The 3-4-7 miracle. Why is this one not super famous?

[Music] welcome to another mathologer video recently while browsing maths on the web and working on a hardcore video i got sidetracked by a very puzzling animation have a look there are 12 points performing a strange dance in certain groups of three the points are waltzing in equilateral triangles at the same time in certain groups of four they're performing a square dance and if that's not enough they're also all tracing out a regular seven-pointed star pretty amazing i had a look around on the web and found a few other incarnations of the stance floating around always more or less viral and without any explanation of the maths at the core of this stance now i've not been able to establish who originally discovered this little miracle and who created this animation please tell me if you know would be nice to give proper credit this mathematical miracle was definitely new to me however it reminded me of some pretty mathematics that i was very familiar with and it did not take me very long to make sense of what is going on here so let me take this thing apart for you a great one for the festive season with that christmas star in the middle and it also meshes in very nicely with a little paradox that i always wanted to cover in a video anyway some very nice visual mattes ahead enjoy so when you have a close look you'll notice that all points are traveling in the counterclockwise direction around the star like this okay now let's have a look at the equilateral triangles a bit of a rolling action going on there don't you think points moving around on straight lines and in circles at the same time sounds familiar well it should if you are regular here on this channel remember the video on the nothing grinder or the one on epicycles and fourier series and a nice spirograph map at the core of these videos here's a quick reminder if a circle rolls inside a circle of twice the size then any point on the perimeter of the small circle traces out a diameter of the larger circle super pretty and super surprising then if you trace the paths of a couple of points on the perimeter of the circle as it is rolling you get this there rolling and rolling but at the same time moving along straight lines there one straight line a second straight line and lots of other straight lines super pretty that's the animation i showed you in the video on the nothing grinder intriguing and really very reminiscent of what we're dealing with in our mystery animation and i already mentioned the spirograph toy well when you go to the wikipedia entry on spirographs one of the first things you see is another very nice animation neat a spirograph can make a five-pointed star mathematically speaking the curves you get by tracing a point on a small circle that is rolling inside a larger circle are called hypertrochoids just in case you don't own a spirograph there's a pile of apps floating around on the web that allow you to play with these curves i also put a little mathematical app together which i've used to make most of the animations in the rest of this video now in essence there are two things you can play with in the setup you can vary the radius of the rolling circle and you can vary the distance of the traced orange point from the center of the rolling circle let's first scroll the radius of the rolling circle a little bit then the trace curve becomes a seven-pointed star okay okay that looks more like a flower with seven petals than a star but we can turn the weed into a star by increasing the distance of the trace point from the center of the rolling circle there great just what we want well almost unlike in our mystery animation at the very beginning this star does not consist of straight line segments right well actually on close inspection of the mystery animation it becomes clear that whoever put it together cheated a little bit by replacing a somewhat curved star like the one over there by a star consisting of straight line segments there have a close look as you can see not all the dancing points are centered on the lines sneaky anyway not a big deal our spirograph star is definitely starry enough for me okay let's roll very nice okay now very slowly so look let's add another orange point on the rolling circle at that tip of the star okay let's keep rolling now it's clear that both orange points will always be moving along the star right pretty magical isn't it let's add another orange point at this tip of the star okay great now we have an equilateral triangle all of whose vertices will always be moving along the star as the gray circle is rolling okay yeah very pretty okay okay that's one of the equilateral triangles that we're chasing getting there now let's count the number of times the radius is pointing to the right until it returns to the starting position okay let's go so there's one okay two three and four times great now the four positions of the disc at which the radius points right are these let's roll these this together ah pretty spectacular right can you see that the four orange points always form a square nice let's add the remaining points to the disks so two more points on each disk connect corresponding points but two more squares voila there are the three squares that we've been chasing lovely let's add the equilateral triangles corresponding to each disk okay there you go that's how you construct our mystery animation [Music] now even if you know how it's done this animation still looks very magical doesn't it but of course the spirograph can do much more than just draw the seven-pointed star and so there should be a lot more fun to be had in this respect for example there's a second kind of seven-pointed star this one here and here is what we get when we replicate the individual steps involved in making our animation [Music] here's another animation based on a ten pointed star which features rotating equilateral triangles [Music] it's sort of clear what's happening here right still with those different stars what determines what numbers and rotating shapes we wind up with again it's sort of clear but a large part of what's really going on here has to do with the so-called coin rotation paradox which i've been meaning to cover in a video for a long time so here we go the coin rotation paradox [Music] so the coin rotation paradox is about two identical coins imagine that one coin rolls around the other one without slipping throughout its journey how many times does the rolling coin rotate around its center what do you think okay quickly decide you think you've got it well let's check the number of rotations is simply the number of times george washington's head is pointing up here we go all right so there's one two it rotates two times did you get this right most people when they see this for the first time guess that it will be just one rotation sort of makes sense the coins are identical so it should be one right this counter intuitive phenomenon the coin rotation paradox even has its own wiki page i'll explain in a second but first let's build some more intuition by looking at two related puzzles this time the diameter of the rolling coin is exactly half that of the stationary coin again we roll it once around the larger curtain how many times does it rotate think about it for a second you think you've got it well let's see all right so once twice three times weird huh again hardly anybody gets this right on the first go okay one more this time the little coin rolls on the inside and actually you should know this one that's the case where a point on the perimeter traces a diameter of the larger circle anyway how many times does the little