Simple yet 5000 years missed ? 

Most modern "science" is full of mathematical nonsense. Like the shape of the Earth. Maths doesn't match reality.

The best math comes from Side Projects. The things you think about when you should be doing something else.

They are "discovering" the things.

how you develop 'wonder' thing? I am not wondering at anything

@@robertveith6383 Not only. They also develop things.

😂

This short format appears to get the thumbs up from many regulars. Nice :)

@@Mathologer Yes, long, short, medium--all good, and a viewer would be foolish to complain about a short video if the alternative is no video at all. Whatever length suits your schedule and the video's content best. Of course it's not ideal to increase the quantity (minutes of videos produced per year, say) at the cost of quality, but I've never noticed that in a Mathologer video, so not an issue in this case!

@@Mathologer Same here! I don't always have time to sit through a lot of the longer ones unless they're a direct interest of mine, but I'll click on these shorter ones any day of the week.

@@leif1075if you can see the area of the blue triangle is exactly 1/3 of the original triangle, you can use the same reasoning for the red and green triangles. In the latter two, you’ll be using the outer red triangle side and the outer green triangle side, NOT the blue triangle side. Each coloured triangle area is 1/3 total area of the original triangle. They are the same area in all three views (blue, red, green), despite the different bases and heights.

@@leif1075He _does_ show that the base of the inner triangle is the same as the outer where that inner triangle is located. _It's the same exact line segment_, so of course it's the exact same length. Each of the three inner triangles are positioned at a different side of the original triangle. So they don't actually have the same base as each other. But they always have the same base as the line segment where their own base is located, hence the bases are always the same. FWIW, your apparent concern -- that the bases aren't all the same length (and again, _all_ the bases aren't all the same length...just when each inner triangle's base is considered with the specific side of the original triangle where that inner triangle is located) -- would also apply to the height of the inner triangle. I.e. the heights are not all the same either; each height of each inner triangle is exactly 1/3 the height of the original triangle when measured from the side that the inner triangle shares, but each of the three inner triangles' heights may well be different as well. The proof in the video demonstrates the equal bases and 1/3 height only for a single inner triangle. You then have to extrapolate the same logic for the other two inner triangles in the diagram. If it helps, make sure you rotate the whole diagram so that both the inner triangle and the original one are oriented with their bases at the bottom of your visualization.

only in exactly one case, the equilateral, the triangle is the same as it's folded counterpart.

@@opensocietyenjoyer You're wrong! You will also get the similar triangle if centroid coincides with one of the Humpty points (projection of orthocenter onto the median). This is because medians to sides will be in ratio sqrt(3):2. In case of equilateral triangle orthocenter already coincedes with centroid, so it's a simple case

@@notEphim Yay.

Hahahah very meta 😂

9:48 apparently he could hear us from the past

Meta I meant @czertify’s comment some how remind me of Pierre de Fermat’s comment, I am not sure if it’s intentional to subliminal but it is just somehow made it even funnier for me

Meta I meant @czertify’s comment some how remind me of Pierre de Fermat’s comment, I am not sure if it’s intentional to subliminal but it is just somehow made it even funnier for me

:)

Please look at the answers i put in the comment of "i l put username later" to link this duallity with the usual midpoint duallity (involving hexagons ABCA'B'C' such that (XY)//(X'Y') \forall X e Y \in {A,B,C} ) This make me wonder if there is not a way to combine opposite triangles and this new duallity in the space of positive triplet that satisfy triangular inequality and that sums is 1 (exept for 000) . The orientation will matter in order to get opposite triangles, we would like to be able to do addition (such that the addition of two opposite gives the emptytriangle (0,0,0)) and a multipplication such that a triangle multiplide with its inverse (defined by the new duality) or maybe the opposite of its inverse gives an equilateral triangle, in such a way that we get a nice structure, why not a field (we will probably get an isomorphism of a well konwns field, I'm thinking about quaternioons because it is the only 3 D field I know and maybe the only one possible, I really don t know much about it lol) Really to many suppositions here so I ll have to stop here not to be ridiculous, but it might be interesting to search something. Note that it is easy to get a tripplet of homogenous coordonates that satisfy triangular inequality (and that are decided equal up to scalar multiplication ) from a triplet of 3 real numbers up to scalar multiplication : take the angles that are obtain with a triangle that vertices are (a,0,0), (0,b,0) and (0,0,c) in the 3-d space. (indeed we get all triangle that angle are all less then pi/2 , which are also triangles s.t. the mesure of angles satisfy the triangular inequality, isnt it nice?^^) I will do a litlle homemade video to talk about this, and I ll give the link^^ Thank you Mathologer for this video and every single other long or short one❤❤❤

Same here. I've been obsessing about identifying the "second best proofs" for theorems for a long time :)

Yes, I was wondering as I watched the video if the folded triangle would be called the “dual” of the original (as you have wjth polyhedra) or has some other name. And, are there other interesting properties of the dual as relate to the original?

Well duel is supposed to mean something very different. You put points at the middle of line segments then draw a line between any 2 points that once shared a point.

