We live on a 4-dimensional Pringle (Non-Euclidean Geometry and the shape of the Universe)

okay pop quiz if you add up the three angles in a triangle how many degrees will you have yeah exactly 180 perfect okay what about a pentagon how many degrees in a pentagon you don't remember do you it's 540 degrees and if you ever forget a polygon with n sides has n minus 2 times 180 degrees so keep those two facts in your head while i introduce to you this origami crane he does not have a tail and what's funny about that is that normally an origami crane is made out of a square piece of paper and there are four corners each one of those corners becomes one of the appendages the head the tail and the two wings so since our crane has three appendages it's probably made from a triangle let's see if it adds up to 180 degrees if i unfold this wing you can see that it makes a 90 degree angle okay we're already left with only 90 degrees for the other wing and for the head so let's look at the other wing and it's also 90 degrees the head is a little bit hard to unfold but believe me it's also 90 degrees so we have a problem our crane is made out of a triangle with 270 degrees i'll let you think about that and while you do i'll introduce you to one of his friends this guy does have a tail and he also has two heads this gives him five appendages two heads two wings and one tail which means that he's made out of a pentagon and like his friend with no tail if we look at each one of his corners they all end in a 90 degree angle which means that he has 450 degrees total that's not enough to be a pentagon so unless i'm tricking you with extra paper or something these two cranes break all the rules of geometry that you learned in school so here's the thing when you learn geometry you learned it from books and white boards where this stuff is true but our cranes are not made from flat pieces of paper our no tailed crane is made from a sphere a triangle cut from a sphere and our two-headed crane is made from a pentagon cut from a shape called a pseudosphere on a sphere every triangle adds up to more than 180 degrees and on a pseudosphere every triangle adds up to less than 180 degrees now i know you're thinking that's cheating the edges of those triangles aren't even straight lines flat geometry is the normal geometry it's the one that we live in yeah it says who a line is the source path between two points and if you're confined to the two-dimensional surface of a sphere then a straight line is always a segment of a circle yeah in three dimensions we could imagine cutting through digging a hole through earth to make an even shorter path but if you live in two dimensions that's all you have as three-dimensional creatures we can't travel or even imagine a fourth dimension the three dimensions is all we have so if our three-dimensional space is curved our straight lines might not be that straight at all we could live in a universe where triangles don't add up to 180 degrees but we just don't realize it because we're so small in this video we're going to use the rules of origami to figure out the shape of our universe the way that we learn geometry in school has stayed pretty much the same since about 300 bc this is when the greek mathematician euclid wrote his treatise elements in the first book of elements euclid gives five axioms of geometry from these you can prove any other fact they are one you can draw a straight line between two points two a line segment can be extended infinitely three if you have a center and a radius then you can draw a circle four all right angles are equal to each other and the fifth one say you have two lines and a third line that intersects both of them if the interior angles add up to less than two right angles then the two lines will always intersect each other now ever since euclid wrote these down mathematicians have had a problem with this fifth one because it's ugly it's long and it's confusing it sounds so arbitrary that they thought maybe they could prove it from the other four to get rid of it we now call the fifth axiom the parallel postulate and trying to get rid of this thing became the focus of mathematicians all over the world but all they could ever do is replace it they would come up with different axioms but there would still be five of them you already know one of these replacement axioms because that's how we started off the video today the angles in a triangle add up to 180 degrees this is the same thing as the parallel postulate let's draw two lines that never intersect also known as parallel now let's draw a third line that goes through both of them according to euclid's fifth these two angles must be 180 degrees if there are any more then the two lines would intersect on the left if they were any less the two lines would intersect on the right now let's draw a fourth line that splits b into two angles clearly b equals c plus d and this angle here must be the same as d you can now see that we've made a triangle with a c and d and if we plug in we get a plus c plus d equals 180 degrees another replacement that mathematicians found is called playfair's axiom it says that if you have a line r and a point p that's not on r then there's only one line that you can draw through p that doesn't intersect r this of course is the line that's parallel to r you can probably see how that's related to the parallel postulate so i'll leave it up to you to prove so in 1820 hungarian mathematician janos boleye was studying the parallel postulate specifically in the form of playfair's axiom his father farkas was also an accomplished mathematician and he thought it was a waste of time he told janos that the topic extinguished all light and joy in his life a little bit dramatic but janus continued to study it boya started by assuming the opposite of playfair's axiom he said let's pretend that if you have a line r and a point p not on r then you can always draw more than one line through p that never crosses r this might have been a good way to start a proof by contradiction but there was never a contradiction but yeah i discovered all these new properties that geometry would have if this were true and they were different from euclid's geometry but they all made sense they were logically sound the only problem was that he couldn't draw it at least not on paper but he could have drawn it on this the shape of this pringle can be approximated by a hyperbolic paraboloid the cross section is an upward parabola