Non Euclidean Geometry

[@jojogothic]

i have had it with these motherfucking shapes in these motherfucking planes. great video though

[@Sparton646real]

+jojogothic that, made my day, thank you, good sir.

[@seandafny]

classic comment. i totally agree

[@ytpancho]

Moloch lmao

[@hawksandwich4742]

Shapes on a Plane, in theaters Friday

[@jilongzhang1851]

that was a nearly perfect hand-draw circle

[@vejymonsta3006]

I can't imagine trying to actually solve problems using non-euclidean geometry. It's just so unnatural.

[@seandafny]

wys

[@LossOfDesire]

That was my first thought too, and I love mathematics. I just recently took a class called Classical Geometries were we did math in Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry. It was a trip! Really interesting but very hard to understand at times. Still if you like math, you should really look into it.

[@ZenoRogue]

While working with non-euclidean geometry is usually harder than with Euclidean, it becomes quite natural with practice :) BTW: why cannot I see most replies to comments in this video (like none of the three replies to this post, basically most of the replies)? This does not seem to happen with other KZbin videos.

[@h-Films]

natural in my opinion. Works so well in my brain.

[@elijahcaudle7365]

I mean we live 3d, adding curvature and topography changes nothing as still a form, right? Only the complex mathematics underpinning it makes a difference which takes a much better man the me to know.

[@cheongziyong8871]

Another example of non-Euclidean geometry is when you fold a piece of paper in half and then roll it up then unfold it, the fold reverses the geometry of the curve and thus causes hyperbolic and ecliptic to be side by side and the paper causes a new curved crease to appear beside the pre-existing half crease.

[@Felevr]

this is too high for my brain

[@is1745]

Shiit I'm too high for this

[@oban6051]

Maybe your brain isn’t high enough for this.

[@IDMYM8]

@@oban6051 shots fired

[@LiberaLib]

i'm trippin' balls right now.

[@greengrassofhome]

LiberaLib Is it LSD or non-Euclidian geometry which is making you trip?

[@ernadmahmic6667]

2:00 AAAAND EVER SINCE I MEET THIS MAN MY LIFE IS NOT THE SAME AND NIIKOLAI IVANOVITCH LOBACHEVSKY IS HIS NAME!

[@matunam250]

..what?

[@charlescole1766]

HPL be like

[@Turin_Inquisitor]

Cthulhu approves

[@a_void]

ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn

[@Bluedragon2513]

Written by Loveheart gang

[@aniksamiurrahman6365]

U ppl know there once lived several Cthulhu avatar among men. One is called Gauss another is Reiman. This teacher-student Cthulhu duo unlocked the door of non-Euclidean world of unlimited possibility.

[@BillStreetStudios]

@@Bluedragon2513 you mean lovecraft?

[@lev7509]

HyperRogue reference?

[@ayushkrishnanchakravarthy5548]

The statement of Euclid's fifth postulate is wrong. The postulate stated in the video is a equivalent postulate used to disprove Euclid's fifth postulate. This was used notably by Girolamo Saccheri.

[@DEJAP001]

This pleases me.

[@DC-zi6se]

Gauss was probably the pinnacle of human intellect, no one comes even close when you think of pure unadulterated intelligence. His brain needs to be in some museum.

[@maxwellsequation4887]

Nah, NEWTON or EINSTEIN was the pinnacle of intellect And then were Euler and Gauss Plus stealing someone's brain.... I would prefer to die alone in a forest than get my brain stolen

[@socearo]

How do you add 180 degrees to the area of a triangle when the two number have different units?

[@zgacc1]

It should be in radians (180 degrees = pi radians); this is only true if the sphere has radius 1 and if the edges of the triangle are parts of a great circle. For a sphere of radius R, the sum of the interior angles is (pi + Area/R). This is due to the Local Gauss Bonnet theorem.

[@socearo]

Area is in meters squared. Angles are in degrees or radians which are unitless. Area cannot be converted to angle because the two measure different things entirely. What do you get when you add the speed of a car to the its mass. Nothing, the question doesn't make any sense. MULTIPLYING them makes sense and gives you the car's momentum. pi+Area/R gives the pi (unitless) + x meters. Alternately what is 5+3x? With out giving x a numeric value the simplest way to represent that is as is 5+3x but change it to 5x+3x and you can use the Distributive property to simplify it to (5+3)x = 8x.

