tessellations: A Beginning
This section will help to intoduce you to importiant facts and techniques associated with tessellations . Make sure to read thoroughly and watch all provided materials as these concepts are crucial to the development of your own tessellation project and to the final quiz .
What is a tessellation?
A tessellation is a pattern made up of one or more congruent shapes that completely cover the plane without
any gaps or overlapping.
any gaps or overlapping.
Practice Time: Answer the following questions to see if you can identify a tessellation.
how big of a plane will a tessellation cover?
All tesselation patterns can be extended in the plane infinitely in every direction.
Why is it called a tessellation?
The Latin root of the word tessellation is tessella, which translates to mean a small stone or tile. Romans used these stones and tiles to
create elaborate mosaic designs like these ones below.
create elaborate mosaic designs like these ones below.
Who uses tessellations?
Over the centuries, artists have worked with tessellations in many different mediums such as tile mosaics for floors,walls and ceilings. They have used them to decorate pottery, tapestries and carpets. They also appear in metal working, wood carvings and stained glass designs and have played a crucial structural role in many objects and buildings.
What Shapes can tessalate?
Any triangle, quadrilateral or hexagon whose opposite sides are parallel and congruent will tessellate the plane by itself. All triangles tesselate because the sum of thier angles is equal to 180 degrees. Any two congruent triangles form a quadrilateral who's interior angles measure 360 degrees. For any regular polygon to tesselate the sum of thier interior angles must equal 360 degrees. Regular heptagons, octagons and pentagons will not tessellate because the sum of thier interior angles does not equal 360 degrees.
Practice Time: Answer the following math questions to see if you can identify which shapes will tessellate.
Can Only flat shapes tesselate?
Both two dimensional and three dimensional shapes will tessellate. However this site will focus on how to create two dimensional tessellations that cover a plane instead of three dimensional tessellations that fill a space.
Four ways to create tessellations
These are by far not the only ways to create a tessellation, however they are four of the easiest techniques to master!
Creating unit Cells
By adding a design to an individual cell, in most cases a square and then arranging those cells in a square cell configuration a tesselation is created.
Modifying by Translation
By modifying one side of a polygon and translating it to the other side of the polygon. Remember that if something is translated it is not roated.
Modifying by rotation at the midpoints
This technique works well with triangles. How it works is that you first locate the midpoint on the sides of your triangle, secondly you modify one half of a side of your triangle, and finally you rotate that modification 180 degrees around the midpoint of the side. You can modify all three sides of your triangle using this technique.
modifying by Rotating Sides
This technique works well with squares and regular hexagons. To make this technique work you must modify an entire side of your polygon before rotating it to an adjacent side.