Geometry Session 7: Solutions

A B C
Homework

Solutions for Session 7, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6

Problem C1

Step 1:

Step 2:

Step 3:

Step 4:

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Problem C2

Using a new design element, the seven patterns are as follows:

1. Translation symmetry only:

2. Glide reflection plus translation symmetry:

3. Reflection over a horizontal line plus translation:

4. Reflection over a vertical line plus translation:

5. Rotation (a half-turn about a point on the midline of the strip) plus translation:

6. Reflection over a vertical line plus reflection over a horizontal line plus translation:

7. Reflection over a vertical line plus glide reflection plus translation:

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Problem C3

Answers will vary depending on the choice of the basic element as well as of the pattern.

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Problem C4

Following the order in which the patterns are originally described, they can be classified as 11, 1g, 1m, m1, 12, mm, mg.

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Problem C5

a.	It would be a combination of translation symmetry (which is present in all patterns), followed by a reflection about a vertical line (hence the m), followed by the 180° rotation about a point on the midline (hence the 2).
b.	See picture:
c.	The symmetry described is equivalent to pattern #7, so it can also be described as having translation symmetry, glide reflection, and reflection over a vertical line. So it could also be classified as mg.

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Problem C6

If you consider symmetry in the basic design element part of the symmetry of the frieze pattern, you may have slightly different answers than the ones here. In the order in which the designs are given, the classifications are as follows: 12, 11 (or possibly m1), m1, 11 (or possibly 1m), 11, 11 (or possibly m1), 11 (or possibly 1m), 11 (or possibly mm), 12, m1 (or possibly mm), m1 (or possibly mm), and 11 (or possibly 1m).

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Session 7 | Notes | Solutions | Video