[@zgacc1]

Ah, my mistake, the formula is pi+Area/R^2.

[@zgacc1]

The one over R^2 comes from what is called the Gauss curvature. I was mistakenly thinking of the curvature of a circle.

[@Teth47]

socearo Math, obviously.

[@mikevandenheuvel1408]

3rd century BCE - you're off by 600 years

[@MaistrePathelin]

This is an aptly-made vulgarisation of a rather complex (at least, to people with little mathematical knowledge -like me) subject. Well done!

[@n8moDraws]

Wonderful ^^ It looks like a lot of work went into the video, you guys have earned a new subscriber!

[@muhammedshameel5684]

The fifth postulate given is the playfair's axioms.Euclid gave a slightly different one.

[@alberoDiSpazio]

I don't know what the big deal is. You have a 2-dimensional triangle wrapped on a 3-dimensionsional sphere, something is going to be warped. What about a 3-dimensional tetrahedron wrapped on a 4-dimensional "sphere"?

[@holoceneevent4534]

Spheres define the infinite space. you cannot have a sphere in any less dimensions.

[@js-yall]

Albert J. Nguyễn holy shit

[@redrealmgaming8859]

Mind blown

[@Rensence]

Then you create an infinite space

[@Kaiveran]

@@metachirality Specifically, you'd get a hyperspherical pentachoron, made of 5 tetrahedral cells, but with wider corner angles than normal.

[@nicholascopeland358]

Very cool! Used it to introduce the unit to my class and they enjoyed it. Nicely done :D

[@spoderman15]

Has anyone ever written a book like _Euclid's Elements_ for these non-Euclidean geometries?

[@spoderman15]

***** I mean by starting out with simple definitions and axioms and then building the geometry proposition by proposition

[@titanarmy4116]

yep

[@spoderman15]

Titan Army got a title?

[@sabercrosby8128]

yeah benoit mandlebrot. Eulidean Geometry doesn't exist in nature. The true nature of reality is fractal geometry

[@titanarmy4116]

Saber Crosby lol, prove it

[@robkim55]

Can you elaborate on the area(s) of research in non euclidean geometry [noneg]

[@foobargorch]

general relativity is the most famous example in the context of math look up manifolds

[@cultivatedjerk5574]

I’m just trying to understand arial combat in Dungeons and Dragons. How did I get here?

[@Jaojao_puzzlesolver]

Fantastic music.

[@carldalton9331]

SINGLE LENGTH OF DEGREE 1. Use one 120 centimetre Right Angle Line as a Diameter Line 2. Multiply this Diameter Line of 120 Centimetres by 3 = 360 Centimetres 3. Therefore the Circumference Length of the Encirclement of the Diameter Line is 360 Centimetres 4. There are 360 Degrees to the Circumference Length of Every Circle 5. Therefore Each Degree of the 360 Centimetre Circumference is One Centimetre long 6. Multiply the 120 Centimetre Diameter Line by 4, & the Square of the 120 Centimetre Diameter Line is 480 Centimetres, with each Right Angle of the Square being 120 Centimetres in Length 7. Therefore the 360 Centimetre Circumference Length of the Circle, is three quarters the length, of the 480 Centimetre Square of the 120 Centimetre Diameter Line SIMPLEST METHOD FOR CALCULATING THE AREA OF A CIRCLE Diameter 120 Centimetres x Diameter 120 Centimetres = 14, 400 Square Centimetres to the Square 1. The area of the square is 14, 400 sq cm's 2. The area of the square divided by 4, is 3, 600 sq cm's 3. Multiply 3, 600 sq cm's by 3 4. 10, 800 sq cm's, to the area of the Circle Sumerian Method 1. Diameter 120 cm's 2. Circumference 360 cm's 3. Circumference squared 129, 600 sq cm's 4. Divide 129, 600 by 12 = 10, 800 sq cm's to the area of the circle ARCHIMEDES 287 - 212 BC Proposition 1. The area of any circle is equal to a right-angle triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle. Archimedes Triangle 1. The base right-angle is equal to the radius 60 cm 2. The height of the right-angle is equal to the circumference of the circle 3. The area of the circle is equal to the above right-angle triangle, which has one side about the triangle that is equal to the 60 cm radius, and the other to the 360 cm circumference of the circle. Check 1, The area of the rectangle is 21, 600 sq cm's 2. The area of the right-angle triangle is 10, 800 sq cm's 3. The three quarter area of the 3 x r squared circle is 10. 800 sq cm's 3 x r squared Diameter 120 Cm x 120 Cm = 14, 400 Sq Cm's to the Square 1. 60 centimetre radius is squared = 3, 600 sq cm's 2. Multiplied by 3 = 10, 800 sq cm's to the area of the circle 3. The area of the circle is three quarters that of the 14, 400 sq cm square B The 14, 400 sq cm area of the square, is divided by 4 = 3, 600 square centimetres This is then multiplied by 3 = 10, 800 square centimetres to the area of the Circle Equating to three quarters of the area, to that of the overall square C 1. 60 cm radius is squared, 3, 600 sq cm 2. 3, 600 sq cm is divided by 4 = 900 sq cm 3. 900 sq cm is then multiplied by 3 = 2, 700 sq cm 4. 2, 700 sq cm is then multiplied by 4, = 10, 800 sq cm's to the circle

