Really did not want to make a 40-minute video, but also don't know how to split it up into two or more videos, so here we are. Theorema Egregium is my all-time favourite theorem, and it was traditionally proved using Christ-awful (a parody of the actual name Christoffel, because these symbols are tedious to calculate) symbols. So when I know of this geometric proof in Needham's book, I have to animate it.
want an even more horrible proof of Theorema Egregium without Christoffel symbols? Check Hubbard's Vector calculus, Linear Algebra and Differential forms.
Very satisfying proof, and I think a lot of it can be credited to how well-organized your presentation is! Also I wasn’t aware Needham wrote a second book!! This is excellent news (well I guess I’m 2 years late) and I can’t wait to get a copy
Thanks you! I am starting down a differential geometry rabbit hole and the Theorema Egregium is one of the things motivating me to study this field. I'm sure I will find this video very useful in the year to come
I read in one of your posts that you probably won't make videos on this and similar topics since they aren't performing well (i.e. fewer views). I am sure I am saying this on behalf of many other people here that please DO NOT STOP making these incredible videos. These videos mean a lot to people like me. And this kind of content deserves to be shared on a platform like RUclips with as many people as possible. All the best for your PhD journey and I look forward to more videos on this and similar topics in the future. Thank you for sharing and spreading the joy of learning maths.
Great video! Early on I could tell this was inspired by Needham’s book. This is a good way to understand this proof without going through most of that book to get there.
Never expected I'd be able to follow ideas behind this important theorem for maps and eating of pizza. Thanks for spreading awareness, and for animating and presenting this lovely proof.
This video is really nicely animated! I also appreciate keeping the definitions+visualisations on screen like at 7:02 instead of jumping to a limit slide
I usually don't comment but this video should be pushed forward. Thank you for the efforts to make this more known. It is a bit heavy for general youtube unfortunately. I needed some time to listen through even though I am a math/physics major. .. Hopefully I come back to this to follow thoroughly. Thank you for put this out.
@Intro's [no such map exists because it must preserve curvature]. That may be true mathematically, but not practically. If you search something like "airplane routes", there are traffic hot spots(places where people want to have a map) and cold spots("lol, I didn't even notice that country was missing") so you could use those places to cut some seams to relax the necessary curvature, like in UV mapping, until the area/angle errors are less than x%. After this whole process you can get an accurate flat map of the whole Earth*. (*With some complaints from Pacific and Atlantic fish, but Greenland should have the right shape and size)
a) Why is the curvature unitless? Shouldn't it be in inverse-square length units? b) If two shapes have the same curvature everywhere, are they necessarily the same? Because I can imagine that maybe the area on the Gauss map is stretched in one direction and compressed in another, cancelling each other out, resulting in the same curvature despite being different shapes c) Once you've proven that straight lines (minimal-length property) are preserved on a hypothetical area- and angle-preserving map, the rest of the proof is redundant. Any triangle on a sphere has a total angle greater than pi (indeed, the difference is the area in units of [earth's radius]^2). On a flat map, if you construct a triangle with straight lines and the three relevant areas, there must be a discontinuity somewhere. However, if you assume that the continuous points form contiguous sections of the map with a non-zero area, then you can put a triangle in that space, and repeat the argument, showing that one of the points aren't continuous, contradicting the premise. This means that given any area of the map, there must be a discontinuity somewhere; that is, they're dense on the map, which makes it completely useless
What a wonderful video! This theorem well deserves the name. And 2-d creatures actually can figure out if the world they're living on is flat or not. :P
31:52 Gauss map preserves parallel transport Geodesics are preservedif areas and angles are Prallel transport preserved if areas and angles are. Holonomy of loop on sphere=area enclosed
I use Geogebra an inordinate amount as well so I'm glad to know I ain't the only one And personally Egregium is probably near my favorite proofs/theorems, but not quite the top, I think Gödel's First Incompleteness Theorem takes that spot for me by being such an intricate beautiful messy wrench in the hopes of philosophy of math
2:41 I computed accurate and timely firing data both manually [Charts n Darts] and using automated systems like a computer, for artillery. And yes, both Curvature of the Earth and Rotation of the Earth was just two functions of the many needed to deliver steel downrange for the Queen. This is why no flat Earther will ever be able to have safe, effective, accurate, and timely artillery fires. With out the functions for curvature and rotation of the Earth, the data would technically be unsafe to fire.
