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
a.
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.