EPT
- stereographic projections and 4-dimensional space
- complex numbers and their relation to fractals
- fibrations in topology
- a couple of geometric proofs
They're aimed at the level of someone who has finished secondary school math. IMO the chapters on complex numbers are the easiest to understand, followed by the chapter with geometric proofs, then the chapters on 4-dimensions, and finally the chapters on fibrations which I found to be the most difficult to understand.
http://www.dimensions-math.org/Dim_regarder_E_E.htm
Below is a brief description of the contents of each chapter.
Chapter 1:
- Introduces stereographic projection using the Earth.
Chapter 2:
- Talks about how a being in a 2-dimensional world might be able to conceptualize objects in 3 dimensions.
- First technique: "slide" the 3-dimensional object through the 2-dimensional world, showing a sequence of cross-sections.
- Better technique: first "inflate" the solid into the shape of a sphere, then use a stereographic projection as shown in Chapter 1.
Chapter 3:
- Introduces five of the six regular 4-dimensional objects and displays and rotates them without using stereographic projection.
Chapter 4:
- Shows the stereographic projections of these five 4-dimensional objects.
Chapter 5:
- A geometric introduction to complex numbers.
- Shows how complex numbers correspond to points in the complex plane.
- Shows how multiplication by i is a 90-degree rotation in the complex plane.
- Defines the modulus and the argument of a complex number.
Chapter 6:
- Shows the geometric effect of transforming complex numbers by adding them, multiplying them, etc.
- Shows the geometric effect of squaring complex numbers repeatedly, leading into the Julia set.
- Explains Julia sets and the related Mandelbrot set (both are fractals).
Chapter 7:
- Visualizes S3 (the unit 4-dimensional hypersphere) geometrically using two complex axes and a circle.
- Visualizes S3 as a set of non-intersecting circles where each circle corresponds to a point on S2 (the unit sphere) projected stereographically, also known as a fibration of S3.
- I've never really taken any topology courses in university, so I had to rewatch parts of this video several times to let it sink in.
Chapter 8:
- Introduction to Villarceau circles on a torus.
- Shows geometrically how each point on the torus has four circles on the surface of the torus that pass through it.
- Shows a stereographic projection of a torus inside a sphere.
Chapter 9:
- Gives a geometric proof that when a non-tangent plane intersects a sphere, the resulting intersection is a circle.
- Gives a geometric proof that the stereographic projection of a circle on a sphere (that does not pass through the "north pole") results in a circle.
- The proofs rely on geometry that one should learn in secondary school.
- IMO these are especially helpful for someone who has an interest in post-secondary math but who has had no exposure to mathematical proofs.
The last video is a "coming in Dimensions II" video.
