PETR'S MIRACLE: Why was it lost for 100 years? (Mathologer Masterclass)
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- Опубліковано 25 вер 2024
- Today’s topic is the Petr-Douglas-Neumann theorem. John Harnad told me about this amazing result a couple of weeks ago and I pretty much decided on the spot that this would be the next Mathologer video. I really had a lot of fun bringing this one to life, maybe too much fun :)
Very good Wiki page on the Petr-Douglas-Neumann theorem en.wikipedia.o...
Napoleon's theorem and the PDN theorem at Cut-the-knot www.cut-the-kn...
Petr's original paper dml.cz/handle/...
Worth a look for eigenpolygon decomposition and other application within geometry, etc.
forumgeom.fau....
forumgeom.fau....
www.researchga...
A fantastic app that does the eigenpolygon decomposition contributed by Steven De Keninck from the Computer Vision Group at the University of Amsterdam
enki.ws/ganja....
Check out Branko Grünbaum notes on "Modern Elementary Geometry" tinyurl.com/5a... and G.C Shephard's "Sequences of smoothed polygons" (paywalled) for how all this fits in the grand scheme of things.
John Harnad's UA-cam video on a complete proof of the Petr-Douglas-Neuman theorem. • The Petr-Douglas-Neuma...
Here is John's (slightly updated) write-up that goes with his video www.qedcat.com/...
If you like this Mathologer video thank John, he lobbied for it :) John's website: www.crm.umontre...
I put a couple of Geogebra Geometry and Mathematica apps that made for this video in this folder www.qedcat.com/...
Here is Geogebra Geometry online www.geogebra.o... (you can also download a standalone version for free)
Nice geogebra classic animation by Ron Vanden Burg
www.geogebra.o... He chooses a different but equivalent approach to Petr, Douglas and Neumann. He attaches regular n-gons instead of the ears. Instead of connecting the tips of the ears, he connects the centers of the added regular n-gons. And instead of using ears with different angles each round, he adds the regular n-gons where each round has a different distribution of vertices that get to each side of the edge. It has a slider to switch from triangle (n=3) to decagons (n=10) and there is a play button to run through the different stages. Here is an updated version www.geogebra.o...
Another really nice app by Christian Baune
www.programath...
Andrew put together the following quick “howto” for tools in geogebra - a tool is a sequence of construction steps that you can reuse (can speed up constructions tremendously) . • geogebra tools
The file he is working on in his video is here
www.geogebra.o...
A UA-cam video animating some instances of PDN
• PDN Teoremi Petr Dougl...
Eigenpolygon Decomposition of Polygons the Microsoft technical report by Pixar founder Alvy Ray Smith alvyray.com/Mem...
Alvy Ray's website alvyray.com
Nice applications:
In electrical engineering: en.m.wikipedia...
Check out the paragraph entitled "Intuition" for an explicit reference to Napoleons theorem.
Being able to find the center of mass of a polygon is another nice application in itself.
Nice remarks:
For a digon we are attaching 2-2=0 360/2-gons to arrive at ... the same digon ... which is automatically regular :)
Adding a 180-degree ear to a segment is the same as bisecting this segment. So the tip of the ear ends up in the middle of the segment.
Visualisation challenges: I sort of had it going in Mathematica just for 10-gons. An app that allows you to pick the vertices of a closed polygon on a canvas and then calculates the intermediate polygons. One problem with the intermediate polygons is that for acute angles the displacement is large and so that can quickly lead to the intermediate polygons growing too large for your canvas. Some rescaling is probably the way to go. Alternatively, since the end result is always the same no matter the order, it makes sense to apply angles in complementing pairs, jump out and in, and only show every second stage of the evolution. Maybe some app that allows to input a smooth curve and then allows to experiment with different polygon approximations to see whether we get some convergence. The decomposition into the special types is a great one to animate. If you’ve got Mathematica I’ve included what I got up to in the file directory I link to above.
Music: A tender heart by the David Roy Collective and Trickster by Ian Post (two slightly different versions)
T-shirt: Rock Paper Scissors Lizard Spock t-shirt (google it, lots of different versions)
Enjoy!
Burkard
This is just the discrete Fourier transform working in the background. Amazing video !
Indeed it is!
That was also my immediate thought after seeing that a polygon is the sum of some ideal polygons! I scrolled down to comment and what's the first thing I see?
Yeah! When we got to the sum of 5 pentagons I started thinking a lot about how it might relate to the Fourier series.
I like how the words "unit" and "Fourier" are interchangeable.
Just like "Lie" and "differentiable".
Are there any connections to a mirror view, something looks familiar?
Worth mentioning quadrilateral as a special case - we have 90 and 180 ears to add (180 are essentially midpoints). When we add 90 degrees first we get Van Aubel's theorem as a special case. When we build 180 ears first (i.e. taking midpoints, giving us a parallelogram) we get Thebault's theorem (which also follows from Van Aubel).
Exactly right :) The wiki page on the Petr-Douglas-Neumann theorem mentions all this and also features some nice diagrams. Worth checking out :)
Also, with the extended version of PDR (the one where we attach 270 and reach a point) we can attach 90 and then 270 reaching a quadrilateral degenerated to the pair of points (which are actually midpoints of the diagonals of the original quadrilateral).
