Tessellations

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• Tessellations

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Warm Up

• A parallelogram with four equal sides is called a
  Rhombus
• A triangle with three equal angles.
• Equilateral Triangle
• Which quadrilateral has four right angles?
  Squares and Rectangles
• Three sides of a triangle measure 5 , 8 , 8
  inches. Classify the triangle by its sides
  Isosceles Triangles
• The angles of a triangle measure 30 and 90
  degrees. Find the third angle
• 60 degrees

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Lets review what we have learned In the last
Lesson
Triangle Three sided polygon. Types of
Triangles Triangles can be classified according
to the length of their sides and the size of
their angles.

• Scalene Triangle
• Isosceles Triangle
• Equilateral Triangle

• Acute-angled Triangle
• Obtuse-angled Triangle
• Right-angled Triangle

Sum of angles in a triangle 180 degrees
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Quadrilateral Four sided polygon.

• Types of Quadrilateral
• Trapezoid - a quadrilateral with two parallel
  sides.
• Rhombus - A quadrilateral with four equal sides
  and opposite angles equal.
• Parallelogram - Quadrilaterals are called
  parallelograms if both pairs of opposite sides
  are equal and parallel to each other.
• Rectangle - A parallelogram in which all angles
  are right angles.
• Square It is a special case of a rectangle as
  it has four right angles and parallel sides.

Sum of angles in a Quadrilateral 360 degrees
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Lets get startedTessellation

• A tessellation is created when a shape is
  repeated over and over again covering a plane
  without any gaps or overlaps.

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More on tessellation

• Another word for a tessellation is tiling.
• A dictionary will tell you that the word
  "tessellate" means to form or arrange small
  squares in a checkered or mosaic pattern.
• The word "tessellate" is derived from the Ionic
  version of the Greek word "tesseres," which in
  English means "four."

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Regular Tessellation.

• Remember Regular means that the sides
  of the polygon are all the same length
• Congruent means that the polygons that you put
  together are all the same size and shape.

• A regular tessellation means a tessellation made
  up of congruent regular polygons.

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Regular polygons tessellate

• Only three regular polygons tessellate in the
  Euclidean plane
• Triangles.
• Squares or hexagons.
• We can't show the entire plane, but imagine that
  these are pieces taken from planes that have been
  tiled

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Examples

• a tessellation of triangles
• a tessellation of squares
• a tessellation of hexagons

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Interior Measure of angles for the Polygon
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Angles division

• The regular polygons in a tessellation must fill
  the plane at each vertex, the interior angle must
  be an exact divisor of 360 degrees. This works
  for the triangle, square, and hexagon, and you
  can show working tessellations for these figures.
• For all the others, the interior angles are not
  exact divisors of 360 degrees, and therefore
  those figures cannot tile the plane.

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Examples of tessellation

• There are four polygons, and each has four sides.

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Square Tessellation

• A tessellation of squares is named "4.4.4.4".
  Here's how choose a vertex, and then look at one
  of the polygons that touches that vertex.

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Regular Hexagon

• For a tessellation of regular congruent hexagons,
  if you choose a vertex and count the sides of the
  polygons that touch it, you'll see that there are
  three polygons and each has six sides, so this
  tessellation is called "6.6.6"

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Tessellation of polygons

• A tessellation of triangles has six polygons
  surrounding a vertex, and each of them has three
  sides "3.3.3.3.3.3".

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Semi-regular Tessellations

• You can also use a variety of regular polygons to
  make semi-regular tessellations. A semiregular
  tessellation has two properties which are
• It is formed by regular polygons.
• The arrangement of polygons at every vertex point
  is identical.

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Examples of semi-regular tessellations
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Useful tips for Tiling

• If you try tiling the plane with these units of
  tessellation you will find that they cannot be
  extended infinitely. Fun is to try this yourself.
• Hold down on one of the images and copy it to the
  clipboard.
• Open a paint program.
• Paste the image.
• Now continue to paste and position and see if you
  can tessellate it.

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History of Tessellation

• tessellate (verb), tessellation (noun) from
  Latin tessera "a square tablet" or "a die used
  for gambling." Latin tessera may have been
  borrowed from Greek tessares, meaning "four,"
  since a square tile has four sides.
• The diminutive of tessera was tessella, a small,
  square piece of stone or a cubical tile used in
  mosaics. Since a mosaic extends over a given area
  without leaving any region uncovered, the
  geometric meaning of the word tessellate is "to
  cover the plane with a pattern in such a way as
  to leave no region uncovered.
• By extension, space or hyperspace may also be
  tessellated

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Your Turn !

• The sum of the measures of the angles of a
  regular Octagon is 1,080
• Determine whether an Octagon can be used by
  itself to make a tessellation.
• No
• 2. Verify your results by finding the no. of
  angles at a vertex.
• About 2.67
• 3. Write an addition problem where the sum of
  the measures of the angles where the vertices
  meet is 360
• 120 60 120 60 360
• 4. Tell how you know when a regular polygon can
  be used by itself to make a tessellation.
• The angle measure is a factor of 360

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Your Turn !
5.One of the most famous tessellations found in
nature is a bees honeycomb. Explain one
advantage of using hexagons in a
honeycomb. There are no gaps 6. The sum of the
measures of the angles of an 11- sided polygon is
1,620. Can you tessellate a regular 11-sided
polygon by itself? No 7. To make a tessellation
with regular hexagons and equilateral triangles
where two hexagons meet at a vertex, how many
triangles are needed at each vertex? Two
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Your Turn !
8. Predict whether a regular pentagon will make a
tessellation. Explain your reasoning. No the sum
of the measures of the angles at a point is
not 360 9. Do equilateral triangles make a
tessellation? yes 10. The following
regular polygons tessellate. Determine how many
of each polygon you need at each vertex.
Triangles and squares 2 squares, 3Triangles
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Its BREAK TIME !!
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6/1/5 Dividing Integers
GAME TIME
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• 1) In the figure, is the object a regular
  tessellation? Find the number of triangles,
  squares and polygons in the figure

Its a semi regular tessellation. Triangle1,
Square3, other Polygon2.
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• 2) In the figure , figure out the regular unit
  which makes tiling?

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3. Determine whether each polygon can be used by
itself to make a tessellation? Verify the result
by finding the measures of the angles at a
vertex. The sum of the measures of the angles of
each polygon is given heptagon 900
degrees nonagon 1260 degrees Decagon 1440
degrees
No 128.6 degrees No 140 degrees No 144 degrees
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Lets Review what we have learnt today

• A tessellation is created when a shape is
  repeated over and over again covering a plane
  without any gaps or overlaps.

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Regular polygons tessellate

• Only three regular polygons tessellate in the
  Euclidean plane
• Triangles.
• Squares or hexagons.
• We can't show the entire plane, but imagine that
  these are pieces taken from planes that have been
  tiled

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Examples

• a tessellation of triangles
• a tessellation of squares
• a tessellation of hexagons

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Interior Measure of angles for the Polygon
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• The regular polygons in a tessellation must fill
  the plane at each vertex, the interior angle must
  be an exact divisor of 360 degrees. This works
  for the triangle, square, and hexagon, and you
  can show working tessellations for these figures.
• For all the others, the interior angles are not
  exact divisors of 360 degrees, and therefore
  those figures cannot tile the plane.

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Semi-regular Tessellations

• You can also use a variety of regular polygons to
  make semi-regular tessellations. A semi regular
  tessellation has two properties which are
• It is formed by regular polygons.
• The arrangement of polygons at every vertex point
  is identical.

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Examples of semi-regular tessellations
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You had a Great Lesson Today!