by the Fourth Grade Challenge Class

Tessellations are repeating polygons designed in such a way that they can be laid down like tiles to fill an area without any gaps or overlapping. Click on the small pictures below to see examples of tessellations designed by our class using the computer program TesselMania. Keep reading for more information about tessellations.

Toucan great dane shark fish hippo
Andrew Ms. McNeely Elena Jeff
pretty kitty s s tropic nutty squirrel blue whale
Kirsten Brady Monica Patrick
stingray sleeping cat wacky elephant frog fish
Nathan Sarah Rachel Walter

Background Information about Tessellations

The design of geometric shapes that individually or in combination cover a flat surface without gaps or overlappings has a long history. The Sumerians (about 4000 B.C.) in the Mesopotamian Valley built homes and temples decorated with mosaics in geometric patterns. The materials used in the mosaics were thin slabs of burned clay, called tiles. When colored and glazed, tiles served not only as part of the structure of buildings but also as artistic decorations.

Later, the Persians showed that they were masters in tile decorations. Similarly, the Moors used congruent, multicolored tiles on the walls and floors of their buildings. Moslem and Islamic tile patterns with striking colors still survive.

Roman buildings, floors, and pavements were decorated with tiles which the Romans called tessellae. The Roman word tessellate is the root of our English word tessellation.

In mathematics, the word tessellation (or tiling) has come to mean the repeated use of polygons and other curved figures to completely fill an infinite plane region without gaps or overlappings.

Regular Tessellations

A regular tessellation is formed by congruent regular polygons. Only three regular polygons (triangle, square, and hexagon) can be used.

Tessellations are described by giving, in order, the number of sides of the polygons at a vertex point.

Triangle Tessellation = 3, 3, 3, 3, 3, 3 
Square Tessellation = 4, 4, 4, 4 
Hexagon Tessellation = 6, 6, 6

Semiregular Tessellations

Semiregular tessellations combine two or more regular polygons so that the same polygons appear in the same order at each vertex point. There are eight possible semiregular tessellations.

A semiregular tessellation is described by giving, in order, the number of sides of the polygons at a vertex point. Begin with the polygon that has the smallest number of sides at the vertex point and move (clockwise or counterclockwise) about the point, writing down the number of sides of the polygons involved.

Here's a semiregular tessellation by Brady. (Click the small picture to see the big one.)


Demiregular Tessellations

In regular and semiregular tessellations, the arrangement of the polygons at each vertex point is the same. In demiregular tessellations the arrangement of the polygons at each vertex point is not the same. They might include a combination of two or three vertex point types. A vertex point in a demiregular tessellation may be regular, semiregular, or nonregular. There are at least fourteen demiregular tessellations.

We drew these demiregular tessellations. (Click the small pictures to see the big ones.)

Kirsten Elena Jeff Andrew Sarah Rachel Patrick Monica Nathan

Nonregular Tessellations

These are tessellations with nonregular simple convex or concave polygons. All triangles and quadrilaterals will tessellate. Some pentagons and hexagons will. The tessellations we made with TesselMania (see the pictures at the top of the page) are all nonregular tessellations.

Nonpolygonal Tessellations

Tessellations formed by plane closed figures that are not all polygons (figures with curves)

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Last updated 12/20/00