[@HiloYT]

VR: a place where physics don't matter, and you can do anything you want

[@EmdrGreg]

I'm not a mathematician. Something basic really bothers me. I can intuitively get that geometries based on spheres or hyperbolas will violate some or all of Euclid's postulates. There may even be many other geometries that are even more fantastic. Every time someone tries to tackle the 5th postulate, it seems the first thing they do is go to non-planar ideas. Does this somehow mean that the 5th postulate cannot be true in Euclidean or plane geometry? It may not be provable as true, but the idea seems so totally reasonable and draws us in so strongly. I guess the easy way to put this is: Is there good reason to suspect that the 5th postulate is false even in plane geometry?

[@ProfeCharley]

no, i means no one succeed on proving 5th postulate using other postulates from euclid; it doesn't mean is false

[@TheGuyRoss]

The 5th postulate is completely true in plane geometry, it is equivalent to a lot of statements that are clearly completely true in the plane. The controversy is more that the other 4 are equivalent and this one is stand alone, only in the plane. It's kind of the same as without the other 4 axioms we cannot prove anything as we need to assume something to have anything going on to work out, so we assume this is true as it clearly is in the plane.

[@zgacc1]

I think your question is due to a misconception. Planar (Euclidean) geometry is by definition the geometry for which the parallel postulate is accepted, in addition to the other four postulates.

[@foobargorch]

no, in fact the 5th postulate is what *defines* flat geometry (the uniqueness in the first postulate can be derived from the 5th postulate and must be removed for non euclidean geometries, for example two poles on a -circle- sphere are connected by an infinite number of line segments so not all points are uniquely connected by a line segment). think of axioms/postulates as fixing constraints in the space of possibilities. if you take them as a given you can explore what the different sets of rules are. more intuitively: the first 3 postulates have to do with the objects in space and on what terms they exist. the fourth relates intersecting lines through a single point and therefore does not constrain the shape of the space this is happening in. adding the fifth postulate relates two lines in a single plane and constrains the shape of that plane (the plane can exist in higher dimensional euclidean space for what it's worth but it will still be a plane), and by the existence of an infinite of arbitrary points and therefore lines and therefore parallel lines implies that the entire space is continuously filled with such planes in every which way. no matter where you go in euclidean space you can construct objects that demonstrate the flatness. changing the fifth postulate gives you different constructions through which you can discover the shape of space, but all the other ones still hold (again, the uniqueness has to be weakened in some cases). you can also replace the fifth postulate with equivalent statements (any logical proposition that that entails the 5th postulate and is entailed by it is formally equivalent and therefore could be seen as an alternate formulation) finally it's worth mentioning that a more modern approach is to start by assuming some axioms as a foundation and discovering the mathematical objects they define, but in euclid's case he apparently if i'm not mistaken he started with the familiar mathematical object (a flat geometric space) and worked back to the rules that defined it explicitly, but the two approaches are equivalent, it's just a matter of perspective.