The term "dual" is very general and is used all over mathematics. It means an operation that yields the same type of object (a triangle in this case) and brings you back to the original if applied again. So no, what you describe is not "THE dual", but just "SOME dual".

​@@pauselab5569 there are many many kinds of dual in loads of situation. A duality is typically some pair of objects that you can swap between by exchanging some property. And I think this triangle situation can work with that. Duals are typically great because, by proving something about one object, you automatically get an equivalent proof for the other, and sometimes it's very easy to get a proof for one but you care about the other.

Thought the same! I wonder what to do with it, what properties are preserved by this type of duality, whether it is the same as some other type of duality, and whether there are analogues of this in higher dimensions!

As a color blind person, I didn't like the dot proof as much. I got the idea, but it wasn't as visual for me

@@kilianvounckx9904 Haha. I’m colorblind as well (red/green). I couldn’t tell which dots were which, mostly.

Maths would still be elegant and beautiful without him (or anybody for that matter), but he certainly does an excellent job of helping a broader audience appreciate it!

@@jrbrown1989my thought exactly! He's brilliant at communicating things in such a way that a broad audience can see their beauty 😊

α β γ δ ε ζ η θ ι κ λ μ ν ξ π ρ ς σ τ υ φ χ ψ ω ϐ ϑ ϒ ϕ ϖ Ϛ Ϟ Ϡ ϰ ϱ ϲ There is some greek letters for you. You can copy and paste them to tidy up the text if you want. :)

My thoughts too. I'd be suprised if the pythagoreans, ancient Indians and who knows before didn't know of this.

Exactly!

Probably quite a few, but possibly/probably never visualised in the way I showed in the video :)

Agreed - it follows easily from the centroid being 1/3 along each median, as demonstrated in this video.

Well, if you put a lid on a pot with boiling water, it is pushed up and clacks sometimes. Probably people noticed that quite some thousand years ago. But not seeing it as trivial and going from there to a steam engine took quite a while. The next step towards many inventions was hidden in plain sight and regarded as trivial, until someone took a really close look and pointed out that it's anything but. So I have the deepest respect for people who have an eye for those things and look behind seemingly trivial things.

Exactly! I am almost 50 years old, and I can distinctly remember playing with these 'folds' in the EXACT same way as shown in the video when I was around 10 years old with my own arts and craft projects at home. In fact, I might still have it laying around somewhere in some boxes at the attic. I suspect when he says _'discovered'_ he actually means either *A)* officially described in some math paper, and/or *B)* a proof was found. Which are VERY different things than _'known/discovered'_ .

@@CookieTube I'd be interested in seeing those folds!

It was probably only published by someone 10 years ago. People playing around with triangles ABSOLUTELY have discovered this many times over history.

​@@LeoStaleylet's raise a glass to all those awesome people throughout history that loved the beauty of form and function. May everyone that wants to share in curiosity and wonder.

Some things are so simple that no one who stumbles upon it would have the nerve to publish it.

Yes it is. This is also a more intuitive way to see the three parts are of equal area: The tiled and then median-cut original consists of 18 triangles, all of which are exactly half of one of the tiles, so they have equal area. And since each of the parts has six half-tiles they are all of the same area. It seems like you should be able to show the rest like this as well, by colouring in the six different types of triangles. (Three median directions to cut a tile, each cutting a tile in half.)

Used to be taught in school. These days at least in Australia hardly any nice geometry is covered in school anymore :(

@@Mathologer really? But that seems like a basic property, not something I'd classify as "nice".

​@qnbitsWhat about ch**trail spotters then? 😮

Nice! So the squares of the sides behave better than the sides themselves. That's sort of a dual version of what I wrote about (the cotangents of) the angles an hour ago.

Glad you enjoyed it!

Yes, scary. Time seems to fly faster and faster. Probably a reflection of the fact that we are getting old :)

Glad you liked it!

go awey

Absolutely :)

Duality is a great way of explaining things that should be used to replace straight single logical proof in certain more complex subjects, such as economics and mechanics. (Axiom: Mankind seeks to satisfy his needs with the least effort:yet mankind''s ambition to meet these needs is unending (Henry George). Action and reaction are equal and opposite (Isaac Newton).

Check out some of his other inventions by following the links in the description :)

Not my design, but check out the link in the description of this video :)

As to the (once) folded triangle, there is no such ratio in general, since it is not congruent to the original one. As to to the folded folded one, the video explains tha the ratio is 1:3 for the lengths and 1:9 for the area.

Left as an exercise for the -reader- viewer...

Yes, that one :)

😳

Correct, they all have the same area :)

u lied

Yes, I've got a video about this ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-XR2u7izwZ04.html

@@Mathologer cool! What a lovely pattern!

do you mean "damn! what a disappointment!" :-)

:)

I actually did two videos on Morley's miracle early on :)

@@Mathologer And I probably saw one or both of them, way back, and forgot :-(

Numberphile (and Numberphile groupies) say that Numberphile wins the -1/12 debate. I'd still maintain that maths wins :)