How Many Regular Mosaics Can You Make?

We challenge you to make as many regular mosaics as possible with these triangles.

This is a regular mosaic:

A regular mosaic is a pattern made up of equilateral triangles. The pattern could continue in all directions. We are showing only a portion of it.

A regular mosaic is a pattern made up by repeating the same regular polygon over and over, with no spaces in between and no overlap of the spaces. A regular polygon is defined as one in which all the sides are equal and the angles formed by adjacent sides are all equal.

For an equilateral triangle, the three sides are all the same length and the three internal angles are each 60^o.

Another regular polygon is an octagon, with eight equal sides.

How many other regular mosaics can you find? To qualify as a regular mosaic, your pattern has to

be made up by repeating only a single regular polygon
have no spaces between the polygons
have no overlapping polygons
use polygons that are all the same size as well as the same shape

When you have arrived at your answer, can you explain why you must be right?

Course:

Math [1]
Geometry [2]

Result/Solution(s)

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.

Math Puzzle [3]