In addition to the practical way of designing tessellations presented last week by Jean Larson, there is a whole field of mathematical theory and practice related to tessellations.  I loved geometry in high school, but the theoretical stuff quickly gets beyond me.  Here is a summary of more practical implications.

A shape is said to tessellate if it can cover a plane without gaps, extending to infinity in all directions.

The regular polygons that will tesselate are:

• Triangles.  All triangles will tessellate.
• Quadrilaterals (4-sided shapes) all tessellate, and all can be divided into triangles, just by drawing from corner to corner.
• Hexagons (regular hexagons) will tessellate, as we know well from English paper piecing.

From there it gets complicated as to which figures will tessellate and which will not, but to go on with practical information:

It’s perfectly OK to draw lines inside your tessellating shapes, which may mean they don’t all look alike anymore.  An excellent example is this pattern by Alison Glass.  The design is composed entirely of equilateral triangles, all the same size, BUT she has drawn lines within some of the triangles to create secondary designs.

Illustration from AlisonGlass.com

It’s OK to use more than one shape to cover a surface, or more than one size of the same shape, as long as the whole pattern can be continued to infinity.  (Who knew?)  Here’s are examples, drawn in EQ8:

This tessellating design is composed of squares of 2 different sizes

This tessellating design is composed of 2 different shapes.

Many of our traditional quilt patterns are actually tessellating designs.  The second example above is just a recoloring of Tumbling Blocks.

There are many, many ways to create tessellating designs, and I’ll direct you to some additional resources next week.  Meanwhile, one of my favorite easy ways to create tessellating designs is something called “pattern blocks“.  The link takes you to a fun site where you can develop patterns consisting of one or more shapes.  This works because the angles of all the pieces are either 30, 60, 90, or 120 degrees.  I just love that the site is intended for kids–it’s all I can do to wrap my head around it!  And I have no idea how to tell which combinations will tessellate except to try.  Here’s one I made on the site that I think will tessellate:

Image imported from https://www.coolmath4kids.com/manipulatives/pattern-blocks

And FYI, the MQG published a brief article on tessellations back in January.  Access it here.  You’ll have to log in with your usual password first.

Next week: more resources to help you create tessellating designs.