The Beautiful Story of Non-Euclidean Geometry
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- Опубликовано: 7 авг 2024
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In this video we are going to explore the origins of non-Euclidean geometry. We look back to Euclid and his infamous book the Elements, where he outlined an axiomatic approach to Euclidean geometry using five postulates. The fifth, the parallel postulate, has always been controversial, a consistent system of geometry that accepted the first four postulates but rejected the parallel postulate wasn't discovered until the nineteen century by Janos Bolyai, Nikolai Lobachevsky, and Carl Gauss who independently worked towards something called hyperbolic geometry, one of multiple non-Euclidean geometries. We specifically will explore the Poincare disk, one model of hyperbolic geometry.
Play around with the Geogebra page I used in the video ► www.geogebra.org/classic/sxak...
0:00 Euclidian Geometry and the Elements
2:12 The Five Postulates
3:21 Should the Parallel Postulate be a theorem?
4:29 Spherical Geometry
5:59 Janos Bolyai discovers Hyperbolic Geometry
6:48 Hyperbolic Geometry and the Poincare Disk
10:49 Resolving the Parallel Postulate Question
11:19 Angles and Triangles
13:44 Brilliant.org/TreforBazett
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I loved this. Great video, thank you.
Here is a little story for you:
I'm 67.5 years young and "retired" from a lifetime of Electrical Engineering. Lots of electronics and software projects. So, it's not surprising that math always came easy to me, except Geometry. I mean, in High School, if I'm remembering correctly, I got "A" in all the math courses, algebra to trigonometry to calculus. But, not in Geometry. Then I went to Engineering school and I got 4.0 GPA in all my calculus courses there.
So here I am after a very successful and creative Engineering career. So, what's the point in the story? How is this related to Euclidian and Non-Euclidian geometry?
Well, something peculiar that I have just noticed about my interests in retirement.
I always wanted to play the piano. So, that became one of the first hobbies that I picked up upon retirement. I bought a keyboard about 3 weeks ago and started to take online lessons. To my surprise, I have not felt that kind of joy with hearing music before. Somehow, now that my hands are creating the sound coming out of the speakers, the music is much more enjoyable. Somehow, I think I had neglected the right side of the brain throughout my life. Learning the piano has stimulated those noodles. But, here is the kicker. Now, for some reason, I had a need to go back to learning more math. But, not calculus necessarily. I went back to learning Geometry. So now I just ordered Euclid's Elements and I have embarked on my adventure into Geometry.
So what's the overall moral of the story? In life, it seems the mind never stops trying to develop itself. I think the enjoyment felt with learning to play the piano has stimulated the right side of my brain to the point that now it is trying to get better at learning something it tried to learn in the past. Anecdotal, but it's an observation never the less.
So, in my opinion, if there is something that you were not good at in life in the past, find a way to continue to stimulate the two parts of your brain for as long as you live. At the very least, you will be rewarded with happiness. I wonder if this would also help with decreasing the chances of getting Alzheimer or dementia later in life?
If you're a painter, artist, or musician, the right side of the brain should be well exercised. At some point, pick up math, logic, engineering. If it is the opposite, take up music, painting, sculpture.
Sorry for the rantings of an old dude. If you have read this far, health, happiness and a long life to you.
Wow, that was so nice
thank you
im late to replying to this but thank you for this beautiful story mr peter
The fifth postulate is a postulate precisely because it's not provable. The entire point is "if we take this for granted, this is the geometry that we get."
That first sentence is killing me
but euclid didn't know that, without such a counterexample it wouldn't be hard to imagine that it is provable by the other 4...
@@DarkPortall Euclid is the one that listed it as a postulate instead of trying to demonstrate it
Such an amazing demonstration of all the green, "parallel", hyperbolic lines! Thank you
Glad you liked it!
This is the best introduction of the non-euclidian geometry that I've ever seen. I was also very impressed that you mentioned Bolyai, who really came up with this idea first. Usually only Lobachevsky is mentioned. When I was a teenager, I read a book about non-Euclidean geometry - "Three destinies - one idea" by Anna Livanova, which impressed me greatly. Unfortunately, I am not sure if it has been translated into English.
Dr Trefor was basically my guiding star back when I was trying to learn my calc II stuff, and I always find your videos so helpful! Now getting to see you explore more in depth and interesting topics and even getting sponsorship, instead of being just a math video my prof provides, is so cool to see! Keep it up !^^
Thanks so much for sharing!
