Creating Tessellation Mathematical Artwork

We created interesting mathematical designs using tessellations.

Tessellation is the process of translating or rotating shapes so that they fit together perfectly. Any shapes that can be laid side-by-side without any space in between can be tessellated. This includes equilateral triangles, rectangles, squares and hexagons among others. Circles don't work because when they are placed side-by-side there are gaps between.

Beginning with an equilateral triangle, we fancied up the edges using a procedure that would allow our new shape to fit together. Tessellations are actually very mathematical. Learning about equilateral triangles, measuring angles, measuring edges, dividing, creating patterns, using rotation and translation, and symmetry are mathematical concepts covered through this creative artistic activity.

For art last year we looked at many different paintings in Come Look With Me: Exploring Landscape Art With Children (Come Look With Me Series). One of the paintings was by Escher, who made extensive use of tessellations. I never thought of trying the technique at home until I read Lucinda at Navigating by Joys' post about tessellations.

After a quick internet search I found this tutorial which clearly explained the technique.

Materials
Scissors
Cardboard, cardboard food container or card stock
Paper
Pencil

We began by creating an equilateral triangle.

Equilateral triangles have three sides the same length, and three 60 degree angles.

Then we replaced one edge with a new line.

The shape was cut out.

The new shape was traced onto the card stock paper.

Next, the creative edge was inverted onto an adjacent edge of the triangle. This was done by rotating the figure using one corner of creative edge's line of the triangle as a fixed point until the other corner of the creative edge's line matched up with other corner of the triangle. The creative edge was traced to replace the second straight edge of the triangle.

The new shape was cut out. In all, the shapes had to be cut out four times.
1. Triangle with 1 fancy edge
2. Triangle with initial fancy edge and inverted fancy edge transferred onto second edge.

The same procedure was used for the third edge, except the line was divided in half to create two parts. One half of the line was fancied and the line was inverted onto the other half of the edge.

3. Triangle with 2 fancy edges and one half of the third edge fancied up.
4. Final shape with the one-half fancied edge inverted onto the other half.