school. Surprisingly, there is a LOT more to this diagram than even expert mathematicians are aware of. And lots of this LOT is really really beautiful and important. A couple of things covered in this video include: Fermat's four squares theorem, Pythagoras for 60- and 120-degree triangles, the four bugs problem done using twisted squares and much more.
00:00 Intro
05:32 3 Squares: Fermat's four square theorem
12:51 Trithagoras
20:29 Hexagoras
22:06 Chop it up: More twisted square dissection proofs
23:42 Aha! Remarkable properties of right triangles with a twist
26:35 Mutants: Unusual applications of twisted squares
30:38 Op art: The four bugs problem
36:01 Final puzzle
36:32 Animation of Cauchy-Schwarz proof
37:16 Thanks!!
Here are a couple of links for you to explore:
My first Pythagoras video from four years ago:
• Visualising Pythagoras...
A collection of over 100 proofs of Pythagoras theorem at Cut-the-knot www.cut-the-knot.org/pythagoras/ (quite a few with animations) I cover proofs 3, 4, 5 ( :), 9, 10, 76, 104. Other proofs closely related to what I am doing in this video are 55, 89, and 116.
A very good book that touches on a lot of the material in this video by Claudi Alsina and Roger B. Nelsen - Icons of Mathematics: An Exploration of Twenty Key Images (2011). Check out in particular chapters 1-3 and chapter 8.3.
Fermat's four square theorem:
Alf van der Poorten's super nice proof arxiv.org/abs/0712.3850
Fibonacci seems to be the discoverer of the connection between Pythagorean triples and arithmetic sequences of squares of length 3 en.wikipedia.org/wiki/Congruum
Trithagoras:
Wayne Robert's pages. Start here and then navigate to "The theory to applied to the geometry of triangles" tinyurl.com/3k6afad4
M. Moran Cabre, Mathematics without words. College Mathematics Journal 34 (2003), p. 172.
Claudi Alsina and Roger B. Nelsen, College Mathematics Journal 41 (2010), p. 370. (Trithagoras for 30 and 150-degree triangles)
Nice writeup about how to make Eisenstein triples from Eisenstein integers
ime.math.arizona.edu/2007-08/0...
More people should know about Eisenstein:
mathshistory.st-andrews.ac.uk...
Other twisted square dissection proofs:
There is an Easter Egg contained in the first proof. Five days after publishing the video only one person appears to have noticed it :) Here is an alternative version of the animation that only uses shifts that I put on Mathologer 2 • Strange Pythagoras
The four bugs problem:
Actually I got something wrong here. Martin Gardner mentioned the four bugs for the first time in 1957 as a puzzle Martin Gardner actually mentioned the four bugs for the first time in 1957 as a puzzle (Gardner, M. November, 1957 Mathematical Games. Nine titillating puzzles, Sci. Am. 197, 140-146.) The 1965 article that is accompanied by the nice cover that I show in the video talks, among many other things, about the more general problem of placing bugs on the corners of a regular n-gon.
If you've got access to JSTOR, you can access all of Martin Gardner's articles through them.
www.jstor.org/journal/scieamer (all issues of the Scientific American)
www.jstor.org/stable/e24941962 (follow the Mathematical Games link)
www.jstor.org/stable/e24931930 (follow the Mathematical Games link)
www.jstor.org/journal/scieamer (all issues of the Scientific American)
Here are a couple of other online resources worth checking out.
tinyurl.com/4erz3zmf
tinyurl.com/ykvvj5sw
Explanation for distance 1: Because the bug that each bug is walking towards is always moving perpendicular to the first bug’s path, never getting closer or further away from the first bug’s motion. So it has to go exactly the same distance as it was at the beginning.
For the mathematics of various bits and pieces chasing each other check out Paul Nahin's book Chases and Escapes: The Mathematics of Pursuit and Evasion.
Solution for the puzzle at the end:
tinyurl.com/3eebxn2k
Today's music: A tender heart/The David Roy
T-shirt: google "Pythagoras and Einstein fighting over c squared t-shirt" for a couple of different versions.
Enjoy!
Burkard
I have never seen any of these proofs - I graduated HS in '56 - got a BSEE in '60 - worked in Aerospace 38 years - retired and became a HS math teacher in 2001 - retired in 2014. Never offered any of these proofs to my students. I think they (and I) were shortchanged.
Love this stuff.
I really have no idea how US schools still manage to produce some mathematically literate people.
Once one of my colleagues brought his 1st or 2nd grade son homework apparently designed to teach subtraction. 3 engineers sat puzzling over (1 UK-educated, 1 Chinese-educated and 1 Polish-educated) until 4th came who's kids were past basic schools and solved it for us.
I knew both of Pytagoras proofs before I was taught them.
From my POV US math teaching is bizarre.
genuine questions, in 14 years of teaching you never thought of going beyond the curriculum you were teaching? like, no curiosity at all?🤔🥺
Every month when Mathologer comes out of his hiding, you know it's going to be a great video
Absolutely! Masterpiece, one after the other!!!
and a cool T-Shirt :)
No, I do not. I expect it to be good, but unless I have watched the entire thing I can not evaluate the quality of the video. There is an important difference between 'expect' and 'know', and that is the same difference as between science and faith, for example.