in one direction and a downward parabola in the other one of the interesting properties of this shape is that the shortest path between two points bends outward away from the center so if we try out playfair's axiom we draw a line r down the center and then we draw a point p above it then we can draw as many lines through p as we want euclidean geometry is not true on this potato chip but bowliaz is it's pretty weird that it took us this long to discover alternatives to euclidean geometry because we're literally living on one try out playfair's axiom on a globe of course keeping in mind that straight lines are great circles like the equator or like longitude lines but not like latitude lines so draw a line r and a point p and what you'll notice is that every line that you can draw is going to cross through r there's no such thing as a parallel line on a sphere now we've discovered three types of geometry one is flat euclidean geometry two is the one that bullied discovered which we now call hyperbolic geometry it takes place on surfaces with negative curvature and then third is elliptical or spherical geometry and it takes place on surfaces with positive curvature so far all the examples i've given you have been in 2d because that's all we can visualize a sphere is a 3d object but the surface itself is 2-dimensional a 2d creature could live on the surface of the sphere but they would never be able to imagine the sphere itself similarly we can't imagine curved 3d space but it is possible and boy i realized this at the end of his book he wrote that we need to do experiments to figure out whether our universe is curved or flat and between 1999 and 2013 we got the evidence we needed to figure this out immediately after the big bang the universe was a hot dense fluid of elementary particles as it expanded it got less dense and since wavelengths increased it got less energetic too and it cooled down when the universe was about 370 thousand years old particles had cooled enough that electrons could become bound to protons to form hydrogen this is the first time the universe had atoms when electrons and protons combine to form hydrogen it releases light at a specific wavelength this light is still around today but because the universe has been expanding it's been redshifted to the microwave spectrum if you point a telescope anywhere in the sky you'll see this leftover radiation so we call it the cosmic microwave background the temperature of the cmb is mostly uniform but if you measure it really precisely there's tiny differences from one place in the sky to another these are called anisotropies and they teach us all kinds of things about our universe when the cmb formed the universe was pretty young but it was already a decent size remember it was about 370 000 years old which means light could have traveled a maximum of 370 000 light years but the universe was already bigger than 370 000 light years wide if a beam of light would have had the entire age of the universe to travel from one side to the other it wouldn't have been able to make it if light couldn't make it then nothing could have made it and that even includes the influences that transfer heat it takes a few minutes for ice to melt in a cup of water and for all the water molecules to reach the same temperature similarly the universe wouldn't have had enough time to reach thermal equilibrium when the cmb formed there are all these patches of slightly cooler and slightly hotter spots and that's where these anisotropies come from we know the speed of light and we know the age of the universe at that point so we can figure out how big these patches should have been we also know the size of the universe then and the size of the universe now so we can use a little bit of trigonometry to figure out how big these patches should look in the sky say you're looking at an object of width x from a distance d away you can draw a triangle between your eye and the two edges of the object then you can split that into two right triangles now tangent of theta over two equals opposite over hypotenuse so x over two d now remember this is space we're talking about so d is huge and x is not as big so it's not going to take up a very big angle in the sky for small angles tangent of a is approximately equal to a so we can safely say that theta over two equals x over two d and therefore our theta equals x over d but what if we live on a big four dimensional sphere remember in elliptical space triangles add up to more than 180 degrees this is the same trick we use to make our no tailed crane so theta should be bigger than what we expect what if we live on a big four dimensional pringle well we know in hyperbolic space triangles add up to less than 180 degrees this is how we made our two-headed crane so theta should be smaller than x over d and that's it all we need to do is measure the temperature of the cmb all over the sky look at the size of these patches and that will tell us whether our universe is curved or flat the cmb was first measured with a high enough resolution to do all this in 1999 by the boomerang experiment then again in 2003 by the wilkinson microwave anisotropy probe and then again in 2013 by the planck satellite using the data from these experiments along with the techniques that we just talked about our current best measurement of the curvature of our universe is 0.0007 with a margin of error of 0.0019 this leaves us confident that our universe is flat for a lot of human history and unfortunately even today there have been a lot of people who thought that the earth is flat and this makes perfect sense from any point that you stand on the earth it looks like a plane in order to get even the smallest piece of evidence that there's any curvature at all you need to travel way far away from your homeland travel across the world and expand your perspective when you zoom in either on the earth or on any surface it's flat you need to look at it from a bigger point of view if all of our learning happens on paper then we're going to be just like euclid the best that we'll ever understand the world is as a giant piece of paper so when you study physics or math or really anything remember to get away from the paper every once in a while or at least use one that's a different shape [Music] you

Channel: Physics for the Birds

Views: 14,288

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Id: NleVVz1Y21Y

Length: 12min 42sec (762 seconds)

Published: Tue Aug 16 2022