![]() 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. Indeed it is, as the following example shows.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. Is it possible to extend a 5.5.10 vertex figure to a full tiling using the simpler 5-rhomb and 10-rhomb shapes from Dürer? But it also includes other complex shapes such as five-pointed stars and combined decagons. Kepler's Aa tiling contains vertices with type 5.5.10 and indeed an entire 5.5.10 rose with a decagon surrounded by a ring of pentagons. It does not appear in Harmonices Mundi, where Kepler described the shape much more prosaically as a "combined decagon". Kaplan says that Kepler called his fused double decagon shapes "monsters" but does not say where the term comes from. The tiling was more complex than any of Dürer's and Kepler himself noted that the "structure is very elaborate and intricate".Ĭraig Kaplan points out in The trouble with five that several mathematicians have proved that Kepler was correct and that his Aa diagram can be continued into a tiling of the plane. ![]() His solution, which he illustrated as diagram Aa in Harmonices Mundi, involved five pointed stars and a peculiar fused double decagon shape. Like Dürer, Kepler could not resist looking beyond regular polygons to see if he could find plane tilings involving pentagons. In Harmonices Mundi he also discussed the fact that pentagons can appear in tilings of the sphere, where they form the basis of the dodecahedron. Kepler knew that pentagons could not occur in edge-to-edge plane tilings of regular polygons. Kepler also experimented with pentagon tilings. You can view Dürer's third and fourth tilings on Wikisource. It is possible that these empty spaces could be filled with rhombs but without more information it is unclear what Dürer had in mind. In his interesting article on pentagon tilings, The trouble with five, Craig Kaplan points out a variant of this third tiling with a central rose (a decagon surrounded by pentagons) that is continued by a spiral pattern of pentagons and 10-rhombs.ĭürer does not show how to complete his fourth tiling, which also has a central pentagon, but merely remarks that the "leftover areas you can then fill with whatever you like". You can view the thick tiling at the bottom right of this Wikisource page.ĭürer's third tiling contains a unique central pentagon and so cannot be periodic. The 5-rhomb and 10-rhomb play an important role in the modern theory of Penrose tilings, where they are often informally called the thick and thin rhombs. We could call this the "thick" tiling because of the thicker appearance of the 5-rhomb. The second tiling also contains pentagon and 10-rhomb prototiles but changes the tiling pattern to add gaps that can be filled by 5-rhombs. The first and simplest combines pentagon and 10-rhomb prototiles and could perhaps be called the "thin" tiling. The "thin" tiling with pentagon and 10-rhomb prototiles is the simplest of Dürer's pentagon tilings.ĭürer illustrates four pentagon tilings in the Painter's Manual.
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