9.2 Symmetries and Tessellations

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Chapter 9

• 9.2 Symmetries and Tessellations

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Symmetry

• Rotation(al) Symmetry
• An object has a rotational symmetry if it can be
  rotated about some fixed point and end up back
  where it started
• Reflection Symmetry
• An object has reflection symmetry if its
  reflection across some line is itself

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An Example

• Notice every vertex must reflect/rotate to
  another vertex, and the same for edges.

4
An Example

• Notice every vertex must reflect/rotate to
  another vertex, and the same for edges.

5
An Example

• Notice every vertex must reflect/rotate to
  another vertex, and the same for edges.

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Note

• General rotations can have their fixed point
  anywhere- but when you are looking for rotational
  symmetries, the fixed point in the rotation
  (I.e., how you spin the figure) is always the
  center of the figure.

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Reflection Symmetry in Quadrilaterals

• What are the reflection symmetries in rectangles?
• What are the reflection symmetries in kites?

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Reflection Symmetry in Quadrilaterals

• What are the reflection symmetries in rectangles?
• What are the reflection symmetries in kites?

For simplicity we will call the lines of
symmetry of a rectangle a side-side line of
symmetry and the one for a kite an angle-angle
line of symmetry
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Reflection Symmetry in Quadrilaterals

• What are the reflection symmetries in rectangles?
• What are the reflection symmetries in kites?
• Can a quadrilateral have a side-angle line of
  symmetry?

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Reflection Symmetries of Quadrilaterals (2)

• What can we observe about quadrilaterals that
  have a side-side line of symmetry?
• An angle-angle line of symmetry?
• Can we classify the shapes that have these types
  of symmetry?

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Rotational Symmetries of Quadrilaterals

• What are the rotational symmetries of a
  parallelogram?
• What other quadrilaterals have rotational
  symmetries?
• What are the characteristics of quadrilaterals
  with a rotational symmetry?

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Rotational Symmetries of Quadrilaterals (2)

• How would you describe the rotational symmetry
  you have found?
• What are possible degree measurements for
  rotation symmetries for quadrilaterals?

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Symmetries in other shapes

• What are the possible angle rotation symmetries
  for a
• Triangle?
• Quadrilateral?
• Pentagon?
• Hexagon?
• What do you observe about reflection symmetries
  for a (2k)-gon versus a (2k1)-gon?

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Homework 6 an Exam 3

• Exam III will be Tuesday April 28
• It will cover 8.2 (coord. Geom) through 10.2
• If you want HW 6 back, stop by on Monday
• Section 9.2 pg. 630
• 5c, d, e, 11, 12, 15
• Due April 23

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Tessellations

• We say a set of figures tessellates the plane if
  they can be placed in a non-overlapping pattern
  that can be iterated to cover the plane without
  any gaps or overlaps.

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Tessellating Triangles

• Does an equilateral triangle tessellate the
  plane?
• How about an isosceles triangle?
• What about any triangle?

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Tessellating Polygons

• Which regular polygons tessellate (by
  themselves).
• We saw that equilateral triangles tessellate
• Squares clearly tessellate (Ill give you that
  one for free).
• What other ones tessellate? Why?

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Tessellating quadrilaterals

• Do all quadrilaterals tessellate?
• What is some evidence supporting that any given
  quadrilateral will tessellate?

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Tessellating quadrilaterals

• Do all quadrilaterals tessellate?
• What is some evidence supporting that any given
  quadrilateral will tessellate?
• The tessellation process place the next copy of
  the quadrilateral by rotating the shape about the
  midpoint of one of the sides still open.