tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfa tessellation covers the entire...
TRANSCRIPT
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Tessellations
Jennifer Li and Maggie Smith
Sonia Kovalevsky DayMount Holyoke College
Saturday, November 10, 2018
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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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39
![Page 7: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/7.jpg)
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
![Page 8: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/8.jpg)
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
![Page 9: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/9.jpg)
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
![Page 10: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/10.jpg)
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
![Page 22: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/22.jpg)
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
![Page 23: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/23.jpg)
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.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 14 / 39
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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.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 14 / 39
![Page 25: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/25.jpg)
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.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 15 / 39
![Page 27: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/27.jpg)
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?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 16 / 39
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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:
Jennifer Li and Maggie Smith Tessellations April 18, 2018 17 / 39
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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:
Jennifer Li and Maggie Smith Tessellations April 18, 2018 17 / 39
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Which regular polygons tessellate the plane?
At each vertex, these angles must add to 360 degrees.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 18 / 39
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Which regular polygons tessellate the plane?
At each vertex, these angles must add to 360 degrees.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 18 / 39
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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
Jennifer Li and Maggie Smith Tessellations April 18, 2018 19 / 39
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Which regular polygons tessellate the plane?
If1
q+
1
n=
1
2
then a regular polygon tessellation is possible!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39
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Which regular polygons tessellate the plane?
If1
q+
1
n=
1
2
then a regular polygon tessellation is possible!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39
![Page 40: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/40.jpg)
Which regular polygons tessellate the plane?
If1
q+
1
n=
1
2
then a regular polygon tessellation is possible!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39
![Page 41: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/41.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39
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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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39
![Page 43: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/43.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39
![Page 44: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/44.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39
![Page 45: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/45.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39
![Page 46: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/46.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39
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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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 23 / 39
![Page 49: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/49.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 23 / 39
![Page 50: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/50.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 24 / 39
![Page 51: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/51.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 24 / 39
![Page 52: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/52.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 24 / 39
![Page 53: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/53.jpg)
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 24 / 39
![Page 54: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/54.jpg)
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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![Page 55: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/55.jpg)
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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![Page 56: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/56.jpg)
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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![Page 57: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/57.jpg)
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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![Page 58: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/58.jpg)
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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![Page 59: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/59.jpg)
More Types of Tessellations
We can make more tessellations by using more than one regularpolygon.
This type of tessellation is called an Archimedean tessellation.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 26 / 39
![Page 60: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/60.jpg)
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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![Page 64: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/64.jpg)
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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![Page 65: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/65.jpg)
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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![Page 68: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/68.jpg)
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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![Page 69: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/69.jpg)
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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![Page 70: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/70.jpg)
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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![Page 71: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/71.jpg)
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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![Page 74: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/74.jpg)
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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![Page 75: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/75.jpg)
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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![Page 76: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/76.jpg)
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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![Page 79: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/79.jpg)
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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![Page 81: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/81.jpg)
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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![Page 82: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/82.jpg)
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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![Page 83: Tessellationspeople.math.umass.edu/~jli/talks/tessellations.pdfA tessellation covers the entire plane (in nite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations](https://reader033.vdocuments.mx/reader033/viewer/2022060419/5f168c9936c94c40cb5de649/html5/thumbnails/83.jpg)
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.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 39 / 39