Difference between revisions of "Tessellation Exploration: The Basics"
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{{Exploration}} | {{Exploration}} | ||
{{Time|45}} | {{Time|45}} | ||
| − | {{Objective|Find a method to | + | {{Objective|Find a method to tessellate the plane with any triangle. Introduce regular and semi-regular tessellations.}} |
| + | |||
| + | ==Materials== | ||
| + | {{printable|Printed version of the Tessellation Exploration: The Basics: [[Image:Tessellations-basics.pdf]]}} | ||
| + | * Printed copy of the Tessellation Exploration: The Basics. | ||
| + | {{clear}} | ||
| + | |||
| + | ==Exploration== | ||
==Tessellations by Triangles and Quadrilaterals== | ==Tessellations by Triangles and Quadrilaterals== | ||
| − | + | <ol> | |
| + | <li>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]] | ||
| − | + | </li> | |
| − | + | <li>Show that a square tessellates the plane and show that a rectangle tessellates the plane. Your tessellations should cover 1/4 of your page. | |
| − | + | </li> | |
| − | + | <li>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. | |
| − | + | </li> | |
| − | + | <li>Why does this imply that all triangles will tessellate the plane?</li> | |
| − | + | </ol> | |
| − | |||
| − | |||
| − | |||
==Tessellations of the Plane by Regular Polygons== | ==Tessellations of the Plane by Regular Polygons== | ||
| − | A {{define|regular tessellation}} is a tessellation made of regular, congruent | + | A {{define|regular tessellation}} is a tessellation made of regular polygons, all congruent to one another. (Recall that a regular polygon has all sides congruent to one another and all angles congruent to one another. This implies the polygon is convex--why?) |
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. | ||
| − | 5 | + | <ol start="5"> |
| − | + | <li>What is the common name for a regular 3-gon?</li> | |
| − | + | <li>What is the common name for a regular 4-gon?</li> | |
| − | + | </ol> | |
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. | 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. | ||
| − | + | {| cellpadding="30" | |
| − | [[Image: | + | |- |
| + | | [[Image:regular-triangles.svg]] || [[Image:regular-squares.svg]] || [[Image:regular-hexagons.svg]] | ||
| + | |} | ||
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. | 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. | ||
| Line 40: | Line 47: | ||
There are non-regular pentagons that tessellate the plane. | There are non-regular pentagons that tessellate the plane. | ||
| − | 7 | + | <ol start="7"> |
| + | <li>Sketch the tessellation for a pentagon that looks like the outline of a house (see below) to illustrate this point. | ||
[[Image:Pentagon-house.svg|center]] | [[Image:Pentagon-house.svg|center]] | ||
| − | + | </li> | |
| + | </ol> | ||
==Semi-Regular Tessellations== | ==Semi-Regular Tessellations== | ||
| Line 49: | Line 58: | ||
A tessellation is a {{define|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. | A tessellation is a {{define|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. | ||
| − | + | <ol start="8"> | |
| − | + | <li>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]] | ||
| + | </li> | ||
| + | <li>Are the following two tessellations semi-regular or not? Explain. | ||
| + | [[Image:Semi-reg-or-not.png|center]] | ||
| + | </li> | ||
| + | </ol> | ||
| − | + | {{handin}} | |
| − | [[ | + | [[category:Tessellation Explorations]] |
Latest revision as of 11:34, 6 February 2009
Objective: Find a method to tessellate the plane with any triangle. Introduce regular and semi-regular tessellations.
Materials
- Printed copy of the Tessellation Exploration: The Basics.
Exploration
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.
- 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.
- 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 polygons, all congruent to one another. (Recall that a regular polygon has all sides congruent to one another and all angles congruent to one another. This implies the polygon is convex--why?)
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.
- What is the common name for a regular 3-gon?
- 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.
 |
 |
|
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.
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.
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.
- Extend the following tessellation. You should draw at least 2 more layers of polygons on each side. Is this tessellation semi-regular? Explain.
- Are the following two tessellations semi-regular or not? Explain.
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Handin: A sheet with answers to all questions.
