10.6 Geometric Symmetry and Tessellations Recall: Two Polygons Were Similar If the Angles Were the Same but the Sides Were Proportional

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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.

Rigid Motions

© 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 8 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) reflection 2) translation 3) symmetry 4) rotation 5) glide reflection Rigid Motions

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

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

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

(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?

Rigid Motions

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

(continued on next slide) © 2010 Pearson Education, Inc. All rights reserved. Section 10.6, Slide 20 Rigid Motions • Solution: 1st: Place a copy of the translation vector at some point, say Y, on the object.

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

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

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 Symmetries

• Example: Symmetries of the star.

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) reflection 2) translation 3) symmetry 4) rotation 5) 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) 90 2) 120 3) 135 4) 180 5) 360 6) 540 For hexagons, n = 6. The interior angles are then 120 degrees.

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. 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. Regular Tessellations

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

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

• 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?

Tessellations

Now try different numbers of squares and corresponding triangles.

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

Octagon: n=8, interior angle = 135