Notecard Tessellation Pattern Art

Tessellations are patterns that cover an entire surface with no gaps or overlaps. Tessellations can be made up of one or multiple shapes, called tiles. In order for a tile to independently tessellate a surface, the sum of the angles at each point must sum to 360 degrees (more on that later!).

Tessellations can be seen at the intersection of geometry and art throughout history, from historic wall and floor tile mosaics to the works of 20th Century Dutch artist M.C. Escher. Examples of M. C. Escher’s tessellation works can be found on the M.C. Escher Foundation’s website.

This Instructable will cover multiple different techniques of creating uniquely shaped tessellating tile stencils using notecards or cardstock, and different ways finished tessellations can be decorated!

I have created and attached templates for tessellating regular and irregular hexagons and a printable worksheet with instructions for this activity and color-in-able tessellation examples. These files are attached under the supplies section of this Instructables.

Supplies:

1. Notecards, Cardstock, or Thin Cardboard to Create Tessellating Shapes
2. Scissors
3. Tape
4. Pen, Pencil, or Marker for Tracing
5. Paper to Trace Tessellating Patterns Onto
6. Markers, Colored Pencils, Paint, ect. For Decorating.

Optional:

1. Ruler/Straightedge
2. Hexagon Template (Attached)
3. Tessellations Instructions Print Out and Coloring Sheet (Attached)

Any triangle can independently tessellate a plane, regardless of if it is equilateral, isosceles, or scalene. In all triangles on a flat plane, the sum of the 3 interior angles of the triangle always adds up to 180 degrees. When 6 congruent triangles with angles A ,B, & C are arranged together meeting at one central point, with 2 triangles oriented with each type of angle towards the central point, the sum of the angles at the point add to 360 degrees, perfectly tesselating the plane.

To create a tessellating triangle drawing, cut a triangle out of cardstock or a notecard. You can draw the triangle first, or cut it out free hand. Trace the triangle cutout on a sheet of paper, then rotate the template, matching one of the edges of the template to a side on the traced triangle of the same size and trace again. Repeat until the page is covered in tessellating triangles!

Like triangles, any quadrilateral (4 sided shape) can be tessellated. The interior angles of every quadrilateral on a flat plane sum to 360 degrees, so when 4 congruent quadrilaterals with angles D, E, F, & G meet at a point where each angle D, E, F, & G meet, the sum of the angles at that point add to 360 degrees, perfectly tesselating the plane.

To tessellate with quadrilaterals, cut a straight edged 4 sided shape out of cardstock or a notecard. You can draw the quadrilateral first, or cut it out free hand. Trace the quadrilateral cutout on a sheet of paper, then rotate the cutout, matching one of the edges of the cutout to a side on the traced shape of the same size and trace again. Repeat until the page is covered in tessellating quadrilaterals!

Translation is moving a shape to a different location without rotating it, reflecting it, or changing its size. It is sliding the shape from one location to another in the x and y directions. Unique shapes that tessellate by translation can be made from parallelograms (in this example I use a rectangle).

Start with a notecard, or cut a rectangle from cardstock. For this activity the accuracy of the starting shape is important, so make sure that the edges of the rectangle that are parallel to each other are the same size.

Cut a shape from one edge of the rectangle. Tape this shape to the parallel side of the rectangle. The parallel side of the rectangle will now have a convex version of the concave shape left on the original side.

These steps can be repeated on the rectangle’s other set of parallel sides.

Trace the tessellating tile on a sheet of paper, then slide the tile, matching one of the edges of the tile to a side on the traced shape of the same size and shape and trace again. Repeat until the page is covered in tessellations!

All parallelograms, including parallelograms, rhombuses, rectangles, and squares, can be used to create tessellating shapes using this method.

In the previous example I cut shapes out of the rectangle from corner to corner. When creating shapes that tessellate by translation, you do not have to cut out sections corner to corner. Once they are cutout, cutout shapes can be translated both vertically and horizontally. They do not have to line up corner to corner. Offsetting cutout shape before attaching them to the parallel edge of the parallelogram creates an offset effect.

Shapes can be cut from 1, 2, 3, or all 4 sides of the parallelogram.

Another way to tessellate a plane is by reflection, flipping the tile over to the other side.

Like translation, unique shapes that tessellate by reflection can be made from all types of parallelograms, but in the example I use a rectangle. In the example one side of the paper is white and the other is gray.

Cut a shape from one edge of the rectangle. Tape this shape to the parallel side of the rectangle. This set of parallel edges of the rectangle will tessellate by translation.

On the other set of parallel edges, cut a shape from one edge, but reflect (flip) the cutout before attaching it to the parallel side of the rectangle. This set of parallel edges will tessellate by reflecting. When tracing the tile on paper, flip the tile along this edge instead of sliding it.

One or both sets of parallel sides of the parallelogram can be reflected to make a tile that tessellates by reflection.

Tiles that tessellate by reflection can be made out of all types of parallelograms including parallelograms, rhombuses, rectangles, and squares.

Cutout shapes can be cut from 1, 2, 3, or all 4 sides of the rectangle. Cutout shapes can be translated both vertically and horizontally. They do not have to line up corner to corner.

Squares can be used to create tiles that tessellate by rotation. To create a tile that tessellates by rotation, cut a shape out of one side of a square, then attach the shape to an adjacent side of the square after rotating the shape 90 degrees. This can be repeated on the other set of adjacent edges. When tracing a tile that tesselates by rotation rotate the tile 90 degrees then match one of the edges of the tile to a side on the traced shape of the same size and shape and trace again.

When creating tiles that tessellate by rotation, if a cutout shape is offset after rotating it 90 degrees before re-attaching it to the base tessellating tile, the tile will no longer tessellate a surface perfectly on its own. However, the tile can tessellate a surface in combination with square and rectangular tiles.

Rotation and reflection and be combined to create tiles that tessellate in a stair step sort of pattern.

Triangles can also be tessellated by rotation. To create an equilateral triangle based tile that tessellates by rotation a cutout from one side is rotated and attached to an adjacent side. On the remaining edge of the triangle a cutout spanning ½ of that edge can be cutout, rotated and attached to the other corner of the same edge.

Regular hexagons and irregular hexagons with 3 sets of parallel sides both can tessellate a plane. Regular hexagons, as well as irregular hexagons with 3 sets of parallel sides, can be used to create tiles that tessellate by translation and by reflection. This is done using the same steps shown for parallelograms, with cutouts from one edge of the hexagon attached to the parallel edge on the hexagon. It can be helpful to have a template when cutting out hexagons, in order to ensure your finished tessellations line up. I have attached a template under supplies. Hand drawn tessellations will probably not tessellate 100% perfect, even with a template, and that's okay!

Regular hexagons can be used to create shapes that tessellate by rotation, similarly to parallelograms, with cutouts from one edge of the hexagon being rotated 60 degrees and attached to an adjacent edge of the hexagon.

There are multitudes of fun arts and crafts projects that can be done with tessellations, and I've pictured a few above including filling the tessellated shapes with patterns or smaller tessellations, drawing animals or monsters M. C. Escher style in the tiles, tessellation landscapes, or coloring in the tiles in a fun demonstration of the Four Color Theorem, that any map can be colored in with 4 or less colors. Let your creativity run wild and have fun tesselating!