In particular, it shows various shapes based on properties of Ellipse, and various linkages. Peaucellier Linkage, and Hart's Linkage, based on properties of Geometric Inversion. (see: Geometric Inversion). What's interesting about these linkages is that they are mechanical means to convert between circular motion and straight line motion, important in the industrial revolution era. (think of the linkage on the wheels of steam-powered trains) The video alse feature curves and solids of constant width. Imagine, you can roll a book on top of several marbles. The marbles are spheres. However, there are also shapes that's not sphere but still lets you roll the book smoothly.
Their website howround.com features these objects in more detail, also with a book How Round Is Your Circle?(2011) John Bryant, Chris Sangwin. amazon
One particul thing i found very interesting in the video is that they showed a drill that is designed to drill a square hole.
See also: Linkage (mechanical).