Jennifer Li and Maggie Smith
Sonia Kovalevsky Day Mount 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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 3 / 39 Tiles
A tile is a geometric shape.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 4 / 39 Tiles
A tile is a geometric shape.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 4 / 39 A tessellation covers the entire plane (infinite). No gaps and no overlaps!
Tiles
Tiles are the building blocks of a tessellation.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39 A tessellation covers the entire plane (infinite). No gaps and no overlaps!
Tiles
Tiles are the building blocks of a tessellation.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39 No gaps and no overlaps!
Tiles
Tiles are the building blocks of a tessellation.
A tessellation covers the entire plane (infinite).
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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39 Polygons
A polygon is a shape that is created by straight line segments.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 6 / 39 Polygons
A polygon is a shape that is created by straight line segments.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 6 / 39 Regular Polygons
In a regular polygon, all angles are equal and all side lengths are equal.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 7 / 39 Regular Polygons
In a regular polygon, all angles are equal and all side lengths are equal.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 7 / 39 Types of Tessellations
A regular tessellation is a symmetric tiling made up of regular polygons, all of the same shape.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 8 / 39 Regular Polygon Tessellations
Equilateral Triangles
Jennifer Li and Maggie Smith Tessellations April 18, 2018 9 / 39 Regular Polygon Tessellations
Squares
Jennifer Li and Maggie Smith Tessellations April 18, 2018 10 / 39 Activity: Regular Polygon Tessellations
Activity Sheet: Tessellate the plane using the regular hexagon.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 11 / 39 Regular hexagons
Vertex
A vertex is a point where the corners of all polygons in a tessellation meet.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 12 / 39 Vertex
A vertex is a point where the corners of all polygons in a tessellation meet.
Regular hexagons
Jennifer Li and Maggie Smith Tessellations April 18, 2018 12 / 39 Examples. n = 3
2 180 180 1 − = = 60 3 3 Each angle in an equilateral triangle is 60 degrees.
Regular Polygons
Each angle of an n-sided polygon equals 2 180 1 − n
Jennifer Li and Maggie Smith Tessellations April 18, 2018 13 / 39 n = 3
2 180 180 1 − = = 60 3 3 Each angle in an equilateral triangle is 60 degrees.
Regular Polygons
Each angle of an n-sided polygon equals 2 180 1 − n Examples.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 13 / 39 Regular Polygons
Each angle of an n-sided polygon equals 2 180 1 − n Examples. n = 3
2 180 180 1 − = = 60 3 3 Each angle in an equilateral triangle is 60 degrees.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 13 / 39 n = 4
2 180 180 1 − = = 90 4 2 Each angle in a square is 90 degrees.
Regular Polygons
Each angle of an n-sided polygon equals 2 180 1 − n Examples.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 14 / 39 Regular Polygons
Each angle of an n-sided polygon equals 2 180 1 − n Examples. n = 4
2 180 180 1 − = = 90 4 2 Each angle in a square is 90 degrees.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 14 / 39 n = 9
2 180 180 1 − = 7 × = 140 9 9 Each angle in a nonagon is 140 degrees.
Regular Polygons
Each angle of an n-sided polygon equals 2 180 1 − n Examples.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 15 / 39 Regular Polygons
Each angle of an n-sided polygon equals 2 180 1 − n Examples. n = 9
2 180 180 1 − = 7 × = 140 9 9 Each angle in a nonagon is 140 degrees.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 15 / 39 Can they fit without gaps and without overlapping?
Which regular polygons tessellate the plane?
How can we tessellate the plane with a regular n-sided polygon?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 16 / 39 Can they fit without gaps and without overlapping?
Which regular polygons tessellate the plane?
How can we tessellate the plane with a regular n-sided polygon?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 16 / 39 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 Each polygon is n-sided:
Which regular polygons tessellate the plane?
At a vertex, there will be q regular polygons that meet:
Jennifer Li and Maggie Smith Tessellations April 18, 2018 17 / 39 Each polygon is n-sided:
Which regular polygons tessellate the plane?
