Tessellations - University of Massachusetts Amherstjli/talks/tessellations.pdf · 2018. 11. 10. · Tessellations Jennifer Li and Maggie Smith Sonia Kovalevsky Day Mount Holyoke College

Tessellations

Jennifer Li and Maggie Smith

Sonia Kovalevsky DayMount Holyoke College

Saturday, November 10, 2018

Tessellations everywhere

Jennifer Li and Maggie Smith Tessellations April 18, 2018 2 / 39

What’s the connection to Math?

Mathematicians REALLY like patterns and symmetry!


Tiles

A tile is a geometric shape.


Tiles

A tile is a geometric shape.


Tiles

Tiles are the building blocks of a tessellation.

A tessellation covers the entire plane (infinite).No gaps and no overlaps!


Tiles




Tiles


A tessellation covers the entire plane (infinite).

No gaps and no overlaps!


Tiles




Polygons

A polygon is a shape that is created by straight line segments.


Polygons

A polygon is a shape that is created by straight line segments.


Regular Polygons

In a regular polygon, all angles are equal and all side lengths are equal.


Regular Polygons

In a regular polygon, all angles are equal and all side lengths are equal.


Types of Tessellations

A regular tessellation is a symmetric tiling made up of regularpolygons, all of the same shape.


Regular Polygon Tessellations

Equilateral Triangles


Regular Polygon Tessellations

Squares


Activity: Regular Polygon Tessellations

Activity Sheet: Tessellate the plane using the regular hexagon.


Vertex

A vertex is a point where the corners of all polygons in a tessellationmeet.

Regular hexagons


Vertex

A vertex is a point where the corners of all polygons in a tessellationmeet.

Regular hexagons


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.


Regular Polygons


180(

1 − 2

n

)Examples.

n = 3

180(

1 − 2

3

)=

180

3= 60



Regular Polygons


180(

1 − 2

n

)Examples.

n = 3

180(

1 − 2

3

)=

180

3= 60



Regular Polygons


180(

1 − 2

n

)Examples.

n = 4

180(

1 − 2

4

)=

180

2= 90

Each angle in a square is 90 degrees.


Regular Polygons


180(

1 − 2

n

)Examples.

n = 4

180(

1 − 2

4

)=

180

2= 90

Each angle in a square is 90 degrees.


Regular Polygons


180(

1 − 2

n

)Examples.

n = 9

180(

1 − 2

9

)= 7 × 180

9= 140

Each angle in a nonagon is 140 degrees.


Regular Polygons


180(

1 − 2

n

)Examples.

n = 9

180(

1 − 2

9

)= 7 × 180

9= 140

Each angle in a nonagon is 140 degrees.


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?











At a vertex, there will be q regular polygons that meet:

Each polygon is n-sided:















At each vertex, these angles must add to 360 degrees.



At each vertex, these angles must add to 360 degrees.



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



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



If1

q+

1

n=

1

2

then a regular polygon tessellation is possible!



If1

q+

1

n=

1

2




If1

q+

1

n=

1

2




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!



1

q+

1

n=

1

2



Then q =2

1 − 23


Is this correct?

Yes!



1

q+

1

n=

1

2



Then q =2

1 − 23


Is this correct?

Yes!



1

q+

1

n=

1

2



Then q =2

1 − 23


Is this correct?

Yes!



1

q+

1

n=

1

2



Then q =2

1 − 23


Is this correct?

Yes!



1

q+

1

n=

1

2



Then q =2

1 − 23


Is this correct?

Yes!


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?



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!



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!



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!




Then q =2

1 − 25

=10







Then q =2

1 − 25

=10







Then q =2

1 − 25

=10






Fun Fact!

There are only three regular polygons that tessellate theplane: the equilateral triangle, the square, and the hexagon!



Fun Fact! There are only three regular polygons that tessellate theplane: the equilateral triangle, the square, and the hexagon!



Fun Fact! There are only three regular polygons that tessellate theplane: the equilateral triangle, the square, and the hexagon!


More Types of Tessellations

We can make more tessellations by using more than one regularpolygon.

This type of tessellation is called an Archimedean tessellation.














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?


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.


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.


Dual Tessellations

Example. The dual is drawn in pink:


Dual Tessellations

Example. The dual is drawn in pink:


Activity: Dual Tessellations

Activity Sheet: Find the dual tessellations. What do you notice aboutthese duals?



A tessellation is monohedral if all tiles are congruent (they have thesame size and shape).

The tiles don’t have to be polygons!










Activity: Tessellation of the plane

Activity Sheet: Draw some monohedral tessellations of the plane withthe given tiles.



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!



Question. Given a collection of tiles, can we create a monohedraltessellation of the plane?














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?


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!


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!


Pentagonal tiling in math research

There are 15 convex pentagons that tessellate the plane monohedrally.



There are 15 convex pentagons that tessellate the plane monohedrally.



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.










Documents

Tessellations - University of Massachusetts Amherstjli/talks/tessellations.pdf · 2018. 11. 10. · Tessellations Jennifer Li and Maggie Smith Sonia Kovalevsky Day Mount Holyoke College