If it is every month I think i missed some videos, going back to watch them
@@SuperJuiceman11 yes true I noticed that too lol
I have indeed been taught this at school. Not in the math class, but during Latin class. Our Latin teacher also knew ancient Greek and had found the Greek text about this interesting enough to teach us about it. I found it very enlightening.
Do they even have Latin class in the country most viewers here come from?
@@jannikheidemann3805 I don't know. I'm from Germany. I grew up in the 90s an 00s.
@@jannikheidemann3805 There are Latin language classes available in some universities in the United States. But Latin Math class is not in the states!
@@jannikheidemann3805 If you mean the United States, then yes, they teach Latin in many high schools, it's gaining in popularity, many Catholics study it independently to deepen their study of the faith, and many more people study the Latin roots of English in preparation for standardized tests like the SAT and GRE.
My school in Britain taught Latin. I think Latin was an essential requirement if you wanted to be accepted as a student in Oxford and Cambridge Universities. A little indicator of a well educated candidate. Not any more of course!
For the bug problem: An equivalent question would be to ask how long it takes for any one ant to reach the ant it is chasing. The chased ant always moves orthogonally to the chaser, so it is neither going towards it nor away from it. Thus the distance between the two ants simply decreases by the speed of the chaser. It thus takes 1 unit of time for the ants to meet.
Wow. Such a simple way of looking at it, that the first reaction is to see if something is missing. But it is not a paradox... thanks.
For added “fun” this shows that the extra path length for the finite cases is the sum of the non-orthogonal movement introduced
other way to look: change the reference system to one of the ants. Now the world is rotating, and the ant goes straight ahead 1 unit.
At 28:42, your generalization is wrong- it should be that cbrt(ABC)
Ah that bug explanation is lovely! Wish I’d given it more thought myself first haha.
_"your generalization is wrong"_ - he does not give a generalisation, so it cannot be wrong either.
Where have you seen that C in the video? The generalization shown is sqrt (AB)
@@raphaellfms The video’s been edited since he put it out- there was an incorrect generalization there. If you look closely, you can see- usually, he fades in and out, but a couple of seconds earlier, he cuts out instead.
@@renedekker9806 See above.
The first proof is by far the best I’ve ever seen, and it was a big lightbulb moment for me. I’ve seen UA-cam videos of teachers doing it with felt triangles taped on the blackboard, but I never got any lessons like that as a kid.
I will be sending my local elementary school a bill as suggested.
The animation in this video is just insane - about the best I've ever seen on UA-cam in fact. I can't imagine all the time it must have required to get all that to flow so smoothly. Well done.
I especially liked the part about the bugs, shows you how to solve such a problem in visual steps - a great problem solving technique. And the animation was just beautiful.
I too enjoyed the animations. The 3Blue1Brown youtube channel animations are also quite impressive. It’s math-oriented, in case you don’t know of it.
I fear most schools prefer not to promote "visual proofs" since it could have pupils do them when the visual is "close enough" yet still false. Meanwhile, the algebraic ones have "rigid" rules, that can be more easily explained and corrected.
That is because every line you draw has a thickness. While in mathematical reality there is no thickness of a line. The mathematical line would become invisible.
@@BartvandenDonk Not only that. Mostly because visual proofs are easy to become deceptive and non generic. The infinite chocolate bar is an example. Things deceptively similar but different, etc. There are books like The Trissectors, A Budget of Paradoxes (by De Morgan) and Mathematical Cranks filled with examples of false visual proofs of wrong theorems or impossible constructions that mathematicians received in letters or in self published books.
i still think that visual proofs are a lot more educational and make math a lot less cryptic, this definitively should be taught at schools as many students run from math teachings only because they cannot see the algebraic connections as anything more than juggling letters and numbers that have nothing to do with reality
I think that good math students can visualize these proofs in their head with perfect lines. A student might become a good math student from a struggling one by seeing one presented and learning the concept and then trying to visualize more concepts himself.
Burkard, rather than heap additional praises regarding the dependable brilliance of your channel's content, I thought I might remark on your delivery: Your pacing, pitch, tone and inflections are such that, rather than lulling my mind to sleep in class, you have a special way of always educing and coaxing my brain forward (stubborn mule that it is).
Wow, thanks!
I mean I cheated or crammed my way through high school I was too bored to try, so this is the first time I’m seeing these design elements explained from a mathematical standpoint which helps me understand them deeper:
It’s amazing how intertwined art, beauty, mathematics, numbers, algorithms, patterns etc.
Truly all connected. I am all that is, because all that is - is within me. 🙏🏻
Today has been something else for me. My wife and I had a break through in our relationship. We came to terms with something incredibly taxing on us.
Second, I discovered a video on the Richat structure in Africa. Stirring a curiosity in my mind I haven't felt in... maybe ever.
Third, through that curiosity I found you. I have never sat through a lecture, not without fidgeting, smoking, going for a snack. Nothing would keep my interest. I barely took a breath watching your explanations of 3,6 and 9 and now this video.
