Puzzle

There are only three regular mosaics. Here are the other two:

To see why these are the only possibilities, look at the point where the shapes meet. In the case of the regular mosaic made of triangles, six triangles meet at any point. In order to fit together, their internal angles have to add up to 360^o. Since there are six triangles, each with an internal angle of 60^o, this works out to 360^o.

For the square mosaic, four squares meet at each point. The internal angles are 90^o, which also adds up to 360^o. Finally, three hexagons, each with an internal angle of 120^o, meet at each point in the hexagon mosaic. Three 120^o angles also add up to 360^o.

The octagon cannot be used for a regular mosaic. Cut out a bunch of octagons and try it. The internal angles of an octagon are 135^o. Two of them add up to 270^o, and three of them make 405^o. There’s no way to get 360^o, and so there’s no way to have octagons meet at a point with no spaces or overlap.

What about a pentagon? The internal angles are each 108^o. No multiple of 108 equals 360.

Could there be some regular polygon with many sides that can be used for a regular mosaic? No, because as you increase the number of sides, the internal angle also increases. With hexagons you get a perfect fit with three of them. Any polygon with more than six sides will have internal angles greater than 120^o, so you can’t fit three together at a point. To fit only two together would require internal angles of 180^o, but a 180^o angle between two adjacent sides means that they form a straight line, so you can’t have a polygon.