Tessellations Tessellations By Kiri Bekkers & Katrina Howat

What do my learner’s already What do my learner’s already know... Yr 9know... Yr 9Declarative Knowledge: Students will know...

Procedural Knowledge: Students will be able to...

Declarative Knowledge & Declarative Knowledge & Procedural 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 shapesHow to identify interior & exterior anglesAngle properties for straight lines, equilateral triangles and other polygonsHow to identify a 2D shapeThey 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 tessellationsCalculate interior & exterior anglesCalculate the area of a triangle & rectangle

Tessellations 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

Geometric Reasoning

Location & Transformati

on

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

Regular Tessellations 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 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

12

3

4

1

2

3

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

12

3

4

12

3

4

5

6

1

2

3

Where the vertices Where the vertices meet... meet...

90* + 90* + 90* + 90* = 360*

120* + 120* + 120* = 360*

60* + 60* + 60* + 60* + 60* + 60* = 360*

Where the vertices Where the vertices meet... meet...

Semi-Regular Semi-Regular Tessellations 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 Where the vertices meet... meet...

Semi-Regular Semi-Regular Tessellations Tessellations

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

Calculating interior anglesformula: (180(n-2)/n) where n = 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* each120* + 120* + 120* + 120* + 120* = 720*

Where the vertices Where the vertices meet... meet...

Semi-Regular Semi-Regular Tessellations Tessellations

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

Some images for inspiration...

Tessellations around Tessellations around us...us...

Tessellations around Tessellations around us...us...

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

Extension Hyperbolic Planes…Extension Hyperbolic Planes…

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

Extension - Working with 3D Extension - Working with 3D shapes…shapes…