Skills Check

image Skills Check

Question 20.1

1. In a tiling of the plane, the tiles

  1. must all be the same size.
  2. must all be the same shape.
  3. may be of different shapes and sizes.

1.

c

Question 20.2

2. The measure of an exterior angle of a regular octagon is _____________.

2.

45°

Question 20.3

3. In a tiling of the plane by polygons, the tiles

  1. must all have the same number of sides.
  2. need not be regular polygons.
  3. can have different numbers of sides.

3.

b or c

Question 20.4

4. A regular tiling can be constructed using polygons with __________, __________, or __________ sides.

4.

3; 4; 6

Question 20.5

5. Regular octagons and squares can form a semiregular tiling of the plane with

  1. two octagons and one square at each vertex.
  2. two octagons and two squares at each vertex.
  3. varying configurations at the vertices.

5.

a

Question 20.6

6. A semiregular tiling has a square, a regular dodecagon (12-gon), and another regular polygon at each vertex. This other polygon has _____________ sides.

6.

6

Question 20.7

7. A tessellation

  1. allows overlapping pieces.
  2. is not the same as a tiling.
  3. covers the entire infinite plane.

7.

c

Question 20.8

8. There are _____________ regular polyhedra.

8.

5

Question 20.9

9. How many semiregular tilings are there?

  1. 5
  2. 8
  3. Infinitely many

9.

b

Question 20.10

10. The smallest number of sides that a polygon can have and not be able to tile the plane is __________.

10.

5

Question 20.11

11. A tiling of the plane can be formed using as a tile

  1. any convex quadrilateral but no nonconvex quadrilateral.
  2. any nonconvex quadrilateral but no convex quadrilateral.
  3. any quadrilateral.

11.

c

Question 20.12

12. How many regular tilings are there?

  1. None
  2. 3
  3. Infinitely many

12.

b

Question 20.13

13. A tiling of the plane can be formed using which of the following as a tile?

  1. Some but not all pentagons
  2. Any pentagon with at least two right angles
  3. Any pentagon with at least three right angles

13.

a

Question 20.14

14. Any quadrilateral can tile the plane using which operations?

  1. Only translations
  2. Translations plus half-turns
  3. Only half-turns

14.

b

Question 20.15

15. Regular pentagons

  1. can’t tile the plane.
  2. can tile the plane, but only if you are very careful.
  3. don’t occur in any tilings.

15.

a

Question 20.16

16. An artist famous for works based on tilings is ________________.

16.

M. C. Escher

Question 20.17

17. A convex irregular polygon

  1. can never tile the plane.
  2. can always tile the plane.
  3. cannot tile the plane if it has more than six sides.

17.

c

Question 20.18

18. The tile below can be used to tile the plane using which operations?

  1. Only translations
  2. Translations plus half-turns
  3. Only half-turns
image

858

18.

b

Question 20.19

19. Which of the following statements is true?

  1. If a polygon fulfills the Conway Criterion, it can tile the plane by translations.
  2. If a polygon fulfills the Conway Criterion, it can tile the plane by translations and half-turns.
  3. If a polygon doesn’t fulfill the Conway Criterion, it can’t tile the plane at all.

19.

b

Question 20.20

20. The ___________ Criterion says that the tile below can be used to create a tiling of the plane using ___________ and ___________.

image

20.

Conway; translations; half-turns

Question 20.21

21. In a nonperiodic tiling of the plane,

  1. the pattern never repeats.
  2. the pattern is not repeated by any translation.
  3. there must be at least three kinds of tiles.

21.

b

Question 20.22

22. Penrose tilings are __________ -periodic.

22.

quasi

Question 20.23

23. A Penrose dart has the property that

  1. opposite angles are congruent.
  2. it is nonconvex.
  3. the edges are all of different lengths.

23.

b

Question 20.24

24. A rhombus always has the property that __________.

24.

opposite sides are equal in length (or parallel, or opposite angles are congruent)

Question 20.25

25. Ammann bars are

  1. an Arab delicacy that comes from Jordan.
  2. jazz venues where musical sequences are played.
  3. sets of parallel bars in a Penrose pattern.

25.

c

Question 20.26

26. Barlow’s law prohibits the existence of crystals with ___________ symmetry.

26.

fivefold

Question 20.27

27. Quasicrystals

  1. do not exist in nature.
  2. are not regular enough to be used in New Age ceremonies.
  3. are symmetric nonperiodic tilings.

27.

c

Question 20.28

28. In a Penrose tiling, the proportion of darts to kites is __________.

28.

the golden ratio

Question 20.29

29. A nonperiodic tiling

  1. is an impossibility.
  2. requires at least two kinds of tiles.
  3. does not have translation symmetry.

29.

c

Question 20.30

30. The process that takes a Penrose pattern into a different Penrose pattern with larger darts and kites is called _________.

30.

inflation