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Tessellations
By Kiri Bekkers & 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 polygon
Parts of a polygon; vertices, edges, degrees
What a tessellation is
The difference between regular and semi-regular tessellations
Functions 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 categories
Manipulate 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
Location &
Transformation
Geometric
Reasoning
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
2
1
3
1
2
4
3
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
2
1
2
1
3
4
3
1
5
2
4
6
3
Where the vertices meet...
Where the vertices meet...
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...
Semi-Regular Tessellations
All these 2D tessellations are on an Euclidean Plane –
we are tiling the shapes across a plane
Calculating interior angles
formula: (180(n-2)/n)
where n = number of sides
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
120* + 120* + 120* + 120* + 120* = 720*
Interior angles = 120* each
We use 180* in this equation because that is the angle of a straight line
Where the vertices meet...
Semi-Regular 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
tessellations...
Some images for inspiration...
Tessellations around us...
Tessellations around us...
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...
Working with 2D shapes
Example by M.C. Escher – “Circle Limit III”