Tessellations - University of Massachusetts Amherstjli/talks/tessellations.pdf · 2018. 11. 10. ·...
Transcript of Tessellations - University of Massachusetts Amherstjli/talks/tessellations.pdf · 2018. 11. 10. ·...
Tessellations
Jennifer Li and Maggie Smith
Sonia Kovalevsky DayMount Holyoke College
Saturday, November 10, 2018
Tessellations everywhere
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What’s the connection to Math?
Mathematicians REALLY like patterns and symmetry!
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Tiles
A tile is a geometric shape.
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Tiles
A tile is a geometric shape.
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Tiles
Tiles are the building blocks of a tessellation.
A tessellation covers the entire plane (infinite).No gaps and no overlaps!
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Tiles
Tiles are the building blocks of a tessellation.
A tessellation covers the entire plane (infinite).No gaps and no overlaps!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39
Tiles
Tiles are the building blocks of a tessellation.
A tessellation covers the entire plane (infinite).
No gaps and no overlaps!
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Tiles
Tiles are the building blocks of a tessellation.
A tessellation covers the entire plane (infinite).No gaps and no overlaps!
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Polygons
A polygon is a shape that is created by straight line segments.
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Polygons
A polygon is a shape that is created by straight line segments.
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Regular Polygons
In a regular polygon, all angles are equal and all side lengths are equal.
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Regular Polygons
In a regular polygon, all angles are equal and all side lengths are equal.
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Types of Tessellations
A regular tessellation is a symmetric tiling made up of regularpolygons, all of the same shape.
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Regular Polygon Tessellations
Equilateral Triangles
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Regular Polygon Tessellations
Squares
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Activity: Regular Polygon Tessellations
Activity Sheet: Tessellate the plane using the regular hexagon.
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Vertex
A vertex is a point where the corners of all polygons in a tessellationmeet.
Regular hexagons
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Vertex
A vertex is a point where the corners of all polygons in a tessellationmeet.
Regular hexagons
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Regular Polygons
Each angle of an n-sided polygon equals
180(
1 − 2
n
)
Examples.
n = 3
180(
1 − 2
3
)=
180
3= 60
Each angle in an equilateral triangle is 60 degrees.
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Regular Polygons
Each angle of an n-sided polygon equals
180(
1 − 2
n
)Examples.
n = 3
180(
1 − 2
3
)=
180
3= 60
Each angle in an equilateral triangle is 60 degrees.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 13 / 39
Regular Polygons
Each angle of an n-sided polygon equals
180(
1 − 2
n
)Examples.
n = 3
180(
1 − 2
3
)=
180
3= 60
Each angle in an equilateral triangle is 60 degrees.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 13 / 39
Regular Polygons
Each angle of an n-sided polygon equals
180(
1 − 2
n
)Examples.
n = 4
180(
1 − 2
4
)=
180
2= 90
Each angle in a square is 90 degrees.
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Regular Polygons
Each angle of an n-sided polygon equals
180(
1 − 2
n
)Examples.
n = 4
180(
1 − 2
4
)=
180
2= 90
Each angle in a square is 90 degrees.
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Regular Polygons
Each angle of an n-sided polygon equals
180(
1 − 2
n
)Examples.
n = 9
180(
1 − 2
9
)= 7 × 180
9= 140
Each angle in a nonagon is 140 degrees.
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Regular Polygons
Each angle of an n-sided polygon equals
180(
1 − 2
n
)Examples.
n = 9
180(
1 − 2
9
)= 7 × 180
9= 140
Each angle in a nonagon is 140 degrees.
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Which regular polygons tessellate the plane?
How can we tessellate the plane with a regular n-sided polygon?
Can they fit without gaps and without overlapping?
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Which regular polygons tessellate the plane?
How can we tessellate the plane with a regular n-sided polygon?
Can they fit without gaps and without overlapping?
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Which regular polygons tessellate the plane?
How can we tessellate the plane with a regular n-sided polygon?
Can they fit without gaps and without overlapping?
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Which regular polygons tessellate the plane?
At a vertex, there will be q regular polygons that meet:
Each polygon is n-sided:
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Which regular polygons tessellate the plane?
At a vertex, there will be q regular polygons that meet:
Each polygon is n-sided:
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Which regular polygons tessellate the plane?
At a vertex, there will be q regular polygons that meet:
Each polygon is n-sided:
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Which regular polygons tessellate the plane?
At a vertex, there will be q regular polygons that meet:
Each polygon is n-sided:
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Which regular polygons tessellate the plane?
At each vertex, these angles must add to 360 degrees.
