Homework 1 - Symmetry

From EscherMath

Read: Pages 1-31 in Visions of Symmetry.

1. In Escher's Castrovalva there are no obvious mathematical symmetries, but Escher's eye for patterns is already evident. Find at least four areas of this print, which have repetition of pattern. Describe the area and the pattern that repeats.
2. For each of the following, copy the F and the line, then draw the reflected image of the F using the line as axis of reflection.

   a.	b.
   c.	d.

3. For each of the polygons below, draw all of the reflection lines. What will the reflection lines look like for a 13-sided polygon? A 20 sided polygon? What is the general pattern?




4. 

   

   Swans Reflecting Elephants. Salvador Dali, 1937

   Examine Salvdor Dali’s Swans Reflecting Elephants.

   1. Explain the title of the painting
   2. What would be different if Dali had decided to paint Swans Rotating Elephants?
   3. Give a (rough) sketch of what the painting paint Swans Rotating Elephants would have looked like. (Simple line drawing is sufficient).
5. For the twenty-six capital letters, determine all symmetries of each letter then group the letters into symmetry classes. All members of the same class should have the same symmetries. Use the following alphabet:
6. Determine the symmetry group for each of the pictures below:

   a.	b.	c.
   d.	e.	f.




7. Discuss the symmetries of the following Escher works. What is the symmetry group of each? Do colors affect the symmetries? Is there symmetry that is almost present?
   1. Alhambra wall mosaic (Visions of Symmetry pg. 9)
   2. Development I
   3. Circle Limit I
   4. Drawing Hands
   5. Möbius Strip I
   6. Whirlpools
   7. Day and Night




8. Choose one asymmetrical motif. Draw your motif, and then use it to create Rosettes with symmetry groups C2, C3, C4, C5, C6 and D2, D3, D4, D5, D6. (You can use Kali to help out, if you like).
9. On page 12 of Visions of Symmetry is a photo of the pulpit in the Ravello cathedral. There is a rectangular pattern in the box shown shaded below.
   1. Make an accurate copy of the pattern (focusing on the white lines). What symmetry group does it have?
   2. Squish the pattern into a square so that it has an order 4 rotational symmetry.
10. How did the politics of 20th Century Europe affect Escher? How did it affect his work?
11. How did Escher become interested in tessellations? What allowed him to master the art of tessellating the plane?