As someone who has felt like he was awful at math, only ever memorizing material. Even in college. I now understand why. The way I learn was never catered to, in the least it seems. I perhaps could've spent my whole life not understanding why we use Pythagoras for things, other than it just works. But this has shown me the beauty in mathematics. No wonder so many strive to this. Wow. I feel like an empty cup, eager to be filled. Thank you for making these, genuinely from the bottom of my curious little heart.
That's great. Glad that these videos work so well for you :)
I don‘t recall seeing any of these proofs in school. Most relevant formulas were simply introduced to us without ever being proven before college.
... which is a crime :(
@@Mathologer from NWT Canada. I assure you, I prove all of these to Gr. 7 and up students.
Same, and even then I never saw it till my elective History of Math class my junior year.
@@davidmeijer1645 Very glad to here that you do. How about your colleagues? Would you say that showing these proofs to kids is something that is commonly done in Canada?
@@Mathologer i would say that the reality is..no. Most teachers are concerned with covering the curriculum, which doesn’t direct teachers to explore these expansive variations on a theme, say this Pythagorean Theorem. That’s why your videos are great…for us teachers to recommend (to the interested student). I went through 3B1B lighthouse explanation of the Basel Problem yesterday with a class of 6 Gr. 10s. I really think it was quite a bit above them, but I’ll see when we next meet. And I think that’s why most teachers are reluctant to delve….concerned that it’s a bit above the students. Have you been amongst high school students lately? I mean, most are at the stage of struggling with something like, how to get the angle, when a trig ratio is known….I.E. learning to use the inverse trig function on a calculator. It’s quite amazing actually how developmental stages are quite uniform across geography and time.
Mesmerizing. Reminds me of being a kid in early years of learning math and just drawing and playing with shapes. Wish I had this kind of instruction back then.
Same!
i wonder the same thing. those students. in ancient greece, didn't have calculators. don't know if they had compases, either. but they solved a lot, calculated Pi, and figured calculus...
It’s been years since I’ve seen these proofs but I was taught the second one first. I love the visual explanations. Great video.
For the bugs distance problem: Fix the frame of reference to one bug. The bug moving towards it, in that frame of reference, just travels along the shortest distance towards it, meaning the covered distance is exactly 1
To complete the proof you need to show that the second bug is moving orthogonal to the first (which follows from the symmetry of the setup)
The hexagonal theorem is what just brought it all into perspective for me! I could not figure out why we were subtracting the “AB” and then the visual for the hex came with the “-6T” and it all clicked with the extra triangles! And then dividing the hex by six and getting the trithagorean buttoned it all up like a beautiful winter coat…
Yes, that one just clicked beautifully. Also one of the pleasures of what I do when I see something like this for the first time :)
Like a beautiful winter coat.
In art we look at the negative space of a sculpture and in technical drawing the extended construction lines - to me this is what helps to visualise the maths.
I learned the derivation of the Pythagorean Theorem when I took geometry. We were on the topic of similar triangles and found that a right triangle could be made up of two smaller right triangles. It was very straight forward and consistent with the topic we were on.
This is one of the many things I enjoy about traditional geometric patchwork design quilt patterns , (like Flying Geese, Irish Chain, Feathered Star, Grandmother's Flower Basket, Disappearing 9-patch,
Bear's Paw, Log Cabin and all it's variations, Pinwheels, etc.) with 1/2 square or equilateral triangles, rectangles, squares, etc. and shuffling them around to create different patterns. Now I have a name for them, and may design quilt patterns with the names of Pythagoras or Trithagoros, or the long ago Chinese mathematician's name! Great visual explorations of these relationships, thank you. (of course I enjoy even more, the random sewing together of odd shaped and sized pieces, called, "crumb quilting"! lol)
(see also, the 'golden ratio' seen so much in nature and used from antiquity in architecture to produce unconsciously, naturally pleasing, building facades.)
You are a brilliant mind. Leaving me smiling from what you teach. I'm literally talking to myself in full conversation when I watch your videos. Thank you for the intelligent video and teaching.
Your video on the many proofs of Pythagoras is one of my favorites, so I’m sure I’m gonna enjoy this one!
A very weird rabbit hole I went down related to the four bugs problem: The function that gives the path length in terms of the step size is f(x) = x/(1 - sqrt(x^2 + (1-x)^2)), and we saw that the limit of f(x) as x -> 0 is 1. What I did was look at the Taylor series of f(x) around x = 0 and what I noticed was the terms up to x^15 all had positive coefficients, but from x^16 on the signs of the coefficients are periodic with period 8 (starting from x^19 the pattern is 4 +'s then 4 -'s).
I think I understand why it's periodic with period 8; it has to do with the roots of x^2 + (1-x)^2 being (1+i)/2 and its conjugate, where the angle corresponding to (1+i)/2 in the complex plane is 1/8 of a turn. And I also sort of have a formula for the coefficients: if the Taylor coefficients of 1/sqrt(x^2 + (1-x)^2) are a[n] and the Taylor coefficients of f(x) are b[n], then b[n+1] - b[n] = a[n-1] - a[n] + a[n+1]/2. But the a[n] sequence itself is probably not expressible in closed form so I'm not sure how to get more info on b[n].
