Deck 20: Tilings

Question 1

Create a tiling with a nonconvex quadrilateral. Is it regular?

Answer 1

No, it's not regular.

Question 2

Create a tiling of the plane using triangles and octagons, if possible.

Answer 2

Question 3

Tile the plane with a nonsquare rhombus.

Answer 3

Question 4

Create an Escher-like tiling that uses only translations and half-turns.

Question 5

Create a nonregular tiling with a right triangle tile.

Question 6

Create a tiling with a parallelogram. Is it regular?

Question 7

Why are regular pentagons not usable to tile the plane?

Question 8

Create a nonregular tiling with octagons and squares.

Question 9

Why are Penrose tilings of interest to chemists?

Question 10

Tile the plane with a (tileable) pentagon.

Question 11

Tile the plane with an equilateral triangle.

Question 12

Use the figure below to tile the plane, if possible.

Question 13

Use the figure below to tile the plane, if possible.

Question 14

Create an Escher-like tiling that uses only translations.

Question 15

Use the figure below to tile the plane, if possible.

Question 16

Choose one of Marjorie Rice's pentagon tiles, and use it to tile the plane.

Question 17

Tile the plane with a (tileable) hexagon.

Question 18

Create an Escher-like tiling that uses only translations and reflections.

Question 19

Why are regular octagons not useable to tile the plane?

Question 20

Create a tiling of the plane using squares and hexagons, if possible.

Question 21

At the vertex of a tiling of the plane there is a regular octagon, a square, and one other regular polygon. Which regular polygon is it?

Question 22

Explain why the tiling below does not form a semiregular tiling of the plane.

Question 23

List all regular tilings.

Question 24

What is a nonperiodic tiling?

Question 25

What are the possible rotational symmetries of a crystal?

Question 26

Which of the following polygons can tile the plane? I: regular pentagon
II: regular hexagon

A) I only
B) II only
C) both I and II
D) neither

Question 27

Which of the following polygons can tile the plane? I: regular octagon
II: nonrectangle parallelogram

A) I only
B) II only
C) both I and II
D) neither

Question 28

Give a numerical explanation of why a regular decagon (10 sides) cannot tile the plane by itself.

Question 29

How many edge-to-edge tilings with regular polygons are possible when we allow different vertex types?

Question 30

If a tiling has only translations, how could it be classified in wallpaper pattern notation?

Question 31

Explain why the tiling below does not form a semiregular tiling of the plane.

Question 32

Can you "tile" space with any regular polyhedron other than a cube?

Question 33

In a semiregular tiling, what is the maximum possible number of sides of a regular polygon?

Question 34

Write the vertex type of this semiregular tiling:

Question 35

What is a regular tiling?

Question 36

Which of the following polygons can tile the plane? I: nonsquare rhombus
II: nonequilateral isosceles triangle

A) I only
B) II only
C) both I and II
D) neither

Question 37

Illustrate a tiling with the L-tetromino that uses both reflection and rotation.

Question 38

Write the vertex type of this semiregular tiling:

Question 39

What is the vertex type?

Question 40

What is a semiregular tiling?

Question 41

The exterior angle of a regular octagon (8-gon) has a measure of:

A) 18°.
B) 30°.
C) 45°.
D) 60°.

Question 42

Regular octagons and squares can form a tiling of the plane that has at every vertex:

A) two octagons and two squares.
B) two octagons and one square.
C) one octagon and two squares.
D) one octagon and three squares.

Question 43

_____ pentagons can form a tiling of the plane.

A) All
B) Some
C) No

Question 44

Regular hexagons and equilateral triangles can form a tiling of the plane that has at every vertex:

A) three hexagons and one triangle.
B) two hexagons and two triangles.
C) two hexagons and three triangles.
D) one hexagon and five triangles.

Question 45

A semiregular tiling has one square, one regular hexagon, and one regular p-gon at each vertex. What is p?

A) 8
B) 9
C) 10
D) 12

Question 46

Semiregular tilings can tile with: I: three different regular polygons.
II: three different vertex combinations.

A) I only
B) II only
C) both I and II
D) neither

Question 47

_____ hexagons can form a tiling of the plane.

A) All
B) Some
C) No

Question 48

Regular dodecagons (12-gons) and equilateral triangles can form a tiling of the plane that has at every vertex:

A) two dodecagons and two triangles.
B) two dodecagons and one triangle.
C) one dodecagon and two triangles.
D) one dodecagon and three triangles.

Question 49

A semiregular tiling has four equilateral triangles and one p-gon at each vertex. What is p?

