Tessellations

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# Tessellations - PowerPoint PPT Presentation

Tessellations. By Kiri Bekkers &amp; Katrina Howat. What do my learner’s already know... Yr 9. Declarative Knowledge: Students will know... Procedural Knowledge: Students will be able to. Declarative Knowledge &amp; Procedural Knowledge.

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### Tessellations

By KiriBekkers & Katrina Howat

### What do my learner’s already know... Yr 9

Declarative Knowledge: Students will know...Procedural Knowledge: Students will be able to...

### Declarative Knowledge & Procedural Knowledge

Declarative Knowledge: Students will know...How to identify a polygonParts of a polygon; vertices, edges, degreesWhat a tessellation isThe difference between regular and semi-regular tessellationsFunctions of transformational geometry - Flip (reflections), Slide (translation) & Turn (rotation)How to use functions of transformational geometry to manipulate shapes

How to identify interior & exterior angles

Angle properties for straight lines, equilateral triangles and other polygons

How to identify a 2D shape

They are working with an Euclidean Plane Procedural Knowledge: Students will be able to...Separate geometric shapes into categoriesManipulate geometric shapes into regular tessellations on an Euclidean Plane Create regular & semi-regular tessellations

Calculate interior & exterior angles

Calculate the area of a triangle & rectangle

### Tessellations

Tessellation:Has rotational symmetry where the polygons do not have any gaps or overlapping

Regular tessellation:

A pattern made by repeating a regular polygon. (only 3 polygons will form a regular tessellation)

Semi-regular tessellation:

Is a combination of two or more regular polygons.

Demi-regular tessellation:

Is a combination or regular and semi-regular.

Non-regular tessellation: (Abstract)

Tessellations that do not use regular polygons.

Transformational Geometry

• Flip, Slide & Turn
• Axis of symmetry
• Shape
• Polygons
• 2D & 3D

Tessellations

Geometric Reasoning

Location & Transformation

### 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...

### 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 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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### 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

All these 2D tessellations are on an Euclidean Plane – we are tiling the shapes across a plane

Calculating interior anglesformula: (180(n-2)/n)wheren = number of sides

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

For a hexagon: 6 sides

(180(n-2)/n)

(180(6-2)/6)

180x4/6

180x4 = 720/6

(720* is the sum of all the interior angles)

720/6 = 120

Interior angles = 120* each

120*

120* + 120* + 120* + 120* + 120* = 720*

90*

90*

180*

### Where the vertices meet...

Sum of internal angles where

the vertices meet must equal 360*

Semi-Regular Tessellations

120*

120* + 120* = ? 240*

What are the angles of the red triangles?

360* - 240* = 80*

80* / 2 = 40* per triangle (both equal degrees)

### Creating “Escher” style tessellations...

Some images for inspiration...

### 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...

Working with 2D shapes

Example by M.C. Escher – “Circle Limit III”