Mathematics has never been so fashionable!

Last year Paul Smith launched a T-shirt whose patterned design, below, was inspired by the Dutch artist and geek hero M C Escher (1898-1972).

One of Escher’s favourite techniques was tessellation – which is the mathematical term for covering an area with tiles that don’t overlap and leave no gap.

Often a tessellation uses only one shape of tile. The charm of a good tessellation is to find an interesting single tile, such as the dove in the Paul Smith T-shirt.

Sam Kerr's designs for Paul Smith flew off the shelves Photograph: Sam Kerr

British artist Sam Kerr designed the Paul Smith T-shirt and he has collaborated with London-based accessories brand Marwood to design tessellations for ties, bow ties and pocket squares that will be on sale later in the year.

Marwood's mathematical neckwear, Autumn/Winter 14. Photograph: Matt Walder Photograph: Sam Kerr

Sam has got so fascinated by tessellations that he has kept on designing them for fun. His hilarious dog tile below fits together in such a way that the animal is always licking its genitals.

"The fact that this tessellation is created by a rubber stamp is crucial to its concept. Rubber stamps are stereotypically quite kitsch and, as a single imprint, the dog reflects this, but it’s not until you repeat the tile that it turns into something obscene."

Photograph: Sam Kerr Photograph: Sam Kerr

Happy dogs. Photograph: Sam Kerr Photograph: Sam Kerr

He has also experimented with tessellations of clothes. Here’s his tiling of shirts, which he thinks might end up as a pattern on clothing. “I was interested in the idea of covering garments with garments, and possibly mismatching them,” he says, “I’m keeping my eyes peeled for an appropriate home.”

A very neat wardrobe. Image: Sam Kerr. Photograph: Sam Kerr

So, how do you make a single tile that can fit together and fill the plane?

The way that Sam makes his tiles is to start with a regular shape that can tessellate – such as an equilateral triangle, a rhombus or a hexagon – and then start modifying it.

His genital-licking dog started life as a rhombus, and the shirt as a hexagon.

Here he shows us how he made a rabbit tessellation. “The rabbits were something I had wanted to tessellate purely for their association with multiplication. The characteristic of being excessive breeders seemed somewhat appropriate for a process involving high numbers of identical figures. I think tessellating rabbits wallpaper would be fitting for a family planning clinic!”

STAGE ONE: Choosing a base shape Photograph: Sam Kerr

"For the rabbit tessellation I used an equilateral triangle with rotational symmetry. What this means is that any alterations made to the left side are copied and rotated 60° to the right side, so that each side is a perfect fit for each other. The bottom side is divided in half with any edits made to one half copied and rotated 180° to the other, again making them perfect fits. These applied rules are ultimately what will allow the tile to repeat infinitely."

STAGE TWO: Editing your base shape Photograph: Sam Kerr

"This second step is to mould your base shape into something recognisable. This stage is very much a case of trial and error and it can often take a while before you see something familiar. It’s frequently the case that one side will begin to resemble something while its identical and corresponding side will oppose what it is you’re trying to create."

STAGE THREE: Developing your tile Photograph: Sam Kerr

"Once a recognisable shape has been achieved, the tile is then developed further to more accurately resemble its intended form. This might involve applying curves and additional characteristics to your silhouette. Once you’re happy with the silhouette you can fill it with as much or as little detail as you like."

STAGE FOUR: Additional colours and tones Photograph: Sam Kerr

"Depending on the base shape, and how many tiles meet at any one corner once your design is repeated, you may require additional colours/tones to allow your tiles to stand out from each other. This ultimately boils down to preference and isn’t compulsory."

STAGE FIVE: Assembling your tessellation Photograph: Sam Kerr

"Once your tiles are finished, all that is left to be done is to repeat them into their formation and your tessellation is complete."

If this has got you inspired, send me your tessellations and I'll put them up on a post at a later date.



I asked Sam a few questions about his work:

When and how did you get interested in tessellations?

I was fascinated by them as a child but it wasn’t until a couple of years ago at age 31 that I revisited tessellations and taught myself the mathematics behind them. This rediscovery came about through a commission to design a T-shirt print based on the work of M C Escher. Although it wasn’t expected of me to produce a mathematically true tessellation I couldn’t resist giving it a go.

How long did it take you to master the technique? Do you use software or by hand?

It took me about two weeks of intermittent trial and error to create my first tessellation. For me, learning the mathematics behind the process wasn’t the hardest part, the real difficulty was in finding my first recognisable figure. Once you’ve gained a grasp of the laws of symmetry the process of tessellating becomes more like a game – a game not dissimilar from spotting familiar shapes in passing clouds. I produced my first tessellation by hand, albeit using Adobe Illustrator, but since then I have also made use of available software to make the preliminary stages a bit faster.

What tips do you have for readers wanting to design their own?

There are plenty of great websites and blogs out there which will tell you all you need to know about how to create tessellations, but try not to resort to software straight away. Creating your first tessellation by hand will help you gain a much better understanding of the process and will also provide you with a greater appreciation for the pre-computer age artists who had no choice but to do it the laborious way.

What reaction do you get from your work?

I feel very fortunate to be able to say that people’s reactions are very positive. One of the greatest attributes of tessellations in general is their ability to spark inspiration and enthusiasm. Every person I have showed my tessellations to, regardless of whether they work in design or even consider themselves to be creative, have their own spirited ideas about where a tessellation could exist. Perhaps the existence of them within nature sparks a subconscious connection within us all?

What place does mathematical art have in fashion?

I see mathematical art as an extension of conventional pattern making, so for me its place within fashion is hugely valid. The reason it’s perhaps less prominent is because there are fewer people doing it, but within that minority are some extremely talented artists doing incredible work. For me, the key to successful mathematical art lies in an even crossover between the two realms. It’s just as important to have a good comprehension of numbers as it is to have a keen eye for design.

PS: If you want to learn more about tessellations I have a section on them in my book Alex's Adventures in Numberland