Thank you so much for your Geogebra file and such brilliantly presented explanation on the hyperbolic plane and lines that I have been searching for so long, your very clear pronunciation and delivery is an example to every speaker, well done and keep it coming.
Thats so interesting. I think it can be so important to talk and learn about the axioms underlying a theory and how and why they came to be. It really brings a perspective to maths or physics (and even philosophy,...) and opens up the possibility to think outside of the "box" aka axiomatic system and find different truths.
Indeed! The implications of different axiomatic systems not being "right" or "wrong" just different I think is really interesting.
Your explanations are perfect, please keep making videos, they're so awesome!
Your videos are always awesome! A little suggestion for even more awesome future videos: you could level down the gain for your mic to avoid clipping. Just normalize the level afterwards.
Thank you for doing this video. I did a project on hyperbolic geometry so this was nice to see.
These kinds of thought provoking videos are really very good, please make more videos like this!
Will do!
Always a great video! I enjoy your contents a lot dr Trefor, so thank you so much for creating enjoyable and insightful contents. Cheers!
Glad you enjoyed!
...Good day Dr. Trefor, Your entertaining presentation has made me very curious about Non-Euclidean Geometry. This forces me, in a positive way, to step out of my comfort zone... Thank you, Jan-W
Glad you enjoyed!
Thanks for teaching me multi variable calculus. Excited to watch this video
Really interesting! Can't believe this doesn't have more views
Good work!
Thanks for the video! I'm studying projective geometry right now. The parallel postulate in projective geometry is that parallel lines do not exist. Between any two distrinct lines there is exactly one point.
Love it! Instead of generalities, specific examples that explain a lot.
Please, more! Explain Lobachevsky's geometry. Tell us what's the difference between Lobachevsky and Boyai. Introduce us to Riemann (in this field). All through slow specific examples. Please add formulas when needed, but explain what they illustrate.
Thanks so much!
Very lucid explanation. Thanks
Liked the presentation. Knew about the first several therms of Euclidean Geometry from High School Geometry. The teacher had observed that the Greek had similar structures for Rhetoric and there were similar parallels in the efforts of reasonableness (?), which I thought was pretty interesting and sort of ironic. Explains some of the pained expressions of politicians that you see on TV news programs that you might see from time to time. Well done and thanks.
very good video. its inspiring
Brilliant video
This is an excellent video
was a bit confused at first, thats my focus... but you tied it so nice together
i was reading this in roads to reality by Roger Penrose yesterday
Oh fun!
Gauss: proves it in 1824
Taurinus: publishes it in 1826
Lobachevsky: formally treated the subject in 1829
Bazett: so let's give priority to Bolyai who did it in 1832
I was doing some Mascheroni constructions (absently making intersecting circles with my compass) and I first discovered concepts in hyperbolic geometry by accidentally drawing a circular horn triangle and trying to find out what it was supposed to be, and then marveling at how a triangle could have 3 zero angles. I might not have believed it if it wasn’t embedded (inscribed?) in a Reuleaux triangle, which has all angles of 120 degrees, and I could see that their vertices could be connected by line segments to make an equilateral triangle, “taking the average,” if you will.
Are there any issues that come up because the space you defined is bounded whereas Euclidean geometry is not?
If we bound Euclidean space wouldn’t there be potentially multiple lines that don’t intersect the original line?
The poincaré disk isn't actually bounded, it just looks like it is because of how we choose to visualise it. In other words, distance is defined in such a way that the distance between any internal point (not on the boundary) to any point on the boundary is infinite. As you approach the boundary, distances that look small actually become bigger and bigger so that you can never actually reach the edge.
For your second question, unfortunately the answer is no. The 5th postulate states that given a line and a point not on the line, there should be only one possible line through the point which never touches the original line when extended indefinitely. That last part is the key - we can't extend lines indefinitely if our space is bounded. Note that this is just the 2nd postulate. A bounded Euclidean geometry would already reject the 2nd (and 3rd) postulates before we even get to talking about whether it even makes sense to reject or accept the 5th postulate (given that we don't have infinite lines). So it doesn't satisfy what we would want unfortunately.
Essentially what you're looking at is a map projection of the hyperbolic plane, and while this projection is angle-preserving ("conformal") it does not map straight lines onto straight lines (there's a different projection that does this, but it doesn't preserve angles) nor does it preserve distances, and in fact it maps the infinite hyperbolic plane to a finite area, the _interior_ of a circle. Note that the boundary circle itself is not part of the space, so you can always go closer towards it but never reach it.
