Which shapes tessellate

Tape together. Trace your stencil on the piece of paper to tessellate your design. Maybe the shapes remind you of your favorite animal—we thought ours looked like chickens—make your tessellation look like that!

McAuliffe Shepard Discovery Center. Activities that explore astronomy, aviation, earth and space science. View fullsize. Colored pencils or crayons optional.

Regular Polygons Print-Out. All regular tessellations:. All semi-regular tessellations:. Can irregular polygons form tessellations?

Consider the following examples:. Escher-Inspired Drawings Materials: Thin cardboard like from a cereal box cut into a rectangle Scissors Tape Pencil Colored pencils, crayons, markers or other coloring material of your choice. Cultures ranging from Irish and Arabic to Indian and Chinese have all practiced tiling at various levels of complexity. Let's explore the wide variety of tessellations we find in nature, functional design and art. In mathematical terms, "regular" describes any shape that has all equal sides and equal angles.

There are three regular shapes that make up regular tessellations: the equilateral triangle, the square and the regular hexagon. For example, a regular hexagon is used in the pattern of a honeycomb, the nesting structure of the honeybee.

Semi-regular tessellations are made of more than one kind of regular polygon. Within the limit of the same shapes surrounding each vertex the points where the corners meet , there are eight such tessellations. Each semi-regular tessellation is named for the number of sides of the shapes surrounding each vertex. For example, for the first tiling below, each vertex is composed of the point of a triangle 3 sides , a hexagon 6 , another triangle 3 and another hexagon 6 , so it is called 3.

Sometimes these tessellations are described as "Archimedean" in honor of the third-century B. Greek mathematician. In the language of mathematics, the shapes in such a pattern are described as congruent. Every triangle three-sided shape and every quadrilateral four-sided shape is capable of tessellation in at least one way, though a select few can tessellate in more than one way.

Penrose tiles - these are based on five-fold symmetry, and they never repeat! Triangles Squares Hexagons. Large grid of triangles Large grid of squares Large grid of hexagons. Squares, triangles Hexagons, triangles Hexagons, squares, triangles Octagons, squares Dodecagons, triangles Dodecagons, hexagons, squares.

Large grid of squares and triangles Large grid of hexagons and triangles Large grid of hexagons, squares and triangles Large grid of octagons and squares Large grid of dodecagons and triangles Large grid of dodecagons, hexagons and squares. Every shape of triangle can be used to tessellate the plane.

Every shape of quadrilateral can be used to tessellate the plane. In both cases, the angle sum of the shape plays a key role. The next simplest shape after the three and four sided polygon is the five sided polygon: the pentagon.

Rather than repeat the angle sum calculation for every possible number of sides, we look for a pattern. In fact, there are pentagons which do not tessellate the plane. Attempting to fit regular polygons together leads to one of the two pictures below:. Both situations have wedge shaped gaps that are too narrow to fit another regular pentagon.

Thus, not every pentagon tessellates. On the other hand, some pentagons do tessellate, for example this house shaped pentagon:. The house pentagon has two right angles. Thus, some pentagons tessellate and some do not. The situation is the same for hexagons, but for polygons with more than six sides there is the following:.

This remarkable fact is difficult to prove, but just within the scope of this book. However, the proof must wait until we develop a counting formula called the Euler characteristic, which will arise in our chapter on Non-Euclidean Geometry.

Nobody has seriously attempted to classify non-convex polygons which tessellate, because the list is quite likely to be too long and messy to describe by hand. However, there has been quite a lot of work towards classifying convex polygons which tessellate. Because we understand triangles and quadrilaterals, and know that above six sides there is no hope, the classification of convex polygons which tessellate comes down to two questions:.

Question 2 was completely answered in by K. Reinhardt also addressed Question 1 and gave five types of pentagon which tessellate. In , R. Kershner [3] found three new types, and claimed a proof that the eight known types were the complete list.