Motion Geometry Part II

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Motion GeometryPart II
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More on Motion Geometry

• Tessellations
• With polygons
• Escher-like Size Transformations
• Dilations (Expansions and Contractions)
• Symmetry
• About a line
• About a point
• Rotational
• Fractals and Chaos

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Tessellations
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Tessellations

• ?A tessellation is an arrangement of congruent
  figures that covers a plane without any gaps or
  overlaps.

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Example

• Floor Tiles

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Pure Tessellation

• ?A pure tessellation is a tessellation consisting
  of congruent copies of one figure.

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What do you think?

• Are these pure tessellations?

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Solution

• The tessellation on the left is a pure
  tessellation because it is made up only of
  squares. The tessellation on the right is not a
  pure tessellation because it is made up of more
  than one shape.

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Fact

• In a tessellation, the combined angles of the
  figures that meet
• at a point must equal .

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Why?

• If the combined angles of the figures that meet
  at a point do not equal , the figures will
  overlap or there will be a gap.

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Why continued

• If the sum of the angles is greater than there
  will be overlapping.

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Why Continued

• If the sum of the angles at point P is
• less than , a gap is created at P.

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What do you think?

• What regular polygons can be used to create a
  pure tessellation?

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These are the only ones.
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How can you prove it?

• Find the number of degrees in each angle of the
  regular polygon you want to test.
• How many of these angles does it take to make
  3600?
• The regular polygon will tessellate if and only
  if this number a whole number.

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What do you think?

• Can any general quadrilateral be used to
  tessellate a plane?

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Reflect
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Reflect again
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Translate

• Each vertex of the quadrilateral is represented
  at the center point.

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Translate the groups of four.
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What do you think?

• Will a Scalene Triangle Tessellate?

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Solution

• Yes.

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Fact

• Although a regular pentagon cannot be used to
  create a pure tessellation of a plane, some
  irregular (convex) pentagons can.

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Heres one.
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How many irregular (convex) pentagons tessellate
a plane?

• Up until 1975, mathematicians had found eight
  types of irregular (convex) pentagons that can be
  used to tessellate.
• Although it had not been proven that these were
  the only ones, mathematicians thought they had
  found them all.
• However, a women by the name of Marjorie Rice
  from San Diego found a ninth one that year. In
  the next two years she found four more.

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Your mother may be smarter than you think!

• Marjorie regularly read her sons Scientific
  American and got interested in tessellations
  after reading an article by Martin Gardner.
• She had no formal training in mathematics beyond
  high school general math.

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Semi-pure Tessellations

• Tessellations that involve more than one type of
  shape are called semi-pure tessellations.

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Application

• Islamic and Christian art

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Semi-regular Tessellations

• If the same combination of regular
• polygons meet in the same order at
• each vertex in a semi-pure tessellation, it is
  called a semi-regular tessellation.
• There are eight different semi-regular
  tessellations.

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How are they named?

• They are identified by naming the number of sides
  of each polygon about a vertex starting with the
  polygon with the least number of sides and moving
  either clockwise or counterclockwise around the
  vertex.

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Alternate way to name them

• An alternate name is found by starting with the
  polygon with least number of sides and continuing
  in order to the polygon with the largest number
  of sides. Exponents are used if a polygon is
  found more than once at each vertex.

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Example

• This tessellation with a square and two octagons
  above is a 4.8.8 semi-regular
  tessellation.

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Can you name these?
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Solution

• The figure on the left is a 3.4.6.4
• or
• The figure on the right is a 4.6.12.

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Name these.
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Solution

• The figure on the left is a 3.12.12
• or
• The figure on the right is a 3.4.3.3.4 or

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Escher-like Tessellations

• M. C. Escher, a Dutch graphic artist, was born in
  1898 and died in 1972.
• Although Escher had no formal training in math or
  science, his creations are mathematically rich.
• He is mostly recognized for his spatial
  illusions, impossible building, and
  tessellations.

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Escher type tessellation with using translations

• Begin with a polygon that will tessellate.
• For example you could begin with a square.

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Step 1

• Draw a square and then create some design along
  one side as shown.

