Using Inkscape to construct the Penrose Tiles (Part 3: Exploring Penrose Tiling (Kites and Darts))

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These videos give step by step instructions on how to use the free Open Source Inkscape software to geometrically construct the Penrose Tiles. In Part 1 a method is given for creating the outside shape of the Kite and Dart. Part 2 covers the edge-matching constraints using circle arcs. In this video (Part 3) two tiling challenges are given. This leads to the discovery of the Ace, Deuce, Jack, Queen, King, Star and Sun patterns.
The Penrose Tiles were discovered by Roger Penrose in 1974. The two examples in these videos were named by John Conway "Kite" and "Dart". They can tile the plane but only in a non-periodic way. They are called Aperiodic Tiles for this reason.
If you attempt a tiling with the Penrose Tiles as you build a larger and larger region you soon find places where no tile will fit. You will need to unpick some of your tiling before you can proceed further.
It has been said of these tilings:
"most patterns, like the universe, are a mystifying mixture of order and unexpected deviations from order. As the patterns expand, they seem to be always striving to repeat themselves but never quite managing it".