How Penrose-drawings were made?

This text is about the geometry of Penrose-drawings. Their base is the Penrose-tiling, or Penrose-parquet - from a Hungarian translation (Géza Perneczki). A branch of geometry is busy with tilings. A geometrical tiling is a set of shapes can tile an infinite plain without gaps. So this branch analyses the pattern of the tilings. For example divides them into classes.

One class of tilings are periodic tilings. For instance a tiling from squares or a tiling from perfect hexagonal tiles.

Periodic tiling from hexagons and squaresPeriodic tiling from hexagons and squares

From the images it is quite clear, why they are called periodic. By rule: a tiling is periodic if outlining a part of it and cutting it, the part can be refitted by shifting into the tiling. The outlined part can be any large, and can be refitted everywhere into the original tiling.

The outlined part can be refitted into the tiling

An other class of tilings are nonperiodic. This does not mean, that their pattern should be chaotic or should not follow any rules. In fact very simple to create a nonperiodic tiling. There is a variation of the tiling with the squares: each square intersected by a straight line, which is not rectangular to the edge of the shape. We have two shapes by this division. They tile periodically on the first figure. Randomly rotating the squares, the tiling is nonperiodic.

Periodic and non-periodic tiling from two kind of quadranglesPeriodic and non-periodic tiling from two kind of quadrangles

This is a very simple example. But one of the important properties seems very well: The shapes can tile both periodically and nonperiodically. Until the beginning of the eighties mathematicians used only this two classes. Nobody could imagine, that there are shapes can tile only nonperiodically. Tiles can tile only nonperiodically called aperiodic tiling. [The first aperiodic tiling was constructed by more than 26.000 elements. The tiles had no really exciting shape (at least from my point of view), they were squares with different gaps and horns at the edges. It is calledWang-dominoes by their inventor.]

Penrose-tiling the pattern forces the aperiodic tilingPenrose-tiling the pattern forces the aperiodic tiling

Penrose-tiling is aperiodic, so the order of the shapes never repeats itself. As it can be seen on the figure it is constructed from two kind of rhombuses. There are a simple rule for them, they can not be fitted by any way. Easy to find a pattern which forces to fit them on the right way. Gaps and horns could be used on their side (inherited from the Wang-dominoes), but the pattern is more clear. The rule is: the light parts has to be fit to light parts, dark to dark ones.

The Penrose-tiling has many exciting properties, but in the drawings I used one of them: the order of the shapes never repeats itself. All of the 100 drawings represents the same part of the same Penrose-tiling. (There is only one Penrose-tiling.) I changed the rhombuses of the Penrose-tiling with small drawings, so the original structure of the tiling is disappears. The small drawings have a special property, the lines always intersect the edges of the rhombuses on the same place. On the figure there is the simplest example. The small drawings shows the diameters of the shapes.

The two shapes of the  Penrose-tilingThe two shapes of the Penrose-tiling

The original shapes of the Penrose-tiling can be replaced with any drawings. In my drawings I used only straight lines, and the lines are always connected at the edges of the rhombuses.

In the example the drawing of the two kind of shapes are identical - just they are adopted to the different ratios of the different rhombuses. (Both drawings shows the diameters.) Sometimes I used different drawings on different kind of shapes.

the original rhombuses can be replaced by any drawings

Tilings and patterns, B.Grünbaum and G.C. Shephard. W.H.Freeman & Co. 1986.

Mathemathical games , M.Gardner. Scientific American, January 1977 p.110-121.

Pentaplexity, R. Penrose. Geometrical combinatories, F.C. Holroyd & R.J. Wilson eds. Pitman 1984.

Illusions found

We know science. It has been with us throughout our school days in the form of subjects, and nowadays it is a programme in the bottom third of cable TV.

Illusions found

About IFS

IFS means Iterated Function System. This is a description of an iterated function system mainly for novices in math.

About IFS