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Step 2

• Translate the design to the opposite side and
  erase the segments of the square that they are
  replacing.

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Step 3

• Repeat on the other two sides.

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Step 5

• Copy the shape and use it to tessellate.

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Step 6

• Add eyes to this design and you have a school of
  fish.

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Escher like tessellation with a rotation.

• Begin with a regular hexagon this time.

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Step 1

• Start by drawing a shape on one side of a hexagon.

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Step 2

• Rotate the shape, using either of its endpoints
  as the center of rotation, until the shape has
  been moved to an adjoining side of the hexagon.

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Step 3

• Continue around the hexagon making shapes and
  rotating them to the next available side until
  you have changed all of its sides.

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Continue
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Tessellate with the shape.

• In order to fit together, the shapes must be
  rotated. H

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Escher Combination

• Use reflections, rotations, and translations to
  create an Escher shape.

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Step 1

• Start with a polygon that will tessellate by
  itself and make a design on one side.

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Step 2

• Reflect the design using a line through the
  midpoint of the design side and parallel to the
  base.

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Step 3

• Translate the reflected design to the
• opposite of the triangle.

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Step 4

• Erase the reflected design and the unnecessary
  sides of the triangle.

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Step 5

• Create a design on half of the remaining side of
  the triangle and rotate it around the midpoint of
  the side.

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Step 6

• Erase the third side of the triangle. Adding an
  eye will make this particular figure look like a
  monster.

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Tessellate
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Dilations
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Comparisons of Transformations

• The flips, turns, and rotations are all forms of
  rigid motion.
• A figure that is mapped by a rigid transformation
  does not change in either shape or size (although
  it may change in orientation).
• In a size transformation, sometimes called a
  dilation, a figure keeps its shape but changes
  its size.

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Dilations

• If the figure increases in size, the
  transformation is an expansion.
• If the figure shrinks in size, the transformation
  is a contraction.
• In either case, the preimage and its image are
  similar. Similar figures have the same shape,
  but a different size.

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Application - Photography
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Applications

• Diagrams, blueprints, maps, etc are also commonly
  found in varying sizes.
• Computers allow you to change the size of letters
  without changing its shape.

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Size Transformations

• ? Size transformations have a
• center and a scale factor.
• If the scale factor is greater than 1, the
  transformation is an expansion. If the scale
  factor is less than 1, it is a contraction.

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Example

• Suppose point O is the center of the size
  transformation and the scale factor is 2. Find
  the image of the triangle.

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What do we know?

• The scale factor of 2 indicates the
  transformation is an expansion.
• The distance from O to must be twice the
  distance from O to A.
• Also O 2OB and O 2OC.
• 3) O, A, and are collinear. O, B, and are
  collinear. O, C, and are collinear.

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Plan

• Count squares. From O to A is up 7 and right 2.
  So is up another 7 and right 2.
• This insures that O, A, and are collinear and
  that the distance from O to is twice the
  distance from O to A.
• Slope may be used in a similar manner to find the
  other points.

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Solution
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Expansion by Construction

• Expand the triangle in the last example by
  construction by

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Plan

• Draw a line connecting O and A.
• Extend it beyond A.
• Use a compass to copy the length from O to A past
  A along the line.

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Transformation - Contraction

• Find the image of the quadrilateral under a size
  transformation of
• (or 21) with center at 0.

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Try It

• By counting squares

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Plan

• Connect each vertex of the quadrilateral with
  point O.
• The new points are the midpoints of the line
  segments.

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Solution
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Try It

• Repeat the same contraction but this time by
  construction.

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Plan

• Connect the each vertex to point 0.
• Bisect the line segments.

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Solution
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Related Field of Mathematics

• An entire branch of mathematics is devoted to
  transformations involving both a change in size
  and shape of a figure.
• These types of transformations are called
  togological transformations and the branch of
  mathematics that deals with these types of
  transformations is called topology.
• Topology is sometimes referred to as rubber sheet
  geometry.

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Fact

• To a topologist, a donut and a coffee cup are
  equivalent.

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Why?