At a vertex, there will be q regular polygons that meet:
Jennifer Li and Maggie Smith Tessellations April 18, 2018 17 / 39 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 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 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 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 2 q × 180 × 1 − = 360 n 2 q × 180 × 1 − n 360 = q × 180 q × 180
¨ 2 q¡ × ¨180 × 1 − n 2 ¨ = q¡ × ¨180 q 2 2 1 − = n q 2 2 1 = + q n 1 1 1 + = q n 2
Which regular polygons tessellate the plane? 2 A total of q angles, each of 180 × 1 − degrees, sum to 360 degrees: n
Jennifer Li and Maggie Smith Tessellations April 18, 2018 19 / 39 Which regular polygons tessellate the plane? 2 A total of q angles, each of 180 × 1 − degrees, sum to 360 degrees: n 2 q × 180 × 1 − = 360 n 2 q × 180 × 1 − n 360 = q × 180 q × 180
¨ 2 q¡ × ¨180 × 1 − n 2 ¨ = q¡ × ¨180 q 2 2 1 − = n q 2 2 1 = + q n 1 1 1 + = q n 2
Jennifer Li and Maggie Smith Tessellations April 18, 2018 19 / 39 then a regular polygon tessellation is possible!
Which regular polygons tessellate the plane?
1 1 1 If + = q n 2
Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39 then a regular polygon tessellation is possible!
Which regular polygons tessellate the plane?
1 1 1 If + = q n 2
Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39 Which regular polygons tessellate the plane?
1 1 1 If + = q n 2
then a regular polygon tessellation is possible!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39 An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Is this correct?
Yes!
Which regular polygons tessellate the plane?
1 1 1 + = q n 2
Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Is this correct?
Yes!
Which regular polygons tessellate the plane?
1 1 1 + = q n 2
Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 Is this correct?
Yes!
Which regular polygons tessellate the plane?
1 1 1 + = q n 2
Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 Yes!
Which regular polygons tessellate the plane?
1 1 1 + = q n 2
Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Is this correct?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 Yes!
Which regular polygons tessellate the plane?
1 1 1 + = q n 2
Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Is this correct?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 Which regular polygons tessellate the plane?
1 1 1 + = q n 2
Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Is this correct?
Yes!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 Activity: Which regular polygons tessellate the plane?
Use the equation for the activities below. 1 1 1 + = q n 2 Activity Sheet: In square tessellation of the plane, how many squares must meet at a vertex?
Activity Sheet: In a regular hexagon tessellation of the plane, how many hexagons must meet at a vertex?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 22 / 39 It’s impossible to tessellate the plane with regular pentagons!
Activity: Which regular polygons tessellate the plane?
Activity Sheet: In regular pentagon tessellation of the plane, how many pentagons must meet at a vertex?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 23 / 39 Activity: Which regular polygons tessellate the plane?
Activity Sheet: In regular pentagon tessellation of the plane, how many pentagons 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 2 10 Then q = 2 = pentagons meet at a vertex... 1 − 5 3 But q should be whole number!
We cannot tessellate the plane with a regular pentagon!
Activity: Which regular polygons tessellate the plane?
A regular pentagon has five sides, so n = 5.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 24 / 39 But q should be whole number!
We cannot tessellate the plane with a regular pentagon!
Activity: Which regular polygons tessellate the plane?
A regular pentagon has five sides, so n = 5. 2 10 Then q = 2 = pentagons meet at a vertex... 1 − 5 3
Jennifer Li and Maggie Smith Tessellations April 18, 2018 24 / 39 We cannot tessellate the plane with a regular pentagon!
Activity: Which regular polygons tessellate the plane?
A regular pentagon has five sides, so n = 5. 2 10 Then q = 2 = pentagons meet at a vertex... 1 − 5 3 But q should be whole number!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 24 / 39 Activity: Which regular polygons tessellate the plane?
A regular pentagon has five sides, so n = 5. 2 10 Then q = 2 = pentagons meet at a vertex... 1 − 5 3 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 There are only three regular polygons that tessellate the plane: the equilateral triangle, the square, and the hexagon!
Activity: Which regular polygons tessellate the plane?
Fun Fact!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 25 / 39 Activity: Which regular polygons tessellate the plane?
Fun Fact! There are only three regular polygons that tessellate the plane: the equilateral triangle, the square, and the hexagon!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 25 / 39 Activity: Which regular polygons tessellate the plane?
Fun Fact! There are only three regular polygons that tessellate the plane: the equilateral triangle, the square, and the hexagon!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 25 / 39 This type of tessellation is called an Archimedean tessellation.
