You have spent this course finding patterns, testing symmetry, tiling shapes and moving motifs with slides, turns and flips. Now it is your turn to become the designer.
Your final project is to create a piece of artwork that is controlled by mathematical rules. The challenge is not just to make something beautiful — it is to make something another person could investigate and describe.
Start with one tiny motif
Draw one small asymmetric shape: perhaps a lightning bolt, fish, leaf, boot, arrow, letter or completely invented symbol.
Before you make a pattern, ask: what happens if I slide it? Turn it? Flip it? Which versions look most interesting together?
Your design brief
Your finished artwork must use at least two mathematical rules. You can choose from ideas you have explored throughout the course.
Slide your motif the same distance in the same direction.
Turn your motif around a chosen centre.
Make a mirror image across a chosen line.
Repeat shapes without leaving gaps or overlaps.
Someone looking at the picture should be able to search for the repetition, mirror line, centre of rotation, tiling rule or other structure you used.
Mix two rules
Here are four possible starting recipes. You do not have to copy one — they are there to show how two simple ideas can combine into something much richer.
Slide + flip: repeat a motif along a strip, alternating its reflected version.
Turn + reflect: build a design around a centre, then add mirror structure inside each section.
Tessellate + rotate: tile the page, but turn the motif as you move from one tile to the next.
Branch + reflect: grow a branching design on one side of a line, then mirror it.
Plan before you decorate
Make a rule card
Before making the final artwork, write down your rules. For example:
My motif: a crooked blue leaf
Rule 1: rotate it by 60° around the centre.
Rule 2: reflect every second copy.
Extra rule: alternate purple and orange.
Then test your rule on a rough sketch. If you cannot follow your own instructions, make the rule clearer before you begin the final version.
Create the artwork
You can work with pencil and ruler, coloured paper, paint, collage, stamps, tiles, a drawing app or anything else that lets you repeat a motif carefully. The mathematics matters more than the material.
Can someone reverse-engineer your pattern?
Do not explain your rules straight away. Show the finished artwork to someone else and ask them to be the pattern detective.
Ask them:
What is the basic motif?
What transformation do you think I repeated?
Can you find a mirror line or centre of rotation?
Can you spot a second rule?
Can you predict what should come next if the artwork continued?
If they can uncover your rules, you have communicated the mathematics through the picture itself.
Write an artist’s label
Artists often display a short label beside their work. Yours should explain the mathematics without turning into a long essay.
Title: ______________________________
My basic motif is: ______________________________
My first rule is: ______________________________
My second rule is: ______________________________
The part I changed or improved was: ______________________________
Go further: invent a constraint
Once your first design works, make the problem harder by adding a restriction. Constraints often make mathematical art more interesting rather than less interesting.
Can neighbouring copies always have different colours?
Everything must come from copies of exactly the same starting shape.
Use repeated turns so that the final copy fits exactly back onto the first.
Can your design cover an area completely like a tessellation?
What have you discovered?
Look back at the course. The individual ideas now fit together: a pattern is not just decoration — it is something with a rule that can be described, tested and continued.
You learned to search for the rule rather than just the appearance.
You investigated which shapes can cover a surface without gaps.
You explored mirror lines and rotational symmetry.
You used translations, rotations and reflections — and discovered that order can matter.
You compared exact mathematical patterns with the approximate structures we see around us.
You combined rules to create a mathematical artwork of your own.
You finished Patterns Everywhere!
Keep your final artwork as a record of the course — and keep looking for patterns. Floors, fabrics, buildings, plants, shells, logos and artworks all become more interesting when you start asking: what is the rule?
Help us improve Sumthing Else
You have reached the end of the course. We would especially love to know which lesson you enjoyed most, what you would change, and what mathematical topic you would like to explore next. Parents and children are both welcome to answer.
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