That sounds like fun. Have to have a close look myself :)
An alternate solution to the bugs problem: Let each of the bugs have velocity 1 unit/s. Split each velocity vector into a vector that points towards the center of the square and one that points 90 degrees to that one. Using the 45-45-90 triangle created with the original velocity vector and two component vectors, we find that the two new vectors have magnitude sqrt(2)/2. Thus the bugs are going towards the center at a rate of sqrt(2)/2 units/s. The center is sqrt(2)/2 units away so it will take the bugs one second to get to the center. Distance = rate*time so distance = 1 unit/s * 1 s = 1 unit.
I think it's even simpler than that. Each bug always moves perpendicular to the motion of its "target" bug, so the target's motion doesn't affect the rate of closure.
Therefore, each bug must travel exactly one unit to reach its target, at the center.
Fred
@@ffggddss Wow nice argument!
@@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.
@@ffggddss I really like that argument. I originally encountered the problem when the bugs were in an equilateral triangle, so I not think that the argument applies to other regular polygons. A fun exercise is to use the method mentioned above to prove that the distance traveled by a bug starting in a regular n-gon configuration with side length 1 is 0.5*sec^2(90(n-2)/n). Another interesting thing to look into is other arbitrary starting configurations of the bugs.
@@manicmaths2861 I put some links to interesting write-ups in the description of this video :)
I've been watching your videos for years, THIS ONE! was the proof I was asked to solve in my grade 12 academic maths course. You brought me to a new realization that I hadn't seen before. My teacher was kind enough to give me a passing grade but I missed the mark. Now I want to make a learning tool that will physically describe the geometric relationship you've described. Thank you for your continued insight.
I love these visual proofs. The animation of the visual proof at 22:40 is extra fun because two of the triangles are re-scaled. But they are re-scaled instead of having a more complicated animation. Well done!
Yes, and it happens five times from 22:17. @Mathologer has the left-most and right-most triangles swapping sizes instead of swapping places. IMHO it detracts from the quality of this video, because the visual proof depends on the pieces only rotating and translating - not expanding.
The next one at 22:40 really confused me when i tried to work it out because the side length of the blue inside square and the side length of the blue isosceles triangle are not actually the same in general.
They do happen to be the same for the 3-4-5 triangle though
@@EricSeverson Did you mean 23:40?
I tend to approach your videos with an open mind and willingness to learn and I am never disappointed. Your teaching methods are wonderful and refreshing as ever. Thank you for all your hard work and dedication. Your love and passion for math is unmatched. We’re kindred spirits, but your grasp of mathematics is astounding. The final problem I believe the answer is 1 if my math is correct. Though I have been wrong before. Thank you for such a wonderful video!❤❤
That's great. Also have a look at this tinyurl.com/3eebxn2k tinyurl.com/
Learned both when i was 13 by my teacher. I loved math more than i can remember! Thank you for your content once again! Hope to see an ecology "ecosystem stability" math video from you, they are quite intriguing!
This a glorious episode! Great job to you and the helpers listed in the credits.
learned the second proof in school, the first in an earlier Mathologer video. Since then I watch all your videos and bought several of your books. I would recommend your videos to everyone who has the slightest interest in maths.
I'm a mandala artist and I find this fascinating. I've been creating eggs before the chicken... :-) Geometry has always been a passion of mine and I thank my high school teacher Mrs. Locke for making geometry interedting and thank you to you too for expanding my understanding !
That's great :)
From here we could go into the Mandelbrot set via the golden ratio... Damn I wish I had a working laptop.
I don't remember which proof I saw first but a few weeks ago I wanted to show a proof to one of my kids and realized I forgot all the proofs I had learned in school. The second (algebraic rearrangement) was the one I worked out first.
@@Mike_Greene yes
That selection of music packed some powerful emotion at the end. Amazing stuff, as usual!
I learned the first one of those proofs first, and the second one before I got out of high school. I had no idea that they aren't taught anymore! 😮 Geometry has always been, as you called it, pretty to me. Thanks for this!
American math student, even though we got up to differential equations I never saw either proof of Pythagoras until much later on UA-cam lol
Don't you also find this very strange? How can they not show this proof in school?
@@Mathologer American schools are focused on the "utility." They teach you what works but not how and it's really sad. I didn't understand how integrals literally functioned until I was in college despite taking 3 years of calculus in high school
@@ZachGatesHere Yes, very sad :(
17:30
Just remember that the triangle can be thought of as a half of a parallelogram of sidelengths A and B.
Thank you
how does that help :(
I attended high school in California and was never shown most of this. It was a real eye-opener. Great video!!
Golden rule. I love it. I’m a humble carpenter. I hang doors and put shelves up. I use a very sketchy and basic knowledge that Pythagoras exists to position things.
