2017-07-20

[public] 83.5K views, 2.84K likes, 21.0 dislikes audio only

Mathologer

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Make sure to watch part 1 of this video before you watch part 2: /youtube/video/rAHcZGjKVvg

In this second part we look at higher-dimensional counterparts of the basic 3D cube shadow theorem of part 1. We'll see how it is used to find the truly wonderful maximal area and volume 2D and 3D shadows of higher-dimensional cubes. Featuring close-to-circular shadows of hypercubes, the amazing rhombic dodecahedron, the even more amazing rhombic triacontahedron, Wolfram Alpha's logo and much more.

The page with my first year hovering-cube assignment lives here: http://www.qedcat.com/misc/cube_assignment.pdf

Pretty much everything that is known about the shadow theorem is written up in the following two papers by mathematician Peter McMullen:

1. Volumes of Projections of unit Cubes, Peter McMullen, Bull London Math. Soc. (1984) 16: 278-280.

2. Volumes of Complementary Projections of Convex Polytopes, Mh. Math. 104, 265-272 (1987)

Just for the sake of completeness here is a 3d .stl file of the maximal 3d shadow of a 5D hypercube: http://www.qedcat.com/misc/5d.stl (one reference that talks about it a little bit is Coxeter's classic textbook, Regular polytopes, p. 256, last paragraph).

As usual thank you very much to Marty Ross and Danil Dimitriev for their help with this video and Michael Franklin for his help with recording this video..

I used Richard Koch's program Hypersolids for the animation of the spinning 4D hypercube: http://pages.uoregon.edu/koch/hypersolids/hypersolids.html

Most of the other animations I programmed in Mathematica. The stills featuring the rhombic triacontahedron were done in Rhino3d, one of my weapons of choice when it comes to preparing files for 3d printing geometrical objects.

Всем любителям математических задачек посвящается!

Как обещано, русский перевод того задания для первокурсников, которое показано в видео на 13:34-13:53: https://yadi.sk/i/xA4NdOU33LVd4H

Enjoy :)