I'm delighted to have been asked to take part in Emily Grosvenor's blog tour for her new children's book, 'Tessalation!'. Emily (@emilygrosvenor) has always had a particular interest in tessellations and was instrumental in organising the first World Tessellation Day on 17th June this year (a wonderful celebration; check out the hashtag #WorldTessellationDay). Her book, funded via a successful Kickstarter campaign, describes the adventures of girl named Tessa, who enjoys hiding herself in the tessellating patterns she finds in the world around her. The illustrations by Maima Widya Adiputri are exquisite, and the book provides the perfect way to introduce young children to the mathematics of pattern-making by tapping into their inherent curiosity about the natural world.
Emily's work got me thinking about the many ways we can use the appealing aesthetics of tessellations to engage the students in our mathematics classrooms. The book itself contains clear step-by-step instructions for making tessellating tiles or 'tesserae' from a starting square. These instructions would be suitable for use with students of all ages (see the central image below) and the method is explained in more detail in this post on the Kids Math Teacher blog.
Many other bloggers on the tour have contributed their ideas for exploring tessellations with children. For secondary students, Brent Yorgey's post presents a clear introduction to the geometrical reasoning behind the mathematics of tessellations. And the post by John Golden (@mathhombre) contains links to his comprehensive resources page covering many aspects of this rich topic.
One of the lesser known ways of working with tessellations is through the medium of paper folding (a favourite pursuit of mine) and I have come across two great resources to support teachers in working with this technique in the classroom.
The first is this fascinating publication from Liz Meenan, which explains how to create various Islamic-influenced tiling patterns (see below) from squares, equilateral triangles, kites and hexagons folded from A-sized paper. Liz has kindly allowed me to use her instructions on this presentation and handout.
The second is this fantastic resource from mathematical origamist David Mitchell, found on his website origamiheaven.com. The whole website is worth exploring, especially if you are interested in mathematical or modular origami. The instructions for folding origami tiles from 'silver rectangles' (A-sized paper is a good approximation) are of particularly high quality, and the easiest ones to fold would be accessible to students from early primary upwards. For example, it would never have crossed my mind that the trapezia created from a simple, single fold would tessellate in such a variety of ways!
David's detailed instructions also discuss the angles, geometry and symmetries of the various tiles, along with investigations of area and perimeter (including work with surds), which make this activity suitable for students across the full age range. Truly worth a look.
I do hope you've been inspired to explore some of these tessellation resources in your classroom. As always, please do share your experiences on Twitter, or in the comments below.
Several people have asked for tips about getting started with geometric modular origami, so I'm hoping this post will be useful. A quick caveat before I begin: I'm no expert, just an enthusiastic amateur with a great deal still to learn! However, the wonderful thing about this form of origami is that beginners can produce fabulous pieces right from the start, as many complex models are built from relatively simple individual units.
But first, what is modular origami and why fold it? Essentially, modular origami is the use of two or more sheets of paper to create complex models, built up from several units, or modules. Construction involves inserting the tabs of one module into the pocket of another. If identical modules are used, the resulting flat or 3D shape contains repeating pattern and symmetry, and can hence be described as 'geometric' in form. This is the kind of modular origami that I most enjoy folding. As I mention on my Origami Gallery page, I find that the process of folding dozens of the same modular unit, over and over again, induces a meditative, flow-like state. This is then followed by the substantial, and often frustrating, challenge of constructing the given model. And, as all maths lovers know, there's not much that's more satisfying than cracking a difficult puzzle! Better still, you end up with a beautiful geometric form to enjoy.
So, a few starting points.
Most origami instructions come in diagrammatic form, so you will need to be familiar with the system of symbols used. These are relatively self-explanatory, and quick to pick up. You can, of course, learn to fold modular origami models from the wealth of Youtube videos shared online, but I personally find that folding from original diagrams feels more authentic, and ultimately more rewarding. Interpreting them is part of the challenge! However, video instructions certainly have their uses, and often prove an invaluable aid to add some visual clarity when encountering particularly tricky folds.
You'll also need some paper! I have to confess that I usually fold from normal 80gsm coloured copier paper; it's cheap and readily available. I do treat myself to 'proper' origami paper sometimes, though nothing special. But to be honest, I find that modular pieces seem to hold together better with copier paper as it has a slightly grainier surface and is therefore a little less slippery. Experts would probably be horrified!
Saying that, I have just purchased this bargain 500 sheet pack of coloured origami paper (on special offer at Amazon at the moment). And recently I found two packs of the most gorgeous rainbow geometric paper (above), which I've yet to use: Rainbow Patterns paper and Geometric Origami paper.
I also often use packs of craft or scrap-booking paper which can be picked up quite cheaply at shops like The Works, The Range or the Craft Superstore. These come in some lovely designs, but as the paper is a bit thicker, and 'frays' a bit more easily under repeated folding, they aren't so well suited to particularly intricate or small-scale models.
In a nutshell, though, pretty much anything goes! More information about different types of paper can be found at Robert J Lang's blog, here.
It is also useful, although certainly not essential, to have access to a small guillotine or paper trimmer, as well as a small ruler, set square or even an old credit card to use as a 'folding bone'. This will help achieve a good, crisp fold, especially important when folding modular units which will need to interlock firmly afterwards. Accuracy is important: small inaccuracies in the individual units can cause problems during the assembly of the final model.
Next, you'll need a project to get started on! A really useful starting point is this page from the Origami Resource Centre, which has links to instructions for many modular designs, handily organised (roughly speaking) by increasing complexity, and hence difficulty. When you are feeling more confident, Meenakshi Mukerji's website has another page of useful links here.
Three important modules to learn to fold are Mitsubo Sonobe's Sonobe unit, Thomas Hull's PHiZZ unit and Robert Neale's Penultimate Module. These can all be used to build models of increasing levels of complexity, or, depending on slight adjustments to the angles in the fold, models with different polygonal faces.
There are also several excellent modular origami books out there. Two that I have found to provide particularly useful starting points are:
Another rich vein of modular origami ideas to tap is the wonderful world of Kusudama: Japanese flower or 'medicine balls'. There are many traditional models to try, but there are also several new designers from across the world, whose designs seem to become ever more intricate. Ekaterina Lukasheva (@kusudamame) and Maria Sinayskaya (@MariaSinayskaya) are two such origamists whose beautiful work is well worth exploring (see the list below). Ekaterina has recently published a new book with step-by-step instructions for making her designs: Modular Origami Kaleidoscope.
And finally, for reference and inspiration, here are the details of some modular origami A-listers (this is by no means intended as a definitive list):
There, that should be more than enough to get you started! If you have any questions do contact me here or in the comments below. And happy folding :)
I teach maths. I'm a bit arty. I like to combine the two.