warm up. aim# 9-7: how do we identify transformations in tessellations, and figures that will...
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Warm Up
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Aim# 9-7: How do we identify transformations in tessellations, and figures that will tessellate?
• A tessellation or tiling, is a repeating pattern of figures that completely covers a plane without gaps or overlaps.
• You can create tessellations with translations, rotations, and reflections. You can find tessellations in art, nature (ex. honeycomb), and everyday tiled floors.
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Identifying the Transformations in a Tessellations
• Identify a transformation and the repeating figures in this tessellation.
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Identifying the Transformations in a Tessellations
• Identify a transformation and the repeating figures in this tessellation.
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Determining Figures That Will Tessellate
• Because the figures in a tessellation do not overlap or leave gaps, the sum of the measures of the angles around any vertex must be 360°. If the angles around a vertex are all congruent, then the measure of each angle must be a factor of 360°.
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Determining Figures That Will Tessellate
• Determine whether a regular 18-gon tessellates a plane.
• a = 180 (n - 2 ) Use the formulas for the measure• n of an angle of a regular polygon.•
Since 160 is not a factor of 360, the 18-gon will not tessellate.
160
18
)218(180
a
a
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Determining Figures That Will Tessellate
• Explain why you can tessellate a plane with an equilateral triangle.
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• A figure does not have to be a regular polygon to tessellate.
• Theorem 9-6:
• Every triangle tessellates.
• Explain why?
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• Theorem 9-7
• Every quadrilateral tessellates.
• Explain why?
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Identifying Symmetries in Tessellations
• The tessellations with regular hexagons at the right has reflectional symmetry in each of the blue lines. It has rotational symmetry centered at each of the red points.
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Identifying Symmetries in Tessellations
• The tessellation also has translational symmetry and
• A translation maps onto itself.
• Glide reflectional symmetry.
• A glide reflection maps onto itself.
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Identifying Symmetries in Tessellations
• List the symmetries in the tessellation.
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Identifying Symmetries in Tessellations
• Solution: Rotational symmetry centered at each red point Translational symmetry (blue arrow)
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Identifying Symmetries in Tessellations
• List the symmetries in the tessellation.
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Creating Tessellations
• Draw a 2.5 inch square on a blank piece of paper and cut it out.
• Draw a curve joining two consecutive vertices.
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Creating Tessellations
• Cut along the curve you drew and slide the cutout piece to the opposite side of the square. Tape it in place.
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Creating Tessellations
• Repeat this process using the other two opposite sides of the square.
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Creating Tessellations
• Rotate the resulting figure. What does your imagination suggest it looks like?
• Is it a penguin wearing a hat or a knight on horseback? Could it be a dog with floppy ears? Draw the image on your figure.
• Create a tessellation using your figure.
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Summary:Answer in complete sentences.
• A pure tessellation is a tessellation made up of congruent copies of one figure. Explain why there are three, and only three pure tessellations that use regular polygons.
• Homework: If you haven’t finished your “tile” with a picture inside of it, that is your homework.