circle rotate around its center let's check [Music] just once surprised two rotations three rotations just one rotation so what's the rule and how can we make sense of these paradoxical numbers well here is my favorite explanation again one rotation two rotations why well let's first uncoil the red perimeter into a straight line segment okay unroll let's roll and count rotations there just one rotation now that's exactly what pretty much everybody would expect given that the coins have exactly the same circumference now coil the red perimeter back into its circular shape with the rolling coin attached to its end prettier there did you notice that while we coil the rolling circle rotates one more time one rotation plus one rotation equals two rotations nice let's look at this one more time uncoil all right roll one rotation coil another rotation two rotations in total perfect now second scenario again here we are dealing with three rotations okay let's just do it again so there's one there's two and there's three rotations where do those three rotations come from okay uncoil roll and count rotations we're predicting two rotations because the circumference of the large coin is two times that of the small coin right okay let's go so one two rotations perfect now just as before coiling adds another rotation there three rotations in total okay perfect third scenario here we're dealing with just one rotation okay let's just do it again it's going going going going going going going going going going going going going going one rotation same coins as just now but why was it three rotations when the little coin is rolling on the outside and only one when it's rolling on the inside well let's see uncoil and count rotations should be two again as before right okay let's check you know one two rotations yes but damn two plus one is still three not one right there plus one three what's going on well there was a difference to before did you notice so far all rotations have been counterclockwise but because the little circle is now rolling at the bottom of the line its rotations are in the clockwise and not in the counterclockwise direction as before let's watch again uncoil now roll check that the coin is really rotating twice clockwise okay here we go yeah twice clockwise now the counterclockwise coil and then two clockwise plus one counter-clockwise rotation from curling nets to one clockwise the universe makes sense again phew [Laughter] in this way you can make sense of any coin rolling puzzle of this type here's just one more complicated example here the radius of the small circle is two-thirds the radius of the larger circle this is the effect that the small circle has to roll two times around the larger circle for the arrow to be pointing up again how many rotations in this case well let's count okay so it's going and going and going and going and it's just one again okay and how do we justify this with our uncoil roll straight and coral calculus well here we go the little circle rolls two times around the larger one this means we have to uncoil the perimeter of the larger circle twice here we go so once and twice okay now for the smaller coin it's two-thirds the circumference of the larger one two-thirds times three equals two so this means the small circle will roll three times on the red line okay two three three rotations clockwise perfect now coil two counterclockwise one and two three clockwise together with two counterclockwise equals one clockwise tada fantastic challenge for you if you roll this little circle twice around the larger circle how many rotations should be an easy one leave your answers in the comments now sometimes people ask me how i make my animations well so that was the mathematical stuff but these coin rolling animations were actually done completely within keynote the apple counterpart to powerpoint don't know what was tricky to figure out the mattes in this video or to put together these animations really really black belt level animations okay to finish off let's use this method to explain some of the numbers that pop up in our animations so back to our original seven-pointed star to draw this particular star we start at some corner then we count one two three and draw in the first edge of our star now repeat one two three draw the next edge and so on because we're always counting to three in this construction our star polygon is usually labeled seven three this three then also translates into the rotating equilateral triangles right roll here we pick up the first new point let's keep rolling here we add another orange point and that's it from now on things repeat okay that explains where the equilateral triangles come from now we need the number of times the radius is pointing right until it returns to its starting position from what we've seen the rule here appears to be seven minus three is equal to four and four corners means squares let's quickly check that this rule really works with our other examples there that's the second seven-pointed star this one is labeled seven two two means we just get a rotating line segment ending in two points and seven minus two is five means that we're also getting rotating pentagons our ten pointed star has symbol 10 3 3 means triangles and 10 minus 3 equals 7 means that we also get rotating seven gons and i'll just say it now this always works as long as the two numbers we're dealing with are relatively prime so don't have any factors in common as in our examples now the question is why does this rule work and how do you prove it of course our coin rolling argument is the key in fact it's easy to check that the two numbers that label our star are one the number of clockwise rotations along the uncalled straight red line and two the number of counterclockwise coils so in this particular example the rolling circle will roll three times around the red circle and rotate seven times while rolling along the uncalled red perimeter and then our coin rotating paradox argument shows that 7 minus 3 equals 4 is really the number of rotations we're after beautiful how it all comes together in the end that was a lot of fun wasn't it here a few challenges for you what's the label of the pentagram star over there and what does the counterpart to our mystery animation look like for this star also here's a super challenge for all the programmers among you can you put together an app that on input of a star label automatically produces the corresponding animation if any of you rise to this challenge you'll enter in a draw for a copy of marty and my latest book which just appeared a couple of days ago oh and one more thing i already talked about this a little in the community section and on patreon just before christmas i expanded with some short animations of proofs and posted them on instagram and my second channel methodology 2. if you have not seen them yet please check them out there are 10 of them to finish off here is your last challenge what is the sum of the seven angles in our seven-pointed star over there one of the short videos i just mentioned helps with figuring out the answer and i'll also include it at the end of this video to wet your appetite for the other short videos okay and that's it for today back to the hardcore video that i was working on when i got sidetracked by that mystery animation [Music] do [Music] [Music] so [Music] so [Music] you

Channel: Mathologer

Views: 450,689

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Id: oEN0o9ZGmOM

Length: 23min 25sec (1405 seconds)

Published: Thu Dec 30 2021

Reddit Comments

[u/neuroblossom]

very interesting post, can you help me connect the dots (huehuehue) to holofractal?