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Which regular polygons tessellate the plane?
At each vertex, these angles must add to 360 degrees.
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Which regular polygons tessellate the plane?
A total of q angles, each of 180 ×(
1 − 2
n
)degrees, sum to 360 degrees:
q × 180 ×(
1 − 2
n
)= 360
q × 180 ×(
1 − 2n
)q × 180
=360
q × 180
�q ×��180 ×(
1 − 2n
)�q ��180
=2
q
1 − 2
n=
2
q
1 =2
q+
2
n
1
q+
1
n=
1
2
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Which regular polygons tessellate the plane?
A total of q angles, each of 180 ×(
1 − 2
n
)degrees, sum to 360 degrees:
q × 180 ×(
1 − 2
n
)= 360
q × 180 ×(
1 − 2n
)q × 180
=360
q × 180
�q ×��180 ×(
1 − 2n
)�q ��180
=2
q
1 − 2
n=
2
q
1 =2
q+
2
n
1
q+
1
n=
1
2
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Which regular polygons tessellate the plane?
If1
q+
1
n=
1
2
then a regular polygon tessellation is possible!
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Which regular polygons tessellate the plane?
If1
q+
1
n=
1
2
then a regular polygon tessellation is possible!
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Which regular polygons tessellate the plane?
If1
q+
1
n=
1
2
then a regular polygon tessellation is possible!
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Which regular polygons tessellate the plane?
1
q+
1
n=
1
2
Question. In an equilateral triangle tessellation, how many trianglesmust meet at a vertex?
An equilateral triangle has three sides, so n = 3.
Then q =2
1 − 23
= 6 equilateral triangles meet at a vertex...
Is this correct?
Yes!
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Which regular polygons tessellate the plane?
1
q+
1
n=
1
2
Question. In an equilateral triangle tessellation, how many trianglesmust meet at a vertex?
An equilateral triangle has three sides, so n = 3.
Then q =2
1 − 23
= 6 equilateral triangles meet at a vertex...
Is this correct?
Yes!
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Which regular polygons tessellate the plane?
1
q+
1
n=
1
2
Question. In an equilateral triangle tessellation, how many trianglesmust meet at a vertex?
An equilateral triangle has three sides, so n = 3.
Then q =2
1 − 23
= 6 equilateral triangles meet at a vertex...
Is this correct?
Yes!
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Which regular polygons tessellate the plane?
1
q+
1
n=
1
2
Question. In an equilateral triangle tessellation, how many trianglesmust meet at a vertex?
An equilateral triangle has three sides, so n = 3.
Then q =2
1 − 23
= 6 equilateral triangles meet at a vertex...
Is this correct?
Yes!
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Which regular polygons tessellate the plane?
1
q+
1
n=
1
2
Question. In an equilateral triangle tessellation, how many trianglesmust meet at a vertex?
An equilateral triangle has three sides, so n = 3.
Then q =2
1 − 23
= 6 equilateral triangles meet at a vertex...
Is this correct?
Yes!
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Which regular polygons tessellate the plane?
1
q+
1
n=
1
2
Question. In an equilateral triangle tessellation, how many trianglesmust meet at a vertex?
An equilateral triangle has three sides, so n = 3.
Then q =2
1 − 23
= 6 equilateral triangles meet at a vertex...
Is this correct?
Yes!
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Activity: Which regular polygons tessellate the plane?
Use the equation for the activities below.
1
q+
1
n=
1
2
Activity Sheet: In square tessellation of the plane, how many squaresmust meet at a vertex?
Activity Sheet: In a regular hexagon tessellation of the plane, howmany hexagons must meet at a vertex?
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Activity: Which regular polygons tessellate the plane?
Activity Sheet: In regular pentagon tessellation of the plane, how manypentagons must meet at a vertex?
It’s impossible to tessellate the plane with regular pentagons!
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Activity: Which regular polygons tessellate the plane?
Activity Sheet: In regular pentagon tessellation of the plane, how manypentagons must meet at a vertex?
It’s impossible to tessellate the plane with regular pentagons!
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Activity: Which regular polygons tessellate the plane?
A regular pentagon has five sides, so n = 5.
Then q =2
1 − 25
=10
3pentagons meet at a vertex...
But q should be whole number!
We cannot tessellate the plane with a regular pentagon!
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Activity: Which regular polygons tessellate the plane?
A regular pentagon has five sides, so n = 5.
Then q =2
1 − 25
=10
3pentagons meet at a vertex...
But q should be whole number!
We cannot tessellate the plane with a regular pentagon!
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Activity: Which regular polygons tessellate the plane?
A regular pentagon has five sides, so n = 5.
