“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” - John von Neumann
Have you heard of tiling or tessellations?
Whether it’s for tiling a street or making designs like M. C. Escher, tessellation involves covering a plane with geometric shapes or polygons, a bit like a puzzle, but as simple and fun as this may seem, there’s a lot of things you need to know about maths before you can get it right.
Here’s Superprof’s guide to tiling like a mathematician, something that GCSE students will need to know. In this article, we'll be looking at tiling in mathematics, how it's done, some examples of famous tiling, and some simple tiling projects you could at home.
What Is Tiling in Mathematics?
You probably see tiling and tessellations regularly in your everyday life. Whether it’s tiled or paved streets, the tiling in your bathroom, or stained-glass windows in a church, there are plenty of examples of geometric shapes and polygons in a pattern that tiles a plane.
Tessellation in maths is covering a plane with one or several different geometric shapes. Typically, when we refer to tessellation and tiling, we’re talking about Euclidean geometry.
A lot of shapes including squares, rectangles, hexagons, parallelograms, pentagons, and triangles can be used to create tessellations and the polygons don't even have to be regular to tessellate, though you'll probably find a regular polygon easier to create a pattern with.
In crystallography (the science looking at crystalline structures at the atomic scale), tiling and tessellation also occur.
You can classify different types of tiling. Euclidean tilings by convex regular polygons are when a single shape can tessellate without leaving a gap. For example, an equilateral triangle, square, or hexagon can be used.
Semiregular tilings can be one of eight possible combinations. Of course, you can still get quite creative with these combinations.
We can talk about isometry when certain tiles or pavings are identical. Isometry is a congruent transformation across a plane. Isometry is when the points of a shape through translation, rotation, or symmetry are moved to a new place but are still the same distance apart.
Some patterns occur with symmetry and isometry and are known as wallpaper groups. There are 17 wallpaper groups in total.
We can also talk about periodic tiling (tessellation) with quadrilaterals and there’s also the idea of tiling in 3-dimensional space, too.
Now you know how tilling works, you can start tiling for yourself.
“We often hear that mathematics consists mainly of 'proving theorems.' Is a writer's job mainly that of 'writing sentences?” - Gian Carlo Rota
How to Tile
As you’ll have understood, this is used in art, architecture, nature, and not just maths lessons.
To try it out for yourself, get two pieces of paper, coloured pencils or felt tips, and some scissors. Choose your geometric shape and cut it out of the paper (you won’t need much mathematical knowledge to do this) and anyone can do this exercise.
Now you have your starting point, you’re going to place it on the other sheet, trace around it, and repeat. Try to find all the types of symmetry available.
Now you’re tiling! As long as your geometric shape doesn’t overlap, you’re fine.
Once you start getting the hang of this without the underlying maths, you can start trying other tilings. Try different shapes to practise. You’ll soon find out which shapes don’t tile. This is a fun way to learn new maths concepts.
Let’s have a look at some famous tilings.
While maths exercises are all well and good, there are some excellent examples in the real world. Penrose tiling, for example, is particularly famous. This is an aperiodic tiling that uses several different geometric shapes and reflection and rotational symmetry.
There’s also Cairo pentagonal tiling, which uses congruent convex pentagons. Unsurprisingly, you can find examples of this on the streets of Cairo. The pentagons have two sides of equal length and one side that’s shorter than them.
Here are some other fine examples of tiling:
- The Archeological Museum of Seville, Spain.
- Chedworth Roman Villa, Yarnworth, UK.
- The art of M.C. Escher
- The mosaics in the Alhambra, Granada, Spain.
- The paving in the Royal Château de Blois, Loire Valley, France.
- The paving in Roman churches.
- And many more!
Are you looking to design the perfect geometric patterns?
Use your imagination and a bit of patience.
Simple Tiling You Can Do Yourself
If you’re looking to do some mathematical tiling, there are plenty of tutorials you can follow. You can download worksheets to make art like Escher. Just look for tessellating fish and rabbits.
Discover different Maths tuition on Superprof.
You can also turn to more mathematical activities with geometric shapes on grid paper. A lot of maths sites will have activities you can follow. You can do them without needing to understand all the underlying mathematics. You can also make your own with pens, paper, and scissors like we showed you earlier.
Once you begin to understand how it works, you can look at the maths behind it. From there you can transform shapes and create patterns that could go on forever.
Don’t forget that you can create rotational patterns, too. There are a lot of interesting possibilities.
Now you know a bit more about tessellation and tiling, you’ll start seeing it everywhere you go. You don’t need to be a mathematician to get started.
You can get started with simple shapes. Grab some paper, draw some basic shapes, work out the angles, and start creating some beautiful tessellations. Tiling is a maths activity that allows you to be creative. Who knew maths and arts and crafts could work together. There are many different options. However, if you want to learn more about the maths involved, consider getting help from a private tutor. From there, you’ll be able to see what you can do.
If you'd like to learn more about tessellations, patterns, symmetry, geometry, just mathematics in general, consider getting in touch with one of the many talented and experienced tutors on Superprof.
You can find tutors specialising in maths for all levels from primary school to university. There are different ways to learn from a private tutor so make sure you choose the type of tutoring that works for you, how you like to learn, and your budget.
Face-to-face tutoring is the most common and usually involves the tutor teaching just one student at their home. Since there's only one student, the tutor can tailor every minute of the lesson to them and ensure they're getting the most out of every minute they're working together. Of course, this bespoke service tends to cost more as you're paying for the tutoring and the time the tutor has to spend planning the lessons and travelling to their students' homes.
Online tutorials can also be taught one-on-one, but since the tutor doesn't have to travel to their students and can teach more lessons each week, they don't tend to charge as much. While these aren't ideal for hands-on subjects and skills, online tutoring is excellent for academic subjects such as maths.
Group tutorials are an excellent choice for those on a budget. With several students attending the same class, the tutor can afford to charge less per student. While you won't get to enjoy lessons that are tailored just to you, you can enjoy paying less for them. If you and some friends, family members, classmates, or colleagues need to learn more about maths, group tutoring could be an excellent and affordable option.
Don't forget that many of the tutors on Superprof offer the first hour of tutoring for free so you can try a few out before deciding on which one is right for you. You could also try out the different types of tutoring if you're not sure which one you'd prefer.
It's always a good idea to outline your requirements before you start looking for tutors. On the Superprof website, you can see what experience they have, what their other students have to say about them, and how much they charge each hour. Before you start getting in touch with tutors and arranging free lessons, we recommend that you narrow down your search to tutors that meet your requirements.