More Types of Tessellations
We can make more tessellations by using more than one regular polygon.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 26 / 39 More Types of Tessellations
We can make more tessellations by using more than one regular polygon. This type of tessellation is called an Archimedean tessellation.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 26 / 39 More Types of Tessellations
We can make more tessellations by using more than one regular polygon. This type of tessellation is called an Archimedean tessellation.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 26 / 39 More Types of Tessellations
Jennifer Li and Maggie Smith Tessellations April 18, 2018 27 / 39 More Types of Tessellations
Jennifer Li and Maggie Smith Tessellations April 18, 2018 28 / 39 Activity: Labelling a Tessellation
Activity Sheet: a) Describe the polygons that surround the red vertex in each tessellation shown below. b) What do you think the labels under each tessellation mean?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 29 / 39 Example. Find the dual of the tessellation below.
Dual Tessellations
The dual of a tessellation is formed by drawing a vertex in the center of each tile, and joining all vertices of tiles that touch.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 30 / 39 Dual Tessellations
The dual of a tessellation is formed by drawing a vertex in the center of each tile, and joining all vertices of tiles that touch.
Example. Find the dual of the tessellation below.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 30 / 39 Dual Tessellations
Example. The dual is drawn in pink:
Jennifer Li and Maggie Smith Tessellations April 18, 2018 31 / 39 Dual Tessellations
Example. The dual is drawn in pink:
Jennifer Li and Maggie Smith Tessellations April 18, 2018 31 / 39 Activity: Dual Tessellations
Activity Sheet: Find the dual tessellations. What do you notice about these duals?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 32 / 39 The tiles don’t have to be polygons!
Types of Tessellations
A tessellation is monohedral if all tiles are congruent (they have the same size and shape).
Jennifer Li and Maggie Smith Tessellations April 18, 2018 33 / 39 The tiles don’t have to be polygons!
Types of Tessellations
A tessellation is monohedral if all tiles are congruent (they have the same size and shape).
Jennifer Li and Maggie Smith Tessellations April 18, 2018 33 / 39 Types of Tessellations
A tessellation is monohedral if all tiles are congruent (they have the same size and shape).
The tiles don’t have to be polygons!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 33 / 39 Activity: Tessellation of the plane
Activity Sheet: Draw some monohedral tessellations of the plane with the given tiles.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 34 / 39 Given a collection of tiles, can we create a monohedral tessellation of the plane?
This can be a hard problem...
There is no general method known!
Types of Tessellations
Question.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 35 / 39 This can be a hard problem...
There is no general method known!
Types of Tessellations
Question. Given a collection of tiles, can we create a monohedral tessellation of the plane?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 35 / 39 There is no general method known!
Types of Tessellations
Question. Given a collection of tiles, can we create a monohedral tessellation of the plane?
This can be a hard problem...
Jennifer Li and Maggie Smith Tessellations April 18, 2018 35 / 39 Types of Tessellations
Question. Given a collection of tiles, can we create a monohedral tessellation of the plane?
This can be a hard problem...
There is no general method known!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 35 / 39 Activity: Which one does not belong?
Activity Sheet: A heptiamond is a shape that is created from seven equilateral triangles glued together. There are a total of twenty-four heptiamonds:
Only one does not give a monohedral tiling of the plane. Can you figure out which one it is?
Jennifer Li and Maggie Smith Tessellations April 18, 2018 36 / 39 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.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 37 / 39 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!
Jennifer Li and Maggie Smith Tessellations April 18, 2018 37 / 39 Pentagonal tiling in math research
There are 15 convex pentagons that tessellate the plane monohedrally.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 38 / 39 Pentagonal tiling in math research
There are 15 convex pentagons that tessellate the plane monohedrally.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 38 / 39 In 2017, it was proven that there are only 15 tilings of the plane using convex pentagons.
Pentagonal tiling in math research
The most recent pentagonal tiling was discovered in 2015:
Jennifer Li and Maggie Smith Tessellations April 18, 2018 39 / 39 In 2017, it was proven that there are only 15 tilings of the plane using convex pentagons.
Pentagonal tiling in math research
The most recent pentagonal tiling was discovered in 2015:
Jennifer Li and Maggie Smith Tessellations April 18, 2018 39 / 39 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 using convex pentagons.
Jennifer Li and Maggie Smith Tessellations April 18, 2018 39 / 39