[@cliveso]

Greg Scott And you can be forgiven for thinking like a non-mathematician. As far as daily life is concerned, our space is Euclidean and no one is contesting that. But pure mathematics doesn't always describe our physical reality. By dropping the 5th postulate, mathematicians have found that non-Euclidean geometry has many interesting properties and profound richness. That alone is enough, and that is pure mathematics. It's not about "true" or "false"─and what does "true" mean anyway? That it conforms to our humanly intuition? Pure mathematics is all about defining some rules, play with the rules and see what fancy things you can get out of them. If in the physical world you encounter a problem, and the rules of the problem happen to correspond to some obscure mathematical concept, then instantly you have a whole set of tools to help tackle the problem with. Many modern technologies rely in this way on antique, abstract mathematical concepts developed purely out of curiosity.

[@theimperfectgod7140]

Perspective is really powerful thing... fascinating 🗿

[@philipfahy9658]

Not a very mathematical proof, but I feel like Euclid's fifth axiom is fairly easily proven. Assuming this all takes place on a flat plane, take that line and draw a perpendicular line through every point along it. Only one of these lines will also intersect the point above the line. Therefore, there is only one line that is both perpendicular to the original line and passes through that point.

[@albertartiles356]

Your proof relies on a statement that is equivalent to the fifth postulate. Say you have a line L and a point p not in L. Then there is a unique perpendicular line M to L that contains p. I am okay with that argument. However then you proceed to say to draw a line N through P that is perpendicular to N. Then without proof you claim that N is parallel to L. This is not necessarily true if we are only assuming the first four axioms. For instance spherical geometry satisfies the first four axioms of Euclid, however, parallel lines do not exist in spherical geometry. Any two great circles, which are the lines in spherical geometry, intersect.

[@spagetychannel5070]

Hyperbolic geometry doesn't break all five of Euclid's postulates. The whole point is that it satisfies all of Euclid's axioms (including his unstated topological assumptions) except the parallel postulate.

[@pronounjow]

Those are some nicely drawn circles!

[@Sunhill991]

Trolden music, sweet learning!

[@Kinsman19]

Knew I'd find a bunch of Lovecraft nerds here. This is awesome.

[@sparklefluff]

What's the name of the background music?

[@philipparker5291]

Forgive me for being a layman with respect to Euclidean geometry, but when it is stated that his fifth postulate cannot be proved, does this specifically mean that it cannot be deduced from the other four postulates? What about extending lines in all directions from the point separate from the parallel line? Wouldn't that show whether the postulate is true or not? Or would this imply that at least one such line would extend indefinitely, and as a consequence not prove the postulate in question? Many thanks in advance! :)

[@devashishnegi8977]

Why did you stopped making any more content, i guess this would have been a great KZbin channel

[@johneonas6628]

Thank you for the video. :)

[@rysea9855]

5:25 they really couldnt be bothered to line it up properly

[@Gl1tchStudios]

I don't understand the last few graphics. When they add more "squares" the angles are obviously not 90 degrees anymore? What did I miss?

[@deepakmsmundayur3472]

Well, actually the angles are supposed to be on a hyperboloid and not on a plane. The figure shown is just a projection i e the one that you see if you just look from above. The angles are still right angles it's just that it appears distorted as you are looking from sideways.

[@aniksamiurrahman6365]

Everyone in the comments is comparing Non-Euclidean geometry with the Cthulhu mythos. But the very starting line of Cthulhu duo talks not about weird worlds but about the inability of humans to comprehend infinity. And there are men who walked this earth, who unraveled the knowledge of infinity among us. Those Cthulhu avatars who walked among us are known by many names: George Cantor, Gotlieb Frege, Ernst Zermelo, Richard Dedekind, Von Neumann, men who knew infinity.

[@shambhavir.7599]

This is so cool! I really like hyperbolic geometry!

[@coopergates9680]

+Shambhavi R. But is hyperbolic three-space more exciting than Euclidean 4-space...