@@hydra147147 I was just about asking about the clepsydra shape made of either a quadrilateral or (let me count in memory: one, two, three...), a pentagon (or a hexagon, etc.).
ThnX!
For each even n (you mention n=4), you have a round with 'flat ears' (180 degrees).
Instead of looking at the 'tips of the ears' for constructing a new n-gon, there is anothe way of looking at it:
in each round add regular n-gons to all sides and construct the new n-gon by connecting the midpoints of these added regular n-gons.
The difference in the rounds (the different degrees in the 'ear wording') is by aligning not two consecutive vertices of the regular n-gons, but such that k vertices ly on one side and n-2-k on the other side, for k=0, 1, ..., n-3.
So it is like every round of constructing a new n-gon is by adding regular n-gons 'sunken one vertex more than previous round'. (And indeed the rounds can be permuted.)
You can see this visualised in the Geogebra application found at Geogebra ... /classic/sdftbe2y.
So there are two equivalent formulations of the Petr-Douglas-Neumann theorem:
- using tips of triangles (ears) with different angles per round; or
- using centers of regular n-gons 'sinking' through the edges per round.
The intriguing idea is that Napoleon's theorem uses the second formulation and Van Auble's theorem uses the second formulation only for the 'first round'.
Why do you call them special cases? They work exactly the same way, don't they? As far as you count a 0º, 0º, 180º triangle as a triangle, of course.
Hell ... at around 23:50 I clearly understood (several times) the term 'the center of mess' until I realized that this has absolutely no mathematical meaning. But in that moment, it made perfect sense to me ... the point clearly was in the center of all that mess.
Center of mess, I like it :)
My intuition suggests that if plain figure have a center of mass, then each line passing through the center of mass divites the figure on two halves with same area.
That's an intuition, not watertight definition, simple counterexample is e. g. a banana.
@@Filipnalepa the banana has a concave shape. You might think that if you were dealing with a convex shape your intuition would have been correct. Unfortunately, it is not the case. However, any line that goes through the center of the mass of a shape, cuts it in two halves that have equal, but oppositely signed "first moments of area". You see, the area of a shape is the integral of (dA). The first moment is the integral of (x dA), where x is the location vector of the infinitesimal area element dA. You can imagine the area as the "zeroth moment" of area. Your intuition told you that the halves had equal zeroth moments, while the reality is they have equal but opposite first moments.
@@Filipnalepa For n>3, there are different kinds of 'center of mass'. There is the vertex center of mass, the edge center of mass and the area center of mass. In this video, Mathologer refers to the VERTEX center of mass. Your intuition looks a the AREA center of mass.
Center of mass = "Average position of the vertices". (Sum them, as position vectors, and divide by the total number.) In geometry, it is called the "centroid".
Today’s topic is the Petr-Douglas-Neumann theorem. John Harnad told me about this amazing result a couple of weeks ago and I pretty much decided on the spot that this would be the next Mathologer video. Seeing is believing. I really had a lot of fun bringing this one to life, maybe too much fun :) Sorry about the lack of Mathologer activity recently. Just survived an absolutely hellish first semester here at my uni in Australia. Hopefully I'll have a bit more time for Mathologer the rest of the year :)
The radii of the regular polygons derived with the PDN theorem can also be derived by treating the reference vertices as complex numbers, with origin at the vertex centroid, and then taking the discrete Fourier transform of the set of points. The modes derived represent the vectors of the radii of the regular polygons. Neumann uses this method to derive his proof of the theorem. In Neumann’s paper on this “Some Remarks on Polygons”, he references “symmetrical components” which is what this DFT method is called in power system theory, which is equivalent to the DFT. Neumann’s father was a power system engineer and gave him the idea to use this technique from power systems. In power systems we use the specific case of the triangle, and the radii of the Napoleon triangles provide a physical interpretation of different types of unbalance that occur in power systems. The radii of the Napoleon triangles also have a deep connection to the Steiner Ellipse which I discuss in one of the videos on my channel. The ratio of the radii of the outer and inner Napoleon triangles are equivalent to the “third flattening” value of the Steiner ellipse.
Hi, 5 minutes into the video and first thing I thought was an exercise in an old projective geometry book of mine that uses the false position method to solve the Cramer-Castillon problem.
Somehow this reminds me a lot of it. I'll see if tomorrow I can link the two somehow.
The geometric flower, beautiful.
Hi, Burkard. would the right name for a 10-gon not just be 'decagon'?? We also say pentagon, hexagon and octagon, after all.
@@nathank7569 Very interesting, especially what you say about Neumann, power system theory and the Steiner ellipse. I sort of know this stuff but will have a closer look now :)
I'm a finn born in 1960. When I got my groung schooling it was given that I need to learn germany as most of the books in the universities in Finland were written in germany. The time (1981) when I entered uni in Helsinki, Finland to study mathematics, all the books were written either in finnish or english.
My physics background led me to guess the name eigenpentagons just before you revealed the eigenpolygon name. Cool!
Makes sense that someone with a serious physics or engineering background would make the right kind of connections early on :)
Yes. Absolutely for me, too.