Great treatment of this subject.
The best video I've seen on this subject. Also the best pronunciation of Bolyai's name I've heard, though János is pronounced yaw-nosh.
Thanks! Also classic, I specifically googled how to pronounce Bolyai but totally wiffed on doing the same for János - oops!
Please make more of these videos
Great content
thank you!
I've got a monstrous Diff Eq final in 12 hours and this is SO much more fascinating than studying..!
hahah good luck:D
More about hyperbolic geometry please
The most wonderful thing about hyperbolic geometry, to me, is that it is the same as the geometry of velocities, or of constant-velocity worldlines through a given event, in special relativity.
Another great video! Have you made or would you consider making a video on elliptical geometry (Riemannian) or add that on to this one?
That would be a fun one, I'd love to do that!
I look forward to it. By the way, your video on the limit of sinx/x = 1 as x --> 0 is the best I've ever seen on that topic! Keep up the great work!
Would you please make a video explaining Wittgenstein's concept of a truth-function in the Tractatus? Also, thanks so much for your videos.
Very beautiful geometry indeed!
the visualization was amazing
I'm building a faux 3d first person shooter video game, and all of this has been a huge part of making a cohesive world with very limited technology. Room over room, teleporters and non Euclidean geometry have become my absolute best friends. Unlimited cool ways to utilize it
Cool!
Nice video. Where are you from bro?
this seems to touch on explaining how relativity/perspective play a role in the fabric of the universe itself
Nice video. A small thing about "lines" in the Poincare disk model: When the two points and the center of the circle line on one line, that straight line is the "line" through them in the model, not an arc of a circle orthogonal to the original circle (which does not exist).
This is precisely the arc of a circle with “infinite” radius. :)
Hey just a heads up, your mic is clipping.
Great vid!
Great video! Your mic is peaking quite a bit in your Geogebra portion.
Thanks! Gah, yes something went stupid with the settings, tried to recover a bit in post but still far from ideal:/
Found you yesterday. Binging since.
What SW did you use for demonstration?
Dear Sir, you have so nicely and precisely explained the subject of Non-Euclidean Geometry. I have a question, why is this geometry called “Hyperbolic” Geometry?
I didn’t make the connection in this video, but locally you can imagine this geometry like living on the surface of a hyperbola, sorta like a pringles chip.
@Dr. Trefor Bazett Thank you Sir
We exist on a cardioid. As all paths converge towards event horizon
And any way from singularity requires a boost. Whereas the surface flow is towards singularity.
I'm wondering if this disc is really an infinite disc, or if it's actually a finite disc like shown in this example...
It is infinite, just that the projection shrinks objects at the boundary.
Thank you very much. Now I see how it is defined. Did Lobatschevskij the same?
And heared that there is a geometry which uses only the first 4 axioms. Is this consistent? If I remember right it is called "absolute geometry".
Best regards
Bernd
Indeed he did! Yes, "absolute geometry" is contained inside of both Euclidean and Non-Euclidean geometry.
Lobachevsky published his work on "imaginary geometry" earlier.
This man literally taught me all of calc 3🤣
For the first time, I think I have understood these postulates.
Great video as usual. I can now see how amazing is the artwork by Escher shown at 0:21 !
Anyone interested in seeing what it would be like to live in hyperbolic space should check out a game called Hyperbolica.
Or Hyperrogue.
2:45 "If you start with a line and then consider some point not on that line, then there is at most one other line that is going to never intersect the original line. That is we are saying that given some starting line you can always construct a parallel line to it."
Did I misquote you just now, and if not, is that what you meant to say?
Indeed. To be more precise, there is a construction using only the first four axioms alone to create one parallel line, so what the axiom is really doing is rejecting the ability to make multiple ones, hence the "at most". Technically this a more modern formulation of the original P5 called "Playfair's Axiom" but it is equivalent to Euclids original P5.
Ahh! Thanks.
@@DrTrefor As you know (but your viewers might not), there are many "equivalent" postulates (I can think of eight) that satisfy P5, but this one (about the unique parallel) is not the one that Euclid chose to articulate in the Elements. For those interested, here is Euclid's P5: "That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." This formulation makes one understand why, for many millennia, mathematicians thought it must be a theorem; it sounds much more like a theorem that needs proving than a self-evident postulate! This video is an excellent introductory presentation, Dr. Bazett. And Geogebra is a great tool to use with the Poincare disc as well! I teach my students how to build the Poincare disc from scratch and prove that it functions as a model of hyperbolic space, and they really enjoy the project. Well done.
