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 Confidential 2

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 .

1) Scalene Triangle 2) Isosceles Triangle 3) Equilateral Triangle 1) Acute-angled Triangle 2) Obtuse-angled Triangle 3) Right-angled Triangle Sum of angles in a triangle = 180 degrees Confidential 3

Quadrilateral: Four sided polygon.

Types of Quadrilateral: 1) Trapezoid - a quadrilateral with two parallel sides. 2) Rhombus - A quadrilateral with four equal sides and opposite angles equal. 3) Parallelogram - Quadrilaterals are called parallelograms if both pairs of opposite sides are equal and parallel to each other. 4) Rectangle - A parallelogram in which all angles are right angles. 5) 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 Confidential 4

Lets get started Tessellation

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." Confidential 6

Regular Tessellation

.

• A

regular tessellation

means a tessellation made up of congruent regular polygons. • Remember :

Regular

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

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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 Confidential 8

Examples

• a tessellation of triangles • a tessellation of squares • a tessellation of hexagons Confidential 9

Shapes

Triangle Square Pentagon Hexagon <6 Sides

Interior Measure of angles for the Polygon Angles

60 90 108 120 <120 Confidential 10

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. Confidential 13

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": Confidential 14

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 Confidential 19

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° Confidential 20

Your Turn !

5.One of the most famous tessellations found in nature is a bee’s honeycomb. Explain one advantage of using hexagon’s 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 Confidential 21

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 Confidential 22

Its BREAK TIME !!

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6/1/5 Dividing Integers

G A M E T I M E

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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

It’s a semi regular tessellation.

Triangle=1, Square=3, other Polygon=2.

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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 Confidential 27

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 Confidential 29

Examples

• a tessellation of triangles • a tessellation of squares • a tessellation of hexagons Confidential 30

Interior Measure of angles for the Polygon Shapes

Triangle Square Pentagon Hexagon <6 Sides

Angles

60 90 108 120 <120 Confidential 31

• 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 Confidential 34

You had a Great Lesson Today!

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