• Study the following sketches and see if you can
  see the transformation. Imagine the donut is made
  of rubber and can be bent and stretched.

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Explanation
Objects in topology are sometimes identified by
the number of holes it has. Do the donut and
coffee cup have the same number of holes?
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Mobius Strips

• A Mobius strip is a surface named after August
  Ferdinand Moebius, a nineteenth century German
  mathematician and astronomer.
• Moebius was a pioneer in the field of topology.
• To make a Mobius strip, start with a strip of
  paper and twist it once and then glue the ends
  together.

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How many sides does a Mobius strip have?
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Solution

• Before it was twisted it had two sides.
• Now it has only one side.
• To test this theory draw a line down the middle
  of the strip.

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Challenges

• First make a guess as to what will happen and
  then try it.
• Cut along the line you drew on your Mobius strip
  in the last question.
• Make another Mobius strip and draw a line that
  stays of the distance from an edge. Cut along
  the line.

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Symmetry
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Symmetry

• ?If there is an isometry (rigid transformation)
  that maps a figure onto itself, the figure is
  said to have symmetry.

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Line Symmetry

• An image reflector can help you find any line
  symmetry in a figure.

• ?If there is a reflection that maps a figure onto
  itself, the figure has line symmetry

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Try It

• How many lines of symmetry does a regular hexagon
  have?

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Solution

• Six

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Try It

• How many lines of symmetry in the figure?

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Solution
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Rotational Symmetry

• Note Every figure would map onto itself if it
  is not rotated at all or is rotated a complete
  turn about its center.

?A figure has rotational symmetry if it can be
mapped onto itself by rotating it some angle
between 00 and 3600 about some point.
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Try It

• What are the degrees of rotational symmetry of
  this pinwheel?

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Solution

• If this pinwheel is rotated 900, 1800, or 2700
  about its center, its image matches the original
  pinwheel.
• The pinwheel has 900, 1800, or 2700 rotational
  symmetry.

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Fact

• A figure with a rotational symmetry of a will
  also have rotational symmetry of na where n is a
  non-zero integer.

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Important Concept

• It is therefore important to find the smallest
  positive angle that maps a figure onto itself
  since all of its other rotational symmetries can
  be found if this is known.

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Try It

• Find the smallest positive angle that can be used
  to map a regular pentagon onto itself.
• Use this to find all of its other positive
  symmetries less than.

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Solution

• Since the pentagon is regular, each of the angles
  about its center is
• So its smallest rotational symmetry is
• Its other rotational symmetries are
• Note

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Sketch
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Point Symmetry

• ?A figure has point symmetry if it has
• rotational symmetry about a point.

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Example

• The following figure has point symmetry about
  point P.

P
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Try It

• Does a regular pentagon have point symmetry? Why
  or why not?
• Does a regular hexagon have point symmetry? Why
  or why not?

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Solution

• A regular pentagon does not have point symmetry
  because it does not have rotational symmetry.
• A regular hexagon has point symmetry because it
  has rotational symmetry.

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Application

• A graphic artist often uses symmetry on designs
  that serve as logos or corporate symbols for
  companies. Do you recognize these sketches of
  famous logos? What corporation does each
  represent? What symmetries, if any, do each
  have?

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Logos
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Solutions

• The logos are the CBS eye, Chevrolet, and
  Volkswagen.
• The Chevrolet symbol has 1800 rotational
  symmetry.
• The CBS eye has horizontal and vertical symmetry
  as well as point symmetry.
• The VW logo has vertical symmetry.

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Sketches
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Fractals and Chaos
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Fractals

• ?A geometric shape that can be subdivided into
  parts that are each reduced-sized copies of the
  whole is called a fractal.

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Koch Edge

• Upon magnification, fractals appear more and more
  complex.

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Magnification 1
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Magnification 2
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Magnification 3
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Koch Snowflake
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Iterations

• ?Iteration is the process of repeating a set of
  instructions or a procedure until a desired
  result is achieved or an approximation results.

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The Sierpinski triangle
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More iterations
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Jurassic Park Fractal
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Rotate
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Connect
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Rotate and Connect
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Continue
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After fifty or sixty iterations
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The End