[uwgqwkaw] Penrose tile jigsaw puzzle

Create a collection of physical Penrose tiles which interlock the way jigsaw puzzle pieces do.  Of course, the mortises and tenons (borrowing terms from woodworking) of the pieces should be shaped and positioned to only allow the acceptable Penrose tile connections.

Unlike a traditional jigsaw puzzle which can be put together in only one way, these pieces can be put together in many different ways, always aperiodic of course.  It's not really a puzzle.

This has probably already been done.  "Penrose chickens" gets close but they don't interlock.

Each shape of Penrose tile should be available in many colors to allow making pretty patterns.

The tricky design part especially for cardboard is to avoid the mortises making pieces too narrow and weak, too easily bent or torn, e.g., creating a thin isthmus.

Make the tenons as large as possible for the best interlocking.  However, a tenon too large induces a weak point somewhere else with a large corresponding mortise.  This tension exists universally in the design of mortise and tenon joints.

We provide details for the kite and dart tiles.

The ratio of kites to darts should be phi, the golden ratio.

Lay a kite so that its largest angle is at the top and thinnest angle at the bottom.  Label its edges starting from the top right, proceeding clockwise: a b c d.

Lay a dart with its point at the top and concave vertex at the bottom.  Label its edges starting from the top right, proceeding clockwise: w x y z.

The permitted Penrose tile connections fall into two families, corresponding to the long edges and short edges.  The permitted long edge connections are bc bz wz cw.  The permitted short edge connections are ay ad dx.  Fortunately, these constraints can be met with mortise and tenon joints; no problem with parity.  We assign mortises and tenons to the two families as follows:

(a tenon) (d mortise), (x mortise) (y tenon)

(b tenon), (c mortise), (w tenon) (z mortise)

Either family can have its parity flipped, but we avoid two mortises on one side of the relatively thin dart.

Of course, the mortises and tenons between two families should be incompatible with each other.

The matching rules would permit edges x and y to connect, but geometry, the concavity of the dart at edges x and y, prevents them from fitting.

Instead of the Penrose tiling based on the pentagon, consider aperiodic tiles based on the square, for example Robinson tiles.  Robinson tiles are very close but not quite like traditional jigsaw puzzle pieces because there is one tile which has special corners.  Can Robinson tiles make striking patterns the way Penrose tiles can?