Tessellation

A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.

Tessellation = tesel·les (català)

Another word for a tessellation is a tiling.

Tiling: When you fit individual tiles together with no gaps or overlaps to fill a flat space like a ceiling, wall, or floor, you have a tiling.

What are Tessellations?

The word ‘tessera’ in latin means a small stone cube. They were used to make up ‘tessellata’ – the mosaic pictures forming floors and tilings in Roman buildings
The term has become more specialised and is often used to refer to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps.

Four Types of Symmetry in a Plane

There are 4 ways of moving a motif to another position in the pattern. These were described by Escher.

  • Translation
  • Reflection
  • Rotation
  • Glide Reflection

Translation:

A translation is a shape that is simply translated, or slid, across the paper and drawn again in another place.

Tassallation translation

Tessellation translation

The translation shows the geometric shape in the same alignment as the original; it does not turn or flip.

Reflection:

A reflection is a shape that has been flipped.  Most commonly flipped directly to the left or right (over a “y” axis) or flipped to the top or bottom (over an “x” axis), reflections can also be done at an angle.

Tessellation reflection

Tessellation reflection

 

If a reflection has been done correctly, you can draw an imaginary line right through the middle, and the two parts will be symmetrical “mirror” images. To reflect a shape across an axis is to plot a special corresponding point for every point in the original shape.

Rotation:

Rotation is spinning the pattern around a point, rotating it. A rotation, or turn, occurs when an object is moved in a circular fashion around a central point which does not move.

A good example of a rotation is one “wing” of a pinwheel which turns around the center point. Rotations always have a center, and an angle of rotation.

Tessellation rotation

Tessellation rotation

 

Glide reflection:

In glide reflection, reflection and translation are used concurrently much like the following piece by Escher, Horseman. There is no reflectional symmetry, nor is there rotational symmetry.

Tesselation Glide reflection

Tesselation Glide reflection

How to make tessellation translation:

You can practise with this link to understand how it works:

http://www.shodor.org/interactivate/activities/Tessellate/

If you use a regular polygon like square, hexagon or rhombus, then you must copy the same line from the opposite sides, for example, first you must add a line along the length of the base. Then you must copy this line and translated it to the top of the square. Then you do the same with the left and right sides.

Tessellation from square patern

Tessellation translation for square

If you are using a hexagon, you should do the same as before, just copy the line from the opposite side:

Tessellation From hexagon

Tessellation translation for hexagon

If you are using a rhombus, you should do also the same, just copy the line form the opposite side:

Tessellation rhombus

Tessellation translation for a rhombus

How to make a rotation tessellation:

If you use a square pattern, first you should draw a line at the top of the square, then you must copy this line to the right side (90 degree clockwise). Then, you should draw the line at the left of the square, then you must copy this line to the bottom side of the square (90 degree counterclockwise):

Tessellation rotation

Tessellation rotation for a square

Using a triangle:

Tessellate rotation from triangle

Tessellate rotation from triangle

As you can see in the image above, section AB is copied to section BC reflectd and rotated, however section AC should be divided into two parts, then draw the line in one half and copied to the other half reflected and rotated.

Tesselate rotation from triangle

Tesselate rotation from triangle

Using a hexagon:

Tessellation rotation from hexagon

Tessellation rotation from hexagon

reptile

 

Tessellation rotation from hexagon

Tessellation rotation from hexagon

These videos show how to do tessellation using paper and scissors, a easy way to make tessellations:

Maurits Cornelis Escher usually referred to as M. C. Escher, was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture, and tessellations.
Although Escher did not have mathematical training—his understanding of mathematics was largely visual and intuitive—Escher’s work had a strong mathematical component, and more than a few of the worlds which he drew were built around impossible objects such as the Necker cube and the Penrose triangle. Many of Escher’s works employed repeated tilings called tessellations. Escher’s artwork is especially well liked by mathematicians and scientists, who enjoy his use of polyhedra and geometric distortions. For example, in Gravity, multicolored turtles poke their heads out of a stellated dodecahedron.

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