People interested in a technical discussion of Penrose tilings are encouraged to find a different source of information (for example, I found John Savard's page to be informative). This site is merely an exposition of some of my experiments with making Penrose tilings using the Golden Gnomon/Golden Triangle recurrence relations.

I first present these recurrences and how they map to pentagons:

Note that I do not draw one side of each shape; the recurrence relation depends on a handedness of the shapes and if this is preserved by the missing side rule, we get Penrose's original darts and kites. Thus two adjacent gnomons form a dart and two adjacent triangles a kite. Each triangle is reduced to two triangles and one gnomon; each gnomon to one triangle and one gnomon. Note also the coloring rule; triangles with a grey base are decomposed into triangles with a grey base and a gnomon with grey. A gnomon with a grey base is decomposed into a triangle with a tan base and a gnomon with tan.

In practice I do not just subdivide; I also expand by φ = (1 + √5) / 2 each time and rotate around the point which the first triangle and the subtriangle created at its base have in common by 2π/5 radians. This guarantees that each subdivision merely adds to, without changing, the previously generated area. The result is a Penrose tiling:

Penrose tilings are peculiar in that they exhibit no translational symmetry (that is, they never repeat) but they might have 5-fold rotational symmetry. If I create ten triangles point-to-point as my seed (or, equivalently, ten gnomons) then I will create two different rotationally symmetric tilings, one with a star in the center and one with a pentagon in the center, depending on the number of expansion steps I use. It also looks as though the tile above has rotational symmetry, as is made more obvious if I superimpose the darts (i.e., the gnomons) on top of the tiles

and then fill in around the darts to accentuate the apparent symmetry around the central star.
A tiling could be made to have this symmetry extend outward forever; however, in this case it does not, as can be seen by zooming out slightly.

Much else could be said; the the coloring makes things which look like, but are not, Ammann bars appear where a single straight line appears to interect only one color of pentagon. However, if looked at closely, there are some few pentagons of the other color on each stripe (if you look far enough; I have not proven this, but empirically any stripe I find if I expand the view I see broken). There are also lines that never pass through stars and other odd features.

Note that in any non-repeating tile no finite set of observations is enough to locate oneself. In general, given any k-radius sub-image of the tile, an identical subimage can be found within a distance of roughly 2k, though there are also other constraints; for example, two stars of the same orientation can never appear closer than do the five surroundign the central star of the images on this page.

Finally, I offer a cool picture (but big; nearly 1Mb) of an infinite Penrose-style pentagonal tiling taken with a spherical lense (implemented as an OpenGL vertex shader).