10.6 Geometric Symmetry and Tessellations

Recall: Two polygons were similar if the angles were the same but the sides were proportional.

We could think of taking the first polygon, moving it, rotating it, and scaling it.

A rigid motion is an action of only moving and rotating an object, without changing its shape.

An object has symmetry if you get the same object back after a rigid motion.

Pop quiz!!!

How many rigid motions can you find which are symmetries for this object?

To study the possible symmetries an object can have, we need to know the types of rigid motions possible.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 8

Rigid Motions

To reflect an object, you need a mirror (axis of reflection) that you are reflecting across.

Notice that the distance of a point and its mirror image are the same distance from the line.

Also, lines connecting corresponding points are perpendicular to the axis of reflection.

Pop quiz!!!

The following is not a rigid motion:

1) reflection2) translation3) symmetry4) rotation5) glide reflection

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 12

• Example: Reflect polygon ABCDE about the axis of reflection l.

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Rigid Motions

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2nd: Draw segments BB', CC', DD', and EE', similar to AA'.

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Rigid Motions

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3rd: Draw the reflected polygon by connecting vertices A', B', C', D', and E'.

Rigid Motions

A translation is a rigid motion where the object is moved without rotating it.

(Finish the drawing)

Every vertex is moved by a translation vector. It is the same for all verticies.

Example: Translate the object by the given translation vector.

Pop quiz!!!

If you translate a polygon with 5 verticies, how many verticies does the new polygon have?

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 20

• Example: Use the translation vector and the axis of reflection to produce a glide reflection of the object shown.

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Rigid Motions

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• Solution:1st: Place a copy of the translation vector at some point, say Y, on the object.

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Rigid Motions

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2nd: Slide the object along the translation vector so that the point Y coincides with the tip of the translation vector.

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Rigid Motions

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3rd: Reflect the object about the axis of reflection to get the final object.

Rigid Motions

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Rigid Motions

Notice that distance PA = PA'Angle APA' = 90 degrees

Distance PC = PC'Angle CPC' = 90 degrees

Recall: A rigid motion moves an object without scaling it or distorting it.

The following are rigid motions:

- reflection - translation - glide reflection - rotation

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 27

Symmetries

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• Example: Symmetries of the star.

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Symmetries

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• One Symmetry:

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Symmetries

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Symmetries• Another Symmetry:

What are the symmetries of this object?

Are there any symmetries from

- reflection - translation - glide reflection - rotation

What are the symmetries of this object?

Are there any symmetries from

- reflection - translation - glide reflection - rotation

Pop quiz!!!

The following is not a rigid motion:

1) reflection2) translation3) symmetry4) rotation5) glide reflection

A tessellation, or tiling, of the plane is a pattern that covers the plane completely by polygons.

It should have no holes and no overlapping pieces.

A regular tessellation, or regular tiling, is built from copies of a single regular polygon.

Which regular polygons tessellate the plane?

Recall the formula for angles in regular polygons.

Pop quiz!!!

The size of an interior angle of a regular hexagon in degrees is

1) 902) 1203) 1354) 1805) 3606) 540

Notice that around a vertex of any regular tessellation, we must have an angle sum of 360°, and we also must have three or more polygons of the same shape and size.

For hexagons, n = 6.The interior angles are then 120 degrees.

Let's try with a regular pentagon.

n = 5 and the interior angle is 108 degrees.

The 4th added pentagon causes overlap! Thus we cannot use pentagons to tile the plane.

If the regular polygon has more than 6 sides, n > 6.

Each time, this formula will give an angle bigger than 120 degrees.

Three of these n-gons will overlap.

So 7-gons, 8-gons, etc cannot tile the plane.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 41

Regular Tessellations

Only three regular tessellations exist: equilateral triangles (n=3), squares (n=4), and regular hexagons (n=6).

Pop quiz!!!

When tessellating the plane with a regular triangle (equilateral triangle), how many triangles meet at a vertex?

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 43

• Example: An architect has tiles shaped like equilateral triangles and squares. All tiles have sides of the same length. Is it possible to produce a tessellation using a combination that contains both types of tiles?

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Plain Tessellations

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 45

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Tessellations

Now try different numbers of squares and corresponding triangles.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 46

Tessellations

A tessellation with 2 squares and 3 equilateral triangles is shown.

Can you tile a floor with octagons and squares if the sides are the same length?

Octagon: n=8, interior angle = 135