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Many ancient cultures have used tessellations.

Johannes Kepler conducted one of the first mathematical studies of tessellations.

E.S. Fedorov proved an aspect of tiling in 1891.

Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative tiling of the Alhambra palace

Islamic art does not usuallyuse representations of livingbeings, but usesgeometric patterns,especially symmetric(repeating) patterns.

“I try in my print to testify that we live in a beautiful and orderly world, not in a chaos without norms, even though that is how it sometimes appears. the nonsensicalness of some of what we take to be irrefutable certainties.”

Most famous creator of tessellations

Born in Holland in 1898 (died in 1972)

Originally studied architecture before becoming interested in woodcuts and printmaking

Did 137 tessellations in his lifetime

House of Stairs

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Tessellations are arrangement of shapes that cover the picture without overlapping and without leaving spaces.

The word “tessellation” comes from the Latin word “tessera” which means “small stone cube”

Tiling is often another term used for tessellation patterns.

Tessellations around us

Formed by TRANSFORMATION(combination of TRANSLATIONS,

ROTATIONSREFLECTIONS

AndGLIDE REFLECTION)

Movements of a figure in a plane

May be a SLIDE, FLIP, or TURN

Another name for a SLIDE

A

BC

A’

C’

B’

A’, B’ and C’ are explained in the next slide...

The figure you get after a translation

Original Image

Slide

A A’

B B’

C C’

The symbol ‘ is read “prime”.

ABC has been moved to A’B’C’. A’B’C’ is the image of ABC.

Finding the amount of

movement LEFT and

RIGHT and UP and DOWN

9

8

7

6

5

4

3

2

1

0 1 2 3 4 5 6 7 8 9

Right 4 (positive change in x)

Down 3 (negative change in y)

A

A’

B

B’

C

C’

Can be written as: R4, D3

(Right 4, Down 3) (x+4, y-3)

Another name for a FLIP

A A’

C C’B B’

Used to create SYMMETRY on the coordinate plane.

Symmetry When one side

of a figure is a MIRROR IMAGE of the other

Axis of Symmetry is a line that divides the figure into two symmetrical parts in such a way that the figure on one side is the mirror image of the figure

on the other side1

2

3

4

12

3

4

5

6

1

2

3

The line you reflect a figure across.

Another name for a TURN A transformation that turns about a fixed

point

B

B’

C

C’

A A’

The fixed point

(0,0)

AA’

C

C’

B

B’

When an image after rotation of 180 degrees or less fits exactly on the original.

90 degreesA

A’

C

C’

B

B’

The figure that results after reflection and translation.

There are three main types of tessellations: Regular Semi-Regular Demi-Regular

A regular tessellation is a pattern only using one regular polygon shape.

May also called Pure Tessellation.

A regular polygon is any many sided shape that has sides of equal length and angles or equal measure.

3. 3. 3. 3. 3. 3 4. 4. 4. 4 6. 6. 6

Divide the whole turn (360⁰) by the number

of exterior angle (= the number of sides) to

find the size of one exterior angle. Then use

the fact that

the exterior angle + the corresponding interior

angle =180⁰

The sum of interior angles of a n-sided

regular polygons { (n-2) 180⁰}⤬ .

Then the size of one of the interior angle

can be found by dividing by number of

interior angle {=n}.

∠ =(n-2) 180⁰ / n⤬

Determine whether a regular 6-gon tessellates the plane. Explain?

Let 1 represent one interior angle of a regular 4-gon.

m∠1=180 (n-2) / n Interior angle ⁰theorem = 180 (6-2)/4 Substitution⁰ =180 Simplify⁰Answer: As 180 is a factor of 360 .so a 6-⁰ ⁰gon will tessellate the plane .

The sum (total) of the angles around any Point is 3 × 120° = 360°.

This fact is true of all such points where the vertices of 3 hexagons meet and thus the hexagons will tessellate.

This tessellation may be

represented by the abbreviated

notation 6^3 (signifying that three

six sided

regular polygons meet at a

common vertex).

Determine whether a regular 16-gon tessellates the plane. Explain?

Let 1 represent one interior angle of a regular 4-gon.

m∠1=180 (n-2) / n Interior angle theorem⁰ = 180 (16-2)/4 Substitution⁰ =157.5 Simplify⁰

Answer: As 157.5 is not a factor of 360 .so a 16-⁰ ⁰gon will not tessellate the plane.

A semi-regular tessellation is a pattern consisting of more than one type of regular polygon.

The vertex arrangement is the same throughout the entire pattern

Shape Sides Exterior Interior

Triangle 3 120o 60o

Square 4 90o 90o

Pentagon 5 72o 108o

Hexagon 6 60o 120o

Heptagon 7 51.42…o 128.57…o

Octagon 8 45o 135o

Nonagon 9 40o 140o

Decagon 10 36o 144o

Hendecagon 11 32.72…o 147.27…o

Dodecagon 12 30o 150o

4. 8. 84. 6. 12 3. 4. 6. 4 3. 3. 4. 3. 4

3. 3. 3. 3. 6 3. 6. 3. 6 3. 3. 3. 4. 4 3. 12. 12

by interior angle theorem …Octagon has 135 degree angle of each side…Square has 90 degree 90⁰+135⁰+135⁰=360⁰

Three equilateral triangle and two square tesselate the plane…60⁰+60⁰+60⁰+90⁰+90⁰=360⁰

Determine whether a semi-regular tessellation can be created from regular nonagons and squares, all having sides 1 unit long.

Each interior angle of a regular nonagon

measures or 140°.

Each angle of a square measures 90°.

Find whole-number values for n and s such that

All whole numbers greater than 3 will result in a negative value for s.

Answer: There are no whole number values for n and s so that

Substitution

Simplify.

Subtract from each side.

Divide each sideby 90.

A demi-regular tessellation is a pattern of regular polygons in which there are two or three different polygon arrangements

Tessellation of an irregular shape can be

obtained by Transformation of other

Tessellating shapes.

Irregular shapes are those that does not

have all sides and angle equal .

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The metamorphoses consist of abstract shapes changing into sharply defined concrete forms, and then changing back again (a bird changing into a fish, a lizard into a honeycomb). 

The most detailed shape can be changed quite a bit

The most detailed shape can be changed quite a bit

Tessellations can be found in quilts, floor tiling, and wallpaper.

snake skin

spider web

Honey comb

Islamic Arch

Islamic Minaret