So...I take it no one actually watched the whole thing. This video has little or nothing to do with astronomy or cosmology in its presentation. As a disclaimer, the authors of this video never say the Earth is a perfect sphere; in fact, in Chapter 1 they acknowledge that the Earth is flatter at the poles (Chapter 1, 1:48-1:51), but only want to draw your attention to a perfect sphere shape so that they can explain some geographic and geometric techniques. Furthermore, outside of Chapter 1, the video never says the word "Earth" again, and in no way attempts to make Physical (as in the science of Physics, not something you can touch physical) interpretations of what a 4th dimension would mean in terms of space-time, so for the people above me who are saying "it's all based on a false assumption", I am right in assuming you didn't watch all 10 chapters of the video. I have my own commentary on this video at the end, but here's an outline to dispel this whole "lawl this is all false" nonsense.
-----------------------------------------------------------------------------------------------------------
Chapter 1
Geography - The study of mapping the Earth
Geometry - The study of measuring the Earth
Introduces the ideas of parallels (biggest parallel is the equator) and meridians (most significant is the prime meridian, through Greenwich, England.)
This whole chapter is devoted to explaining the idea behind stereographic projections. A stereographic projection takes all the information from the surface of a 3 dimensional object and maps it down to a 2 dimensional space. Starting from the highest point on a sphere (north pole for the Earth example), lines are drawn through the north pole, THROUGH the inside of the sphere, and reappear at another point on the surface of the sphere. These lines are then extended onto a plane which is perpendicular to the south pole (i.e., if you set a ball on a table, then the table is the perpendicular plane to the south pole), and all of the information is drawn out onto the table. The projection (table) image preserves all shapes, but doesn't preserve the measures (preserves a figure's geography, not its geometry). Such projections are called "conformal".
Chapter 2
This chapter deals with showing us how 3 dimensional objects are perceived from a 2 dimensional perspective. That is, how would you explain what a 3 dimensional solid (polyhedron) is to someone who only sees the world in 2 dimensional solids (polygons). The video gives two methods:
1) By passing a polyhedron through a plane, you observe all of its cross-sections. These cross-sections can vary based on the angle at which you pass the polyhedron through the plane (for example, if you stand a cube flat on a square face, you always see a square cross-section from a 2D perspective; however, if you stand a cube on its edge, you see a line, then a rectangle, a square as the middle passes through the plane, back to a rectangle, and then a line before it disappears again). As the narrator points out, this method is particularly difficult to understand, because it requires a lot of intuition about all the possible cross-sections of a polyhedron.
2) Inscribe a polyhedron into a sphere, so that all the vertices of the polyhedron touch the surface of the sphere. Then, round out the edges of the polyhedron until they too touch the surface of the sphere. Color each face of this "inflated" polyhedron differently, and stereographically project this new sphere (divided into faces) onto a plane. As pointed out in Chapter 1, this method will preserve the geography of the polyhedron, not its geometry. So in this method, counting the number of different colors on the plane will allow you to know the number of faces of the polyhedron you began with. This is much easier than method 1, as all of the information of a 3 dimensional object is translated in one unique way into 2 dimensions, whereas the former method had many non-unique ways of taking 3D info to 2 dimensions.
This section is presented to us because we can understand readily both 2 and 3 dimensions. The narrator then hopes to give you some kind of understanding of what a 4th dimension would be like (geographically and geometrically) by taking the same analogy of stereographic projections from higher dimensions into lower dimensions.
Chapter 3
This section deals with the 4 dimensional analogue of the "cross-section" method outlined in Chapter 2. When we passed a 3 dimensional polyhedron though a plane, we could observe 2 dimensional cross-sections, which varied based on the angle at which the polyhedron was moved (remember the different sections of a square if you started flat or on an edge). The narrator then continues this analogy into 4 dimensions. If you could (hypothetically) pass a 4 dimensional object through our 3 dimensional space, we would observe changing 3 dimensional cross-sections of this 4D object. So, by this reasoning, there exist 4 dimensional objects whose 3 dimensional cross-sections are tetrahedrons, cubes, dodecahedrons, icosahedrons, etc. As with the analogy in 2 dimensions, this method doesn't give a good feel for what the higher dimensional object looks like, as the cross-sections can and do vary based on how they pass through the lower dimensional space.
