(You can read more about this substitution system in the article Pentaplexity by Roger Penrose.)įigure 19: An aperiodic Penrose tiling, based on two shapes.įrom the point of view of the five-fold tiling problem, Penrose's system above seems less satisfying than our first example because it relies on two shapes that don't have five-fold symmetry: the rhombus and the half-pentacle. Pentagons, dividing each into a pentagon, a pentacle and a half-pentacle or "paper boat". ![]() Penrose observed that the spiky rhombi can be simplified by inserting In the next generation, the rhombi grow into spiky shapes. As demonstrated in figure 16, the same six-for-one substitution can then be applied to each of the six smaller pentagons, leaving rhombic tiles surrounded by six pentagons. Original pentagon, in the form of five isosceles triangles with angles of 36° at their peaks. The packing leaves some empty space in the Penrose observed that a pentagon could be packed with a flower-like arrangement of six smaller pentagons, as in the first step of figure 3. Penrose tilingsĪ different substitution system, also based on the regular pentagon, was devised in the twentieth century by Sir Roger Penrose, and led to an amazing new discovery. We could suppress the monsters by subdividing them as in figure 12. ![]() We can avoid adding such an odd shape to our vocabulary by subdividing every monster into pentagons, pentacles and rhombi:įigure 15: The spiral arrangement from figure 8, with rhombi subdivided as shown in figure 9, and the other shapes subdivided as shown in figure 13. It eliminates the rhombi of the original tiling, but introduces monsters (which have only two-fold symmetry) in their place. Our substitution certainly does not solve the five-fold tiling problem. Kepler represents a pleasing bridge spanning the hundred years between these two great thinkers. The hidden connection demonstrated here between the drawings of Dürer and In the twentieth century, several mathematicians showed formally how this construction may be carried out. The text accompanying the drawing suggests that Kepler had an idea of how to extend the patch to cover the whole plane. It uses the fused decagons, which Kepler called monsters. The patch of tiles he called Aa, shown in figure 11, Kepler drew several arrangements of five-fold shapes, perhaps implicitly in search of an answer to the five-fold tiling problem. It can be found in Harmonice Mundi, Kepler's seventeenth century treatise on astronomy and geometry. Pentacles, shown in red in the top right of the figure, would overlap.Īpplying the substitution to Dürer's radial arrangement leads to an especially interesting tiling. Now we find that there is no easy way to add another layer of shapes. The gaps left behind by the pentacles can in turn be filled with more pentagons, this time in two sizes. We can fill these gaps by introducing pentacles (five-pointed stars) asĪn additional shape. We know we can attach pentagons to each of its sides, leaving five 36° gaps. We'll start in figure 3 with a regular pentagon, a very simple shape possessing five-fold symmetry. ![]() Putting aside for now any deep mathematical reason why such a tiling should or should not exist, let us attempt to construct one a piece at a time. The problem is also a source of many wonderful geometric designs, with more waiting to be discovered by anyone willing to experiment. Greatest thinkers of mathematical history, it is still only partially solved. Though easily stated, the problem turns out to be surprisingly subtle. Given the rich variety of five-fold shapes available, it seems possible that we might find a tiling based on them. Is it now possible to find a set of shapes with five-fold symmetry that together will tile the plane? We refer to this question as the five-fold tiling problem. Rotations about p through 1/5th, 2/5th, 3/5th and 4/5th of the circle - in other words through multiples of 72° - leave the tile unchanged. That is, every tile contains a centre of rotation p such that Let us drop the restriction that the tiles be identical copies of one polygon, and ask only that each individual tile have five-fold symmetry. Figure 2: Three pentagons arranged around a point leave a gap, and four overlap.īut there is no reason to give up yet: we can try to find other interesting tilings of the plane involving the number five by relaxing some of the constraints on regular tilings.
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