What a joy to see this topic covered with such clarity! I wanted to make my own video about it a few years ago, and I went so far as to create animations in Desmos and tease the topic at the end of my video on Napoleon's Theorem. I came across the idea of using DFT's to explain Petr's theorem in a paper by Gregoire Nicollier, "Some Theorems on Polygons with One-line Spectral Proofs" (Forum Geometricorum Volume 15 (2015) 267-273). It was a lot of fun learning about it and trying to animate it. I just couldn't quite simplify it enough to make it accessible to my audience. So, thanks for the gift of seeing "my video" realized by a real pro!
Glad you liked this video. Definitely also had a lot of fun putting this one together :)
The fact that such high-quality content is freely available here on the Internet is really amazing. Thank you so much! This was beautiful.
Today is my birthday, this is not less than a gift, thanks!
Happy birthday :)
If it was my birthday, This would have been the best birthday gift I would ever get
Maybe I'm late but happy b-day!
I studied electrical engineering and at University we learned something called symmetrical components theorem, of wich we use the specific simpler case for triangles. In reality we're not taught it as a triangle, but rather, as the three components of our three phase electrical grid (be it current or voltage).
You see, in the electrical grid, the 3-phase system is not always symmetrical (with 3 phasors of the same length and 120° apart) because of asymmetrical loads and asymmetrical faults and such. So it would be very complicated to model and visualize it as a 3 phase system because of the symmetries. So instead, we model it as a sum of 3 symmetrical systems (analogous to those 5 symmetrical base pentagons). And seeing them as triangles would be like one equilateral triangle, one "mirrored" equilateral triangle and one "triangle" with all 3 vertices in the same location.
It's sad the in my university we didn't go into much depth with this theorem, we just saw a glimpse of it and were immediately learning the application. But it's really nice to now see a video about it explaining everything in detail in an easy to comprehend format.
Glad that what I do here clicked with you :)
I hadn't seen the theorem before. When you explained the phenomenon, I said to myself, 'wow, cool application of circulant matrices'. I didn't trouble to pause the video to try to work it through, but as soon as you showed the five components, is was "oh wow, that's just a Discrete Fourier Transform, summing roots of unity" and everything else was obvious in that perspective. (Seeing the title of Alvy Ray Smith's paper clinched it!) I'm curious whether that's the approach Petr, Douglas and Neumann _all_ took, or whether one or another of them had some more painful route to the conclusion. But not curious enough to try to plow through a paper written in Czech.
Are you curious enough to convert an image to text using Optical Character Recognition and then use Microsoft Word translate or Google translate from Czech to English?
Yeah, I also thought of Fourier when he talked about the sum of polygons. I thought this as an analogy, but apparently, this is really the same maths.
As you started explaining the "decomposition", I was reminded of fourier transforms. However, when you then mentioned that this in fact *is* a fourier transform, I was quite surprised. Not at all obvious that these two things really are the same (to me anyway). Fascinating!
The similar triangle thing:
if r1+r2+...+r5 = R (the decomposed red points as complex numbers add up to the red point in the starting pentagon) and b1+...+b5 = B (for the black points) you can just notice that the construction of any of these "ears" can be done by taking the vector from the r's to the b's (b_i - r_i) rotating and scaling it by some amount which is the same as multiplying by some complex number m, and then adding it back to the red position. So the coordinates of the new points is r_i + m(b_i - r_i), adding those up gives you r1+...+r5 + m(b1+...+b5 -r1-...-r5) = R+m(B-R) the same construction on the original pentagon.
Very good :)
It's really crazy that I just heard about "Napoleon's theorem" and looked for a deeper explanation, and here it is Mathologer posted one just 6 days ago!!! Love your work!!
Glad you liked this video. Also check out the description lots of extras :)
Greetings from Czech Republic and Thank You for educating me about the achievement of my fellow countryman I was not aware of.
Fun fact. Before the iron curtain came down I had to do my military service in the West German army. There I was part of a unit that specialised in radio surveillance of Czech and Russian military radio transmissions. Before I joined the army I knew Russian but they also had me take lessons in Czech. Forty something years later (mostly living in Australia) I don't remember much apart from the numbers and tongue twisters like Strč prst skrz krk though. That paper brought back memories :)
@@Mathologer Thank You for sharing this fact. It is probably getting even funnier, as I live really close to the border of Bavaria, like 15 kilometers from the town Furth im Wald and there is a hill next to it - Schwarzriegel - where, as per my knowledge, some of such surveillance was being perfomed during Cold War, it would have been a nice coincidence if You had served there :-).
@@Mathologer First of all, your Czech pronunciation is pretty good, and this is one of my favorite videos on your channel. Great job!
I have a similar story: I once interviewed a Norwegian guy for a job, and surprisingly, he spoke some Czech. Apparently, he was doing military service in a Norwegian interrogation unit assigned to potential Czech PoWs. I told him that there was an alternative universe meeting where he would do some interviewing/waterboarding on me, and he said no, only Americans do waterboarding. :)
Once again, thank you for your work. Great channel!
@@jiribay2706 I was part of a mobile unit based in Donauwörth :)
I believe a person in the Swiss Patent Office published a number of papers in 1905 that garnered some attention and may have elbowed out other interesting papers.
That's also what John said :)
Yeah, what was his name? Alvin Engels?
@themathhatter5290 no i think it was gilbert something
Lemme check the Nobel prizes... the high point of that guy's career was the photoelectric effect?? Sorry, not impressed.