Interesting topic! Last video I watched about this was from the guy who's building Earth in Minecraft (he and the other builders spent a lot of time picking the best projection for the job, since Minecraft map is flat).
Here's my attempt at the problem. We want an angle preserving and distance preserving map. In other words an isometry between the 2 sphere and the plane. Note that isometries are continuous. Metric spaces have a metric that induces a topology making them topological spaces. Preserving the metric implies preserving the topology induced by the metric and preserving topology means the map is continuous. This means that we want to find a homeomorphism between the 2 sphere and the plane but this is impossible because they are not homeomorphic spaces.
Great video, but I didn't understand the proof for why lengths and geodesics are preserved in such a map. In particular, if a triangle on one manifold is mapped to another shape on another manifold, why is it a triangle and why are the side lengths the same? This seems to not be correct in general if the curvature is different, and you can't rely on it being the same because that's what you want to prove, or do you deal with triangles which are small enough for this to not matter? Thanks!
Is it possible to create a compromise map to be more mathematically rigourous? the Nat Geo projection, for example, looks nice but it's neither area nor angle preserving, can we somehow quantify that deviation?
Of course you can. First, project the errors in angles and areas into the same unitless space, each scaled to the importance you assign to errors in each. You also need to select the norm to combine these errors, and it should be well-behaved w.r.t. differentiation, so L₂ (the sum of errors squared) is the most practically used. The choice of the norm affects the weights, but you can always adjust them to the norm. You'll get an error at each point in the domain, but you need a single scalar error to minimize, so you need to "average" the error, possibly weighted, if you want some regions be more precise in whichever you've emphasised when choosing weights, areas or angles, at the expense of other regions. Generally, you integrate your weighted scalar over the whole domain, normalized by total area, E = ∫ w(P) e(P) ds/S = 1/S ∫ w(P) e(P) ds, where e(P) is the normed error at point P (the tip of *P,* the position vector), and w(P) the weight to make some regions more precise at the expense of the others: the larger w(P), the more “important” is to make the error smaller at P. Then you can minimise the scalar error E of the map M(P) by the usual variational calculus.
Depends on what you mean by "map" you could cut literally peel the map off the globe and cut it many times and paste it onto a flat surface. Does the definition of map really say there must be a contiguous connections?
You still cannot paste it onto a flat surface without stretching the surface in some way, though. In that case, either areas or angles will still not be preserved.
@@mathemaniac Well, you can preserve them within a tolerance e.g. if you require maximum 1 deg of angular distortion you have a trade off between the percent by which the area is distorted and the number of cuts you need to make (possibly cutting continents in half or more for some %).
Example: start at the North Pole, head towards the equator, go 1/4 of the way around the equator, and then return to the North Pole. This is a triangle with three right angles. Such a shape cannot exist in Euclidean geometry, so distortion can’t be avoided.
Is there any results about what's the minimum angle distortion necessary for preserving area? Often when I meet those kind of "no free lunch" kind theorems where you need to choose between two ideal scenarios, the first thought in my head is always: ok, what's the optimal trade-off then? Instead of trying to be perfect in both respects, what's the smallest distortion I need in one of them I need to show in order to be perfect in the other?
“what's the smallest distortion I need in one of them I need to show in order to be perfect in the other?” - By “perfect” you probably mean exact? Then, the answer is, as large as it takes to be exact in the other. You have to choose one of the two projections shown at the beginning: one exactly preserves angles, the other areas. An infinitesimal deviation in one leads to deviation in the other, however small. In your lunch analogy, you're asking how much, at most, you can take off my lunch and add to your, such that mine doesn't change but yours gets larger. The answer is, of course, zero.
@@cykkm my understanding is that there are many projections that preserve area, and that Gall-Peters is just one example. Similarly there are many projections that preserve angles, and Mercator is just one example. I assume that among the projections that preserve area, the amount of angle distortion is not necessarily the same for all of them. If that's true, one of them will be the one who distorts angles the less.
