Difference between revisions of "Tessellation Exploration: The Basics"

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==Tessellations by Triangles and Quadrilaterals==
 
==Tessellations by Triangles and Quadrilaterals==
# 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.
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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.
  
 
[[Image:Parallelogram.png|center|300px]]
 
[[Image:Parallelogram.png|center|300px]]
  
# Show that a square tessellates the plane and show that a rectangle tessellates the plane. Your tessellations should cover 1/4 of your page.
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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.
  
# 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.  
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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.  
  
  
# Why does this imply that all triangles will tessellate the plane?  
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4 Why does this imply that all triangles will tessellate the plane?  
  
  
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A {{define|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.
 
A {{define|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.
  
# What is the common name for a regular 3-gon?
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5 What is the common name for a regular 3-gon?
  
# What is the common name for a regular 4-gon?
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6 What is the common name for a regular 4-gon?
  
  
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There are non-regular pentagons that tessellate the plane.  
 
There are non-regular pentagons that tessellate the plane.  
  
# Sketch the tessellation for a pentagon that looks like the outline of a house (see below) to illustrate this point.
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7 Sketch the tessellation for a pentagon that looks like the outline of a house (see below) to illustrate this point.
  
 
[[Image:Pentagon-house.svg]]
 
[[Image:Pentagon-house.svg]]
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# Extend the following tessellation. You should draw at least 2 more layers of polygons on each side. Is this tessellation semi-regular? Explain.
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8 Extend the following tessellation. You should draw at least 2 more layers of polygons on each side. Is this tessellation semi-regular? Explain.
  
 
[[Image:Semi-regular-(3,3,3,4,4).png|center]]
 
[[Image:Semi-regular-(3,3,3,4,4).png|center]]
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9. Are the following two tessellations semi-regular or not? Explain.
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[[Image:Semi-reg-or-not.png|center]]

Revision as of 13:00, 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.

5 What is the common name for a regular 3-gon?

6 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.

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

Pentagon-house.svg


Semi-Regular Tessellations

We can also look at tessellations formed by a combination of several regular polygons. A tessellation is a semi-regular tessellation if it is composed of regular polygons of two or more types so that the arrangement of polygons at all the vertices is the same.


8 Extend the following tessellation. You should draw at least 2 more layers of polygons on each side. Is this tessellation semi-regular? Explain.

Semi-regular-(3,3,3,4,4).png


9. Are the following two tessellations semi-regular or not? Explain.

Semi-reg-or-not.png