Warm-Up
Each figure is an isometry image of the figure above. Tell whether
their orientations are the same or opposite. Then classify the
isometry.
opposite;
opposite;
reflection
glide reflection
same;
same;
translation
rotation
Objectives: 1 Identify transformation in tessellations, and figures that will tessellate; 2 Identify
symmetries in tessellations.
A tessellation, or tiling is a repeating pattern of figures that
completely covers a plane, without gaps or overlaps.
Some examples:
A figure that creates a
tessellation is said to
tessellate.
rectangles
Octagons and Squares
Different Pentagons
Objectives: 1 Identify transformation in tessellations, and figures that will tessellate; 2 Identify
symmetries in tessellations.
Identify a transformation and the repeating figures in each tessellation below.
1.
2.
1. rotation; one fish
2. translation;
horse and rider
Objectives: 1 Identify transformation in tessellations, and figures that will tessellate; 2 Identify
symmetries in tessellations.
Regular Tessellations
A regular tessellation is a pattern made by repeating a regular polygon.
There are only 3 regular tessellations:
33 3
333
Triangles
3.3.3.3.3.3
4
4
4
4
Squares
4.4.4.4
Hexagons
6.6.6
For a
regular
tessellation,
the pattern
is identical
at each
vertex!
Look at a Vertex ...
Three hexagons meet at this vertex,
and a hexagon has 6 sides.
So this is called a "6.6.6" tessellation.
Objectives: 1 Identify transformation in tessellations, and figures that will tessellate; 2 Identify
symmetries in tessellations.
Semi-regular Tessellations
A semi-regular tessellation is made of two or more regular polygons.
The pattern at each vertex must be the same!
There are only 8 semi-regular tessellations:
3.3.3.3.6
4.8.8
3.6.3.6
3.12.12
3.3.3.4.4
3.4.6.4
3.3.4.3.4
4.6.12
Objectives: 1 Identify transformation in tessellations, and figures that will tessellate;
2 Identify symmetries in tessellations.
four rectangles in
a square shape
two  rectangles
Objectives: 1 Identify transformation in tessellations, and figures that
will tessellate; 2 Identify symmetries in tessellations.
Because the figures in a tessellation do not overlap or leave gaps,
o the sum of the measures of the angles around any vertex must be 360.
Objectives: 1 Identify transformation in tessellations, and figures that will tessellate;
2 Identify symmetries in tessellations.
If the angles around a vertex are all congruent,
o then the measure of each angle must be a factor of 360.
Determine whether a regular 15-gon tessellates a plane. Explain.
Because the figures in a tessellation do not overlap or leave gaps, the sum of the
measures of the angles around any vertex must be 360°.
Check to see whether the measure of an angle of a regular 15-gon is a factor of 360.
Because 156 is not a factor of 360, a regular 15-gon will not tessellate a plane.
Determine whether each figure will tessellate a plane.
12. Rhombus
Yes, all quadrilaterals
tessellate.
15. regular hexagon
13. acute triangle
Yes, all triangles
tessellate.
16. regular dodecagon
14. regular decagon
NO
17. regular 15-gon
NO
Objectives: 1 Identify transformation in tessellations, and figures that
will tessellate; 2 Identify symmetries in tessellations.
A translation maps the
tessellation onto itself.
A glide reflection maps the
tessellation onto itself.
Identify the repeating figures and a transformation in the
tessellation.
4.8.8
A repeated combination of an octagon and one adjoining square will completely
cover the plane without gaps or overlap. The arrow shows a translation.