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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?

The Maths of Life and Death – The God Equation

Aperiodicolleague Kit Yates has recently had a new book out: The Maths of Life and Death. He’s kindly agreed to share a sample chapter with us, explaining the God Equation: it’s used by NICE to decide whether to fund new drugs.

In my new book, The Maths of Life and Death, I explore the true stories of life-changing events in which the application (or misapplication) of mathematics has played a critical role: patients crippled by faulty genes and entrepreneurs bankrupt by faulty algorithms; innocent victims of miscarriages of justice and the unwitting victims of software glitches. I follow stories of investors who have lost fortunes and parents who have lost children, all because of mathematical misunderstanding. I wrestle with ethical dilemmas from screening to statistical subterfuge and examine pertinent societal issues such as political referenda, disease prevention, criminal justice and artificial intelligence. I aim to demonstrate that mathematics has something profound or significant to say on all of these subjects, and more.

High definition

We asked #bigmathoff competitor Lucy Rycroft-Smith to tell us a little about her latest project – CM Define It, an app aiming to collect and define mathematical vocabulary, which launches today.

When you teach mathematical vocabulary, how do you define its meaning?

Are you exact, choosing your words specifically?  Do you give a written definition?  Do you give multiple explanations?  Do you use diagrams?  Metaphors? Connect to previous vocabulary?

As part of our work at Cambridge, creating a Framework for mathematics learning, we are creating a network of semantic links across nodes in our mathematical layer – and we initially thought we could just import a mathematical glossary from somewhere else to populate this.  But we found so many inconsistencies, technical errors, and definition loops in many existing glossaries that we decided to make an app to ask the mathematical community what they thought, with the aim of developing a crowdsourced, multi-layered collage which takes into account different layers of mathematical experience.

MathsJam’s “Back of an Envelope” Fermi Challenge

Aperiodipal and MathsJam regular Rob Eastaway organised an inter-MathsJam competition for last month’s events, challenging Jams to make Fermi estimates on the back of an envelope. The prize was a copy of his new book, Maths on the Back of an Envelope. Here Rob gives a summary of the entries he received, and shares his favourites.

Regular attendees of MathsJam will know that in September, Katie Steckles kindly allowed me to hijack the evening (in the nicest possible sense) by posing some envelope-related challenges, in celebration of the publication of my new book Maths On The Back Of An Envelope. In addition to some envelope-related puzzles, there was also an open challenge to Maths Jam groups to come up with their own back-of-an-envelope problems, with the chance to win the book as a prize.

Alan Turing is the face of the new £50 note

The Bank of England has announced, following a public poll, that the new £50 note will feature mathematician, cryptographer and computer science pioneer Alan Turing. While this might seem like unambiguous good news, the issues it raises are more complicated than they first appear. Here’s a guest post from LGBT+ mathematician Calvin Smith with his thoughts on the decision.

Image result for £50 note

It’s obviously fabulous that Alan Turing is being recognised on the new £50 note, but this joy at seeing a gay mathematician given this recognition is tainted with the memory of his cruel treatment by the society of the day and the ongoing persecution of queer and trans people today.

It’s absolutely right that we celebrate the achievements of Turing the mathematician, and I’m hoping that this also creates an opportunity to talk more about the amazing work he did away from his well-known code-breaking in the field of mathematical biology, where he developed models which help us to understand the formation of shapes and patterns in biology. As a gay man working in mathematics I’m also hoping that the additional prominence given to Turing’s work acts as a further catalyst to drive inclusion in STEM subjects and that more young queer people consider study and careers in these areas.

If we are committed to doing the best science and solving modern problems then we need the most diverse set of thinkers available and for each of them to be able to bring their whole authentic selves to work. There are some wonderful organisations like Pride In Stem and events like the LGBT STEMinar which bring together and give voice to a diverse mix of queer scientists who can otherwise feel invisible or over-exposed in their local work environment. Stonewall’s work in making workplaces (especially universities) more LGBT+ inclusive cannot be underestimated as a driver for positive change. Hopefully the greater awareness and visibility afforded by Turing can drive improvements: the future of science is fabulously queer and intersectional. Fire the glitter cannon!

However, setting aside my maths-joy at seeing another mathematician celebrated I’m also filled with a cocktail of bitterness and seething rage at this announcement. Turing and society’s treatment of him is hugely symbolic and cannot be underestimated. Turing’s appearance on a bank note does not excuse society its historic treatment of him of other victims of homophobia. Just because we now have same-sex marriage does not mean the fight for queer inclusion is over. I grew up under Section 28 telling me that homosexual relationships were “pretend families” and while I am now a proud gay dad to two lovely scamps it is worth reflecting that Section 28 was only repealed in 2003, same-sex adoption introduced in 2005. Many of our recent victories seem fragile and culturally we still have a long way to go to eradicate the homophobia, lesbophobia, biphobia and transphobia which continues to lead to assaults on queer people to this day. Turing is also a call to action and constant vigilance so that we don’t turn the clock back on acceptance without exception, on inclusion for all, and for no outsiders.

July is LGBT wrath month: a Turing banknote cannot be allowed to become a pinkwashed sticking plaster over the underlying issues of intolerance, and we need LGBT+ people and their allies to continue the fight so that it remains a call to action – and not the undeserved pat on the back which leads to complacency and the further loss of queer and trans lives.


This is a guest post from Philipp Legner, the creator of Mathigon an interactive maths education platform.

Every year, thousands of students around the world ask themselves why they have to learn mathematics. Calculators can do long division. You can look up the quadratic formula on the internet. And when will you ever need calculus in everyday life? It seems like they have a point.

In fact, the maths curriculum has not changed significantly in the last 50 years. Its primary focus is on memorising rules and procedures which can be used to solve standardised exam questions. I created Mathigon because I strongly believe we need to change this – not only to make mathematics more enjoyable for students, but also to teach different skills that are much more useful in life: problem-solving, abstraction, logical reasoning, creativity, and curiosity.