@muskyoxes That established the existence of photons and was instrumental in the foundation of quantum mechanics. You know, that field of physics which says that God plays dice with the Universe.
It begs the question about just how much amazing maths is out there but has just been lost to time. Amazing video thanks Mathologer!
Always nice when you upload a new video!
I feel the same way ...
I see an immediate application of this theory. It could be used in power system engineering. The three-phase voltages and currents for any imbalanced system could be broken down into three balanced components. The first balanced component would represent positive rotational torque, the second component would represent a negative rotational torque, and the third component would represent a non rotational component. Since these components are themselves balanced we could call this idea Symmetrical Sequence Components.
You can take the transformation matrix shown in the video at 27:30 and create a 3x3 version of this to use for three-phase systems - but it would be good to change the Greek letter used for the rotational operator to the small letter alpha and then try and come up with a catchy name for this 3x3 transform. Could I suggest that we call it a Fortescue Transform and back date it to 1918.
Wow ... sincere thanks ... wasn't aware of this at all ... Great video, as usual ... Cheers
Wasn't aware of this myself until quite recently. Well I've known Napoleon's theorem for 50 years but not this generalisation. Mathematicians really have a lot to learn when it comes to promoting their results. That's why so many theorems do not get named after their original discoverers :)
I'm forever grateful for the visuals lol as always. Thanks so much for being such a great teacher. I'm not even into maths, but i love learning. So many answers to grown up questions can be found in these videos of yours
That's great :)
Since this video was published, I've added lots of extra information in the description. In particular, check out this fantastic app that does the "eigenpolygon" decomposition in a very stylish way contributed by Steven De Keninck from the Computer Vision Group at the University of Amsterdam enki.ws/ganja.js/examples/coffeeshop.html#AONlCLF1r&fullscreen
Also check out this animation contributed by Ron Vanden Burg. Instead of ears he attaches n-gons
www.geogebra.org/classic/sdftbe2y
Thanks @mathologer. And I hardly dare to tell that I created yet another update. I added buttons to use the 'ears' version, or use the 'regular polygon' version (or both). So you can compare the two equivalent methods.
Published at Geogebra id fyrrk28m.
I must say, I appreciate Steven's morphing of the polygon to the next round/cycle much. But I couldn't find a way to add that using Geogebra.
If math is a garden, you consistently show us its prettiest flowers and sweetest fruits.
Glad you think so :)
I’ve been studying linear algebra and have been really interested in its applications. When I got to the end and realized that he was going to use linear algebra to tie it all together I got sooo excited! Super cool video and I’m excited to delve deeper!
It's always beautiful how different fields of mathematics combine. Like geometry and linear algebra in this case.
Definitely made my day the first time I learned about this connection :)
@@Mathologer
I was a bit embarrassed about how long it took me to catch on. But when you showed the 'base' polygons, I thought 'Oh, it's like linear combinations and Eigenvectors!"
An application which immediately comes to mind is in creating a novel measure of ‘randomness’ for a set of points. Or at least to gain better intuition for those measures (or tests) of ‘randomness’ already existing which make use of the discrete Fourier transform. Great video, lots to consider! 🤗
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
Integration has always been some of the most interesting math to me, the process of paring down what makes a difficult problem unique, until you're left with something familiar and simple
I think your czech is quite good ;-) I have read the original paper in the link and although I have understood each word, your explanation in english was much clearer to me.
Fun fact. Before the iron curtain came down I had to do my military service in the West German army. There I was part of a unit that specialised in radio surveillance of Czech and Russian military radio transmissions. Before I joined the army I knew Russian but they also had me take lessons in Czech. Forty something years later I don't remember much apart from the numbers and tongue twisters like Strč prst skrz krk though. That paper brought back memories :)
@@MathologerNice coincidence; fortunately, I was only 10 when commies went away and I could enjoy my puberty in free and wild 90’s. The most difficult thing for everybody in the world including Slovakians is to correctly pronounce Ř, the “Put your finger through the neck” twister is piece of cake in comparison.
@@oldadajbych8123 Yeah, nowadays I think I can nail Ř well enough (perhaps 10% of perfection) but originally I just said RZ. Horrible, huh, but at least it produced a lame pun. A colleague of mine was a Czech-enthusiast (thanks to him for introducing me to Czech music) and his surname is Hirsimäki. So, I called him Jirzi Mäki and he liked it.
These videos always impress upon me how unimaginative and utterly unoriginal my mental workings are. Marvellous!
Mathologer love ❤ fav channel
I just love videos like this one. Starting out with some surprising "magic", continuing with eye-opening connections and leaving me with the fuzzy warm feeling of "understanding" how some parts of what I knew already connect to each other. Thanks. (I also like the final 'how to build this in GeoGebra" time lapse.)
That's great :)
19:32 Can't believe you missed using the color sequence Red, Black, Blue, Yellow, Pink. This would be a delightful easter egg to the Power Rangers fans.
I'll keep it in mind for next time I have to pick a sequence of colors :)
snooker fans are horrified
But what about the imaginary green Power Ranger? 🤨
@@perpetualbystander4516 So many color constraints to satisfy ... harder than the math(s) :)
@@MatthewGilliard 🤣You won the Internet for today.
Your videos are always an inspiration.
Thank you.
♥️
You can´t challenge me with visualisations. Am doing it all in my head!