Then q =2
1 − 25
=10
3pentagons meet at a vertex...
But q should be whole number!
We cannot tessellate the plane with a regular pentagon!
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Activity: Which regular polygons tessellate the plane?
A regular pentagon has five sides, so n = 5.
Then q =2
1 − 25
=10
3pentagons meet at a vertex...
But q should be whole number!
We cannot tessellate the plane with a regular pentagon!
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Activity: Which regular polygons tessellate the plane?
Fun Fact!
There are only three regular polygons that tessellate theplane: the equilateral triangle, the square, and the hexagon!
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Activity: Which regular polygons tessellate the plane?
Fun Fact! There are only three regular polygons that tessellate theplane: the equilateral triangle, the square, and the hexagon!
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Activity: Which regular polygons tessellate the plane?
Fun Fact! There are only three regular polygons that tessellate theplane: the equilateral triangle, the square, and the hexagon!
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More Types of Tessellations
We can make more tessellations by using more than one regularpolygon.
This type of tessellation is called an Archimedean tessellation.
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More Types of Tessellations
We can make more tessellations by using more than one regularpolygon.
This type of tessellation is called an Archimedean tessellation.
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More Types of Tessellations
We can make more tessellations by using more than one regularpolygon.
This type of tessellation is called an Archimedean tessellation.
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More Types of Tessellations
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More Types of Tessellations
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Activity: Labelling a Tessellation
Activity Sheet:a) Describe the polygons that surround the red vertex in eachtessellation shown below.
b) What do you think the labels under each tessellation mean?
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Dual Tessellations
The dual of a tessellation is formed by drawing a vertex in the centerof each tile, and joining all vertices of tiles that touch.
Example. Find the dual of the tessellation below.
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Dual Tessellations
The dual of a tessellation is formed by drawing a vertex in the centerof each tile, and joining all vertices of tiles that touch.
Example. Find the dual of the tessellation below.
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Dual Tessellations
Example. The dual is drawn in pink:
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Dual Tessellations
Example. The dual is drawn in pink:
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Activity: Dual Tessellations
Activity Sheet: Find the dual tessellations. What do you notice aboutthese duals?
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Types of Tessellations
A tessellation is monohedral if all tiles are congruent (they have thesame size and shape).
The tiles don’t have to be polygons!
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Types of Tessellations
A tessellation is monohedral if all tiles are congruent (they have thesame size and shape).
The tiles don’t have to be polygons!
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Types of Tessellations
A tessellation is monohedral if all tiles are congruent (they have thesame size and shape).
The tiles don’t have to be polygons!
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Activity: Tessellation of the plane
Activity Sheet: Draw some monohedral tessellations of the plane withthe given tiles.
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Types of Tessellations
Question.
Given a collection of tiles, can we create a monohedraltessellation of the plane?
This can be a hard problem...
There is no general method known!
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Types of Tessellations
Question. Given a collection of tiles, can we create a monohedraltessellation of the plane?
This can be a hard problem...
There is no general method known!
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Types of Tessellations
Question. Given a collection of tiles, can we create a monohedraltessellation of the plane?
This can be a hard problem...
There is no general method known!
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Types of Tessellations
Question. Given a collection of tiles, can we create a monohedraltessellation of the plane?
This can be a hard problem...
There is no general method known!
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Activity: Which one does not belong?
Activity Sheet: A heptiamond is a shape that is created from sevenequilateral triangles glued together. There are a total of twenty-fourheptiamonds:
Only one does not give a monohedral tiling of the plane. Can youfigure out which one it is?
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Tessellations using nonregular pentagons
We saw that regular pentagons do not tessellate the plane.
BUT...some pentagons that are not regular do tessellate the plane!
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Tessellations using nonregular pentagons
We saw that regular pentagons do not tessellate the plane.
BUT...some pentagons that are not regular do tessellate the plane!
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Pentagonal tiling in math research
There are 15 convex pentagons that tessellate the plane monohedrally.
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Pentagonal tiling in math research
There are 15 convex pentagons that tessellate the plane monohedrally.
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Pentagonal tiling in math research
The most recent pentagonal tiling was discovered in 2015:
In 2017, it was proven that there are only 15 tilings of the plane usingconvex pentagons.
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Pentagonal tiling in math research
The most recent pentagonal tiling was discovered in 2015:
In 2017, it was proven that there are only 15 tilings of the plane usingconvex pentagons.
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Pentagonal tiling in math research
The most recent pentagonal tiling was discovered in 2015:
In 2017, it was proven that there are only 15 tilings of the plane usingconvex pentagons.
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