Many thanks to Branko Grunbaum for assistance with this page.Īll these tilings can be play with online at the Wolfram|Alpha Pentagon Tiling page.ĭoris Schattschneider's interactive tiling page is here. Peter J.The 14 Different Types of Convex Pentagons that Tile the Plane However, just as neither Emil Mackovicky nor Harmonia Mundi are strongly suggestive of the original Penrose Pentagons: rather, a rectangular cell is being repeated, althoughĪt about the same time, Kepler, who corresponded with Dürer,Įxperimented with arrangements of pentagons, and some of his drawings in Overall pattern having pentagonal symmetry, like the following:īut as you can see, the parts that are repeated are not Pentagons had been devised long before by Dürer. So the lack of once-enlarged boats, stars, and double pentagons Thus, we do have a choice which order of diamond to use, and Illustrates that the larger pentagons only create diamonds Here, unlike Dodecafoam, which is a true fractal. This illustrates the fractal nature of this tesselation,Īlthough it is fractal only upwards, not downwards as shown One must go to encounter an enlarged double-size pentagon.įinally, an incomplete diagram on a larger scale, with only Since the basic shapes that contain double-sized pentagons are theīoat and the star, and one-step enlarged equivalents for them are Thus, we do have a usable recurrence relation, even if a complex one, Switches to using a different version of the diamond as theīase when enlarging one step, instead of attempting to replaceĪll the diamond's internal components by enlarged versions of Larger equivalents of the diamond are rearranged, one simply But there is an equivalentįor each one after two enlargement steps, andīecause of the way in which the internal components of the Unlike the diamond and the pentagon, there are no equivalentsĪfter only one enlargement step. This illustration shows these: note that for these shapes, Need larger versions of the double-size pentagon, the star,Īnd the boat, as well as the diamond and the pentagon. To be able to continue this process indefinitely, we It is a larger pentagon, with each sideĮxactly twice as long as the sides of the original pentagon. Note that the fifth basic tile shape makes its appearance Versions of the diamond, and this illustration shows Of themselves, composed of a boat, a pentagon, and a star.Īs we repeat the process further, we will need even larger Iteration, we replaced the small diamonds by a bigger version Pentagon by a bigger version of itself, in the last So far, we have repeated the process of expanding the Only one more basic tile type remains to be seen. Types would look nicer, even if it made the tiling more Respectively, but I thought having them as additional tile Those two shapes could be built up from five and three diamonds Penrose termed a "boat" when heĭescribed the first of his famous aperiodic tilings thus, I In addition to wedges, this has two other component shapes:Ī star, and an incomplete star which I originally called "hat" Repeating this again, one gets the next larger pentagon. Gaps have now paired to form an additional tile, a smallĭiamond, that we will need in addition to a pentagon. Replacing its component pentagons by its original shape, Larger one by substituting the larger pentagon into itself, If one takes that larger pentagon, and forms a yet Next comes five pentagons, surrounding a sixth one On the smallest scale, one has one pentagon. To build larger and larger pentagons which are mostly composed Pentagons, the first step, with this technique, is to try In trying to make a tiling which is mostly composed of Here is part of my first attempt at such a tiling: This is illustratedĪ three-dimensional fractal tesselation of dodecahedrons.Ī less impressive technique, but still capable ofĭrawing inspiration from the other two, is called It is also possible to make a fractal tesellation, withĭifferent sizes of pentagons, down to the infinitely small, The Penrose tilings will be discussed on the Public taste than on the merits of these tilings. On these tilings, but that is more of a comment on the intelligence of This hasĪpparently happened without producing a flood of cheap toys based This patent, granted on January 9, 1979, has expired. This produces a Penrose tiling.Īlthough these tilings are covered by U. One is to use tiles with special rules (or jigsaw puzzle-like Produce a pattern with pentagonal symmetry. There are various techniques which can be used to
0 Comments
Leave a Reply. |