I don't think the fifth postulate has been disproved, rather it must be qualified that it is True only on flat planes. I'm assuming the curved lines in the program act as if they are on a curved surface, but since the image is on a 2-d screen an accommodation must be made. Am I wrong? Are curved lines allowed in Euclidean geometry?
Dr. is this the geometry that describes the space-time continuum? I've heard sir R. Penrose and I intuitively see a connection between his theory (and how he sees infinity) and this presentation. Not a mathematician here, just curious. Please see Escher's MC Angels and Devils.
The universe isn’t hyperbolic, but in some sense it could have been and that would have been consistent
@@DrTrefor is it possible, in some sense, to map out eucledian geometry into the hyperbolic space. My guess, after seeing your video, is that a line in the eucledian space might as well be an area in the hyperbolic. Maybe time is just a property of an special geometry, and the distinction between what it was, is and will be, is but a mapping function. Also, I cant believe I have the honor to talk to you 🤣 you're an incredible educator
What about the right angles postulate?
Because angles are defined by taking the tangents and then Euclidean angles, it is true for the same reason as for Euclid
Also, while spherical geometry does not satisfy Euclid's first four postulates, elliptic geometry, in which you take a sphere but then define the "points" to be *pairs* of antipodal points on the sphere (and the "lines" to be great circles), does. This space is also called a projective plane.
Around 8:50, your circle gets degenerate (correct word?) when A and B are in the Euclidean sense on a diameter of the circle. Your circle's radius tends to infinity as B approaches the diameter line A is on.
That is ok for our purposes. Basically we are considering the straight line a "circle" of infinite radius.
What if there is another cool non-Euclidean geometry that can be discovered? I mean, we can just make any 3D shape and use its surface as a new geometry but what if there is something way more creative?
Also, since spherical geometry subtracts one square from the corner neighbourhood and hyperbolic geometry adds squares to it, I think I need something that will affect the corner neighbourhood in some other way. But I can’t just think of something besides adding and subtracting. Maybe multiplying? Nah, that’s pretty much a hyperbolic thing
Are parallels constructible?
🙏🏽!
How does geometry on a sphere correlate with euclidian geometry
God, Euclid and his elements created mathematics, in the end he wasn't wrong either, in fact he was right, he did well to include the fifth postulate, what a great Euclid
Amazing indeed! And even better that his story wasn’t the only story to tell
Euclid missed multiple other postulates, to the point that the very first construction (an equilateral triangle) is invalid, as the Euclid's postulates don't imply that the circles intersect. It's easy to notice if you consider geometry on Q² - it fulfils all the Euclid's postulates, yet it contains no equilateral triangles.
Of course, it is true, among others the truth, but that case of Q^2 is not a counterexample I think, since not at every point, there is a circle of given radius, for example, of radius 3, the 3 postulate fails, And that already leaves Q^2 without Euclidean geometry, I think that, it can be demonstrated with topology, assuming a bijection(homeomorphsim) of the Euclidean line, to the real numbers, the third postulate, is thatanyway good observation
So, when Euclid was writing and thinking the Elements, he actually did not confine his geometry onto a plane, did he? I used to think he did, and thought his geometry was completely planetary.
Can you please present the same way the Lobachevsky geometry.
TIL.... Geogebra. Thanks Dr.
What are the new definitions of angles and straight lines ?
He literally answers that in the video
Does this mean you can map R'Yleh?
U r good.
oh so the sum of angles of geometric shapes in hyperbolic geometry depends on where we draw them and how big they are?
Exactly!
This universe is so obviously hyperbolic. All paths lead to a black hole eventually.
And then pops out again in deep voids.
The inversion of the circle. Only the circle and the periphery join.
This is the snoop diggity.
Can I someone tell me why it's called "hyperbolic" geometry? What does this word really mean here? Why is it simply called non-Euclidean geometry or spheric geometry?
If the angles at which the euclidian lines are intersecting the circle boundary are all 90 degrees, as per the postulate, then all the visually different lines referenced to argue against it are actually just the same line, no? It's just the perspective changing. The fact that "any" one of those green lines is valid is irrelevant when they're all the same line
A pedagogical tour de force.