Good luck with this visualising this one in your head :)
@@Mathologer , Danke für deine großartigen Videos...
Happy teacher, happy students, happy results. Thanks a lot. ❤❤❤❤❤.
28:17 I came up with a proof of complex numbers: if you know two points a, b of the triangle as complex numbers, to make the third one you just take the vector from A to B (A.K.A b-a), then to get side AC you rotate and scale it (A.K.A multiply by a complex number (named z)) , and then you add it to point A to get C. overall the function you get is (b-a)z+a=(1-z)a+zb. overall a linear function, so you get this linearity, that when you add up two vertices, the sum of the third vertices is the third vertex of the sum.
Yep, complex numbers are definitely the way to go :)
So,which witch is which?
I have proven in my comment that this mentioned theorem is not TRUE :) - by a non-trivial counterexample (trivial counterexamples are the degenerate ones when a sum adds up to a point or an edge). Similarity of triangles allows one to be a REFLECTION of the first one - like in your example the reflected triangle will be rotated by a (-z) instead of (z) and the sum will not add up to a similar one!
Regarding the "application" question: I'm currently visiting a class on complex analysis (/calculus) and finding a "complex circle" description is essentially the alpha and the omega of what we're doing so far. Integrating and deriving off of parametrised curves in the complex plane is almost trivial, and as such being able to find a closed curve of any polynomial shape is really useful.
It also becomes useful when trying to do calculations with rather complex shapes like e.g. the wing of an airplane, and being able to just transform it down into a simple shape (e.g. something with a circular cross section). Doing the maths on the simplified shape is usually the way people approach these calculations in computational mechanics, and as such it has a huge amount of "behind the scenes" real life application.
Interesting. Playing on a plane was fun, yet I wonder what would be the generalizations of this into multiple dimensions. I also wonder how it would change on a curved surface.
My guess for three dimensions is that since every vertex would have one extra degree of freedom, the resulting polygon in 3D would not be a regular polygon, but there would be some plane to which its 2D projection would be a regular polygon.
i suppose if you try the same thing on the surface of the Earth, you end up at The Pentagon.
Just started watching. It is amazing with the n-side-gons. Thanks for making this video!
Hope you enjoy the rest of it too!
Seeing this as a computer science person the most fun application I could think of is steganography. You could take a uniform tiling of the plane e.g. with regular hexagrams, and then slightly perturb them with small values in their other frequency components. The data would be hidden in these small values. Then you can go and tile a real floor with them and now you have a secret message in your floor.
I think it also has a lot of potential as a technique for lossy compression of polygonal meshes, by quantizing different elements of the frequency vector representation differently (e.g. dedicating more bits to the regular components, which would tend to produce lossy meshes that move slightly more toward regularity, which would be visually pleasing).
That's a nice application :)
Nice example!
2:47 THAT is such a true elementar point to the moment we are living in right now. The tools are simply exceeding exponentially our capabilities to design their (the tools) own structure. Like a production vacuum that pulls all the elements needed (imagination, creativity, actually material things, their arrangement, their singularity), and then more is produced over the previous things. Life momentum.
I've also just spent a couple of days programming some animations with the help of AI "copilots". So much fun and so empowering :)
Very interesting, thank you!
My pleasure!
Does anyone else see a cardioid forming as the number of sides increases? Id be interested in Mathologer revisiting this with a connection to the interior points of the Mandelbrot Set. Naturally that would also connect to the Logarithmic Map.
It also feels like there is a connection between the regular polygon at the end with the Dot Product and the regular star with the Wedge Product. Then the whole thing would describe the Geometric Product.
I have a habit every time you release a video, of taking a cold bath, then I turn my air conditioner on to remain cold to make the video more fun, and then I watch the video and try to do every challenge in it, just to make the best out of this video.
That's great :)
@@Mathologer why do you stay up so early
edit: tomorrow is a sunday so you will probably not go to work
bro thats a whole ritual, although ngl his videos are just a treat
@@TheMichaelmorad Actually it's Sunday morning in Australia and it's a tradition. For some reason, early on I decided that this is a good time to publish my videos and I've stuck with it over the years :)
Another amazing maths video, well paced and brilliant reveal. I had no idea that was going to connect and there are many more concepts that can be built from this one phenomenon.
Glad this one worked so well for you :)
My brain ain't braining
Even happens to me sometime :)
That's a good thing.
Don't get brained.
All n-polygons in a 2D plane, can be described by a sequence of n complex points.
Using a discrete Fourier transform they can be decomposed into their spatial frequencies (which are also complex). The inverse Fourier Transform of each of these n points, will result in a regular convex or star polygon.
I wish if I could give you 100 likes at once.... Lots of Love from Kerala. Thank you for making such wonderful maths videos ❤💚💙
In case anyone needed to know (although it's obvious to anyone who's seen it), the matrix at 27:45 is the discrete fourier transform. In general it could be any size, and the roots of unity need not be complex roots of unity - they could come from any field, or similar structure with the right nice properties.
Also, any results about adding ears conserving the sum are a simple result of treating any directed edges as vectors. Everything is a linear combination, so it's no suprise that everything has the linearity property.
