REGULAR SHAPES
Just three will do
Only the equilateral triangle, the square and the regular hexagon tile a plane on their own. Six, four or three of them meet at each corner.
Test a shapeStart
THE MATHS ON YOUR FLOOR
No gaps,
no overlaps.
Only three regular shapes can tile a floor on their own.
Triangles, squares and hexagons fit because their corners add up to a full turn. From there the story runs through centuries of star patterns to a single tile, found in 2023, that covers a plane but never repeats.
360°around every corner
Will it tile?Keep in mind
- Corners must make 360°
- Stars start with circles
- Some tilings never repeat
- Tessellation, in plain English
THE SHORT ANSWER
Angles decide
what fits.
A tiling covers a flat surface with no gaps or overlaps.
Three kinds of pattern tell the story.
STAR PATTERNS
Built from simple forms
Geometric patterns in Islamic art grow from the circle and the square. Star patterns come from squares and triangles drawn inside a circle.
How they are built
NEVER REPEATING
The 2023 hat
In March 2023 a single shape called the hat was shown to tile a plane in a pattern that never repeats: the long-sought "einstein".
Meet the hatWILL IT TILE?
Pick a shape,
see if it fits.
Choose a shape to see whether copies of it can cover a flat surface with no gaps or overlaps. The bar shows how readily it tiles.
Regular polygons: Plus magazine, University of Cambridge. Hat tile: Smith, Myers, Kaplan and Goodman-Strauss (2023).
TWO NUMBERS WORTH KNOWING
From thousands
to one.
How many shapes it once took to force a pattern that never repeats,
and how many it takes now.
20,426 tiles
Tiles in the first non-repeating set
The first known set of tiles that could only tile without repeating used 20,426 square-edged Wang tiles. Later work cut the number down to sets of just two tiles, including Penrose's in the 1970s.
1966 · BERGER
1 tile
Tiles needed today
The hat, announced in March 2023, is a single shape made of eight kites that tiles the plane but never periodically. A mirror-free relative, the spectre, followed about two months later.
2023 · THE HAT
"Einstein" here comes from the German ein Stein, one stone, not from the physicist. Aperiodic tilings also help explain quasicrystals, discovered in 1982 and recognised with the 2011 Nobel Prize in Chemistry.
FOUR THINGS TO TRY
Make a tiling
at the kitchen table.
All you need is card, a ruler, scissors and a pencil. Each step shows one of the ideas on this page in your own hands.
Step 1
Add up the corners
Cut out several equal squares, triangles and hexagons. Push corners together and see that they close up exactly at 360°.
Step 2
Try a regular pentagon
Cut out a few regular pentagons and try to fit them around one point. A gap or an overlap always appears.
Step 3
Start a star from a circle
Draw a circle, then a square and a triangle inside it. Rotate and overlap them and a star pattern begins to emerge.
Step 4
Shift and compare
Slide a tracing of a square tiling one tile across and it matches perfectly. That is what periodic means, and what the hat never does.
THE EINSTEIN PROBLEM
One shape,
no repeats.
A square tiling is periodic: slide it one tile across and it looks the same. Mathematicians long asked whether one single shape could force a tiling that never repeats. In the 1970s Penrose found non-repeating tilings using two shapes, which now decorate buildings including Oxford's mathematics department.
In March 2023 a mathematical hobbyist, working with three mathematicians, announced the hat. Turned around it looks like a T-shirt. Its tilings need some mirror-image copies, so a few months later the team found the spectre, which manages without them.
Questions
Questions on the way
What is a tessellation?
A tiling, or tessellation, covers a flat surface with copies of shapes leaving no gaps and no overlaps, like the tiles on a bathroom wall or a floor.
Why can only three regular shapes tile a floor?
The corners meeting at each point must add up to 360°. That only works with equilateral triangles (6 at a point), squares (4) and regular hexagons (3).
How are Islamic geometric patterns made?
The Metropolitan Museum of Art says they are generated from simple forms such as the circle and the square, combined, duplicated and interlaced. Star patterns derive from squares and triangles inscribed in a circle.
What does aperiodic mean?
A tiling is aperiodic if it never contains arbitrarily large patches that repeat by sliding. Adding one odd tile to a square grid does not count.
Is the hat really just one shape?
Yes, though some copies in a hat tiling are mirror images, at a ratio of about 7 to 1. The related spectre tiles the plane without repeating and without mirror images.
Destination
Where the facts come from
- Plus magazine (University of Cambridge): Maths in a minute, Tiling troubles
- Plus magazine: Tip of the hat, celebrating the aperiodic monotile
- University of Cambridge, Faculty of Mathematics: the hat aperiodic monotile
- Smith, Myers, Kaplan and Goodman-Strauss: An aperiodic monotile (arXiv:2303.10798)
- The Metropolitan Museum of Art: Geometric patterns in Islamic art
Updated