Difference between revisions of "Tessellation Exploration: The Basics"

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A tessellation is a regular tessellation if it is constructed from regular convex polygons of one size and one shape. There are exactly three regular polygons that tessellate the plane.
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A tessellation is a regular tessellation if it is constructed from regular convex polygons of one size and one shape. There are exactly three regular polygons that tessellate the plane: the equilateral triangle, the square, and the regular hexagon.
  
  
 
[[Image:Regular-tessellations.png]]
 
[[Image:Regular-tessellations.png]]
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Note for instance that the regular pentagon does not tessellate the plane! The figure below shows that when we try to arrange the pentagons around a vertex, then we will always have a gap or an overlap.
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[[Image:Regular-pentagon-fail2.svg]]
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There are non-regular pentagons that tessellate the plane.
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# Sketch the tessellation for a pentagon that looks like the outline of a house (see below) to illustrate this point.
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[[Image:Pentagon-house.svg]]

Revision as of 12:48, 18 May 2007


Time-45.svg

Objective: Find a method to tessellates the plane with any triangle. Introduce regular and semi-regular tessellations.

Tessellations by Triangles and Quadrilaterals

  1. Convince yourself that the parallelogram below will tessellate the plane. Draw your tessellation on a separate piece of paper. It should cover 1/4 of your page.
    Parallelogram.png
  2. Show that a square tessellates the plane and show that a rectangle tessellates the plane. Your tessellations should cover 1/4 of your page.
  3. Draw an acute, an obtuse and a right triangle. Now convince yourself of the fact that two congruent copies of the same triangle fit together to form a parallelogram or a rectangle.
  4. Why does this imply that all triangles will tessellate the plane?

Tessellations of the Plane by Regular Polygons

A regular tessellation is a tessellation made of regular, congruent, convex polygons.


A semi-regular tessellation is a tessellation made of regular polygons of two or more types so that the arrangement of polygons at each vertex is the same.

  1. What is the common name for a regular 3-gon?
  2. What is the common name for a regular 4-gon?

A tessellation is a regular tessellation if it is constructed from regular convex polygons of one size and one shape. There are exactly three regular polygons that tessellate the plane: the equilateral triangle, the square, and the regular hexagon.


Regular-tessellations.png

Note for instance that the regular pentagon does not tessellate the plane! The figure below shows that when we try to arrange the pentagons around a vertex, then we will always have a gap or an overlap.

Regular-pentagon-fail2.svg

There are non-regular pentagons that tessellate the plane.

  1. Sketch the tessellation for a pentagon that looks like the outline of a house (see below) to illustrate this point.

Pentagon-house.svg