11.1

. In each diagram in Exercises 7–10, identify the motion or combination

of motions that would produce the image. Justify your choices.

8).

Rotation, around an imaginary point below the images.

10). the top of the image connects

Translation, since the image stays in the same orientation.

24.Describe the rotational symmetry, if any, of each of the

figures in (a) through (f ). Use the “trace and turn” test

if helpful, the instructor said we have to give the degree if their rotational symmetry

Each triangle creates a rotational symmetry.

There are 8 triangles.

360 degrees / 8 = 45 degree

There are four rotations here:

360/4 = 90 degree symmetry

Each point creates a symmetry. Count them: 24 triangles:

360 deg / 24 = 15 degrees

This doesn’t have rotational symmetry.

If you consider colors, this doesn’t have rotational symmetry.

There are 5 repeating points in this pattern.

360 deg / 5 = 72 degrees

Figures that correspond under a translation, rotation, or

reflection are said to be translation congruent, rotation

congruent, or reflection congruent. The instructor said we can use the letters A to Z to give me examples on each question. Take question a for example, O is the letter that has all three congruence. You can translate, rotate and reflect the letter O. It remains the same..

a. All three congruences

HIOXY

b. Translation and rotation congruences only

NSZ

c. Translation and reflection congruences only

ABCDEKMTUVW

d. Rotation and reflection congruences only

none

e. Translation congruence only

FGJLPQR

f. Rotation congruence only

none

g. Reflection congruence only

none

h. None of the three congruences

none

11.2

4). How many rectangles will fit about a point when tessellating

the plane? How do you know?

There will be 4 rectangles at each point. The angle in a corner of a rectangle is 90 degrees, and a circle around a point is 360, allowing for 4 rectangles.

10). Why does a regular pentagon not tessellate the plane?

The sum of the interior angles is 540 degrees, and there is no way to fit that into 360 degrees, which is required for tessellation.

14). Why is it impossible for a regular polygon with more

than six sides to tessellate the plane?

The angles of the polygons that meet at a point must be 360 degrees in total, and once you are past a six sided polygon, this cannot happen. For more than 6 sides, the interior angles are less than 60 degrees each.

22). Why is the tessellation shown not a regular tessellation?

The squares do not all meet at the vertices. They are offset.

11.3

4). What are the defining characteristics of a golden triangle

The triangle has two equal legs and a shorter leg. The equal legs are the golden ratio multiplied by the length of the short leg.

12). For Exercises 12–16, tell if the figure is a star polygon, a polygon,

or other, and how many sides it has.

{9/3}

No figure

11.4

8). Show that Euler’s formula holds for the polyhedra described in

Exercises 8–11.

A right pentagonal prism

V-E+F = 2

10-15+7 = 2

14). Describe the different axes of rotational symmetry for

each of the figures shown in Exercise 7. Assume that the

triangular faces of figure (a) are equilateral triangles and

the nontriangular face of figure (c) is a square

A: it can be rotated around a center axis. There are three points of rotation: 360/3 = 120 degree
B: can be rotated in five ways: 360/5 = 72 degrees
C: can be rotated in four ways: 360/4 = 90 degrees

16). a. Describe the different types of planes of symmetry

and axes of symmetry of a regular octahedron.

They go through the faces, edges, and vertices of the octahedron.

b. How many planes and axes of symmetry does a regular

octahedron have?

There are 13:

3 go through opposite faces

4 go through opposite vertices

6 go through the midpoints of opposite edges

c. Make a sketch and compare the symmetry properties

of a regular octahedron and a cube.

The symmetry properties of cubes are the same as octahedrons.

40). Answer the following questions to determine why Euler’s

formula continues to hold for the polyhedron formed by

cutting corners off of an octahedron, as in Figure 11.35(b):

a. For an octahedron,V=6, F= 8, and E= 12.

b. When you slice off one corner of the octahedron,

you (gain) 3 vertices, (gain) 1

faces, and (gain) 4 edges.

c. Therefore, the total change in V is 3 the total

change in F is 1 and the total change in V+F is

4

d. The total change in E is 4

e. What does the comparison of the total change in V+F

to the total change in E tell you?

The change in V+F must be the same as the change in E.

Review.

10). Describe the symmetry properties of the following figures

A: reflection about the middle

B: rotational

12). Name three regular polygons that will tessellate the plane.

Square, triangle, hexagon

18). How do the axes of rotational symmetry of an octahedron

compare to the axes of rotational symmetry of a

cube?

Cubes and octahedrons have the same symmetries! See #16 above.