12/19/2023 0 Comments Tiling shapes geometryThe Aperiodical © 2023 Peter Rowlett, Katie Steckles and Christian Lawson-Perfect. Tags: arc, concave, facet, nonperiodic, pattern, pentagon, periodic, puzzle, radial, tiles, tiling About the author What other main shape would you try as a starting point? What happens when we use both the concave pentagon and the tricurve as main shapes in the same set? overall provide more challenge and play value.have the aesthetic appeal and interest of connected arcs and.have main shapes that are simpler to describe and construct.have more complex diamond shapes, due to their 36° arcs.have the additional part count of the 36° lens shape.Likewise, to keep the effect of the concavity of the smallest arc, the faceted equivalent of the tricurve needs 12 sides and four unique angles –whereas the much simpler tricurve can be described with two angles (36° and 72° – the 108° is the simple sum and redundant).Ĭompared to structurally equivalent tilings with faceted tile shapes, the above arc-sided sets: Since we sometimes connected at the midpoint of pentagon’s concave side, we’ll need to describe the shape as having ten faceted sides. The larger arcs of the two main shapes would now be more complex to describe and construct. Surprisingly, this reduction by faceting makes some things a little more complex. The 36° lens shape disappears, reducing the set part count and the count of the lens pieces in the tiling. We could of course reduce these sets and their tilings by replacing all 36° arcs with straight lines (facets). The underlying diamonds with corners of 36° and 144°, or 72° and 108°, are two rhombus shapes used in a version of the Penrose tiles. This is not surprising since all the shapes incorporated 36 and 72 angles. Also we can choose shapes to make tiling (as a puzzle) more challenging for instance, if we modify the concave-sided pentagon so one of its sides is a convex arc, tiling will require more thought and thus be more interesting.īoth main shapes above are of course compatible with the minor shapes. There is a pattern of arcs interwoven with the pattern of shapes this may be seen as full or partial circles, or in the patterns of the arcs as they branch and connect. In both shape sets, part of the complexity of the final tiling is in the use of the arcs. We can lay these out in various ways to get different types of gaps, as shown here:ĭesigning a small tiling set involves making tradeoffs between shape complexity, part count, and aesthetic appeal. But let’s replace the sides of a regular pentagon with concave arcs of 72°. The set of tiles needed to help tile the plane with regular pentagons is well known. The regular pentagon of course can’t tile by itself. Let’s look at two main shapes: the first is based on the pentagon the second is a tricurve. My simple approach here is to start out with one interesting main shape and see what other (minor) shapes are needed to fill in the gaps, by trial and error then try to refine and optimize that set to make it, in a sense, efficient.įor this post I’ve avoided the frameworks of the self-tiling regular triangles, squares and hexagons. “Simplest” covers not only size of set and the shapes, but also the least total information needed to describe or construct the shapes. “Interesting” includes variety, complexity, challenge and aesthetic appeal. Lately I’ve been playing with groups of curved tiling shapes, asking a question common for me: how to get the most play value as an open-ended puzzle? This means getting the most interesting possibilities from the simplest set. In an earlier post elsewhere I covered some basic arc-sided shapes that tile by themselves. Tim has previously written guest posts here about tiling by tricurves, and is now looking at ways of tiling with other shapes. You're reading: Irregulars Small Sets of Arc-Sided Tiles Small Sets of Arc-Sided Tiles | The Aperiodical The Aperiodical
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