Chapter 4
This section deals with the 4 dimensional analogue of the stereographic projection method. Just as before when we "inflated" a 3 dimensional object in a sphere, then projected that sphere onto a plane and observed the results, the narrator (with some help of your imagination) places a 4 dimensional object (which you cannot see) into a hypersphere, and stereographically projects it down onto 3 dimensional space. The results are really intricate, conformal 3D objects, all with great symmetry, and are really well rendered by this video. Again, this section does not claim to show you a 4 dimensional object, because we have no way of viewing it because our senses are bound in 3 dimensions; however, they try to give you an idea of how 4D objects interact in 3 space by showing you these stereographic projections, much as the 3D objects interacted with the plane in 2 space. Really worth watching this section for the animations.
Chapter 5
A really great introduction to complex numbers. Not much to say, the section is really self-explanatory and well done. One thing worth noting is that now points in a plane can be described with one complex number (as explained in the video), whereas before we needed 2 real numbers to describe a point in the plane. That is, with complex numbers, a plane is dimension 1, instead of dimension 2. By extension, the end of this chapter shows that the stereographic projection of a sphere into the complex plane (1D) is a 2 dimensional line (remember before the projection of the Earth (3D with real numbers) was a 2D plane). Hinting that now we can describe 4 dimensional, real number valued objects in 2 dimensional complex space.
Chapter 6
A look at complex transformations, Julia sets of complex transformations, and the Mandlebrot set (a set of complex numbers for which the Julia set exists), and the resultant fractal sets they produce. Really beautiful stuff, but this section does not go into great detail about fractals (which is a pity, since a lot of image storing software today utilizes fractal decomposition to store pictures without quality loss). Worth watching for the pictures, but probably beyond the scope of non-math people.
Chapter 7
An extenuation of the idea of mapping 4 dimensional real valued objects into 2 dimensional complex space. Deals with Hopf Fibrations, which are these really neat objects made up of infinite amounts of interlocked, non-intersecting rings (which is really a 2 complex dimensionsal simplification of a 4 real dimensional hyper-sphere). This is Topology 101, and the pictures are really wonderful, with good explanations. Not for the Math faint-of-heart.
Chapter 8
A further look at Hopf fibrations, as well as Villarceau circles on a torus. Simply put, a torus is a shape you would get if you took a circle, and rotated it around a line (think dount shape). On a torus, there are four families of Villarceau circles (explained in detail in the video as bitangential, parallel, and meridian). When these circles are seen together, they produce a Hopf fibration. What's interesting is that we have found with Villarceau circles a 2 real dimensional way to describe a 2 complex dimensional object (Hopf fibration). But by extension from Chapter 5, this means we have some more insight into the world of 4 dimensions, since a 2 dimensional complex space is analogous to a 4 dimensional real space. It's all pretty uphill from here, and most people don't cover this level of topology until graduate school.
Chapter 9
A pretty well thought out proof explaining why stereographic images work. Nothing to explain, as the video outlines the proof in detail using geometric theorems in both 2 and 3 dimensions (real dimensions), and makes the analogue to 1 and 2 complex dimensions. Easier to understand than Chapters 7 or 8, but still requires a good background in math to appreciate.
------------------------------------------------------------------------------------------------------------
I really wanted to outline this video because it is EXTREMELY well done. Chapters 1-3 are pretty accessible to anyone despite your background in Math. Chapters 4-6 are good for those with some college level Math background, while 7-9 are really for senior level undergrad/grad students. I got a little upset when someone earlier in this thread said everything in this video is a lie...because it's just simply not the case, but I'm sure hardly anyone has watched the whole 2+ hours of this anyways.
I've already watched it 3 times through, and its wonderfully done, and they are releasing a "Dimensions II", which will cover more Topology and Dynamics. Math rox!
