- 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.
- One of Escher's earliest works to feature symmetry was Paradise. Carefully examine and analyze the details of this print. Discuss the mathematical symmetry of his composition. If falling short of being mathematically symmetric, are there parts of the design that are at least balanced? (Explain/describe) Are there areas that are not balanced?(Explain/describe) Are there hints of mathematical objects or is the print completely organic in nature? (Explain/describe).
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. Your resulting drawing should have two F's (possibly overlapping) and one line, and the whole picture should have reflection symmetry.
a. b. c. d.
For each of the following, copy the F and the dot, then draw the 180° rotated image of the F using the dot as rotation center. Your resulting drawing should have two F's (possibly overlapping) and one dot, and the whole picture should have rotation symmetry.
a. b. c. d.
- 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?
Path of Life I and Path of Life II are prints made by Escher in 1958.
- Do they have rotational symmetry?
- Do they have reflectional symmetry?
Examine Salvdor Dali’s Swans Reflecting Elephants.
- Explain the title of the painting
- What would be different if Dali had decided to paint Swans Rotating Elephants?
- Give a (rough) sketch of what the painting paint Swans Rotating Elephants would have looked like. (Simple line drawing is sufficient).
- Determine the symmetry group for each of the pictures below:
a. b. c. d. e. f.
- For each of these Escher works, answer the following questions:
- Does the figure have reflectional symmetry?
- Is there rotational symmetry? If so, what is the rotation order?
- Based on your previous answers, what is the symmetry group of the figure?
- 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:
Place the letters in the appropriate column C1 D1 C2 D2 C4 D4 Other
- Escher's The Scapegoat has no symmetry in the mathematical sense. However, some symmetry is almost present. Discuss the symmetry that is almost present. How does that near symmetry affect your interpretation of the print? (You might find wikipedia:Scapegoat helpful background reading).
- 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?
- What is the symmetry group for Escher's Snakes? Snakes required three woodblocks, one each for red, green, and black. Since Escher took advantage of symmetry to avoid extra carving, how many total impressions did he need for each print of Snakes? This video of Escher printing Snakes may help, as well as Escher's Snakes related work on pages 188-192 of Magic of M.C. Escher.
- Choose one asymmetrical motif (keep it simple, though!). 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 software such as Kali or iOrnament to help, if you want).
This problem is to show that a finite picture cannot have two 90° rotation symmetries with different centers. Take a full sheet of graph paper, and draw two 90° rotation centers near the middle of the sheet. Now outline one of the squares of the graph paper, near the two centers. This is your first square, so put a 1 in it (see picture).
Now rotate the square around one of the centers. Outline the new square, label it 2. Rotate again, around the other center, and label with a 3. Repeat until the square leaves the page.
Find a symmetry of this poem by Lewis Carroll:
- I often wondered when I cursed,
- often feared where I would be -
- wondered where she'd yield her love,
- when I yield, so will she.
- I would her will be pitied!
- Cursed be love! She pitied me ...
- On page 20 of Visions of Symmetry, there is a story of a sign at a swimming pool: NOW NO SWIMS ON MON. This sign has C2 symmetry. Find other phrases with C2 symmetry. Can you find a phrase with D1 symmetry? How about D2 symmetry? You might look at some of Scott Kim's Inversions.
- Ernst Haeckel's 1904 book Kunstformen der Natur is a beautiful collection of images of life. There are 22 numbered diatoms in the fourth plate, Diatomea. Identify the symmetry group of each one.
Sol LeWitt's 1983 Wall Drawing #398 is a monumental image of seven "stars".
- What are the symmetry groups of the stars?
- Mathematically, what is missing from the picture?
- Draw another star that would fit naturally into the picture.
- Donald Lipski's installation Got Any Jacks? is made of nearly 100 fish replicas, and hangs at Miami International Airport. In this image, which Rosette symmetry groups can you find?