Remarkably, some of it is actually related to the Hodge Theory!

*Animation of the square in
the
plane.*

The configuration space of planar realizations of the
"square" linkage

(polygon which has four equal sides), is not a circle but
the union
of

three circles; each two of these circles share a common
point.

The above animation illustrates three different irreducible
components

of the configuration space. The realizations which look like
parallelograms

correspond to one of these circles. The "folded"
realizations
correspond

to the two other circles.

Look here for a beautiful animation of the Peaucellier linkage.

Look here for computer animation of the proof of Kempe's linkage theorem.

Animation of a buckyball (carbon nanostructure).

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