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Small Sets of Arc-Sided Tiles

Tim has previously written guest posts here about tiling by tricurves, and is now looking at ways of tiling with other shapes.

In an earlier post elsewhere I covered some basic arc-sided shapes that tile by themselves. 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. “Interesting” includes variety, complexity, challenge and aesthetic appeal. “Simplest” covers not only size of set and the shapes, but also the least total information needed to describe or construct the shapes.

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.

For this post I’ve avoided the frameworks of the self-tiling regular triangles, squares and hexagons. Let’s look at two main shapes: the first is based on the pentagon; the second is a tricurve.

Using Pentagons

The regular pentagon of course can’t tile by itself. The set of tiles needed to help tile the plane with regular pentagons is well known. But let’s replace the sides of a regular pentagon with concave arcs of 72°.  We can lay these out in various ways to get different types of gaps, as shown here:

Note in many cases a point is hitting midpoint on a neighboring arc. Many of these gaps can be filled with simple lens shapes of 36°, 72° and 108°:

The remaining gaps need to be filled with partial lenses: 72° or 108° lens with one or more chunks gone. To fill the remaining gaps as is would require at least four more shapes. But we can reduce this number by backing up and combining the smaller tiles. If we start with the 36° lens and add a 4-side concave diamond (with corners of 36° and 144°, and 36° concave arcs) we can get the 72° lens and any partial 72° lens. In order to make the 108° lens we need to use another concave diamond, with corner angles of 72° and 108°.  This also lets us fill out the end of the elongated 108° lens shape.

So now the part count is four shapes (above): one major and three minor, and these let us fill the gaps:

If this were a real puzzle we would probably complain about the large number of little 36° lens pieces. Can we use less of these? The 36° lens only needs to be separate from the 4-concave diamonds in the cases where the lenses would overlap. We can permanently attach two of the 36° lenses to the thin diamond; and attach three 36° lenses to the wide diamond. So now our minor shapes look like this:

 and the tiling looks like this:

This set of tiles seems a reasonable solution (although other similar sets are possible). Now rather than simply filling gaps, we can start exploring various tilings:

Using Tricurves

The second main shape is a 36°-72°-108° tricurve, which is quite different. The tricurve already has great flexibility for tiling by itself periodically, non-periodically, and radially (as shown in previous post). So any additional parts should add to the possibilities – and it doesn’t take much. Even adding a single 36° or 72° lens at the center of a radial tiling opens many possibilities:

Since the underlying geometry is similar let’s start out with our original three minor shapes: the 36° lens and the two 4-sided concave diamonds. These let us create a very wide range of tilings:

Some of these patterns are of course not sustainable for tiling the plane. The additional complexity allowed comes partly from a means to fill gaps between adjoining convex sides or concave sides. Each of the three minor shapes by itself can add to tricurve tiling complexity, as can the use of any two of the minor shapes. Also the minor shapes can tile without the main shape –which the pentagon minor shapes can’t do (Why not?).  

Because of the ways a tricurve can tile with itself, there are many more opportunities for odd-shaped gaps that can’t be covered with the three minor shapes. With the tricurve the tiling is much more open-ended that with the pentagon above. There are no doubt various other minor shapes that could be added to fill gaps, but we’ll stick with these three for now. This whole set is interesting since it consists partly of nested lens shapes:

Also the tricurves—or either of the concave-diamond shapes for that matter—can make a circular hole, which can be filled with a circle made of the set or just the minor pieces:

Thoughts on tiling set design

Designing a small tiling set involves making tradeoffs between shape complexity, part count, and aesthetic appeal. In both shape sets, part of the complexity of the final tiling is in the use of the arcs. 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. 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.

Both main shapes above are of course compatible with the minor shapes. This is not surprising since all the shapes incorporated 36 and 72 angles. 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.

We could of course reduce these sets and their tilings by replacing all 36° arcs with straight lines (facets). The 36° lens shape disappears, reducing the set part count and the count of the lens pieces in the tiling.

Surprisingly, this reduction by faceting makes some things a little more complex. The larger arcs of the two main shapes would now be more complex to describe and construct. Since we sometimes connected at the midpoint of pentagon’s concave side, we’ll need to describe the shape as having ten faceted sides. 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).

