• Home
  • Textbooks
  • Contemporary Abstract Algebra
  • Frieze Groups and Crystallographic Groups

Contemporary Abstract Algebra

Joseph Gallian

Chapter 28

Frieze Groups and Crystallographic Groups - all with Video Answers

Educators


Chapter Questions

01:46

Problem 1

Show that the frieze group $F_{6}$ is isomorphic to $Z \oplus Z_{2}$.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
05:33

Problem 2

How many nonisomorphic frieze groups are there?

Chris Trentman
Chris Trentman
Numerade Educator
03:49

Problem 3

In the frieze group $F_{7}$, write $x^{2} y z x z$ in the form $x^{n} y^{m} z^{k}$.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
03:49

Problem 4

In the frieze group $F_{7}$, write $x^{-3} z x y z$ in the form $x^{n} y^{m} z^{k}$.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
02:07

Problem 5

In the frieze group $F_{7}$, show that $y z=z y$ and $x y=y x$.

John Nicolle
John Nicolle
Numerade Educator
06:11

Problem 6

In the frieze group $F_{7}$, show that $z x z=x^{-1}$.

Stephanie Carter
Stephanie Carter
Numerade Educator
01:58

Problem 7

Use the results of Exercises 5 and 6 to do Exercises 3 and 4 through symbol manipulation only (that is, without referring to the pattern). (This exercise is referred to in this chapter.)

Sydney Trombly
Sydney Trombly
Numerade Educator
01:49

Problem 8

Prove that in $F_{7}$ the cyclic subgroup generated by $x$ is a normal subgroup.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:49

Problem 9

Quote a previous result that tells why the subgroups $\langle x, y\rangle$ and $\langle x, z\rangle$ must be normal in $F_{7}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:18

Problem 10

Look up the word frieze in an ordinary dictionary. Explain why the frieze groups are appropriately named.

Anupa Sharad Medhekar
Anupa Sharad Medhekar
Numerade Educator
01:12

Problem 11

Determine which of the seven frieze groups is the symmetry group of each of the following patterns.

David Collins
David Collins
Numerade Educator
04:36

Problem 12

Determine the frieze group corresponding to each of the following patterns.
a. $y=\sin x$
b. $y=|\sin x|$
c. $|y|=\sin x$
d. $y=\tan x$
e. $y=\csc x$

Gregory Higby
Gregory Higby
Numerade Educator
03:10

Problem 13

Determine the symmetry group of the tessellation of the plane exemplified by the brickwork shown.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:12

Problem 14

Determine the plane symmetry group for each of the patterns in Figure $28.17$.

David Collins
David Collins
Numerade Educator
01:01

Problem 15

Determine which of the 17 crystallographic groups is the symmetry group of each of the following patterns.

Narayan Hari
Narayan Hari
Numerade Educator
03:55

Problem 16

In the following figure, there is a point labeled $1 .$ Let $\alpha$ be the translation of the plane that carries the point labeled 1 to the point labeled $\alpha$, and let $\beta$ be the translation of the plane that carries the point labeled 1 to the point labeled $\beta$. The image of 1 under the composition of $\alpha$ and $\beta$ is labeled $\alpha \beta$. In the corresponding fashion, label the remaining points in the figure in the form $\alpha^{i} \beta^{j}$.

Ankit Singh
Ankit Singh
Numerade Educator
01:40

Problem 17

The patterns made by automobile tire treads in the snow are frieze patterns. An extensive study of automobile tires revealed that only five of the seven frieze patterns occur. Speculate on which two patterns do not occur and give a possible reason why they do not.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
00:52

Problem 18

Locate a nontrivial glide-reflection axis of symmetry in the $\mathrm{cm}$ pattern in Figure $28.16$.

Holly Miner
Holly Miner
Numerade Educator
03:19

Problem 19

Determine which of the frieze groups is the symmetry group of each of the following patterns.
a. $\cdots$ D D D D $\cdots$
b. $\cdots \mathrm{V} \wedge \mathrm{V} \wedge \cdots$
c. $\cdots$ L L $L L \cdots$
d. ... V V V V $\cdots$
e. $\cdots$ N N N N \cdots.
f. $\cdots \mathrm{H} \mathrm{H} \mathrm{H} \mathrm{H} \cdot \cdots$
g. $\cdots$ L $T_{L} T \cdots$

Arjun Tibrewal
Arjun Tibrewal
Numerade Educator
00:52

Problem 20

Locate a nontrivial glide-reflection axis of symmetry in the pattern third from the left in the bottom row in Figure $28.17$.

Holly Miner
Holly Miner
Numerade Educator