Transcript Geometry - BakerMath.org

Tessellations
You ask,
“What is a tessellation, exactly?”
 A tessellation is any repeating pattern of
interlocking shapes.
 In English, a way to tile a floor with no
overlapping pieces and no gaps.
Everything has rules, even tessellations!
 There can be no overlapping and no gaps.
 (Heard that one already!)
 The polygons must be regular.
 The sides of the figure must all be the same length
 The interior angles of the figure must be the same
measure

Each vertex must look the same.
 What’s a vertex?
Where the
“corners” meet
What type of polygons will work??
 Triangles?
 Squares?
 Pentagons?
 Hexagons?
 Heptagons?
 Octagons?
Finding the Interior Angle
180 ( n  2)
n
 Where” n” is the number of sides
Triangles

Yes, they do!
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The interior angle of each
triangle is 60o
At each “vertex” the sum
of the measure is 360o
Squares

Yes, they do!


The interior angle of each
square is 90o
At each “vertex” the sum
of the measure is 360o
Pentagons

No, they don’t!

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The interior angle of each
pentagon is 108o
At each “vertex” the sum
of the measure is 324o
 Do you see the gap?
Hexagons

Yes, they do!

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The interior angle of each
hexagon is 120o
At each “vertex” the sum
of the measure is 360o
Heptagons

No, they don’t! ?

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The interior angle of each
heptagon is 257.1428…o
At each “vertex” the sum of the
measure is 771.428…o
Do you see the overlap?
Octagons

What do you think? ?

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The interior angle of each
octagon is 135o
At each “vertex” the sum
of the measure is …
You’re right!
 Octagons won’t tessellate.
 In fact, any polygon with more than 6 sides won’t
tessellate
 So, can you remember which polygons tessellate?
 Triangles
 Squares
 Hexagons
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Fairly Simple Tessellations…
Semi-Regular Tessellations

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3, 6, 3, 6
3, 3, 3, 3, 6
These tessellations are made up of triangles and hexagons, but
the configuration of the vertices are different. That’s why they
are named differently!
To name a tessellation, simply work your way around the vertex
naming the number of sides of each polygon. Always put the
smallest possible number first!
Semi-Regular Tessellation??

This tessellations is also made up of triangles and
hexagons, can you see why this isn’t an “official” semiregular tessellation?

It breaks the vertex rule, can you tell where?
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More Semi-Regular Tessellations…
What’s M.C. Escher?
He was a Dutch graphic artist (18981972) who made prints involving
tessellations, impossible figures or
worlds, polyhedra, and unusual
perspective systems.
“Convex and Concave”
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“C
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“Bats and Owls”
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“Ascending & Descending”
“Encounter”
“Hand with Reflecting Sphere”
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“Sky and Water I, 1938”
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“Metamorphose II, 1940”
Some more Escher tessellations…
and more…
Are you ready to tessellate?