Simple yet 5000 years missed ?

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Welcome to another Matholger video. When  it comes to mathematics, most people think   that all the really good and simple stuff has  already been discovered a long, long time ago.   Right? Triangles, Pythagoras, all this stuff has  been picked over for thousands of years. How can   anything truly nice and easy have gone unnoticed  until now? Well, not so! :) Let me tell you about   a wonderful little discovery about triangles that  was only made 10 years ago. Here we go. Start with   any triangle. Mark the midpoints of the sides.  And draw in the three medians of the triangle   that end in these midpoints. These guys here.  And looks are not deceiving. The three medians   always intersect in a point. And so the medians  chop the triangle into six smaller triangles,   usually all different in shape and size. This  fact has been known since time immemorial and   mathematicians have been staring at this diagram  for just as long. Now highlight two neighbouring   triangles, those two. Insert a hinge right there  at that midpoint. Fold up. Nice. Let’s do the same   with these two triangles here. Actually, wait,  before I fold things up a second time, can you   already guess where all this is going? Yes? No?  Well, if we fold things up now we get EXACTLY the   same triangle as before. Have a look. And now you  can definitely guess what comes next, right? Yes,   put in a hinge, fold up and you get the same new  triangle a third time. Tada, and that’s the new   discovery. Very simple and very beautiful, isn’t  it? Well, there is a bit more. Let’s call the new   triangle the folded triangle. The common side of  the red and blue up there, that’s a median of the   folded triangle. Right? Pretty obvious. And, there  is the second median. And the third. Okay, now   fold up the folded triangle. Great. Now compare  this new folded folded triangle to the original   triangle. Notice anything special? Yes, the folded  folded triangle is just a scaled down version   of the original triangle. Interesting and very  beautiful isn’t it. And it really looks like this   was only discovered for the first time 10 years  ago. Amazing, right? But of course as with most   mathematics, a lot more beauty is hiding in the  `why’. And so, let me also show you why all this   works. Lot’s of nice AHA moments ahead and so make  sure to stick around. Actually we’ll prove more   than just the medians meeting in the controid.  We’ll also show that the centroid splits every   median into a long and a short part. and that  the long blue part is always exactly double the   length of the short green part. There. The blue is  exactly twice as long as the green. Again, that’s   true for any median, the blue is always twice  as long as the green part. There blue is twice   as long as the green. And again, blue is twice as  long as the green. Here is a nice way to see why   all this is the case. Scaling our triangle down  by a factor of 3 gives this little triangle. And,   you’ve probably all seen this before, we can now  tile the large triangle with copies of the little   triangle like this. Okay. Here is one of the  medians of the little grey triangle at the bottom.   From the tiling it’s clear that if we extend this  median up we end up with the median of another   little triangle. That one here. There extend. And  extending further we get the median of the little   triangle at the very top. And now of course,  together the medians of the three little triangles   form the median of the large triangle AND this  median passes through the special point right in   the middle of the tiling, AND the blue part of the  median is twice as long as the green part. AND,   of course, the same is true for all three medians.  Easy :) Okay, so medians all meet in the point   right at the center of the tiling and are all cut  by this point in the ratio of 1:2. Proof complete.   Super nifty, right? :) Actually with these two  properties under out belt, it’s very easy to now   also prove that the three ways of folding give the  same folded triangle and that, furthermore, the   folded folded triangle is a scaled down version of  the original triangle. Here we go. If these three   parts really all fold up into the same triangle,  then obviously all three parts, red, green and   blue have to have the same area, exactly one third  of the area of the original triangle. Let’s first   see why that is true. Easy, The area formula for  triangles is 1/2 base times height. Well then,   the blue triangle and the original triangle have  the same base. And because of the 1:2 ratio in   which the medians are cut by the centroid, it’s  clear that the blue triangle’s height is 1/3 that   of the original triangle. And, therefore, also  the area of the blue triangle has to be exactly   1/3 of the area of the original triangle. And of  course the same is true for the red triangle, and   the green triangle. Nice. Have you ever done any  origami before? For example, have you ever folded   a paper crane? Well when you unfold the crane you  get this interesting crease pattern which really   is the blueprint for the crane design. Similarly,  our median intersection pattern is a blueprint for   everything that happens in our triangle folding  adventure. Everything that we come across in   terms of little triangles, distances and angles  is present in this blueprint. Let’s have a closer   look. Let’s first highlight the angles around the  centroid. These two opposite angles are the same.   