@@RafaelSCalsaverini I think I understand what you mean now, sorry about the confusion. Indeed, that's correct: if you have a map preserving one variable, you can transform it to be _worse_ in the other than it is, according to some metric of "goodness" (in the sense of projecting the variable into a metricized set); a reverse transform could improve it. Therefore, _improvable_ maps preserving one variable exist, according to the same criterion. The Mercator projection isn't _practically_ optimal, as it results in an infinitely tall image. Think of angles near the poles: if you're approaching the, w.l.o.g., North pole at a constant angle, you'll never reach it; instead you follow a logarithmic spiral, shortening the distance to the pole at a constant multiplicative rate per revolution _r,_ strictly between 0 and 1-think of infinitesimal similar right triangles whose adjacent sides are the globe's parallels and hypotenuses are infinitesimal elements of your path, in a plane, ignoring curvature of the globe, as you are looking at an arbitrarily small patch around the pole, to see why (parallels become concentric circles around the pole; r is the tangent of the angle). Correspondingly, to preserve this constant angle in the image, assuming left and right sides of the map glued together into a cylinder, the path will be a helix extending upwards to infinity. Real maps have to be either truncated at a certain latitude or distorted, losing the angle-preserving property. The G.-P. projection doesn't have such a defect, as areas monotonically decrease to asymptotic zero as you approach a pole on a monotonic trajectory, which does cause a blow-up (although angle error does blow up). For “optimality,” you have a freedom to choose how to define it: the error is a scalar field over the domain, but you define “optimality“ in the sense of the minimal _single scalar_ error. “Averaging,” i.e. integrating the error per area element (which is preserved, we're lucky!), as its L₁ or L₂ norm both seem to be a sensible way to compute a scalar error. I have a hunch that integrating raw, signed error might be unhelpful: because of the sphere's symmetry, it's likely zero. See if the integral of the (divergent) angle error near poles per unit area converges, using either of the two norms. If it is, then G.-P. is optimal in the sense of the unweighted integral, the “average error,” without any arbitrary choices except the norm. If it's divergent in L₁ but not in L₂, even better: the latter is arguably the most natural norm choice in Euclidean geometry.
Finding an optimal map obviously depends on the aims, i.e. what best fits the intended purposes for its use. In case that can be confined to some dedicated part of the globe, a map projection is chosen that best fits that part. About every country has elected some particular projection method (together with a choice of parameter values) to produce its offical topographic maps. Combining such local maps as patches in an atlas is familiar to every school kid. That fundamental idea is also how mathematics started to approach the study of more intricate "manifolds".
As far as I understood in chapter 3.4 was shown that holonomy of the curve on the sphere is equal to the area enclosed by the curve. But how do we decide which part of the sphere exactly is enclosed?
@@mathemaniac But say that I walk in some circle around the poles. I can argue that I'm walking counterclockwise around a small circle around the south pole or that I'm walking clockwise around an enormous circle around the north pole. How can one decide which way I'm walking?
Despite the earth not being perfectly flat, the reason all maps are wrong is because the king does not want we-the-sheeple to know how reach his castle and he wants to conceal land from us ----- duh. Let the sheeple argue amongst themselves over the geometric theorems and projections we shine on the screen.
majority of the videos/lectures are great! but unfortunately the artificial voice is a disgusting hermaphrodite voice... it would be more enjoyable if a male or a female voice narrated the video....
Yes, it's called a globe, and you can have it on your phone (Google Earth). These are old problems or mathematical curiosities. They have no bearing on our modern world.
Have you ever thought about how the maps in Google Earth and ubiquitous GPS based navigation systems are constructed? Look up "UTM" and learn how relevant this video's subject really is!
Really did not want to make a 40-minute video, but also don't know how to split it up into two or more videos, so here we are. Theorema Egregium is my all-time favourite theorem, and it was traditionally proved using Christ-awful (a parody of the actual name Christoffel, because these symbols are tedious to calculate) symbols. So when I know of this geometric proof in Needham's book, I have to animate it.
All-time favorite theorem? That's a big statement 😂
Well, it's true what can I say.
I like longer videos better, thank you for your effort ❤
Excellent presentation! Even better as a single video. Thank you!
want an even more horrible proof of Theorema Egregium without Christoffel symbols? Check Hubbard's Vector calculus, Linear Algebra and Differential forms.
Very satisfying proof, and I think a lot of it can be credited to how well-organized your presentation is! Also I wasn’t aware Needham wrote a second book!! This is excellent news (well I guess I’m 2 years late) and I can’t wait to get a copy
Thanks you! I am starting down a differential geometry rabbit hole and the Theorema Egregium is one of the things motivating me to study this field. I'm sure I will find this video very useful in the year to come
I read in one of your posts that you probably won't make videos on this and similar topics since they aren't performing well (i.e. fewer views). I am sure I am saying this on behalf of many other people here that please DO NOT STOP making these incredible videos. These videos mean a lot to people like me. And this kind of content deserves to be shared on a platform like RUclips with as many people as possible.
