Tilings and

Tesselations A regular tiling must: Pentagons Hexagons Triangles This principle also works for irregular tilings and tessellations with more than one shape. Tile a 2-D surface leaving absolutely no gaps or overlaps.

Have tiles that are regular polygons and all the same.

Have each vertex look the same with 360°. A triangle has an interior angle of 60º. A regular pentagon has interior angles of 108º. Warm-up: Use the following figure to create a pattern that leaves no gaps or overlaps You should have something that looks like this: A tiling is.. When you fit individual regular polygons together with no gaps or overlaps to fill a flat surface. A tessellation is... A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. These shapes are not necessarily regular polygons.

Tessellations can have more than one shape that is repeated to create the pattern 60+60+60+60+60+60=360º If we were to make a tiling with this figure, each vertex would look something like this: We can make a tiling with triangles! Squares A square has an interior angle of 90º. Let's try to do a tiling...

Each vertex looks like this: 90º+90º+90º+90º=360º We can also do a tiling with squares. 108+108+108=324º We can NOT tile a plane with pentagons. The vertex of a tiling made with hexagons looks like this: 120+120+120=360º We can make a tiling with a hexagon. If we do a tiling with pentagons, each vertex would look like this: Any figure with more than 6 sides will create overlaps and therefore is unable to form a tiling. So... We can make tilings with:

triangles

squares

hexagons because... Their interior angles are factors of 360º At each vertex, the interior angles of all the shapes that meet must add up to 360º for it to be tiling or tessellating a plane. Learning Objective: Students will demonstrate understanding of a tessellation by constructing a pattern that tiles a plane with one or more regular polygons. This is also known as a tiling. For this tiling we are only using one shape ,therefore, this is a regular tiling. A regular hexagon has 120º in each interior angle. Exit Slip: If we have a square, three equilateral triangles, another square and another triangle in a vertex, can we make a tiling? What if we have a triangle, two hexagons, and another triangle?

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# Tilings and Tessellations

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