15:00 Also; adding the 10th multiple-angled ears (in the case of the 10-gon; 5th multiple-angled ears, in the case of the pentagon), a.k.a. 360°-ears, a.k.a. line segments, to the point, will result, in another regular 10-gon (or pentagon, or n-gon). Obvious, I know. I just thought I’d point that out, lol. 😅
Ah, another master class by Mathologer. I recreated my own Geogebra project of this "screwed-up boring pentagon" on my phone. I found out that even if all vertices of the boring pentagon are collinear, the end result is still the same as the one you explained. And, one of the sides of the regular pentagon produced is parallel to the line of the collinear vetrices. I haven't tested the regular pentagram one. Once again, thank you, and have a good night sleep.
Glad you were inspired to try this yourself :)
Thank you,@@Mathologer. You inspire me a lot. Please wish my lovely daughter a success for competing in the Indonesian Math Olympiad for Elementary Schools on the provincial level. Thank you again.
We summoning Satan with this one🗣️🗣️ 🔥🔥💯💯
Let's see what happens.
I think you got your angles and angels mixed up again there, Father Ted.
Originally,Satan was not associated with the 5-point star; only Pythagoras. Wo sind sie Doktor Faust?
The whole idea of "summoning" satan with pentagrams/pentacles comes from horror movies heavily misrepresentating thing.
First, the pentacle/pentagram, if pointed up, is a protection symbol, and has been used by many religions, including Christianity, Pagans, and others.
Second, if pointed down, the symbol is linked to Satanist, which doesn't really have the right name since we don't even believe in Satan, and most Satanist are basically atheists with a few more rules (which are mostly common sence, ex. dont hurt kids; dont hurt innocents; if someone invites you, respect them; and so on) and the few that do believe in an entity believe in either Lucifer (which is NOT the one from the Bible) or Baphomet, which represents balance as they are both female and male, light and dark, good and bad, and so on.
So, occultists think that pentagrams are magical. I'm more impressed by how math-gical they are.
You've just asked (24:27) what happens if you apply some ears repeatedly, in a combination of ears. Looking at the "eigenpolygons", I notice they're not exactly axis-aligned, although three of them are close and the other is almost diagonal-aligned for one of its points, but I'm guessing that's a distraction and what we're doing is, for an original p-gon with corners represented by complex numbers z(n) for n in p = {0, ..., p-1}, a decomposition of form z(n) = sum(j in p: q(j)*cycle(j*n/p)) where cycle(t) = exp(2*pi*i*t) is the unit-period traversal of the unit circle; q(0) is the centroid (cycle(0) = 1 so j = 0 just contributes q(0) to the sum), the phase and scale of other q(j) configure the orientation of the p-gon traversed j steps at a time. The real fun must be in proving that, for any z, there is a q that makes this work; but, give or take normalisation (my guess is 1/p), I'm guessing q(j) = sum(j in p: z(n)*cycle(j*n/p))/p will do the job. Then application of each of the "ears" is linear in the two end-points of the edge to which it's applying the ears, so maps onto doing the same to each cycle(j*n/p) polygon contributing to the sum. The effect on these regular polygons is a simple scaling, scaling each q(j) by a j-dependent (complex) factor that's zero when the ear is either j/p or 1 -j/p turns (depending on the direction of our traversal, I suspect). If you apply k/p turn ears for each k from 1 through p-1 you'll thus annul all q(j) except q(0) and be left with the centroid. Repeated application of a given "ear" angle thus just repeats that scaling of each q(j) by the same j-dependent factor, so doesn't annihilate any q(j) that the first application didn't. So if you apply k/p turn ears for all but one of the k from 1 through p-1, you'll get a regular p-gon; and repeating some of those ears will only change the scaling. Now to resume watching and see if I'm wrong ...
Interesting, I noticed years ago that if you free hand draw the best pentagram that you can manage and keep connecting the points of the pentagon in its middle into another pentagram and keep doing this you will quickly end up with horribly crooked pentagrams. I hypothesized (🧐) that what was happening was basically an amplification of whatever small inaccuracies were already in the pentagram that I started with and that if I had drawn a perfect pentagram the inner pentagrams would also be perfect forever.
There is actually a whole pile of papers about doing exactly what you did there a couple of years ago :) As far as I remember it all started with E. R. Berlekamp, E. N. Gilbert, and F. W. Sinden, A polygon problem, American mathematical MONTHLY, vol. 72 (1965) 233-241. As far as I can remember the main result is that if you start with a convex polygon, up to scaling, things converge to another convex polygon all of whose vertices are on an ellipse :)
@@Mathologer danke schön
I did it in the other way, by connecting the corners with lines, and it _still_ made crappy pentagrams!
@@asheep7797 :)
The long awaited Mathologer video, finally!!!
So it’s like a Fourier analysis of a wave?
Exactly! :)
One of the best visual and oral explanation of ANYTHING I have ever seen
I'm kind of confused at the 180 degree step for the 10-gon. Shouldn't it do nothing since the triangle with 180 degree would be a line?
Well spotted. We just bisect the sides of the last polygon in a 180 degree step :)
@@Mathologer Oh right. With that pointed out I now noticed that the highlighted colour for the angle matches the input figure not the result of the step hence the confusion on my part. Thanks for the explanation!
This happens at the fourth step of the octagon also.
A good application for Petr's miracle would be the plot of a crime drama that involves a random pentagon to start, places where crimes were done, with the center point and vertices changing with each application of this theorem, and it takes a math whiz of a detective to figure it out and predict where the final crimes will take place. Something like that.
