Dimensions video series
I just finished watching the challenging Dimensions video series - 9 videos, each 13 minutes long, which teach you how to visualize objects in 4 dimensions. I watched each video as my evening treat for the past week.
Videos 1 and 2 are basic, and you might find parts of them a bit boring. But they're important for the later videos – especially the bit about how rolling a sphere around affects its stereographic projection.
![FliB4](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sPNeo5hPXQtHEpQf9_tC2hQzVNCM-IC8VBjfuFt0BN-LBv_nk9WD7JkXuhGgWmPDEKgjZI6ThOJWcaYEjaAYVwjsO2uK-xiM6u5_5gkMsxM-l6Ihv_3AT44CgCDUS9lNg=s0-d)
In videos 3 and 4, we start to get into the good stuff: the fourth dimension. For example, if we want to visualize the 4-dimensional equivalent of a cube (called a "hypercube", with 16 nodes and 32 edges), we can get a feel for it by "stereographically projecting" it into 3-dimensions:
![FliA2](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tVC3IpQCYbXWlB9NU0eLsr58pHNidnW3mxGXys0grIw2GWNcbELjkOkWW7aQ-D8gEHqDHA1CYHBPof5ibDHsG1GF45CQxVC0Pw4byRToKvVV0ohcUqmV8k-GsimkePIBY=s0-d)
Wow!
Next, videos 5 and 6 delve into complex numbers. It was neat to finally get a visual explanation of what the Mandelbrot set represents. It's tricky - it's the set of points for which the Julia set (the blue shape on the right) appears. Outside of the Mandelbrot set, the Julia set disappears.
![FliAA](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sMfBwQZ2ny7F3FFxT8nxH1Dk13wDwsPUaEfAwfJ8Jy18j0J5ojPtr-e9euyDS35aeU-elwGQlmeNXtfWGe8yS2jsGhRyzcxB7w-tneDxxFy60htCymhawgl4pVkSSdnhYHDQ=s0-d)
Video 6 finishes with a beautiful zoom deep into the Mandelbrot set.
![FliA1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_va5FK4HA5l-ta0tI82lsGBG4nGo8FnYkrL8p33XWvD_trMzX5_FK6FQhnNGjBFPYY5M5quMOG9W_p5VwR4skRqcB4z9nqP0_CXFzXnKB4dsvsCWRhL_jOsQLjEWRb-EDg=s0-d)
Videos 7 and 8 were the most difficult and had me pressing the pause button quite a bit to try to understand what was said. For example, in Video 7 we learn how to divide up a 4-dimensional sphere into circles ("decomposing S3 into its Hopf circles"). Many replays later, I think I get the general idea.
![FliA4](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tIqwBBkH9C7WLiaafR8Hv2-zDYGp17yawjzZroJzcrGDWUWLeT1UGgESOFrUPqwET6pgi2OFhfBwlS8Dj96gMjVXGWedg7egz3OsMBiFN7CzBejBCpK3jcAvdpgKx7mEw=s0-d)
Video 8 almost totally lost me, but I think I get it now. If you highlight a torus (a doughnut shape) in a 4-dimensional sphere (yes, a torus can be a subset of a 4-dimensional sphere) and then stereographically project the torus into 3 dimensions, you get a deformed torus - and it gets REALLY deformed as you roll the sphere around in 4-D land:
![FliA5](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tcJe_s3Epl_R6EiV2wXrLrYPfZMZfOKPI6uIFd9KeY79_gar8x1SQi-3umxWr_BU6ohZ8DYNmEqnhHXi3DAK4Dt1TPEYBXe04dUlJpFv3JyyLCPAX5_ySgjRao-eiwZEmdcQ=s0-d)
![FliA7](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uoH8Wm862EtHjHqCZFikOo1_IyYr7-p8aTn25uPS3Qn_w_hoqhev31jvUOU32pNj66omKE6i4AuKiG9Xr7VBGw0RuTWMXZz4vcV1VisEFVhfcw9Ce6AxZcbJL3Ch08Hg=s0-d)
![FliA8](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tnKAoHOE63UXxAcmweNWLXCMAoav_I-V6ALKNdjGOKkuetj-BB9CUyqFyffhp73r-ySnYzsqgLbnrmdw-6ZRyoYq8pZZ4q46WUXlDoHRkYEqIgIOfj4HGVNpF4Hfbrb7O7=s0-d)
These videos have amazing animated visualizations that you'll find nowhere else. And they have a great soundtrack to boot. Go watch!
Videos 1 and 2 are basic, and you might find parts of them a bit boring. But they're important for the later videos – especially the bit about how rolling a sphere around affects its stereographic projection.
In videos 3 and 4, we start to get into the good stuff: the fourth dimension. For example, if we want to visualize the 4-dimensional equivalent of a cube (called a "hypercube", with 16 nodes and 32 edges), we can get a feel for it by "stereographically projecting" it into 3-dimensions:
Wow!
Next, videos 5 and 6 delve into complex numbers. It was neat to finally get a visual explanation of what the Mandelbrot set represents. It's tricky - it's the set of points for which the Julia set (the blue shape on the right) appears. Outside of the Mandelbrot set, the Julia set disappears.
Video 6 finishes with a beautiful zoom deep into the Mandelbrot set.
Videos 7 and 8 were the most difficult and had me pressing the pause button quite a bit to try to understand what was said. For example, in Video 7 we learn how to divide up a 4-dimensional sphere into circles ("decomposing S3 into its Hopf circles"). Many replays later, I think I get the general idea.
Video 8 almost totally lost me, but I think I get it now. If you highlight a torus (a doughnut shape) in a 4-dimensional sphere (yes, a torus can be a subset of a 4-dimensional sphere) and then stereographically project the torus into 3 dimensions, you get a deformed torus - and it gets REALLY deformed as you roll the sphere around in 4-D land:
These videos have amazing animated visualizations that you'll find nowhere else. And they have a great soundtrack to boot. Go watch!