A) 3
B) 4
C) 5
D) 6

Question 50

A semiregular tiling has one equilateral triangle and two regular p-gons at each vertex. What is p?

A) 8
B) 9
C) 10
D) 12

Question 51

A semiregular tiling has two regular octagons and one regular p-gon at each vertex. What is p?

A) 3
B) 4
C) 5
D) 6

Question 52

A scalene triangle ABC tiles the plane. What is a possible configuration of the angles at the vertices? I: three types of vertices: 6 As, or 6 Bs, or 6 Cs
II: one type of vertex: 2 As, 2 Bs, and 2 Cs

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 53

Squares and equilateral triangles can form a tiling that has at every vertex:

A) three squares and two triangles.
B) two squares and three triangles.
C) one square and four triangles.
D) one square and five triangles.

Question 54

The exterior angle of a regular pentagon has a measure of:

A) 18°.
B) 30°.
C) 60°.
D) 72°.

Question 55

A semiregular tiling has one square and two regular p-gons at each vertex. What is p?

A) 5
B) 6
C) 8
D) 10

Question 56

_____ nonconvex quadrilaterals can form a tiling of the plane.

A) All
B) Some
C) No

Question 57

Which of the following polygons can tile the plane? I: scalene triangle
II: equilateral triangle

A) I only
B) II only
C) both I and II
D) neither

Question 58

The exterior angle of a regular hexagon has a measure of:

A) 18°.
B) 30°.
C) 45°.
D) 60°.

Question 59

_____ convex quadrilaterals can form a tiling of the plane.

A) All
B) Some
C) No

Question 60

The exterior angle of a regular decagon (10-gon) has a measure of:

A) 18°.
B) 36°.
C) 45°.
D) 54°.

Question 61

A non-convex quadrilateral ABCD tiles the plane. What is a possible configuration of the angles at the vertices? I: two types of vertices: 2 As and 2 Cs; 2 Bs and 2 Ds
II: one type of vertex: A, B, C, D

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 62

What is a possible way to tile the plane with a general isosceles triangle? I: six at a vertex
II: four or eight at a vertex

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 63

Which of the following form the faces of a regular polyhedron? I: six equilateral triangles
II: eight equilateral triangles

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 64

A regular polyhedron with six faces has what regular polygon as faces?

A) triangle
B) square
C) pentagon
D) hexagon

Question 65

Can the tile below be used to tile the plane?

A) no
B) yes, with translations only
C) yes, with translations and half-turns only
D) yes, but reflections must be included

Question 66

Is 3.10.16 a possible vertex type for a vertex surrounded by regular polygons?

A) Yes
B) No

Question 67

If 4.8.n is a possible vertex type for a vertex surrounded by regular polygons, what is n?

A) 6
B) 8
C) 11
D) 12

Question 68

If 4.5.n is a possible vertex type for a vertex surrounded by regular polygons, what is n?

A) 10
B) 18
C) 20
D) 22

Question 69

Can the tile below be used to tile the plane?

A) no
B) yes, with translations only
C) yes, with translations and half-turns only
D) yes, but reflections must be included

Question 70

Can the tile below be used to tile the plane?

A) no
B) yes, with translations only
C) yes, with translations and half-turns only
D) yes, but reflections must be included

Question 71

What is a possible way to tile the plane with a pentagon with three 90° angles and two equal obtuse angles? I: three at a vertex
II: three or four at a vertex

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 72

Can the tile below be used to tile the plane?

A) no
B) yes, with translations only
C) yes, with translations and half-turns only
D) yes, but reflections must be included

Question 73

A regular polyhedron with 12 faces has what regular polygon as faces?

A) triangle
B) square
C) pentagon
D) hexagon

Question 74

Is 3.3.4.12 a possible vertex type for a vertex surrounded by regular polygons?

A) Yes
B) No

Question 75

What is a possible way to tile the plane with a parallelogram with 60° and 120° angles? I: four at a vertex
II: three or six at a vertex

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 76

Which of the following form the faces of a regular polyhedron? I: eight squares
II: four equilateral triangles

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 77

What is a possible way to tile the plane with a pentagon with two 90° angles and three equal obtuse angles? I: three at a vertex
II: three or four at a vertex

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 78

A regular polyhedron with 20 faces has what regular polygon as faces?

A) triangle
B) square
C) pentagon
D) hexagon

Question 79

Which of the following form the faces of a regular polyhedron? I: six squares
II: 10 pentagons

A) I only
B) II only
C) both I and II
D) neither I nor II

Question 80

What is a possible way to tile the plane with a right triangle? I: six at every vertex
II: four or eight at every vertex

A) I only
B) II only
C) both I and II
D) neither I nor II