[@therealohead]

Bruh we got us some Kevin MacLeod

[@realityversusfiction9960]

THE LENGTH OF A CIRCLES EDGE Using a 120-centimetre length of diameter multiply this by 3 1.The circle's edge length is 360 cm long 2. The circle's edge has 360 degrees of subdivision 3. The circle's edge has 360 degrees and each degree is 1 centimetre long THE SUMERIAN METHOD - CALCULATING THE AREA OF A CIRCLE Using a 120-centimetre length of diameter multiply this by 3 1. The Circles Edge is 360 cm long 2. 2. Multiply the 360 centimetres "Edge Length" by itself = 129, 600 square centimetres 3. 3. Divide 129, 600 by 12 = 10, 800 Square Centimetres to the Area of the Circle ARCHIMEDES: PROPOSITION The area of any circle is equal to a right-angled triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle. Archimedes Triangle The Circle in question has a 120-centimetre Diameter length 1. The base right-angle is equal to the radius of 60 centimetres 2. The area of the circle is equal to the above right-angle triangle, which has one side that is equal to the 60-centimetre radius, and the other to the 360-centimetre circumference of the circle 3. The 360-centimetre height of the right-angle is equal to 6 x the 60-centimetre radius length 4. (1r) 60 centimetres x (6r) 360 centimetres is 21, 600 square centimetres the area of the rectangle 5. Half of the rectangle is 10, 800 square centimetres 6. The area of the triangle is half of the 1r x 6r rectangle 7. Half of the 1r x 6r rectangle is 1r x 3r 8. (1r) 60 centimeters x (3r) 180 centimeters = 10, 800 square centimeters THREE TIMES THE RADIUS SQUARED 1. The Diameter of the Circle is 120 centimetres 2. The diameter x 120 centimetres gives, 14, 400 square centimetres to the square of the diameter 3. The 60-centimetre radius x 60 centimetres yields 3, 600 square centimetres to the square of the radius 4. The square of the radius x 3 gives, 10, 800 square centimetres to the area of the Circle SUMERIAN AREA: 10, 800 square centimetres ARCHIMEDEAN AREA 10, 800 square centimetres THREE TIMES THE RADIUS SQUARED AREA: 10, 800 square centimetres FOUR QUADRANTS 10,800 square centimetres Four Identical Results Is Not A Coincidence Twelve Steps From The Cube, To The Sphere Calculating the surface area and volume of a 6-centimetre diameter sphere, obtained from a 6-centimetre cube. 1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimetre’. 2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of the perimeter to the square face = Length 24 cm, Square area 36 square cm. 3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm. 4. Divide the cubic capacity by 4, to obtain one-quarter of the cubic capacity of the cube = 54 cubic cm. 5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm. 6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm. 7. Divide the cubes surface area by 4, to obtain one-quarter of the cubes surface area = 54 square cm. 8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm. CYLINDER TO SPHERE 9. Divide the Cylinders cubic capacity by 4, to obtain one-quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm. 10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere. 11. Divide the Cylinders surface are by 4, to obtain one-quarter of the surface area of the Cylinder = 40 & a half square cm. 12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere Confirmation by Weight Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grammes, prior to it being turned on a wood lathe into the shape of a sphere The Cylinder of the Cube would weigh 120 grammes The waste wood shavings would weigh 40 grammes Given that the Cylinder weighed 120 grammes The waste wood shavings would weigh 30 grammes. Note: And ironically you can also obtain this same result by volume, using Archimedes Principle www.fromthecircletothesphere.net.

[@SNoCappidona]

If you are looking for the edge length of the circle (circumference?) Why not multiply the radius by (pi)(r)^2. In the sumerian example for area... the units don't match. 360cm x 360cm/12cm = 10,800 cm... not squared cm. How can this be the area? Doesn't calculus make everything else irrelevant?

[@novakastmusic]

What, you haven't fit more space into a finite space already?

[@Juwar1974]

I'm a little confused by how the angles are measured. If the lines of an angle are straight, I understand its measurement. But if the lines of the angle are curved, how is it still 180 degrees?

[@randompersonwithcommonsense]

I'm surprised no one is giving props about beautiful that pronounciation was at 1:57

[@adamrpierce]

This was excellent. Thank you.

[@Spartan322]

I would've thought distorting the planes would in the end invalidate it as a square, that seems like redefinition of terms instead of changing the perspective.

[@nullmoore9943]

In a way... It *is* a square, but from the foundation of a plane that isn't flat. It's a square, but not by the euclidian definition (because it's non-euclidian geometry). So when you represent something that's non-euclidian in euclidian space, it gets distorted like a cube represented on a piece of paper does.