Compared to structurally equivalent tilings with faceted tile shapes, the above arc-sided sets:

  • have the additional part count of the 36° lens shape
  • have more complex diamond shapes, due to their 36° arcs
  • have main shapes that are simpler to describe and construct
  • have the aesthetic appeal and interest of connected arcs; and
  • overall provide more challenge and play value.

Further possible investigations:

What happens when we use both the concave pentagon and the tricurve as main shapes in the same set?

What other main shape would you try as a starting point?

Making Tricurves

Tim Lexen has written a series of posts on the topic of Tricurves: Bending the Law of Sines, Combining Tricurves and Phantom Tiling. In this latest post, Tim has been working with our own Katie Steckles to turn Tricurves into real objects to play with.

When you discover an interesting mathematical shape or object, there’s a strong instinct to play with it – maybe by drawing sketches and doodles to test the limits of the idea. But in the case of Tricurves, drawing an accurate shape takes a little time, and it doesn’t lend itself well to idle experimentation.

Producing a physical version of a shape, in enough quantity to allow for experimentation, makes it much more tangible. In our own respective locations, we’ve each made use of laser cutting facilities to produce wooden Tricurve tiles to play with, and we encourage you to join in.

“Transposition”, a sliding block puzzle by Jacob Siehler

Start and finished states of a Transposition puzzle

This is just a quick post to tell you about a nice puzzle game I spotted on Mathstodon.

It’s called Transposition, and it’s a sliding block puzzle in the vein of the popular game Rush Hour. You’re given a grid that’s almost full of rectangular blocks, and you have to slide them around each other until the two coloured blocks have swapped places.

The puzzle was invented by mathematician Jacob Siehler, who says he used a computer search to generate a pool of puzzles, given the rules of the game. I took quite a while to solve all 5 “easy” puzzles – as with any logic puzzle, you need to play about for a while to get a feel for the mechanics. I hadn’t appreciated at first that the grey blocks don’t need to be in their starting places when you solve the board – only the coloured blocks need to in the right positions.

There are 26 puzzles at the moment, ranging from “easy” to “very hard”. Have a crack at it! I really enjoyed it.

Play: Transposition, by Jacob Siehler

Finding an equation that has the same solution when rotated

(x+8)/6=9/(5+x) or, flipped, (x+5)/6=9/(8+x)
Solve this equation for x. Then rotate 180 and solve for x again.

I made this. Here’s how…

A puzzle for another day

A few months ago, my faculty’s PR person sent an email round asking if anyone would like to write a puzzle for the Today programme’s “Puzzle for Today” slot, to be broadcast during the programme’s trip to Newcastle in Freshers’ Week. A colleague said this might be the kind of thing I’d like to do, which it was, so I started thinking, and eventually came up with a brand new puzzle which I thought would work well.

If you listened to the Today programme this Thursday morning, you’ll have heard not my name, but that of Dr Steve Humble, who’s got a lot more experience doing this kind of thing. Turns out, they wanted something more ‘visual and interactive’, so asked him instead. I think that was a polite way of saying they just didn’t like my puzzle. Oh well!

Steve chose a classic puzzle that coincidentally appeared on Twitter about a month ago, prompting much discussion. It’s a good puzzle, much better than the one I came up with, but I don’t think Steve was completely right to say “It is possible that you can always create a winning game” – that’s only the case if there are an even number of coins, but his statement said “around ten coins”. I suppose he might’ve meant that, starting from having a handful of coins, you can decide to only use an even number of them.

The upside is that I can now talk about the puzzle here, where someone might actually enjoy it.

Curvahedra Geometry

Longtime friend of the Aperiodical, artist, mathematician and #BigMathOff semifinalist Edmund Harriss has come up with a new puzzle/toy/exploration set, developing his Curvahedra system. We asked him to explain the maths behind it in this guest post.

Curvahedra is a flexible system of connectors that can make all sorts of different things, combining puzzles (and self-created puzzles) with art. You can get your own to play with, explore, prepare for Christmas (they make great decorations, wreaths and presents) at our online store, and get 15% off with the discount code APERIODICAL.

As this is the Aperiodical, you might be most interested in how it can be used to explore mathematics. In the big math off I talked about the basic ideas behind the system, Gauss’ famous Theorema Egregium and Gauss-Bonnet theorems. A really simple version of this comes from just considering triangles, that can be built up to make this: