Exploring Tessellations: Understanding and Creating Patterns
Delve into the world of tessellations by learning how to construct patterns using regular polygons. Discover the concept of regular tessellations and their role in forming intricate designs without any gaps. Explore which regular polygons tessellate and how they contribute to fascinating geometric arrangements like honeycombs.
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TESSELLATIONS TESSELLATIONS Objective: To understand and construct tessellations using polygons
360 n Starter Activity Starter Activity 1. What s the size of an exterior angle of a regular: 2. What s the size of an interior angle of a regular: a) square? 180 90 = 90o a) square? 360 4 = 90o b) pentagon? b) pentagon? 360 5 = 72o 180 72 = 108o c) hexagon? c) hexagon? 360 6 = 60o 180 60 = 100o
Recap External angle Size of 1 external angle 360 n = Internal angle Size of 1 internal angle =180 external angle
What shapes are used to make up the honeycomb? Can these shapes be arranged so that there are no gaps between them?
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? White marble
Do tessellate Equilateral Triangles:
Which regular polygons tessellate? White marble
Do tessellate Squares:
Which regular polygons tessellate? White marble
Regular Pentagons: Don t tessellate
Which regular polygons tessellate? White marble
Do tessellate Regular Hexagons:
Which regular polygons tessellate? White marble
Regular Octagons: Don ttessellate: This is called a semi-regulartessellation since more than one regular polygon is used.
Which regular polygons tessellate? White marble
Regular Polygon Size of each exterior angle Size of each interior angle Does this polygon tessellate? Equilateral Triangle 360 3 = 120o 360 60 Yes = 6 180 120 = 60o Square 360 4 = 90o 360 90 180 90 = 90o = 4 Yes Regular Pentagon 360 5 360 108 = 72o 180 72 = 108o = 3.33 No Regular Hexagon 360 6 360 120 = 60o = 3 180 60 = 120o Yes Regular Octagon 360 8 360 135 = 45o 180 45 = 135o = 2.67 No Regular Decagon 360 10 360 144 = 36o 180 36 = 144o = 2.5 No
There are only 3 regular tessellations. Can you see why? 60o 120o 60o 60o 90o90o 90o 120o 60o 90o 60o 120o 60o 3 x 120o = 360o 6 x 60o = 360o 4 x 90o = 360o 108o 108o 135o 135o 36o 108o 90o 2 x 135o = 270o 3 x 108o = 324o Consider the sum of the interior angles about the indicated point.
Non Regular Tessellations A non-regular tessellation is a repeating pattern of a non- regular polygon, which fits together exactly, leaving NO GAPS. All triangles and all quadrilaterals tessellate.
Drawing tessellations Show that each of these shapes tessellate by drawing at least 8 more around each one. b) ex. a) d) e) c) f) g)