[@tanmayg7824]

Actually the fifth postulate is Plafair's axiom!

[@baileyesimone]

Great video..thanks for sharing!

[@cinema2748]

Euclidean geometry basically works in 2D

[@ratchet5980]

What is the song used in this?

[@kingarthurthethirdthst3804]

+Ratchet5689 Kevin MacLeod - Scheming Weasel

[@stripedhyenuh]

Kevin Macleod - Scheming Weasel

[@TheRealGigaVoltage]

Kevin MacLeod ~ Scheming Weasel

[@Despotic_Waffle]

Kevin MacLeod - Scheming Weasel

[@randomsubject2021]

Life.

[@MatthewGoodbred]

Great video, it helped me a lot. Keep up the good work!

[@elijahcaudle7365]

This was actually well put together but hearing a lil girl sounding person from old lady was a jarring transition

[@kaitlynperta7213]

increase the scale of the hyperboloid? what does that even mean?

[@raydencreed1524]

Kaitlyn Perta I know at least that hyperboloids come in 3 different varieties. You can have what is essentially an infinitely tall cylinder that bows inward at the middle, and bows outward more and more the higher or lower you go. The second type looks sorta like an infinitely large saddle, and the third is the one shown in the video. For the last type, as it extends off in both directions, it does so by tending towards some well-defined slope. So maybe what they mean is that the slope increases? I don’t really know, just a suggestion.

[@deepakmsmundayur3472]

Increase the curvature perhaps?

[@srpenguinbr]

What can it be used for?:

[@valarikjeffeerson7226]

general relativity is based on this

[@AndyChamberlainMusic]

Just to be clear: Euclidean, the spherical non-euclidean, and the hyperboloid non-euclidean geometries are all assuming that a "line" is the shortest route from one point to another point, and that a "triangle" (or any other polygon) is defined by the number of sides that it has. Is that correct?

[@AndyChamberlainMusic]

and how do you define a right angle on a sphere? Is the right angle now defined by the square?

[@albertartiles356]

@@AndyChamberlainMusic These are all awesome questions. A a line segment from a point a to a point b is the shortest path from a to b. Angles and lengths are defined via Riemannian metrics, Each of these spaces have associated Riemannian metrics that dictate their geometry.

[@NickDHammond]

Great vid, thank you!

[@cjramosm]

I found the background music too distracting. Otherwise great presentation.

[@DrewBorrowdale]

Very well done and informative 4/5 But needs more Shoggoths

[@js-yall]

I knew this was gonna be trippy so I clicked here

[@benjaminamkhanitsky1372]

I understood everything up until the last part, can you explain how those stretched out squares are still right angles?

[@alexanderm5728]

They're warped by the way the hyperbolic geometry is made to look on your flat computer screen. If you were to go to the centre of any one of those squares, that particular one would look square-shaped (although none of the others would).

[@albertartiles356]

They are actually not. Rectangles do not exist in hyperbolic geometry. This means in particular squares do not exists. That was just something they did not get quite right in the video.

[@jimmystavridis9200]

its lit

[@kevinstefan98]

Hyperbolic geometry is not "geometry on the surface of a hyperbola". The hyperbolic plane can't even be embedded in Euclidean 3-space (at least not continuously). The hyperbolic plane can be modeled on a hyperboloid, but shapes will be stretched out greatly. The only way to model the hyperbolic plane without distorting shapes is with a warped coral-like surface like this: theiff.org/images/hyperbolics/01.JPG Also, hyperbolic geometry doesn't break all of Euclid's postulates. Only the fifth one.

[@cparks1000000]

Can you give a reference to the fact that the hyperbolic plane can not be embedded (isometrically) in Euclidean 3-space?

[@celtman58]

Playfair form of parallel postulate given.

[@sabercrosby8128]

great vid. love the music what is it?

[@scotthappy6885]

I love myself a geometry

[@zech6857]

All these people struggling to wrap their head around the idea of non-Euclidean geometry and I'm struggling to find out if you're a guy or a girl

[@skyadventurer7574]

Shows how much I know, I don’t even get what he means by “changing the scale” the only way I can view it is by seeing the shapes being distorted making acute angles. Idk man this is all messing with my head I didn’t even make it through high school math

[@SMASivaPriyanka]

It's amazing

[@itsprivate3061]

why they dont teach interesting subjects like this on school

[@lazarbukva9794]

What is the name of song?

