**Chapter 8**

**Polygons**

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Polygons are often used in many types of architecture, artwork,
and graphic design. Floor tiles are usually grids of squares, but
sometimes are hexagons or other polygons. **Tessellations** are a
special type of tiling pattern, but usually using more complex
polygons. A tessellation is a graphic design composed of congruent
images that interlock to fill the page, with each shape fitting
perfectly into a sort of "jigsaw puzzle" pattern. A Dutch artist by
the name of Escher has done some beautiful designs using tessellated
polygons.

"Maurits Cornelis Escher was born in Leeuwarden, June 17, 1898. He studied drawing at the secondary school in Arnhem, by F.W. van der Haagen, who helped him to develop his graphic aptitude by teaching in the technique of the linoleum cut. From 1919 to 1922 he studied at the School of Architecture and Ornamental Design in Haarlem, where he was instructed in the graphic techniques by S. Jessurun de Mesquita, whose strong personality greatly influenced Escher's further development, as a graphic artist."

Escher created many beautiful linoleum and wood cuts, using a variety of techniques and subject matter. His work is of great interest to graphic artists and to mathematicians, especially his tessellations.

The information above came from an intriguing website on Escher. To visit this site, and see more of Escher's fascinating work, click on the link below:

Look closely at the tessellation below, and you will see that it is composed of interlocking shapes, all congruent but in 2 different colors. Each shape fits in the spaces between the other shapes, with no space left over. This may, at first, seem quite simple to do - but if you think about it a bit, you will realize that not all shapes will "tessellate"; the shape has to be carefully designed so this will work.

Only certain polygons will tessellate. If we look at regular polygons, then the sum of the angles at any one vertex would have to be 360°, as shown below with regular hexagons, on the left. The example on the right shows regular pentagons, and we see that they will not "tessellate"; they do not fit together.

Based on your knowledge of the interior angles of regular polygons, can you figure out which regular polygons will tessellate?

*If you discovered that only equilateral triangles, squares, and
regular hexagons will tessellate, then you are absolutely correct!
Their interior angles (triangle: 60°, square: 90°, and
regular hexagon: 120°) are the only ones that will divide evenly
into 360°.*

If we look at non-regular polygons, then there are an unlimited number of shapes that will tessellate, even very irregular shapes such as the ones you saw in the tessellation above. Some more common shapes will tessellate, such as parallograms, and that is why you can use a parallelogram grid as a base for a creative tessellation project, as explained in the Geometer's Sketchpad Tessellation Activity for this chapter.

Create a tessellation of your own, using the method described in The Geometer's Sketchpad activity called "Create Tessellation", in chapter 7 of the GSP Activities. Color it in Sketchpad including Polygon Interiors. (Remember, to constuct the interior of a polygon, select the vertices in consecutive order, then choose Polygon Interior in the Construct menu.)

Go to Chapter 9 Similar Triangles