All the best for your PhD journey and I look forward to more videos on this and similar topics in the future.
Thank you for sharing and spreading the joy of learning maths.
Another great video! You've really been pumping out top notch videos recently, thank you!
This was such an elegant and intuitive way to lay out the argument, thank you!
Great video! Early on I could tell this was inspired by Needham’s book. This is a good way to understand this proof without going through most of that book to get there.
One of my favorite theorems! Thanks for the video and the wonderful proof
That was breathtakingly beautiful, thanks for sharing this proof!
Never expected I'd be able to follow ideas behind this important theorem for maps and eating of pizza. Thanks for spreading awareness, and for animating and presenting this lovely proof.
This video is really nicely animated! I also appreciate keeping the definitions+visualisations on screen like at 7:02 instead of jumping to a limit slide
I usually don't comment but this video should be pushed forward.
Thank you for the efforts to make this more known.
It is a bit heavy for general youtube unfortunately.
I needed some time to listen through even though I am a math/physics major. ..
Hopefully I come back to this to follow thoroughly.
Thank you for put this out.
@Intro's [no such map exists because it must preserve curvature]. That may be true mathematically, but not practically.
If you search something like "airplane routes", there are traffic hot spots(places where people want to have a map) and cold spots("lol, I didn't even notice that country was missing") so you could use those places to cut some seams to relax the necessary curvature, like in UV mapping, until the area/angle errors are less than x%.
After this whole process you can get an accurate flat map of the whole Earth*.
(*With some complaints from Pacific and Atlantic fish, but Greenland should have the right shape and size)
This was great, I finally feel like I understand the course I took about this last semester!
Very nice visualization!
a) Why is the curvature unitless? Shouldn't it be in inverse-square length units?
b) If two shapes have the same curvature everywhere, are they necessarily the same? Because I can imagine that maybe the area on the Gauss map is stretched in one direction and compressed in another, cancelling each other out, resulting in the same curvature despite being different shapes
c) Once you've proven that straight lines (minimal-length property) are preserved on a hypothetical area- and angle-preserving map, the rest of the proof is redundant. Any triangle on a sphere has a total angle greater than pi (indeed, the difference is the area in units of [earth's radius]^2). On a flat map, if you construct a triangle with straight lines and the three relevant areas, there must be a discontinuity somewhere. However, if you assume that the continuous points form contiguous sections of the map with a non-zero area, then you can put a triangle in that space, and repeat the argument, showing that one of the points aren't continuous, contradicting the premise. This means that given any area of the map, there must be a discontinuity somewhere; that is, they're dense on the map, which makes it completely useless
What a wonderful video!
This theorem well deserves the name. And 2-d creatures actually can figure out if the world they're living on is flat or not. :P
31:52 Gauss map preserves parallel transport Geodesics are preservedif areas and angles are Prallel transport preserved if areas and angles are. Holonomy of loop on sphere=area enclosed
I use Geogebra an inordinate amount as well so I'm glad to know I ain't the only one
And personally Egregium is probably near my favorite proofs/theorems, but not quite the top, I think Gödel's First Incompleteness Theorem takes that spot for me by being such an intricate beautiful messy wrench in the hopes of philosophy of math
Awesome video!
2:41 I computed accurate and timely firing data both manually [Charts n Darts] and using automated systems like a computer, for artillery. And yes, both Curvature of the Earth and Rotation of the Earth was just two functions of the many needed to deliver steel downrange for the Queen. This is why no flat Earther will ever be able to have safe, effective, accurate, and timely artillery fires. With out the functions for curvature and rotation of the Earth, the data would technically be unsafe to fire.
Interesting topic! Last video I watched about this was from the guy who's building Earth in Minecraft (he and the other builders spent a lot of time picking the best projection for the job, since Minecraft map is flat).
I've known about this theorem but resources like Wikipedia are so dry. Your video was excellent at explaining the Theorema Egregium!
Enjoyed it.
Amazing. Thanks
Getting Berry phase vibes from the holonomy discussion :D
Excellent video!
Here's my attempt at the problem. We want an angle preserving and distance preserving map. In other words an isometry between the 2 sphere and the plane.