When challenged I mention Pythagoras. I urge anybody to learn more about this. It makes the world more understandable.
personally, never seen either twisted square proof. but I think I am one of these children you speak of :)
So, are you going to ask for your money back ? :)
Nice one! I have also done the versions of AM-GM and Cauchy Schwarz like you did here - love this diagram for those visual proofs! Used it to animate Priebe’s and Ramos’ sine of a sum too. Was gonna create a mashup of them to show they are all the same but here you’ve done it better. Thanks!
Cool!
I'd be interested to read your take on Zog from Betelgeuse animated satirical math lampoons.
Hi~!
@@SuperYoonHo 😀
@@MathVisualProofs 🖐🖐how's ur day goin' sir
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Very interesting. I wasn't taught either proof in school. Being an amateur math enthusiast, I actually did manage to figure out the algebraic proof for myself a while ago. But I've never seen the first proof before, very elegant and amazing. Thank you.
I consider myself very intellegent. I do excell at certain things but when I come to the Mathologer channel, I find out how much I truly do not know. I like how you explain things yet I have trouble absorbing it. I am in awe of your intellect but also I am in awe of the people who comment on your videos. That in our dumbed down society there are still people that fully understand what is hidden to most of us mere mortals is just amazing to me. I am drawn to these videos but much like a moth is drawn to a flame. I see and then I crash and burn! I am hoping for a breakthrough moment when I shout Eureka! Keep up the good work and I will keep coming back and try to grasp it all!
I had seen the twisted squares in a textbook before, but I didn't connect them to the proof of the theorem until I received a book specifically on the Pythagorean Theorem called Hidden Harmonies. It had a full chapter of different proofs, half of which led back to the twisted squares. I didn't quite understand the rest of the book at the time, but it really is quite fun to see the history of humans proving the same thing dozens of different ways.
Last puzzle: The area of the middle square is 0.2, since you can divide the big square of area 1 in five equal squares of the desired area. 1/5=0.2
Don't you mean 0.2 of the big square. According to the given scale that's area 1 unit^2
Thanks for another fantastic video! I really liked the first proof in the introduction because during school I only learned the second algebraic proof. It was neat how you could prove identities about sin, AM-GM, etc using these diagrams!
Thank you very much :)
I first became familiar with Pythagoras theorem while surveying field sizes with my father ( a crop farmer in lower Michigan, USA). I was privileged as a seven-year-old boy to be the hold-down boy for the end of the rod-chain. I would look Dad square in the eyes as he circled round me, ending with kneeling with the rod stretched tightly across the ground, and Dad pounding a peg in at my next point. We measured many fields, but Dad had five sons, so the privilege got spread around. I always loved this time, because it was both intimate with my father, and mathematically practical--my teachers were surprised in grade school how well I did with geometry, almost all aspects which I experienced with practical applications on the farm with my father. He's now 96, and we still write to one another often, though now it is his love of philately that drives the mail truck!
Thank you so much what a great video
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.
That's it :)
You are really the best educator I have ever encountered and make supreme use of this medium. I have an advanced degree in chemical engineering and have worked with r&d at three Fortune 500 companies, so I have encountered some fantastic people until now, yet you are the best.
Glad you think so :)
Look at the 👕 Pythagoras and Einstein fighting for C^2 😂😂
So, according to Einstein, e = mc^2, so that e/m = c^2.
But according to Pythagoras, a^2 + b^2 = c^2.
Therefore, e/m = a^2 + b^2, so that e = m(a^2 + b^2).
😜🤪😝😛
I don't remember doing anything in primary school involving indices including ^2 for 'squared', we did areas of shapes by counting squares on squared paper, nothing about algebra either. It was all new in secondary school. The first year there (ages 11-12) was basically doing catch up for what we should have/could have all learnt in primary school, but without a 'national curriculum', each junior school had probably taught different things and missed some things out. I think things have probably improved since then with maths teaching, but some (10-12 year olds) still seem to struggle with the basics - because of not learning times tables, or because new maths concepts are not introduced in a simple way, avoiding using complicated language.
Same in the Netherlands, sadly. No equations, no Pythagoras or pi, not even long division. Arithmetic education got wrecked here since the 80's.
Absolutely beautiful visuals as always. Please keep it up!
Glad you enjoyed the video :)
Your laugh absolutely makes my day! The sound of the joy of learning and knowledge.
At 17:37: an A-A-A equilateral triangle clearly has area A²F and nestles into the corner of the larger A-B-? 60 degree triangle. So notably, the two have the same height. Now scale our A-A-A triangle horizontally (i.e., along the base, not the height) by B/A. This new triangle has area A²FB/A = ABF. It's not the same size as our large triangle, but it does have the same base (A*B/A = B) and the height wasn't changed by our horizontal scaling. Any two triangles with the same base and height have the same area, hence the A-B-? 60-degree triangle has area ABF, QED.
Great video!
I was always terrible at math and great at science and now since I'm 40 years old I'm trying to learn all sorts of things I love your channel you explain things nicely and calmly 😊
If only we had teachers like you. Maths would be a subject way more people can enjoy.
