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.

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.

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 %).

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?

“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".

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.

Now trying the new thumbnail, and see if it works.

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

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!

this is a real person speaking in the video

It's probably better if you didn't use loaded language