All is silent and peaceful as your bedroom-turned-spacecraft sails smoothly through space. Out the window you see bright stars and spiral galaxies, and you see the same view on your computer screen as you zip along.
NASA, ESA, and J. Lotz and the HFF Team (STScI)
Suddenly a brilliant green dot appears on the screen. You reach for the computer mouse on your desk and click on the dot. The glistening green dot grows into a multicolored, sparkling sphere. You click on the sphere again and it grows larger.
Now it looks like a bright, multicolored soccer ball! Its surface is covered with a pattern of five-sided and six-sided patches, which are pentagons (five-sided) and hexagons (six-sided).
You click on one of the pentagons, and it expands to fill the entire screen. Now you see that it is made up of a curious pattern of diamond shapes. Some of the diamonds are fat; others are skinny.
This curious pattern is made from just two types of tiles: fat and skinny diamonds.
All at once a loud roar fills the room, and you feel your stomach rising, as if you were descending in an elevator. Then suddenly it's quiet.
You stand up, walk over to the window, and stare out in disbelief. You seem to have landed on a very weird planet. There are no rocks and no hills. You see no Martian desert and no mysterious jungle.
The scene outside your window looks more like the floor of a giant's bathroom than a natural landscape. The flat "ground" is covered with diamond-shaped tiles, some fat and some skinny, arranged in the same unusual pattern that appeared on your screen.
You step outside and walk along on the tiled floor until you come to a sign.
Beside the sign is a map made up of a pentagon surrounded by hexagons.
Directions from Penrose Plaza.
Using the map as a guide, you make your way toward Checkerboard City, stepping from one fat or skinny diamond tile to the next. Before long, you come to a vast checkerboard.
Turning to the right, you head in the direction of Triangle Terrace. Sure enough, there you see a terrace paved with triangles. Then you proceed through Octagon Square and Honeycomb Haven.
Ambling on, you at last arrive at the Construction Zone. It's a total mess! The floor is bare and black, and tiles of different shapes and colors are scattered all around: hexagons, squares, triangles, and diamonds.
Just for fun, you decide to play around with the pieces. Using only the squares, you find that you can easily cover the ground, leaving no bare spots. You then try covering the ground with triangles.
A tiling pattern made up of equilateral triangles.
Then you cover it with hexagons. No bare spots!
A tiling pattern made up of regular hexagons.
Instead of using just one type of piece, you try to cover the ground using a mixture of squares, triangles, and hexagons. Can you fit the shapes into a pattern without leaving any gaps?
From what you saw at Octagon Square, you already know that you can use a combination of regular octagons and squares to cover a flat surface without leaving any bare spots, a process called tiling the plane. Another name for a tiling is tessellation.
A tiling made up of regular octagons and squares, with four octagons surrounding each square.
Squares and matching equilateral triangles fit together in two different ways to form a tiling.
Equilateral triangles and matching regular hexagons also fit together in two different ways.
There is just one way to combine equilateral triangles, squares, and regular hexagons.
TRY IT!
Stroll around your house or check out your neighborhood. What sorts of tiling patterns do you see? How would you go about sorting them into different types?
Examples:
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