Seeking Symmetry

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In 1922, and again in 1936, M. C. Escher visited the Alhambra, in Granada, Spain, a fourteenth century Moorish palace renowned for its decorative artwork. He (with his wife) made sketches of numerous wall tilings, sketches that he later used as sources for his own mathematical art.

Objective: Imitate Escher with a pilgrimage to a site that can provide some of the inspiration that he found at the Alhambra. Groups will take photographs of symmetric patterns, classify their symmetry groups, and complete an assignment.

Site

In St. Louis, we are lucky to have some of the finest mosaic artwork in North America, at the Cathedral Basilica of St. Louis. The cathedral is lavishly decorated with symmetric patterns, including mosaics, floor patterns, furniture, ironwork, tapestry, and other adornments both inside and outside the building.

St. Louis is also privileged to be the home to the truly unique City Museum. While the City Museum is well known as a vast playground, it also houses a vast array of architectural relics. Symmetric patterns can often be found in the detailing of urban buildings and the City Museum has a vast array of examples to be explored.

Every city has it's symmetries. For example, here's a Virtual Field Trip of some U.S. Locations created during a math workshop.

Instructions

Find examples of rosette, frieze, and wallpaper patterns at the site. For each interesting pattern, take a photograph or make a rubbing and note the location and symmetry type on the checklist. There may not be time on the trip to identify the symmetry group of every pattern, so shoot first and ask questions later. Take a photograph of your group at the site.

After the trip, place photos on a photo sharing website. Carefully identify the symmetry group of each pattern shown in the photos and fill out an entry on the checklist.

Possible Assignments

Poster

Create a poster exhibiting the best of the patterns in the site.

Choose three patterns to use on the poster - one rosette, one frieze, and one wallpaper. Photographs distort shapes and angles, and usually contain more than simply the symmeric pattern of interest. To emphasize the symmetry, make an accurate sketch of the idealized pattern from each photograph. Each sketch should be at least 5x8 inches, on graph paper or white copier paper (not lined binder paper!), and done in pen or dark pencil. On each sketch, mark all symmetries of the pattern.

The poster should feature:

  • A title.
  • Names of group members, and a group photograph taken at the site.
  • Three photographs. (At least one rubbing if going to the City Museum.)
  • Three sketches.
  • Descriptions of the symmetries in each sketch, including the name of the symmetry group.
  • Descriptions of where to find each pattern at the site.

All of this should be stuck to a good sized piece of poster board.

As these posters are intended for display, put in the effort to make something you're proud of.

Scavenger Hunt

Score points for finding interesting types of symmetry.

Basic scoring:

  • Rosette Patterns
    • Dihedral groups: D1, D2, D4, D6, and D8 score 10 points for one example. All other dihedral groups score 10 points for each example.
    • Cyclic groups: C1, C2, and C4 score 10 points for one example. All other cyclic groups score 10 points for each example.
  • Frieze Patterns
    • pmm2 scores 10 points for one example. All other frieze groups score 10 points for each example.
  • Wallpaper Patterns
    • p4m scores 10 points for one example.
    • cmm and pmm score 10 points for each example.
    • p6m and cm score 30 points for each example.
    • All other wallpaper groups score 50 points for each example.

Bonus scoring:

  • Five different D5’s 100
  • Three different p6m’s 100
  • All seven kinds of Frieze symmetry 100
  • More different wallpaper groups than anyone else 50
  • Biggest cyclic groupin the class 50
  • Biggest odd dihedral group in the class 50
  • Best photo of your group 50

For full credit on this assignment, a group should score 150 points. The group with the highest overall score should receive a fabulous prize.

Checklist

Rosette Group Photo ID Location Score
Frieze Group Photo ID Location Score
pmm2
pm11
p1m1
p112
pma2
p1a1
p111
Wallpaper Group Photo ID Location Score
p4m

Handin: The checklist, all photographs, and the completed assignment

Sample Fieltrip Assignments

The article above outlines several ideas for fieldtrips. Below are some samples of the actual assignments that have been used.

Related Sites