Wonderful video as always! Although I can’t help but mention the Father of Geometry himself whenever I see the general theorem shown at 15:06. Euclid gave proofs both of the standard Pythagorean Theorem (Proposition 47 of his first book), and of the more universal similar figure theorem (Proposition 31 of his sixth book). Very interestingly, the more universal theorem in fact does not depend upon the Pythagorean Theorem, as it is proved independently. This suggests that somehow the more universal proof is in a way the cause of the already universal Pythagorean Theorem. All in all, these are wonderful things to marvel at!
Have you seen an animation of Euclid's proof. If not there is one on the Cut-the-knot site (just google Cut-the-knot Pythagoras)
36:34 Challenge accepted (Spoils the answer)
Consider that the small right angle triangle at the top right is similar to the same triangle but including the trapezium below it. The ratio of sides is 0.5:1=1:2 (by considering the ratio of hypotenuses). Therefore, big triangle area = 4×small triangle area. White area = 16×small triangle area then.
Now, add the small triangle at the bottom and you get a right triangle with sides 0.5 and 1, so area = ½×0.5×1=0.25. It also happens that's the area of 5 small triangles.
5∆=0.25
∆=0.05
16∆=0.8
Hence, area of the square is 1-0.8=0.2
Hope my proof is pretty
My proof:
let x be the length of the sides of the blue square.
The big square has area 1.
The two big triangles on the right and left of the blue square have area 2(½½1)=½.
The two small triangles at the top say have legs of length y and z then considering the diaganoal of the big triangle at the top we get z+x+y = √5/2. Hence z+y = √5/2 - x
The two trapezoids (trapeziums) below and above the blue square have area 2x(z+y)/2 = x(√5/2-x) = (√5/2)x - x^2
putting this together we get 1 = ½ + x^2 + (√5/2)x - x^2 so x =1/√5 and finally x^2 = 1/5
This is very well constructed as far as an answer with the proof goes! Than you for the hard work.
I'm getting a bit confused with this solution. I followed a similar method to @Prometheus. To summarise there are 16 small triangles that make up the white area and four of those same small triangles which make up the area of the smaller square in question. The area of each small triangle is 0.05 so 4*0.05 = 0.2 = 1/5 = the area of the blue square. So far all is good.
However if we then reverse engineer this problem to calculate the sides of the smaller triangle I get a bit confused. The hypotenuse of the smaller triangle is 1/2 (the long side of it spans 1/2 the length of the original square). If I refer to the other two sides of this small triangle as y and z where y is the longer side and z is the shorter side (and z = 1/2y), then the inner square that we were finding the area of should be y*y? So y^2 should be the area of the inside square = 1/5 = 0.2.
That means that the shorter side of the smaller triangle z should be 1/2y and its square z^2 should be 1/4y^2 which is I think 1/20 (0.05).
But then if we take Pythagoras Theorem that the square of the hypotenuse = sum of the square of the other two sides then y^2 +z^2 should be = 1/2 (because the hypotenuse is half the length of the original square).
But 1/5 +1/20 = 1/4 not 1/2. I'm obviously getting something wrong here. I'd appreciate any help in resolving this.
I'm a math teacher in Germany - I teach both of the proofs shown in the intro in my classes.
The first proof is usually discovered by the students themselves - i bring cardboard triangles to class and let the students re-arrange those themselves until they discover the proof.
Can be done even by students who are usually not that strong regarding mathematics.
For the riddle at the end: instead of rearranging triangles I came up with algebra and the cartesian coordinate system. You get the corner points of the blue area with solving linear equations like -1/2x+1=2x --> x=2/5 y=4/5 and so on. You get the difference by subracting two of these points from another and then you have data to work with Pythagoras. The width of the square is then sqrt(5)/5 and its area 1/5. ^^ Spectacular video by the way. A++
P.S. after that I came to the conclusion that the areas of the smaller triangles are 1/20 and therefore fit 4 times into the blue square. So all the trapezes together make up 3/5.
I am a math uni student and I did not know like half of these proofs... This was amazing. The sin(a+b) formula in particular is just so beautiful.
I absolutely agree with you, it is a crime that schools don't introduce proofs. Even the most simplistic proofs, like the ones you showed.
The only time they are put into my curriculum is if you take the hardest math course, which only 5% of the cohort takes. 95% of students aren't even introduced to proofs.
As far as I am concerned there is no justification for calling a subject :mathematics" if you don't show any of this
@@Mathologer If pupils just learn ways to shove numbers around there is nothing that elevates thier mathematical abilities above those of a computer, which will always be faster than them.
What they miss out on is learning to strategically and creatively think about numbers and the world those might describe.
@@jannikheidemann3805 Exactly. Following a given algorithm is calculation, not mathematics, and reducing maths to caulculation defies the very purpose of mathematics, which is essentially an ever-expanding toolkit for solving problems.
Geometry was all proofs.
Proving the visually obvious. Seemed to be a grand waste of time and effort.
Spoiler for anyone interested in the square within a square puzzle : ua-cam.com/video/5VaQVXECaVM/v-deo.html
Always great to wake up to a new Mathologer video!
Very cool.
My favourite Pythagorean triple is 20, 21, 29 because it so close to an isosceles triangle and thus if you actually make it physically out of wood or meccano with inherent measurement errors, it will be more accurately a right angle than say 5,12,13 would be, subject to the same measurement errors.
