Tessellations Objective: Wednesday, 09 November 2016 Tessellations Objective: To understand and construct tessellations using polygons MC Escher´s work Create a non-regular shape and tessellate

What shapes are used to make up the honeycomb? Can these shapes be arranged so that there are no gaps between them? A regular tessellation is a repeating pattern of a regular polygon, which fits together exactly, leaving NO GAPS.

What does this have to do with tessellations? A regular tessellation is a repeating pattern of a regular polygon, which fits together exactly, leaving NO GAPS. So the bees honeycomb… is a regular tessellation of hexagons

Which regular polygons tessellate?

Equilateral Triangles: Do tessellate 

Which regular polygons tessellate?

Squares: Do tessellate 

Which regular polygons tessellate?

Regular Pentagons: Don’t tessellate

Which regular polygons tessellate?

Regular Octagons: Don’t tessellate: This is called a semi-regular tessellation since more than one regular polygon is used.

Non Regular Tessellations A non-regular tessellation is a repeating pattern of a non-regular polygon, which fits together exactly, leaving NO GAPS.

Drawing tessellations example Does this shape tessellate?

Drawing tessellations example Does this shape tessellate?

Drawing tessellations How about this one?

Non Regular Tessellations There are tessellations all around us!!!

Geometric shapes & Art Maurits Cornelis Escher (usually referred to as M. C. Escher) born in 1898 in Leeuwarden in Holland, was a Dutch graphic artist. He is developed Tessellating shapes as an art form Let’s take a look at some of his work…

MC ESCHER BY DISTORTING BASIC SHAPES HE CHANGED THEM INTO ANIMALS, BIRDS AND OTHER FIGURES 18

Your turn Step 1: Draw a line between two adjacent corners on one of the sides of the square. Your line can be squiggly or made up of straight segments. Whatever its shape, your line must connect two corners that share one side of the rectangle. Step 2: Cut along the line you drew. Take the piece you cut off and slide it straight across to the opposite long side of the rectangle. Line up the long, straight edges of the two pieces and tape them together. Step 3: Now draw another line that connects two adjacent corners on either side of the shape. Cut along this new line. Take the piece you cut off and slide it straight across to the opposite side of the shape. Line up the straight edges and tape them together.

Your turn Step 4: On your grid paper, carefully trace around your pattern shape. It may help to position the squared-off corner (formerly the edge of the index card) in one corner of the grid.

Plenary 360 n What’s the size of an exterior angle of a regular: What’s the size of an interior angle of a regular: a) square? b) pentagon? c) hexagon? a) square? b) pentagon? c) hexagon? 180 – 90 = 90o 360 4 = 90o 360 5 = 72o 180 – 72 = 108o 360 6 = 60o 180 – 60 = 100o

Consider the sum of the interior angles about the indicated point. There are only 3 regular tessellations. Can you see why? 60o 60o 120o 90o 90o Consider the sum of the interior angles about the indicated point. 120o 6 x 60o = 360o 4 x 90o = 360o 3 x 120o = 360o 108o 135o 36o 90o 2 x 135o = 270o 3 x 108o = 324o

Regular Polygon Size of each exterior angle Size of each interior angle Does this polygon tessellate? Equilateral Triangle Square Regular Pentagon Regular Hexagon Regular Octagon Regular Decagon 360 3 = 120o 360 60 = 6 Yes 180 – 120 = 60o 360 4 = 90o 360 90 180 – 90 = 90o = 4 Yes 360 5 = 72o 360 108 = 3.33 No 180 – 72 = 108o 360 6 360 120 = 60o = 3 Yes 180 – 60 = 120o 360 8 360 135 = 45o = 2.67 No 180 – 45 = 135o 360 10 = 36o 360 144 = 2.5 180 – 36 = 144o No