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.
@@Mathologer Very extensively detailed! Change and adapt some things to make it look like your own idea and you can submit it to a Hollywood producer.😎
Mathologer bingo:
“This was shown to me by a friend of mine”
“… that not even most experts are aware of!”
“Why was this lost/forgotten/undiscovered for X years?” Or “why is this not taught” on the thumbnail
Banter with Marty
Shirts
Also, "....can you guess??..I'm pretty sure some you have already figured it out by now" * meanwhile my brain has crickets buzzing in it *
Also, often leaves Riemann Hypothesis level problems for the viewer to figure out at the end.
(I'm mostly kidding, I really like his videos and await for them to be uploaded ...)
The last few minutes of music were just fantastic
Trickster by Ian Post, two slightly different versions :)
Thanks for your videos.
19:00 I was starting at this plot, after you mentioned the paper on eigenpolygons, when it suddenly hit me
that's a discrete Fourier transform!!
of course! the degenerate pentagon point is the constant term, and each of the others is a constant (size and angle) times the phase e^2πik/5
and adding the ears is a linear operation, so it plays nicely with the DFT, and had the property that it knocks out the pentagon with phase/angle opposite to the ear!
That's it, glad you had this AHA moment :)
Me: "Wait, am I missing something? It's not obvious to me that the "slap on ears" operation distributes over the components. Am I being dumb?"
Video, almost immediately after I have that thought: "I should say something about that."
:)
He is remarkably good at that sort of thing.
Mathologer: "What are some natural questions to ask at this point?"
Me: "I mean, 'Why the hell does this work?' ... but probably without the expletive, I suppose. And you're probably looking for something more specific."
Mathologer: "Why the hell does this work?"
Me: "Fair."
This is beautiful and elegant. Thank you!
Long waited 💪
This one video is.... absolutely amazing....
Great nice job !
This is very much nit-picking, but clickbaity tiles like _"Why was X lost/forgotten/ignored/missed for Y years?"_ does the channel a disservice. Like question in headlines, this makes it look like poor journalism (despite it being the opposite). Furthermore, the interesting thing that the video focusses on is the maths itself, not the sociology around it, which makes the title somewhat inaccurate. Again, not a major issue, but such great educational content deserves better than a gutter-level, click bait title - no matter what a UA-cam brand consultant might say.
Unfortunately the algorithm is such that titles like this are required to show in the YT algorithm in the first place.
Well, one of the reasons why the original paper was forgotten for 100 years was exactly because it was lacking a catchy title, no neon pictures, etc. In fact, this is one of the main reasons why so many results in maths are not named after their discoverers. The view from the high horse is great, but ....
SCATHING COMMENT DISGRACES SHADY MATH CHANNEL
You are right, it is very much nit-picking
It's mathologer, he has a reputation for good videos and personally it worked on me since the title was interesting along with the thumbnail
thank you this is a great visualization of Fourier transform and has enlightened me
Tenagon ? I prefer Decagon. Maybe because I'm Greek ?
10-gon. Well, it's got a gon in it. I should have used a 17-gon to give you guys the opportunity to point out that a 17-gon is REALLY a heptadecagon :) Or what about a 93-gon? As any primary school kid will be able to tell you this is a ... enneacontatriagon :) But, it gets better 666-gon = hexakosiohexekontahexagon for all of you who are suffering from hexakosioihexekontahexaphobia. Here is a nice easy one 1000000-gon that's a megagon. Sounds like a transformer....
@@Mathologer Maestro!! If primary school kids in Australia play with enneacontatriagons then they're in another level , indeed. Megagon is very easy . It's " Ecatomiriogon". "Mega" is Greek too and means BIG , by the way. I would very much enjoy a visual presentation of you , of Gauss proof of Decaheptagon (Heptadecagon is wrong). OK ... 17-gon. Many respects from Greece , Maestro. We are watching you (lol).
I did one video on impossible constructions. Did you already watch that one. I did not cover Gauss's contributions but it's all the same maths.
I am going to go out on a limb here and say that this is your most amazing presentation to date.
Glad you think so. Just wondering how many Mathologer videos have you watched ? :)
@@Mathologer I have a Ph.D. in Mathematics, but I can't count that high. :)
@@goboy6882 Meaning, you can NO LONGER count that high (I'm just an amateur mathematician, but I've said that about myself sometimes).
This seems like the geometric equivalent of the fourier series, if you ask me
You are on the right track! :)
I love your math videos ❤ Lots of Love 😊❤
The intersection may be in these vids:
Why can't you multiply vectors? (or how Algebraic Geometry told me to stop worrying)
ua-cam.com/video/htYh-Tq7ZBI/v-deo.html&ab_channel=FreyaHolm%C3%A9r
Smooth interpolation in one dimension
ua-cam.com/video/vD5g8aVscUI/v-deo.html&ab_channel=EpsilonDelta
Be awesome, math people
5 minutes gang
What fo you mean? 🤔
Mean I was here 5 minutes after it uploaded
28:30 consider your two original triangles A and B. Since they are similar, one is just a rotated/scaled version of the other, a.k.a. its points have been multiplied by a single complex number. We can write this as A=cB, where c is the complex scale factor. Now when we add the two together we can simply substitute: A+B=cB+B=(c+1)B. The new triangle can be expressed as triangle B multiplied by a single complex number, and thus it is a rotated/scaled version of triangle B. This proves that it is similar to triangle B!