Note that isometries are continuous. Metric spaces have a metric that induces a topology making them topological spaces. Preserving the metric implies preserving the topology induced by the metric and preserving topology means the map is continuous.
This means that we want to find a homeomorphism between the 2 sphere and the plane but this is impossible because they are not homeomorphic spaces.
The Gall-Peters protection: because nothing says "not Eurocentric" like having the only regions with an accurate shape be at Latitude +- 45°
This is my favorite video.
As Stephen Wright says, "I have a map of the world, but it is actual size."
Great video❤
Great video, but I didn't understand the proof for why lengths and geodesics are preserved in such a map. In particular, if a triangle on one manifold is mapped to another shape on another manifold, why is it a triangle and why are the side lengths the same? This seems to not be correct in general if the curvature is different, and you can't rely on it being the same because that's what you want to prove, or do you deal with triangles which are small enough for this to not matter? Thanks!
This is long so I will watch it a bit at a time.
Oooh topology, cool 🖖🤓👍
So basically, without Theorema Egrium, we wouldn't have General Relativity!
The parallel transport you showed is only for tangent vectors
Is it possible to create a compromise map to be more mathematically rigourous? the Nat Geo projection, for example, looks nice but it's neither area nor angle preserving, can we somehow quantify that deviation?
Of course you can. First, project the errors in angles and areas into the same unitless space, each scaled to the importance you assign to errors in each. You also need to select the norm to combine these errors, and it should be well-behaved w.r.t. differentiation, so L₂ (the sum of errors squared) is the most practically used. The choice of the norm affects the weights, but you can always adjust them to the norm. You'll get an error at each point in the domain, but you need a single scalar error to minimize, so you need to "average" the error, possibly weighted, if you want some regions be more precise in whichever you've emphasised when choosing weights, areas or angles, at the expense of other regions. Generally, you integrate your weighted scalar over the whole domain, normalized by total area, E = ∫ w(P) e(P) ds/S = 1/S ∫ w(P) e(P) ds, where e(P) is the normed error at point P (the tip of *P,* the position vector), and w(P) the weight to make some regions more precise at the expense of the others: the larger w(P), the more “important” is to make the error smaller at P. Then you can minimise the scalar error E of the map M(P) by the usual variational calculus.
Depends on what you mean by "map" you could cut literally peel the map off the globe and cut it many times and paste it onto a flat surface. Does the definition of map really say there must be a contiguous connections?
You still cannot paste it onto a flat surface without stretching the surface in some way, though. In that case, either areas or angles will still not be preserved.
@@mathemaniac Well, you can preserve them within a tolerance e.g. if you require maximum 1 deg of angular distortion you have a trade off between the percent by which the area is distorted and the number of cuts you need to make (possibly cutting continents in half or more for some %).
Example: start at the North Pole, head towards the equator, go 1/4 of the way around the equator, and then return to the North Pole. This is a triangle with three right angles. Such a shape cannot exist in Euclidean geometry, so distortion can’t be avoided.
How do you zoom in on a torus like that at 5:31?
Is there any results about what's the minimum angle distortion necessary for preserving area?
Often when I meet those kind of "no free lunch" kind theorems where you need to choose between two ideal scenarios, the first thought in my head is always: ok, what's the optimal trade-off then? Instead of trying to be perfect in both respects, what's the smallest distortion I need in one of them I need to show in order to be perfect in the other?
“what's the smallest distortion I need in one of them I need to show in order to be perfect in the other?” - By “perfect” you probably mean exact? Then, the answer is, as large as it takes to be exact in the other. You have to choose one of the two projections shown at the beginning: one exactly preserves angles, the other areas. An infinitesimal deviation in one leads to deviation in the other, however small. In your lunch analogy, you're asking how much, at most, you can take off my lunch and add to your, such that mine doesn't change but yours gets larger. The answer is, of course, zero.
@@cykkm my understanding is that there are many projections that preserve area, and that Gall-Peters is just one example. Similarly there are many projections that preserve angles, and Mercator is just one example.
I assume that among the projections that preserve area, the amount of angle distortion is not necessarily the same for all of them. If that's true, one of them will be the one who distorts angles the less.
@@RafaelSCalsaverini I think I understand what you mean now, sorry about the confusion. Indeed, that's correct: if you have a map preserving one variable, you can transform it to be _worse_ in the other than it is, according to some metric of "goodness" (in the sense of projecting the variable into a metricized set); a reverse transform could improve it. Therefore, _improvable_ maps preserving one variable exist, according to the same criterion.