Much of this just doesn't sit well with me. It seems severe spatial limits are placed on this system and consequentially lines are defined differently. Further there are strict limits on the length of finite lines and consequentially no infinite lines. So is the absence of a fifth postulate truly the only difference between this and Euclidean geometry? Do I assume that employing tangents is a useful teaching tool and not actually necessary to define angles? And am I to suppose that by "valid" you mean consistent?
In Non Euclidean we don't talk about straight lines so how we are talk about angles which formed by two rays !!!
Also, how we talk about parallel lines which is a features of straight lines only !
I would say we are “defining” what we mean by words like angles and parallel and so forth in this new context, so they are well defined even if different from our familiar usage
@@DrTreforso what is the definition ?
That's Playfair's postulate, not Eulid's.
If there is a consistent geometry with the 4 axioms but disproves the fifth... doesn't that mean euclidean geometry is not consistent as its 4 axioms do not agree with the fifth?
I’d say it means Euclidean geometry is not the ONLY geometry. The five Euclidean axioms are consistent but aren’t the only consistent choice.
Does hyperbolic geometry assume the four euclidean axioms excluding the firth, or did I misunderstand that? Thank you for responding, your videos are really informing.
@@wqltr1822 Yes
Isn't the plane old 3D space an example of a space where the parallel postulate is false? There are infinity many lines which pass through a random point which are parallel to the original line. This is because there is always a plane which is parallel to a line. And there are infinite lines on a plane?
The parallel postulate is within a plane. If you have a line and a point (not on the plane) in 3D then there is a unique plane containing the line and the point and a unique parallel line within that plane.
For 3 dimensions you have unique parallel planes but that can be constructed by considering all lines within the plane and their parallels through the off plane point - no new axiom needed.
Thank you for a nice presentation. However it is confusing when you describe an arc as a line. A line has no curvature whereas an arc is a section of a circle. Ultimately it is possible that Euclidean geometry coupled with some of the geometry of Archimedes can represent all shapes in nature. All shapes in nature are a product of the structural alignment of the charges of protons and electrons in atomic and molecular configurations. Perhaps this indicates there is no need for non-Euclidean geometry. The goal is to establish the natural geometry, and non-Euclidean geometry seems to be a diversion from this goal. This notion is supported by Johanne Bolyai in his Science of Absolute Space: Section 15. "Weighing Sections 13 and 14, the System of Geometry resting on the hypothesis of the truth of Euclid's Axiom XI is called sigma; and the system founded on the contrary hypothesis is S." Then in the last paragraph of Section 34 Bolyai states: "It remains, finally (that the thing may be completed in every respect), to demonstrate the impossibility (apart from any supposition), of deciding a priori, whether sigma, or some S (and which one) exists. This, however, is reserved for a more suitable occasion." Bolyai knew that mathematical reasoning alone can not decide what the geometry of nature or matter is because it is the work of the physical scientists that will reveal this. Thus far organic chemistry and inorganic chemistry reveal the tetrahedron to be the fundamental structure of natural design. It appears that Euclidean geometry has pulled ahead of non-Euclidean geometry and that the latter is a function of the former only this has not yet been documented.
soo.. this is why we could be living in a hyberbolic shaped universe, but still use Euclidean geometry, because the resolution we dealing with, the difference is pretty much 0
1:04 "postulates and the idea here is this is a statement that is so completely obvious that we're all just going to agree that it's true" the "so completely obvious" please don't limit your mind in this way, axioms should be more like contextual frame work, the speaker establishes to convey the idea, think about how this changes with out it "the idea here is this is a statement that is that we're all just going to agree that it's true" a idea does not have to be obvious, a example to this is the imaginary number that is a mathematical framework way to deal with a square root of a negative number most upon first encountering imaginary number many do not regard it as obvious.
How does finding a different geometry that rejects the 5th axiom even relate to Euclidean geometry? Let alone prove that the 5th axiom can’t be proven from the other 4 in Euclidean geometry.
What’s fascinating is that Euclid knew or intuited that the Fifth Postulate _had_ to be a postulate and could not be derived from the other four postulates.
More likely is that he didn’t assume the affirmative in the absence of a proof.
A very rebellious son I see…
The only thing making me uncomfortable with applying the postulates to the hyperbolic space is that the lines can't truly extend indefinitely, only to the boundary. Why isn't this breaking the rules?
Edit: thinking about this more, maybe I can answer my own question. The extension truly is indefinite because the space is an open set, not a closed one. You can always draw the line getting closer to the boundary, but it isn't allowed to actually terminate at the boundary.
