Comment hidden because you are ignoring dag.(show it anyway)

Good stuff.

Great! This *music is perfect for chatting up a gorgeous girl with my witty banter about constant width. =P

Adding video to channels (Music) - requested by GenjiKilpatrick.

I was curious what software he was using in this video, so started poking around at howround.com and eventually found the link to it -
http://www.geogebra.org/

(it's free)

Is it bad when you want them to explain the explanation?

It's a shape that, because it "wobbles" around it's centre axis, rather than rotating around its centre axis like a wheel, it can present a consistent cross sectional width between two parallel planes. In reality, there are three points it rotates around at any particular time, depending on what point it's at.

Think of it as 3 pie shapes. It rotates on a point at the book, and when it finishes it's arc on the table, THAT becomes the new pivot, and the book travels on the arc, then the table, then the book... etc.

What's neat is you can do this with any odd-sided polygon. It won't work on even numbers.

Yeah, I'm deleting that last part as there's too many conditions for that to be simple.

Yay, GeoGebra! I used that as an educational tool when I taught maths, and it is a very neat little program. This guy does some things I didn't know was possible with it.

FYI - The video shows that having a constant diameter (cross-section) is not enough to show the shape is round. This was discovered to be a problem in getting segments of the Space Shuttle solid rocket boosters to fit together. (The SRB segments would flex out of shape when they were transported across country lying sideways on railcars.) They had to be forced back to a round shape to fit together, with the O-rings in the gaps. As described in Richard Feynman's book, "What Do You Care What Other People Think?", NASA would measure diameters at different points. But Feynman knew about the funny shapes in this video, and knew that diameter measurements did not prove roundness. The technicians on site always had to keep inspecting the segments as they came together to get them to fit together.