Math and Art Collide with Tessellations

I love art, and math has one such art form that draws my attention.  It is called tessellation.  The patterns tessellation can create are most recognizable in quilts, but there are many ways to use them in real life.  Another recognizable use of tessellations is in floor tiles.  Below is an example of a beautiful quilt.

Basic tessellations are designed with a few basic rules:
1.  Each piece must be a regular polygon.
2.  Each vertex must be the same.
(Remember, this is a basic design, not a technical one.)
3.  The total of the vertex points must total 360 degrees exactly.
Squares work.  There are many shapes that tessellate alone or in combination with other shapes.

Print out your own sheet and create a fun color combination.  Here, see how triangles and hexagons work together to create tessellations?

My oldest son has a book about the Dutch artist, M.C. Escher.  His pieces of this art are so amazing and mesmerizing.  Here is one of them:

This is called "Horseman"

Another is entitled "Lizard":

If you would like to watch a video on how to create your own tessellation in the style of Escher, check out this video:

Polygons Please!

Polygons are exciting shapes to work with.  They offer so much variety, that they are never boring.  Two points that are connected is the beginning of a polygons.  Add enough line segments to enclose a shape and you have a polygons.
Polygons are named according to the number of sides.  My kids have fun with listing all the names they can remember.  Three line segments are needed to create the polygons with the least number of sides.  This would be a triangle.  Here is a chart to help you out:

Name	Sides	Shape	Interior Angle
Triangle (or Trigon)	3		60°
Quadrilateral (or Tetragon)	4		90°
Pentagon	5		108°
Hexagon	6		120°
Heptagon (or Septagon)	7		128.571°
Octagon	8		135°
Nonagon (or Enneagon)	9		140°
Decagon	10		144°
Hendecagon (or Undecagon)	11		147.273°
Dodecagon	12		150°
Triskaidecagon	13		152.308°
Tetrakaidecagon	14		154.286°
Pentadecagon	15		156°
Hexakaidecagon	16		157.5°
Heptadecagon	17		158.824°
Octakaidecagon	18		160°
Enneadecagon	19		161.053°
Icosagon	20		162°
Triacontagon	30		168°
Tetracontagon	40		171°
Pentacontagon	50		172.8°
Hexacontagon	60		174°
Heptacontagon	70		174.857°
Octacontagon	80		175.5°
Enneacontagon	90		176°
Hectagon	100		176.4°
Chiliagon	1,000		179.64°
Myriagon	10,000		179.964°
Megagon	1,000,000		~180°
Googolgon	10100		~180°
n-gon	n		(n-2) × 180° / n

For polygons with 13 or more sides, it is OK (and easier) to write "13-gon", "14-gon" ... "100-gon", etc.

Now for the fun activity to create polygons:
                                                        Sandbox/Beach/Snow Polygons
Supplies:
Protractors
Straight Edge (a yard stick or ruler)
Print out of the chart above

Head to the sand or snow to create your polygons.  Depending on the age of the children, you may or may not need to assist them.  Have each child draw as many polygons as they can. 

Remember to take pictures of this family math lesson.

When everyone has had some practice drawing polygons, see what creations can be made by drawing combinations of new polygons.  A few ideas to get you started could be a silly car, strange animals, a Farris wheel, flower, etc.
Another idea is to create a progressive drawing in the sand or snow.  One person draws a polygons of their choice.  The next person adds something, and so on until the group feels it is done. 


Here is a class video about polygons:

For snowflake polygons, check out this page:  Polygon Snowflakes
For a fun inside project, check out this page:  Star Polygon Wrappings