How about 696, 697, 985?
@@colinpountney333 Nice but I haven't a hope of memorising that.
@@donaldasayers The 96 97 98 bit is memorable (I think). But you could just go for 7 7 10 to an engineering tolerance.
@@colinpountney333 (5, 5, 7) would appear to be ever so slightly better engineering-wise than (7, 7, 10), but (12, 12, 17), (17, 17, 24), and 29, 29, 41) are seemingly progressively ever better!
I always love your videos Mathologer, going to grab a laptop to watch this on full screen and yeah, I'm one of those who never got any proofs.
I've missed the "Ready to be amazed?"
Your videos are perfect, when you think you can raise an objection you address the issue immediately and even when I think know where you are going to 2 or 3 steps in advance you frequently surprise me. I like that.
Glad that this video worked so well for you :)
Great video, as always!
I think that, at 28:43, the AM-GM inequality for 3 numbers should really be
(ABC)^1/3 =< (A+B+C)/3
Correct :)
5:18 is the first proof I ever saw (or that I remember). For some reason I think it was featured in the “Cosmos” book but I have it in storage and I can’t check right now; I’m probably misremembering.
Interesting, so which one of all the proofs in this video do you like best?
@@Mathologer proof #1 at 2:00 is perhaps the nicest of the Pythagoras proof presented. But overall the sin(a+b) proof at 30:30 is the one that really made me applaud with giddiness, what a delightful approach!
Art as the inspiration for math, now that is something!
Thank you so much for making videos and sharing! It is so beautiful!
You are so welcome! :)
3:41 1046-256 BCE
Look into Anatoly Fomenko's Empirico-Statistical Analyses regarding Isaac Newton's and other chronologists' timelines.
It's more plausible that Chinese Imperialists added fifty-nine years to Li Qingyun's life, thus extending their chronology and thereby the longevity of the empire's mandate.
He claimed to be 197 when he died. They said he was 256 and that he lied about his ago pretending to be younger. He was a mountain herbalist. The imperialists were autocrats, totalitarians. Fomenko's New Chronology holds that this type of invented history is more common than not, that Pharaoh ruled Egypt through the 1700s, that 2022 CE (7530 Slavic Aryan Vedic Calendar) is 869 AD, and that The Great Wall of China is still being built.
Or maybe you had a typo, and it was 146-256 BCE? I didn't look it up yet, I was just reminded of all the above.
Thank you for excellent reference!
1046-256 BCE People just kept adding to it over a long period of time. I had heard of Fomenko's New Chronology. Interesting but in the end a crackpot theory I'd say :( Having said that I am a fan of his mathematics and his maths illustrations :)
I applied at my school to get my money back, and now they charged a processing fee.
:)
I remember trying out the areas of equilateral triangles and semicircles to see if Pythagoras' Theorem still held true. I became fascinated with mathematics and was so proud to tell my teacher what I had found out! It gave me so much confidence in the subject, it became my firm favourite.
That's great :)
Based on your T-shirt the following proof of the Pythagorean theorem came to my mind:
Consider a right angle triangle with an hypotenuse 1, and call its area m. Scale it by a factor of c. Then the area of the new triangle is
E = m*c².
Split it into two triangles by its altitude (which is not any of its sides). These triangles will be right angle as well, and their hypotenuse-which length I will call a, and b-are the short sides of the triangle with area E. If you check their angles you can see that these two new triangle are also similar to the original triangle, so we can get them by scaling it by a factor of a and b, respectively, and therefore their areas are
F = m*a, and
G = m*b².
Finally, consider the fact that E = F + G,
substitute: m*c² = m*a² + m*b²,
and divide by m (what is strictly positive, because it was an area of a triangle).
This proof was told me by one of my maths teachers as a handy proof for matura exam, and he referred it as "Einstein's proof". As I typed it in, I recalled that probably you had a short animation on this very same idea, maybe with some other notation. (Indeed: ua-cam.com/video/r4gOlttnJ_E/v-deo.html But the choosing of the letters like this was my idea :) Even though I know, probably someone else has done it long before me…)
About the question at 3:00, both proofs were shown to me in high school, and I don't really remember which came first. But I remember that I was told to always emphasize that the quadrilateral in the middle is indeed a square because of the sum of the angles of a triangle is π.
I had not seen this one before. Thank you very much for sharing :)
For the last puzzle, the small square's area is 1/5.
The big square is made of 4 triangles, 4 trapeziums, and the small square.
If you fit the triangles to the trapeziums, you get 4 squares sharing a side with the small square, therefore they all have the same area.
The large square with an area of 1 divides into 5 equally sized small squares, so each small square's area is 1/5
I can see a couple other ways to solve it, and there's probably more that I haven't thought of. I noticed the small triangles are similar to the large triangles, that could be useful.
That is a clever way to solve it (and probably one Burkard would be thrilled to animate)! I can confirm though that the small triangles and the original triangles are similar (and that the fact is useful) as that is how I reached the same result of 1/5 for the area of the small square.