Nice and simple🎉
If they were rotated/scaled then YES - but they only have SIMILAR characteristics - so one can be reflected,rotated and scaled vs another and then it doesnt hold ;)
@@TymexComputing reflection is a transformation which preserves dimensions, so that's only a tiny (but admittedly necessary) addition to the original statement
Nice, I intuitively knew that you were doing an IFT but seeing all the parallels and the path you took to get there was really informative
Glad you enjoyed it!
@@Mathologer As always :)
Me too! Every time I do an IFT my brain goes KUL! It has to be the Bose's effect on TGM...
Absolutely incredible
Amazing! One of the best mathologer videos ever
Glad you think so :)
This is so cool! I immediately paused the video and programmed it to see for myself. Thx for making my day
Please share your program with the rest of us :)
Super good thanks mathologer! I especially enjoyed the "making of", cool to see geogebra in action.
Try to make a geogebra app for Napoleon's theorem or for the quadrilateral case. Really a lot of fun :)
Another wonderful insight into the fascinating World of Prediction.
Love the T-shirts too, have several. They went over big when I was at SpaceX.
The geogebra animation was great!! Thank you for it, and please use again when you can!
If you have not done so yet, you should try putting one of these little Geogebra Geometry apps together yourself. A lot of fun :)
A question that immediately comes to mind, and I'm only 1/3 into the video so maybe the answer is coming: If I choose n points on a closed smooth curve, does the final n-gon approach a specific circle as n goes to infinity? To discard cleverly weird constructions, let's fix a smooth parameterization F(t) of the curve for 0 ≤ t < 1 and choose points F(k/n), 0 ≤ k < n. Does the limit circle exist? Does it depend on the parameterization, or on the curve only?
EDIT: Oh. Fourier. The circle would just be the constant term + base frequency term?
Not sure, I don't think anybody has nutted out the details yet :)
Hey, you're coming up to one million subscribers! Rightly so, a brilliant channel.
Will still be a while but I am not in a hurry :)
Those aren't pentagons; they are petr-gons! Absolutely fascinating subject and video!
:)
Me, seeing the visualisation: cool i should put that in geog-
Me at 2:36: oh you did it awesome
Some absolutely beautiful maths in this one. I rather love the concept that all n-sided shapes are the interferance pattern of n perfect shapes.
Glad you like it!
......a matter of simple matrix multiplication....
Yeah right, I'll just break out the calculator.
One of your best ones yet. 'Loger! I just wish I could keep up. 🏃♂
Thanks for the informative and joyful videos! Your enthusiasm is contagious.
I believe the asymmetry---in the final decagon size---demonstrated with your random decagon example could be used to communicate the concepts of 'left' and 'right' to an alien civilization, without relying on physics' favorite weak interaction. Assuming Petr's Theorem only works when 'ears' are drawn to the left for angles 180°, could we use this to establish a shared understanding of direction?
-Pick a random decagon.
-Choose a direction of motion.
-Choose an orientation ("Alien Right" or "Alien Left") so Petr's Algortihm produces a regular decagon.
-Repeat with the same Alien orientation but reversed direction.
-The "Alien orientation" with respect to the direction resulting in the larger regular decagon is equivalent to our "left."
Is this correct?
the scissors and rock NEED to switch places on your t-shirt, it would make the graph so much more beautiful
I actually have three versions of this t-shirt. Two of them get it right but I am mostly wearing the one in the video, mainly to prod people like yourself into springing into a little math action yourself :)
MIND BLOWN !!!
I tried using a direct proof with complex numbers, however I am grateful there exists such an elegant proof with the decomposition. The fact that all the intermediate steps can be completed in any order makes me think a direct proof would not be nice now.
This reminds me of Jansen's linkage planar leg mechanism but in reverse, where the angles of the central rotary axle can be adjusted minimally for maximum dynamic leverage in the extended polygon structures at the outer limb. The animations of the polygons twisting and moving in response to the different starting polygons, make this comparison particularly striking, see Theo Jansen's 'Strandbeesten' sculptures as an example
Interesting, hadn't looked at this for a while. I can see what you mean :)
I have never heard about this before. I think it’s beautiful!
Glad you think so :)
@@Mathologer It is, it kind of defies the concept of entropy. It brings order from chaos.
A thought i had about your program is you cna make this a fun form of encryption for maps, or locations and even messages.
The process is you make a line and pick a number like 18 points along that line. Then the distances of the line and how many numbers are chosen would determine the pattern and shape and size, this could in turn create a map. Further increase its complexity with how you right it down which could force you to alternate or counter or clockwise simply by the number orientation, laying a few roles that its always in sequential order, but how they are listed determines the other shape. TO further make a complex encryption you could take it further and put a curve into the system, ,plus you can denote the terms of measurements also. Interesting how distance effects location... so its a interesting way you could encrypt a very topographical map with recognizable features.
A test could be attempting to encode a location at the school in such away. It may require two orthogonal lines with perpendicular intersection, and one letters and the other numbers.
Excellent video! Thank you!
The Mathematica standardform i symbol at 26:57 brings back memories!