1. Tessellations
Confidential

2. 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

3. 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
Confidential

4. 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
Confidential

5. 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.
Confidential

6. 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

7. 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.
Confidential

8. 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

9. Examples a tessellation of triangles a tessellation of squares
a tessellation of hexagons
Confidential

10. Interior Measure of angles for the Polygon
Shapes
Angles
Triangle 60
Square 90
Pentagon
108
Hexagon
120
<6 Sides
<120
Confidential

11. 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.
Confidential

12. Examples of tessellation
There are four polygons, and each has four sides.
Confidential

13. Square Tessellation
A tessellation of squares is named " ". Here's how: choose a vertex, and then look at one of the polygons that touches that vertex.
Confidential

14. 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

15. Tessellation of polygons
A tessellation of triangles has six polygons surrounding a vertex, and each of them has three sides: " ".
Confidential

16. 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.
Confidential

17. Examples of semi-regular tessellations
Confidential

18. 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.
Confidential

19. 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
Powerpoint presented created by Anitha Rao
Confidential

20. 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

21. 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

22. 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

23. Its BREAK TIME !!
Confidential

24. 6/1/5 Dividing Integers
GAME
TIME
Confidential

25. 1) In the figure, is the object a regular tessellation
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.
Confidential

26. 2) In the figure , figure out the regular unit which makes tiling?
Confidential

27. 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 degrees
No 140 degrees
No 144 degrees
Confidential

28. 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.
Confidential

29. 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

30. Examples a tessellation of triangles a tessellation of squares
a tessellation of hexagons
Confidential

31. Interior Measure of angles for the Polygon
Shapes
Angles
Triangle 60
Square 90
Pentagon
108
Hexagon
120
<6 Sides
<120
Confidential

32. 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.
Confidential

33. 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.
Confidential

34. Examples of semi-regular tessellations
Confidential

35. You had a Great Lesson Today!
Confidential