By Kiri Bekkers & Katrina Howat Tessellations By Kiri Bekkers & Katrina Howat

Learning Object

Declarative Knowledge & Procedural Knowledge Declarative Knowledge: Students will know... Differentiate between the different types of Tessellations Functions of transformational geometry - Flip (reflections), Slide (translation) & Turn (rotation) Algebraic formula relating to internal angles of shapes. How conservation of area and various types of symmetry can be applied to tessellations. Procedural Knowledge: Students will be able to... Separate geometric shapes into categories i.e. polygons (regular & non regular), triangles Create a non-regular, tessellating polygon Use rules relating to conservation of area. Create regular & semi-regular tessellations Prove congruence of various polygons using triangle and angle properties. Define axis of symmetry, and identify various types of symmetry within a pattern.

Tessellations Tessellation: An infinitely repeating pattern of shapes which completely covers a plane without overlapping or gaps, while displaying various types of symmetry. Regular tessellation: A pattern made by repeating a regular polygon. (only 3 regular polygons are capable of forming a regular tessellation) Semi-regular tessellation: Is a combination of two or more regular polygons. Non-regular tessellation: (Escher) Tessellations that do not use regular polygons.

Regular Tessellations A regular tessellation can be created by repeating a single regular polygon...

Regular Tessellations A regular tessellation can be created by repeating a single regular polygon... These are the only 3 regular polygons which will form a regular tessellation...

Transformational Geometry... Reflection (Flip) Rotation (Turn) Translation (Slide)

Axis of Symmetry Axis of Symmetry is a line that divides the figure into two symmetrical parts in such a way that the figure on one side is the mirror image of the figure on the other side 1 2 3 1 2 4 3

Axis of Symmetry n(sides) = n(axis of symmetry) Axis of Symmetry is a line that divides the figure into two symmetrical parts in such a way that the figure on one side is the mirror image of the figure on the other side n(sides) = n(axis of symmetry) 1 2 1 2 3 4 3 1 5 2 4 6 3

Where the vertices meet... Sum of internal angles where the vertices meet must equal 360* 90* + 90* + 90* + 90* = 360* 120* + 120* + 120* = 360* 60* + 60* + 60* + 60* + 60* + 60* = 360*

Semi-Regular Tessellations A semi-regular tessellation is created using a combination of regular polygons... And the pattern at each vertex is the same...

Where the vertices meet... Sum of internal angles where the vertices meet must equal 360* Semi-Regular Tessellations Exterior Angle Interior Angle Side Vertex (1) – Vertices (plural) All these 2D tessellations are on an Euclidean Plane – we are tiling the shapes across a plane

Calculating interior angles of a regular polygon formula: (n-2) x 180 Calculating interior angles of a regular polygon formula: (n-2) x 180* / n where n = number of sides We use 180* in this equation because that is the angle of a straight line 120* For a hexagon: 6 sides (6-2) x 180* / 6 4 x 180* / 6 4 x 180* = 720 / 6 720 / 6 = 120* (720* is the sum of all the interior angles) Each Interior Angle = 120* each 120* + 120* + 120* + 120* + 120* = 720* 90* 90* 180*

Possibilities... All together 21 possible combinations of regular polygons 11 combinations (blue highlighted) create regular or semi-regular tessellations

Non-Regular (Escher style) Tessellations This type of tessellation can made up of regular or non regular polygons, as well as abstract shapes. The internal angles surrounding the vertices must still equal 360 degrees… Therefore, most abstract tessellating shapes are designed from a regular polygon.

Creating “Escher” style tessellations... Some images for inspiration...

Tessellations around us...

Digital Resource 1: Conservation of Area…..An interactive program allowing students to create irregular polygons which can then be tessellated. http://www.shodor.org/interactivate/activities/Tessellate/ Evaluating the math content: Positives: Interactive, visual, immediate, easy to use, would be a good teaching tool Negatives: Doesn’t explain ‘why’, no terminology, doesn’t explain properties of tessellations or of the shapes

Demonstration..... Using the techniques detailed in the previous web resource, we have created our own tessellation shape conserving area of an equilateral triangle This is a fun activity which would be suitable for any high school year level.

Digital Resource 2: This webpage explains how to create an Esher style (or non-regular) Tessellation using a square grid. http://www.paec.org/fdlrstech/escher.pdf Evaluating the math content: Positives: Step by step, skill levels from basic to advanced, focuses on the geometry of tessellations, starting from equilateral triangle, includes ideas for digital application Negatives: Doesn’t discuss angles, no explanation of type of planes, doesn’t extend into a 3D level

Demonstration..... Using the techniques detailed in the previous web resource, we have created our own Esher-style tessellation. This is a fun activity which would be suitable for any high school year level. Year 10 students should be asked to find both internal and external angles at vertices, identify axis of symmetry for both individual shapes as well as the entire tessellation, and demonstrate congruence.

Digital Resource 3: “Hyperbolic Tessellations” http://aleph0.clarku.edu/~djoyce/poincare/poincare.html Evaluating the math content: Positives: Extremely comprehensive explanation of the maths involved in tessellating on the Euclidian, Hyperbolic and Elliptical planes. Lots of examples of symmetry. Great teacher resource. Visual examples. Algebraic formulas, expressions and explanations. Negatives: Possibly too advanced for Grade 10 students

Extension Hyperbolic Planes… Extension - Working with 3D shapes… Extension Hyperbolic Planes… The Hyperbolic Plane/Geometry – working larger than 180* & 360* Circular designs like Escher’s uses 450* - a circle and a half... Example by M.C. Escher – “Circle Limit III”

I have discovered such wonderful things that I was amazed… Out of nothing I have created a strange new universe. Janos Bolyai, speaking about his discovery of non-Euclidean Geometry