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Tessellations. By Kiri Bekkers & Katrina Howat. Learning Object. Declarative Knowledge & Procedural Knowledge. Declarative Knowledge: Students will know... Differentiate between the different types of Tessellations

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tessellations

Tessellations

By KiriBekkers & Katrina Howat

declarative knowledge procedural knowledge

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.

tessellations1

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

Regular Tessellations

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

regular tessellations1

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

Transformational Geometry...

Reflection (Flip)

Rotation

(Turn)

Translation

(Slide)

axis of symmetry

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

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axis of symmetry1

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)

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where the vertices meet

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

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 meet1

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

slide13

Calculating interior angles of a regular polygon formula: (n-2) x 180* / nwheren = number of sides

We use 180* in this equation because that is the angle of a straight line

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* + 120* = 720*

90*

90*

180*

possibilities

Possibilities...

All together 21 possible combinations of regular polygons

11 combinations (blue highlighted) create regular or semi-regular tessellations

non regular escher style 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

Creating “Escher” style tessellations...

Some images for inspiration...

digital resource 1

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
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

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

demonstration1
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

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 Hyperbolic Planes…

Extension - Working with 3D shapes…

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”

slide24

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