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[Videos]Visualizing 4D, complex numbers, and more

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Forum index > Blogs
  Bill307   Canada. August 25 2008 09:51. Posts 8583Profile Blog 
Posted on Slashdot, the following videos show you how to visualize a few different aspects of mathematics:
- 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.
Last edit: 2008-08-25 09:52:03


*****
Old Post

  MeriaDoKk   Chile. August 25 2008 10:17. Posts 1545Profile Blog 
Just saw Chapter 1, was really interesting ^_^.
Thanks for sharing.
Old Post

  Jyvblamo   Canada. August 25 2008 10:42. Posts 4760Profile Blog 
Heh, watched this earlier today when I saw it on Digg. Excellent videos.
I posted in BuGzlToOnl\'s thread and all I got was this stupid t-shirt...
Old Post

  talismania   United States. August 25 2008 11:23. Posts 880Profile Blog 
I'm trippin balls wtf
Old Post

  JeeJee   Canada. August 25 2008 11:26. Posts 2527Profile Blog 
yep saw this on digg, i watched the first 4.
i still can't find a way to think about it though, aside from projecting 3d clones from each of the 2d faces out into 3 space

=(
(\o/)  Life is good ^^
 /_\   aka feelShinbi (requesting a name change since 27/05/09 ☺)
Old Post

  ieatkids5   United States. August 25 2008 11:38. Posts 3171Profile Blog 
This sounds really cool, thanks. I'm gonna watch these instead of doing my calculus summer homework.
Old Post

  Cloud   Sexico. August 25 2008 12:07. Posts 4631Profile Blog 
Love your blog Bill
BlueLaguna on West, msg for game.
Old Post

  skyglow1   New Zealand. August 25 2008 12:22. Posts 3891Profile Blog 
*Head explosion*
Old Post

  Mastermind   Canada. August 25 2008 12:26. Posts 2945Profile Blog 
The first one was a little slow at the beginning, but got interesting in the second half. I hope they dont all go at such a slow pace. I will watch the rest later.
Old Post

  skyglow1   New Zealand. August 25 2008 12:51. Posts 3891Profile Blog 
What's up with the guy in the first complex numbers video screwing up so many times when reading off the numbers?
Old Post

  JeeJee   Canada. August 26 2008 01:15. Posts 2527Profile Blog 
On August 25 2008 12:26 Mastermind wrote:
The first one was a little slow at the beginning, but got interesting in the second half. I hope they dont all go at such a slow pace. I will watch the rest later.


yeah
2 and 4 were really good
3 was a bit retarded, i think 4 is just a better version of 3

that's all i've seen so far. 1 was just silly imo
(\o/)  Life is good ^^
 /_\   aka feelShinbi (requesting a name change since 27/05/09 ☺)
Old Post

 
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