pajaros

	BIRDS
	Geometry in the art of M.C. Escher

1. TILES

When we want to reconstruct a tessellation it is often difficult to produce elaborately-shaped tiles. Is it possible to come up with a basic pattern, (based on a smaller polygon within the tile) which can be used as the basis from which the tessellation can be created?

1.- Try and work out the shape of the tile and the closed design on the tile which when put together with other identical tiles would produce the tessellation illustrated in this window.

Increase the solution parameter to 1 to see the answer.

2.- This design forms part of the Nazari tessellation called bone.

Make copies and construct the tessellation. hueso.gif (2272 bytes)

3.- Can you work out the design on each individual square tile, which once moved on the plane appropriately form the bone pattern? Could the pattern be formed using triangular tiles?

2. 3^rd DESIGN RULE FOR THE CONSTRUCTION OF ESCHER'S TESSELLATIONS.

Start with a parallelogram or hexagon with parallel opposite sides. You just need to apply the following rule to create a shape which can be used to form a tessellation: Any piece which is cut from one of the sides of the shape should undergo a parallel translation to the opposite side.

4.- Follow the rule explained above and describe the process you need to go through in order to create a bird from a square.

Click on the Animate button to watch the complete process.

5.- Note that you do not have to cut out the whole length of the side.

6.- Design your own pieces by using the rule above and a parallelogram or hexagon as your initial polygon.

7.- Is there a connection between the way the piece is made and the way the pieces are positioned in the tessellation?

8.- Should the direction in which we move the cut out pieces always necessarily be perpendicular to the side of the polygon? Try creating a piece from a parallelogram.

3. CONSTRUCTING THE TESSELLATION

9.- Which directions should we translate the piece in in order to form the rest of the pattern?

Click on the Animate button to see the solution.

10.- Work out the connection between these directions and the necessary translations used whilst creating the piece.

11.- What is the modulus (length or magnitude) of the vector which represents the translation?

12.- Do these kinds of tessellations include movements which are different to the translations used?

	Enrique Martínez Arcos

	© Spanish Ministry of Education and Science. Year 2001