1. 13 Chapter Congruence and Similarity with Transformations
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2. 13-4 Tessellations of the Plane
Regular Tessellations
Semiregular Tessellations
Tessellating with Other Shapes
Creating Tessellations with Translations
Creating Tessellations with Rotations
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3. Tessellations of the Plane
A tessellation of a plane is the filling of the plane with repetitions of figures in such a way that no figures overlap and there are no gaps.
Study of Regular Division of the Plane with Reptiles by Maurits C. Escher.
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4. Regular Tessellations
Tessellations with regular polygons are both appealing and interesting because of their simplicity.
Tessellation with squares
Tessellation with equilateral triangles and hexagons.
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5. Regular Tessellations
Which regular polygons can tessellate the plane?
To understand this, look at the tessellation using hexagons.
Around a vertex of any regular tessellation, we must have an angle sum of 360° (3 × 120 = 360). The interior angles of a hexagon measure:
120°
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6. Regular Tessellations
If we have three pentagons surrounding the vertex of a tessellation, the sum of the angles around the vertex is × 108° = 324° < 360° . We do not have enough pentagons to surround the vertex.
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7. Regular Tessellations
On the other hand, if we include a fourth pentagon, the sum of the angles around the vertex exceeds 360°. So, there cannot be a regular tessellation using pentagons.
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8. Regular Tessellations
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9. Regular Tessellations
Since 360 divided by a number greater than 120 is smaller than 3, and the number of sides of a polygon cannot be less than 3, no regular polygon with more than six sides can tessellate the plane.
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10. Semiregular Tessellations
When more than one type of regular polygon is used and the arrangement of the polygons at each vertex is the same, the tessellation is semiregular.
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11. Tessellating with Other Shapes
Although there are only three regular tessellations (square, equilateral triangle, and regular hexagon), many tessellations are not regular. One factor that must always be true is that the sum of the measures of the interior angles around a vertex must equal 360°.
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12. Tessellating with Other Shapes
A regular pentagon does not tessellate the plane. However, some nonregular pentagons do.
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13. Tessellating with Other Shapes
The following two tessellations were discovered by Marjorie Rice, and the problem of how many types of pentagons tessellate remains unsolved today.
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14. Creating Tessellations with Translations
Consider any polygon known to tessellate a plane, such as rectangle ABCD (a). On the left side of the figure draw any shape in the interior of the rectangle (b). Cut this shape from the rectangle and slide it to the right by the slide that takes A to B (c). The resulting shape will tessellate the plane.
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15. Creating Tessellations with Rotations
A second method of forming a tessellation involves a series of rotations of parts of a figure. Start with an equilateral triangle ABC (a), choose the midpoint O of one side of the triangle, and cut out a shape (b), being careful not to cut away more than half of angle B, and then rotate the shape clockwise around point O.
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16. Creating Tessellations with Rotations
Complete the tessellating shape and tessellate the plane with it.
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