Tuesday, February 21, 2017

Reflection, Rotation, Transformation and Tessellation



Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps.

tesstri.giftessellation


Regular Tessellations

regular tessellation is a pattern made by repeating a regular polygon.
There are only 3 regular tessellations:
tessellation regular triangles
Triangles
3.3.3.3.3.3
 
tessellation regular squares
Squares
4.4.4.4
 
tessellation regular hexagons
Hexagons
6.6.6

Look at a Vertex ...

tessellation vertexA vertex is just a "corner point".

What shapes meet here?
Three hexagons meet at this vertex,
and a hexagon has 6 sides.
So this is called a "6.6.6" tessellation.
tessellation notation 6.6.6

Semi-regular Tessellations

semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same!
There are only 8 semi-regular tessellations:
tessellation 3.3.3.3.6
3.3.3.3.6
 
tessellation 3.3.3.4.4
3.3.3.4.4
 
tessellation 3.3.4.3.4
3.3.4.3.4
tessellation 3.4.6.4
3.4.6.4
 
tessellation 3.6.3.6
3.6.3.6
 
tessellation 3.12.12
3.12.12

right arrow
To name a tessellation, go around a vertex and write down how many sides each polygon has, in order ... like "3.12.12".
right arrowAnd always start at the polygon with the least number of sides, so "3.12.12", not "12.3.12"

In order for a shape to tesselate, the interior angles must divide evenly from 360.

Other Tessellations

There are also "demiregular" tessellations, but mathematicians disagree on what they actually are!
And some people allow curved shapes (not just polygons) so we can have tessellations like these:
tessellation curvy
Curvy Shapes
 
tessellation circles lines
Circles
 
tessellation eagles
Eagles?

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