[@AdityaPrasad007]

I think saying "unique line segment" is wrong (kzbin.info/www/bejne/gKfWkJqKp7Vmrck), because you can have loads of line segments which form the same circle. The actual third postulate would be, 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

[@trevorphilips256]

thanks

[@drs_8108]

I new it was old vid because the music and the voice then I seen it was made 6 years ago

[@YouTubeDoesntSupportMyUsername]

this music is banging 💪💪💪💪

[@reedrichards8677]

but thats dimensional engineering, a key timelord discovery

[@chewie-v9546]

Not at all a math person, but this was really fuking interesting

[@serpentartist1348]

And Nikolai Ivanovich Lobachevsky was his name, hey!

[@РоманГогешвили]

I don't get it. The hyperboloid you shown in the video clearly has positive curvature. I personally thought that it only applied to surfaces wtih negative curvature. So how does this work? Why is a positively curved surface supposed to have propeties of a hyperbolic space? Damn it

[@albertartiles356]

You are entirely right. The hyperboloid shown there is not the same as hyperbolic space. We need to see that hyperboloid but instead of giving the metric inherited from 3-dimensional Euclidean space we need to give it a different Riemannian metric. With this new metric it will be negatively curved, but then the picture we have above will not be an isometric embedding of hyperbolic space.

[@downformaintenance]

that was very useful and well explained, thanks

[@digaddog6099]

I want someone to take a 4d sphere, project a 3d space over it, and make it a shooter

[@תומרשאול-ת5מ]

I liked it very much :)

[@alansmithee419]

I... Have a friend in Minsk who has a friend in Pinsk Who's friend in Omsk has friend in Tomsk With friend in akmolinsk Who's friend in alexandrovsk Has friend in pnepopaplovsk Who's friend somehow is solving now The problem in dnepopaplovsk. (Damn these Russian towns)

[@gustavowadaslopes2479]

These non Euclidean geometry just apear to be "weird geometry" based around different types of groups of of functions that form "lines", with everything else being basically about projections and comparisons to Euclidean geometry. Really interesting, but as many things in math, seems more exploratory than pratical. Nah, they always find a use for it.

[@samuraifugitivo]

Gustavo Wadas Lopes ptotally wrong buddy, without non euclidian geometry it would be impossible to do space travel, ask nasa which geo,etry they use to launch rockets and probes.

[@EyeLean5280]

Beautifully done, but spherical geometry was known in Ancient Greece and the Islamic Golden Age.

[@rishabhpurohit2663]

isn't a square defined by having right angles at every corner?

[@ashmarie5049]

Non Euclidean geometry makes me nauseous

[@jamesperry5208]

nice video

[@calcamez]

Cool!!

[@tobi1245ify]

So is it sort of like manipulating the shape of your 2d plane into a 3d object and finding new ways for 2d objects to try and interact with each other?

[@nullmoore9943]

In essence, yes. It's difficult to explain/understand because the world we live in is euclidian. Anything non-euclidian is literally a whole different reality.

[@arpitdhongade23]

5:28 That ain't look like a square tho

[@multiseiberpurz8981]

I got it at 5:00 but you forgot too call the rectangles squres!

[@PowerfulAtom111]

Bruh this is just how things naturally are in Wyoming

[@PkSage89]

thanks for the vid

[@hauntologicalwittgensteini2542]

This scares me... AND MAKES ME WANT TO LEARN MORE

[@Harrison3579]

Anyone here because of school? I’m just here because of KZbin recommendations

[@theimperfectgod7140]

Recommendations indeed 🗿

[@John-lf3xf]

Space time isn’t Euclidian

[@99bits46]

we know 5th postulate is true.

[@MOHAMEDButtonGames]

Why is this in my recommendation.

[@martink2875]

Aaaaargh, beautiful maddening chaos from different dimensions that my simple human mind just can't handle. "Thank you very much" my mind has been raped, while being caressed... I won't be the same ever again. Uaaaaiiig, phn'lui fhgtagn.