So are these. and these and so on. Okay, now any  other angle visible in this diagram is composed of   the angles in the middle. For example, let’s take  one of the angles at the bottom. Put in a parallel   to the bottom line through the centroid And so the  angle we are interested in is equal to the orange   angle plus the green angle. Another example, this  angle here. After drawing this parallel here it’s   clear that our angle is equal to the grey angle  plus the blue angle. Doing this angle chasing for   the whole diagram gives this. Pretty spectacular,  hmm? Reminds me a little bit of an Australian   aboriginal dot painting. Okay, get ready for an  AHA moment. Let’s do the folding. Okay, these are   the three folded up triangles. They all look the  same. But to be sure that they really are, let’s   check the angles at the vertices. There all the  same and the same. And since we already checked   that all three triangles also have the same  area, we can be sure that all three triangles are   congruent, that all three triangles are the same.  Okay, now let’s fold a second time We already know   that these triangles are all the same. But we  still need to check that this folded folded   triangle is similar to the original triangle. For  this we have to check that corresponding angles   are the same. Beauuuutiiiiiful :) Proof complete  :) Now, let me just show you the folding action   again in a slightly different way. As you’ll  see, in a way, folding a triangle is really   just turning it inside out, with a bit of scaling  down thrown into the mix. And then folding again   turns things inside out again and returns us to  the original triangle. We are back to where we   started from. So, folding is really just a nifty  way of turning a triangle inside out. Here are   a couple more interesting observations. First,  isosceles triangles always fold into isosceles   triangles. and an equilateral triangle folds  into another equilateral triangle. What about   right triangles. Well let’s see. Folding a right  triangles does not give a right triangle. However,   since folding turns triangles inside out we can  still find the right angles in the middle of the   folded triangle. There are the right angles.  I can see some of you serious triangle lovers   bouncing up and down in your seats :) Being well  versed in the ways of the triangle, you will have   immediately spotted another way of seeing pretty  much at a glance why the three folded triangles   are the same. Here is that second proof that  quite a fwe of you will have come up with on   your own. This one just takes a couple of seconds.  Color in the long median parts like this. Let’s   fold So two of the sides of this folded triangle  coincide with two of the coloured parts yellow and   orange. What can we say about the remaining black  side at the bottom? let’s unfold again. Well,   we know that this green segment is half as long  as the blue. Right, remember that? :) But then   the remaining side of the folded triangle consists  of two copies of the green and so the remaining   side is as long as the blue side. nice. And so  the different sides of this folded triangle are   just the three coloured long median parts of the  original triangles. Since this is the same for   all three folded triangles, all have to be the  same. How nice is that :) Finally, here is the   article by Lee Sallows about his discovery in the  December 2014 issue of the Mathematics Magazine.   Just one page :) And so it really is still  possible even today after thousands of years   of triangle spotting to discover new simple  and beautiful mathematics in this area. And   that’s all for today’s mini Mathologer. Flat  out with work at uni work at the moment and so   didn’t have time for a Kurosawa length Mathologer  video. Still a very nice topic I think and I hope   you enjoyed it :) Oh, and by the way, which of  the two proofs do you prefer and why? The dot   painting leisurely one which really takes in  a lot the sights off the beaten track? Or the   one-glance one at the end that goes straight  for the kill? Let me know in the comments :)
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Channel: Mathologer
Views: 199,310
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Length: 13min 48sec (828 seconds)
Published: Sat Feb 24 2024
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