So the part I didn't get into in this video is that we define distance in such a way in the poincare disk such that you can never get to the boundary circle in finite amount of time. It really is a circle "at" infinity in this model.
el quinto postulando está mal interpretado y un triángulo es una figura geométrica que está compuesta de líneas recta
Y para que interpreten bien los postulados recuerde que una línea recta la puede extender infinitamente como concepto pero si usted limita la superficie plana donde construye la línea recta ya es otra cuestión tener en cuenta la forma de la hoja en el espacio como limitante de los conceptos mentales
La versión que el profesor comenta es una afirmación equivalente al quinto postulado original conocida como El axioma de Playfair, a veces usado porque es más fácil de explicar y entender.
No lo menciona en el video pero la forma de medir los segmentos de recta en el Disco no es la misma que la que usamos comúnmente, con la nueva métrica, la distancia de un punto a la orilla del círculo es infinita, así que en esencia las líneas son infinitas.
Al final todo se ajusta como se quiere.
@@43hi voy a tratar de ser más claro cuando se nos enseña que es una línea recta o que es una línea curva y nos dan ejemplos y características de ellas por ejemplo una curva elíptica no es igual a un circunferencia y si te piden dibujar un segmento de línea recta en una superficie plana lo podrás hacer pero sobre la superficie curva de una esfera no pues las líneas rectas no se puede hacer su representación gráfica en esta superficie hay que tener algo muy claro los conceptos mentales los usamos para tratar de la mejor forma de interpretar la realidad pero no implica que de cierta forma se hacer que a esta pues nosotros mismos asemos parte de ella
@@jhonnyangarita409 Si bien la matemática se inspiró por muchos años de las interpretaciones físicas, la matemática moderna no necesita de éstas para generar o re-interpretar sus conceptos.
En este caso, línea es aquello que satisface los postulados, sin importar que para nuestra perspectiva sea diferente.
Se que suena raro decir que esos segmentos de círculo son rectas, pero es que lo correcto es decir que son rectas Hiperbolicas, obviamente no son como las que siempre nos enseñaron.
Y al final, esto solo es un Modelo para visualizar dicha geometría, si vivieras en el espacio Hiperbolico verías esos arcos como nosotros vemos nuestras rectas.
@@43hi esa es la cuestión para hacer teoría hay que ser imaginativo pero realista no es poco razonable saber que en el espacio se puede desplazarse en línea recta o hacer desplazamientos en cualquier otra forma distintas o sea infinidad de formas de desplazamientos en el espacio y pensar analiza lo que hay en el espacio con sólo el concepto de línea recta por ejemplo que tan inteligente es pensar hacerca de la superficie de una esfera con nuestro concepto mental de línea recta
por ésto es que los conceptos de topología son muy cuestionables cómo decir que la cinta de Moebius tiene una sola superficie yo me imagino que esté señor no se la imagino
si no qué la tuvo entre sus manos y realizó sus experiencias en ella
una cinta de papel por más que tú la gires en el espacio nunca va a dejar de tener dos superficies y lo que sucede cuando la recorres y se regresa al mismo punto de partida es porque ése recorrido también se hace en el espacio
@@43hi yo pienso que a los matemáticos cómo que les afectó la historia de Alicia en el país de las maravillas
K so I’m both stoned and only halfway through the video but I think you need lines before parallel lines
What is non-Euclidean geometry
Thales
Why didn't you mention Lobachevski's name for at least once?
I mean talking about non-euclidian geometry and omitting his name is just like talking about German classical music and not mentioning Bach. It raises some brows
I'm learning geometry from Kermit the Frog.
😅
why is it called hyperbolic geometry if it is based on circles? Wouldn't it make more sense to call it circular geometry?
At 3:14, Dr. Bazett knows he lives in a 4-dimensional space, and he is even holding a 4-ball analog in his hand, but yet he still accepts that Euclidean's first axiom is true. This sounds suspiciously like something a flat-earther would say. ;-)
Man, y’all really got confused about sliding a line? Y’all think sliding a line over would make it intersect with the original
Why did you omit the coolest thing about non-Euclidian geometry - the fact that we live in a non-Euclidian universe ?
This is the first comment
This is my first reply!
And thus a counting system was invented.
you cannot create angles with curves. Using Euclidean straight lines to define something in hyperbolic geometry means you're operating in Euclidean geometry.