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Tessellations

  • Introduction
  • Results & Video
  • Materials
  • Procedure
  • Preparation & References
  • Downloads
hexagon

A tessellation is a two-dimensional repeating pattern of a basic shape with no overlaps and no gaps between the repeating units. The artist M.C. Escher incorporated tessellations into many of his art works. Using some techniques described at www.tessellations.org, kids will design their own tessellations on Family Science Night.

There are several classes of tessellations. A regular tessellation is a pattern created by repeating a regular polygon. There are only 3 regular tessellations, which are made by squares, triangles and hexagons. Every point of int ersection in a regular tessellation has the same pattern.

trianglesqre

A semii regular tessellation is made of two or more regular polygons. Each vertex must bet the same. There are only a few semi regular tessellations as well. These tessellations are from http://www.mathsisfun.com/geometry/tessellation.html..

semi1 semi2

There are other tessellations with curved lines and complex shapes. Mathemeticians sometimes disagree as to whether these are true tessellations but they can make interesting patterns. The artist M.C. Escher famously integrated complex tessellations into his art. A gallery of these can be found at www.tessellations.org/eschergallery1thumbs.shtml

 

Examples of tessellations

Here are examples of tessellations with a variety of methods.

These 2 images were made by sliding (translating) hexagons. The patterned arrangement of the hexagons is the same but the coloring is more elaborate in the first image.

hexagon_color1

tessellation with curved lines

This example was made with the Slice Method

tesselation1slice method

Materials

  • paper or card stock for creating the pattern
  • rulers
  • tape
  • glue sticks
  • colored card stock, colored pencils or paints for creating colored patterns
  • larger paper for drawing the patterns

Procedure

There are several methods for the creation of tessellations.

Paper Cut Method

  1. Start with a shape that will tesselate, like a square.
  2. Label the corners clockwise from the top left, with P A R T.
  3. Draw a line from one side to the opposite. Then do again from the remaining 2 sides.parttrap





  4. Now pull apart the pieces and put them back together with the corners in the middle, but spell T R A P (above).
  5. tess1
  6. This shape can be tiled to make a tessellation pattern. Trace the shape on paper, then move the cutout shape so that it fits without spaces to the orginal trace. Repeat as many as copies as desired.

 

 

Slice Method

The Slice Method starts with a basic shape that can be tesselated and modifies it by cutting pieces from one side and placing it on the symmetry related side.

  1. Start with a shape that will tesselate, like a square.
  2. Cut an interesting piece out of one side of the shape.
  3. Tape the shape to another side of the starting shape that is symmetric to the starting side. On a square it can be any of the 3 other sides. For unlimited tiling, the shape should be taped to the opposite side.
  4. slice
  5. (optional) Repeat step 2 &3 for the remaining 2 sides of the square.
  6. Use this new shape to tile a piece of paper. We traced on one color paper and then cutout shapes from a different color and pasted in the pattern.
  7. slice

Tessellation Artist

This website application is great for quickly creating tessellation patterns from regular polygons. Students specify the spacing, rotations and size of the geometric shapes to create interesting patterns.

  1. Go to the tessellation Artist website.
  2. Pick a shape and then click in the middle of the main window. Drag until you have the size shape that is desired.
  3. Adjust the values in the top left window to change the tiling specifications.screen grab

Role of each Tiling Direction

1. Spacing in the first row between shapes.

2. How much shapes are shifted up and down within a row.

3. The number of shapes in a row that are added the left and right of the first shape.

4. The horizontal offset for shapes in adjacent rows.

5. The vertical shift between rows.

6. The number of rows added above and below the first row.

Notes

We used 2" x 2" squares to start the Slice and the Paper Cut method. With this size starting square, tesselated patterns fit well on 8.5" x 11" paper. Using 3" x 3" did not give many full copies on this size paper.

References

Another excellent website with ideas can be found here.

A great website that integrate lessons from geometry is mathisfun.com, where they have a basic program for the creating tessellations from regular polygons.

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