The Mercator projection isn't _practically_ optimal, as it results in an infinitely tall image. Think of angles near the poles: if you're approaching the, w.l.o.g., North pole at a constant angle, you'll never reach it; instead you follow a logarithmic spiral, shortening the distance to the pole at a constant multiplicative rate per revolution _r,_ strictly between 0 and 1-think of infinitesimal similar right triangles whose adjacent sides are the globe's parallels and hypotenuses are infinitesimal elements of your path, in a plane, ignoring curvature of the globe, as you are looking at an arbitrarily small patch around the pole, to see why (parallels become concentric circles around the pole; r is the tangent of the angle). Correspondingly, to preserve this constant angle in the image, assuming left and right sides of the map glued together into a cylinder, the path will be a helix extending upwards to infinity. Real maps have to be either truncated at a certain latitude or distorted, losing the angle-preserving property.
The G.-P. projection doesn't have such a defect, as areas monotonically decrease to asymptotic zero as you approach a pole on a monotonic trajectory, which does cause a blow-up (although angle error does blow up). For “optimality,” you have a freedom to choose how to define it: the error is a scalar field over the domain, but you define “optimality“ in the sense of the minimal _single scalar_ error. “Averaging,” i.e. integrating the error per area element (which is preserved, we're lucky!), as its L₁ or L₂ norm both seem to be a sensible way to compute a scalar error. I have a hunch that integrating raw, signed error might be unhelpful: because of the sphere's symmetry, it's likely zero. See if the integral of the (divergent) angle error near poles per unit area converges, using either of the two norms. If it is, then G.-P. is optimal in the sense of the unweighted integral, the “average error,” without any arbitrary choices except the norm. If it's divergent in L₁ but not in L₂, even better: the latter is arguably the most natural norm choice in Euclidean geometry.
Finding an optimal map obviously depends on the aims, i.e. what best fits the intended purposes for its use. In case that can be confined to some dedicated part of the globe, a map projection is chosen that best fits that part. About every country has elected some particular projection method (together with a choice of parameter values) to produce its offical topographic maps. Combining such local maps as patches in an atlas is familiar to every school kid. That fundamental idea is also how mathematics started to approach the study of more intricate "manifolds".
Comment to boost this in the algorithm, don’t mind me :P
As far as I understood in chapter 3.4 was shown that holonomy of the curve on the sphere is equal to the area enclosed by the curve. But how do we decide which part of the sphere exactly is enclosed?
So that it is on the left when you traverse the boundary counterclockwise.
@@mathemaniac But say that I walk in some circle around the poles. I can argue that I'm walking counterclockwise around a small circle around the south pole or that I'm walking clockwise around an enormous circle around the north pole. How can one decide which way I'm walking?
is the password not working for anyone else?
The one thing I'm confused about is how can you define the Gauss map without relying on an embedding
I have defined the Gauss map normally as you would, and it depends on the normal vector of the point at the surface, i.e. depending on the embedding.
Could someone please tell me the file download password, I can't seem to find it...
I didn't know Needham had another boook, I like Needham's books
oh that's because it was released in 2021
Just add a point to the plane... then u can get a perfect map
Despite the earth not being perfectly flat, the reason all maps are wrong is because the king does not want we-the-sheeple to know how reach his castle and he wants to conceal land from us ----- duh. Let the sheeple argue amongst themselves over the geometric theorems and projections we shine on the screen.
I am going to hit the like button under every video of yours
Use an atmosphere to draw the map. Problem solved
You can't see anything on the preview... You should have gone with the other one, with a single map.
Now trying the new thumbnail, and see if it works.
hyallo!
38:40
hi
First!
literal misinformation
@@shanathered5910 Nobody cares
@@rdococ not mad. it's just funny to say.
@@rdococ I'm not mad, it's just funny to me
majority of the videos/lectures are great! but unfortunately the artificial voice is a disgusting hermaphrodite voice...
it would be more enjoyable if a male or a female voice narrated the video....
this is a real person speaking in the video
It's probably better if you didn't use loaded language
Yes, it's called a globe, and you can have it on your phone (Google Earth).
These are old problems or mathematical curiosities. They have no bearing on our modern world.
Have you ever thought about how the maps in Google Earth and ubiquitous GPS based navigation systems are constructed? Look up "UTM" and learn how relevant this video's subject really is!