I grew frustrated quickly with trying to calculate each line segments length and just did that but, as noted earlier, the only way to check that is to precisely cut out a trapezium and smal triangle and see if they perfectly overlap the small square, whereas the proof is fool "proof." This is why I and so many other people have trouble with plane geometry; such is logic.
Prosecutor: I can't believe they acquitted someone so OBVIOUSLY guilty!
Judge: Of course he's guilty and of course that's obvious--but you failed to PROVE it, counselor.
@@herbpowell343Sorry, I don't understand what you're trying to say? There's no need to calculate any numbers or do any perfect cutting, just the given values are easily enough to prove it.
It's given that the edges of the large square are each length 1, and the outside edge of each triangle is 1/2. This means the outside edge of each trapezium is also 1/2. If you match each triangle to each trapezium along that edge, forming a quadrilateral, it must be a rectangle since all the corners are right angles (two given in the trapezium, a third from the angles that form each corner of the large square, and if three are equal then the last must be as well), and the quadrilateral's length and width are both equal to the height of the trapezium, which is equal to the length of each side of the central square. Due to having all right angles and equal sides, each quadrilateral must be a square, and since they all share sides with the central square, all of the squares' areas are equal. Since the total area is 1 and is evenly split into five, the area of each little square must be 1/5.
I do like that quote at the end. Is it an analogy you came up with, or a specific reference to something? Either way, it certainly applies to maths in general quite well.
Ty
I love to watch your videos on an a Sunday afternoon. Great stuff. Oh and the Science Centre in Ontario Canada had a great way to show the
Pythagorean Theorem. The three three triangles were watertight in a display and able to rotate. As they did the coloured water went from C to the smaller A and B filling them up, rotating again to fill C. Lovely.
Very good video as usual. Thank you mathologer!
In elementary my math teacher told the class circles had no area 😭😭😭
I mean even if their areas are incalculable you can just do the Archimedes thing and bound the area between a polygon inside the circle and another outside? (Admittedly Archimedes was trying to work out the exact length of the circumference but I'd say my point still stands)
Was she trying to say that a circle was just the bounds? That's a mean trick anyway if that's what she meant.
I saw the first twisted square proof first, and it was from either you or numberphile lol. Am in America for context
You are truly a great science communicator - a wonderful teacher. I humbly thank you
its funny how u should say science communicator instead of math communicator, but Pythagoras tends to do that
in the last couple of years watching people like yourself - I was at school hundreds of years ago - and yes, beautiful
I was never taught the proof.
:(
1:00 or so. The way this animation explains it is mind blowing. As a kid I learned it and was like how did this come to be… these graphics are really helpful as usual.
I DID see a similar animation/explanation once before, but not as I was a kid and had all the questions about it, therefore I get excited to be able to visualize something I have burned into my skull for decades but didn’t completely understand it.
Prior to completing a calculus course taught via uncannily mundane and l laborious methods, I loved math. Watching this video felt like reuniting with a friend I hadn't seen since I was 15. Thank you!!
That's great! Mission accomplished :)
The solution to the final puzzle is 1/5.
Perhaps the most visually pleasing way is to rearrange the 4 small triangles so they each fit onto one of the trapeziums to form 4 identical squares.
We know that these are also identical to the middle square with unknown area because the ratio of original lengths create similar triangles.
The result is the large square of area 1 is separated into 5 smaller squares of equal area, one of which is the middle square.
I have never seen these proofs before. I am not the brightest bulb but I admire and appreciate mathematics very much. It is always so satisfying to take a concept and make it easy to understand or to amplify the idea in some way. There is a creativity and sophistication into seeing the relationships between the logic of the pure math and its applications.
Brilliant explanation, so simple and effective. The first one is more awesome.
Brilliant, brilliant, brilliant!
This was great! The bugs things was especially beautiful!
I know the second proof by heart, it has sentimental value for me. The video was absolutely beautiful!
Glad you enjoyed it!
lol I knew you were going to show that first diagram. At least subconsciously. Just before pressing play I got an incredible urge to draw/measure various such diagrams. Been doing a lot of playing around with right triangle geometry lately or triangles in general so it's right up my alley.
Never have seen this. Loved it!
Fascinating. :) The music suggestion is excellent as well. Thank you.
In my retirement I have been working occasionally as a substitute public school teacher. (Math, Science, Spanish, only subjects I know well). I like to show these proofs to the students and jokingly quote your "get your money back" comment.
One day I was talking with a math faculty member and told him of your channel and some of the things I like to share with the students to demonstrate that math is fun. He had never seen a proof of the Pythagorean Theorem , and in fact had believed it was an axiom.
So thanks for your channel, you do more good than you know.
Brilliant video as always! Thanks a lot, professor!
I have seen both versions in my coaching classes. The coaching class taught people for Regional Math Olympiads, kinda like qualifiers for Indian teams in IMO. The setup was in their entrance test material, and we had to prove PT using triangle congruence. They had also mentioned the visual proof.
Amazing. Thanks for sharing knowledge
I'd never seen either of the two proofs you showed in the beginning of the video. I remember a proof that was way more complicated and hard to understand. These are beautifully simple.