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_sHZJLSFcx0eRNB8nc2OGh2pbDSeu_AfrikTJSW_SeLzlRN6CvyL2cx7JO4tvwjf978JlC0HLjCB1BVEUMRsnlG-csNwo9EPg7I_bchBG7jsrhTxfR_U22TpzBCReybZlI=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_vkponijy1EqqS40yAmm25mb50HjYRqZjY9LxcmRvqid7pYVU9vrxIv8K1z3Xdf1PkOrxx8JLzv_aQKVfownT-Az7U1cX5yAtbpvZGjQZSu2O5ilzHw5d_7SgIFTZEfoEg=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_scu61MYy1a3bzUcZ0faXMLw8iOLul28e6eCtuoWmga9NGWhbZOHHZcCumFU7TdISjq8pF4-KQYL1Qd99d8x6egZCyn5v8qPqqUzw5DcUxeK27OHoqsqrt_Brfj834zTKtt0A=s0-d)
Video 6 finishes with a beautiful zoom deep into the Mandelbrot set.
![FliA1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_seDAQ_UDkiYsnU_G8mZgCYVF2y5MnNt6YmyJXDx9LU6JCnOEnUQgeueG-WWj_hMFypbQiyzBSe0ZxOpSC3IStO0M9bXgt_o1g-HYhEsLrrd2agnIAB5zTbadYDV5uUEyc=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_uDyMjAU9FziaF_ecld9dao6sAHKF-sz0VUXILL_NsucBlnIUhMcL84JZJFvInFSUKSuK7z8wc5j6DkuZiExGHQR1npjDe0fxJLZeYuU2mv9veCoAYr2YG-3Cmf6p-pkYg=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_vI2f7vNiuLIIP0oVQ4dezpBfxsUr6vX-jD2HEtOJz9Rz2XbByvCxTy-ajqtSbp1WaA34cSlk4yUQsIDUYfkJ84_6XDfWV94BAlzYsC6MfOBbbnpQgMfGcVCa_scKC7bmOJrQ=s0-d)
![FliA7](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ucGq2JhWKNyRlfZXnAJc13GTtmN4R5q7F7NYi6h2vMgV_ppCNakTxWUsFKJKRix-YXAx1jy16YIe_loeJkG3ea37ShA-xTfAZvRqafkWI_exCK5-IcUw4bUsvvTXBQ-g=s0-d)
![FliA8](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v298afy55ADdp4VT6U4_jwHAvSjTekJ8qsHmkhzHo6sioPMh6HUJyXGRXjwd3-or1GBa2J52eIwImt_mz3grw5febkF2PFpy_4kK_9rfkNtpY1L_9XyJzT9TZZ6--hkwTU=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!