Tessellations

Tesselations

A tiling or tessellation of a plane is the covering

of a plane using one or more geometric
shapes, called tiles, with no overlaps and no
gaps

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Tesselations

It is derived from Latin word “tessella” - a small

cubical piece of clay, stone or glass used to
make mosaics. It means “small square”.

The word “tessares” in Greek means four.

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Tesselations

It is a pattern in which geometric figures repeat

and the repeated figures fit perfectly together.

They form a pattern that can stretch in every

direction on the coordinate plane.

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Tesselations
Tessellations are also called

Pavings - outdoor floor, superficial surface

covering

Parquetings - flooring composed of wooden blocks

arranged in a geometric pattern

Mosaics - piece art made from small pieces of

materials
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Types of Tesselations

➢ Regular Tessellation

➢ Semiregular Tessellation

➢ Irregular Tessellation

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Regular Tessellation

➢ a tessellation
formed by only one
type of regular
polygon

➢ Example:
honeycomb
Image Credit: Peer2Peer Finance News

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Regular Tessellation
Shapes that form regular tessellations

Square

Equilateral Triangle

Regular Hexagon

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Regular Tessellation
Shapes that will not tessellate

gaps

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Regular Tessellation

In order for a regular polygon to be tileable, its

interior angle must divide evenly into 360°.
The interior angle of a regular polygon with n
sides is
n−2
= (180 )
n

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Regular Tessellation

3− 2 4−2 6−2
= (180 ) = 60 = (180 ) = 90 = (180 ) = 120
3 4 6

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Semi-regular Tessellation

➢ a tessellation made
from two or more
polygons

➢ also called
Archimedian
tessellation
Image Credit: lostmathlessons.blogspot.com

Brick Driveway Pattern

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Semi-regular Tessellation

Tiled floor in the Archeological Museum of Seville,

Spain,
using square, triangle and hexagon prototiles (Wkipedia)

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Irregular Tessellation

➢ can be formed
from almost any
kind of geometric
shapes

Image Credit: https://www.behance.net/gallery/68241055/Tesselating-Patterns

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Irregular Tessellation

Jigsaw Puzzle

https://craftcue.com/tessellations-for-kids
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Irregular Tessellation

➢ M. C. Escher, a Dutch
artist, is famous for
making tessellations with
irregular interlocking
tiles, shaped like animals
and other natural
objects. Image
Credit:
flickr.com

Maurits Cornelis Escher (1898-1972)

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Irregular Tessellation

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Irregular Tessellation

“We adore chaos

because
we love to
produce order.”

Maurits
Cornelis Escher

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Tessellations in Nature

Honeycomb Cracked Mud

Photo Credit: http://www.spacemakeplace.com/tessellation-patterns/
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Tessellations in Nature

Image Credit:
Image Credit: geoinnature.blogspot.com
edtech2.boisestate.edu
Pineapple
Turtle Shell
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Tessellations in Nature

Image Credit: wikipedia

Colchicum flower Prickly Pear Cactus
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Tessellations in Art

Escher's Butterfly Birds and Horses by Escher

https://mathstat.slu.edu/escher/images/
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Tessellations in Art

Sky and Water Eight Heads

by Escher by Escher https://mathstat.slu.edu/escher/images/
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The art of creating tessellation has been known
since the origin of the history of civilization.
When people began to build their houses and
fortifications, they tried to fill space or plane.

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The first houses, churches and castles were
built by broken stones which made up random
tessellations.

In Gothic architecture, window-panes (mosaics

with colored glass pieces connected by lead
strips) have occurred.

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Different sorts of parquetings, complicated
covering of ceilings or room's walls by tiles, or
pavings of paths and places are samples of
tessellations documenting human creativity.

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Tessellations in Architecture

Image
credits:landlopers.com
Al Hambra Palace in Spain

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Tessellations in Architecture

Credits: e-architect
Ravensbourne College in London

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Tessellations in Architecture

Cellular Tessellation Pavilion.

Photo by Patrick Boland photography
Bond Architectural University in Australia archinect.com)

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Tessellations in Architecture

Image Credits
Hazrate-Masomeh’s Mosque, Iran fastcodesign.com

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Tessellations in Architecture

Image Credits:
Museum of Modern Art in Warsaw e-architect.co.uk

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Tessellations in Weaving Designs

Photo by Evan Zodl

(https://www.flickr.com/
Basket Weave Tessellation photos/ezorigami/7445
142252)
Designed by Joel Cooper and Folded by Tom Crain
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Tessellations in Weaving Designs

Photo Credit:
http://sarahhjackson
Tote Bag Fabric .blogspot.com/

by Sarah Jackson
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Tessellations in Weaving Designs

Photo Credit:
handeyemagazine.com
Woven Fabric from Bhutan https://www.pinterest.ph/
pin/19787697734775566
7/
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Where to Find Tessellations?
Tessellations in Weaving Designs

Photo Credit:
knitapotamus.blogspot.com/
Knitted Fish Blanket

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Tessellations in Board Games

https://gandhiappliances.com/ commons.wikimedia.org

Carrom board Chess board

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Make a Tessellation

➢ Exploit symmetries
Translation
Rotation
Reflection

➢ Start with a regular tessellation

Square, triangle, hexagon

➢ Apply transformations

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Department or Mathematics and S tatistics, College of S cience and Mathematics
Translation

1. Draw a square on a blank piece of paper and cut it out.

2. Draw a curve joining two consecutive vertices.

3. Cut along the curve you have drawn and slide the cutout
piece to the opposite side of the square. Tape it in place.

4. Repeat this with the other two opposite sides of

the square.

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Department or Mathematics and S tatistics, College of S cience and Mathematics
Translation
Start with a square

https://mathstat.slu.edu/escher/index.php/Tessellations_by_Recognizable_Figures

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Department or Mathematics and S tatistics, College of S cience and Mathematics
Translation
This four steps demonstrates how Escher may have created his “Pegasus type” tessellation

https://mathstat.slu.edu/escher/index.php/Tessellations_by_Recognizable_Figures

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Translation

https://mathstat.slu.edu/escher/index.php/Tessellations_by_Recognizable_Figures

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Department or Mathematics and S tatistics, College of S cience and Mathematics
Reflection
Start with a triangle

https://mathstat.slu.edu/escher/index.php/Tessellations_by_Recognizable_Figures

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 Rotation
3
Start with a
triangle

Tessellations.org

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Web Sources
▪ https://en.wikipedia.org/wiki/Tessellation
▪ https://lostmathlessons.blogspot.com/2017/05/tile-patterns-of-regular-
polygons.html
▪ https://mathstat.slu.edu/escher/index.php/Tessellations_by_Recognizable_Fig
ures
▪ https://craftcue.com/tessellations-for-kids
▪ https://mcescher.com/
▪ http://phonetimes.co.kr/esejong/phone/news_paper.htm?uid=415&code=speci
al&BOOK_CODE=TP0003
▪ https://www.ck12.org/book/ck-12-middle-school-math-concepts-grade-
8/section/6.16/
▪ http://www.tessellations.com/SymmetryTess.html
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End