Tesselations and Platonic Solids Assignment

During the next two weeks we will be learning about Escher-like tesselations and how to apply them to 3D Platonic solids.

Part 1: Tessalation using rotation, translation and

1. Complete the Worksheets handed out in class: Tesselations by translation, rotation and reflection.
2. Research: As is a part of any design project, this one begins with research. To better understand Tesselations, Crystallography, Escher and Platonic solids you will need to research the subjects. Use the school's library, public libraries and the internet. To help you get started, see a link the instructor's website: www.stevepasos.com. The type of questions you might ask your self during your research are:
• What is isometry?
• What is meant by "motifs of diminishing size"?
• Escher pursued themes of transformation in works he called "Image Stories" which involved images transforming from one state into another. How could you create a tranforming tesselation?
• Who is G. Polya and what significance does his 17 different playne symmetry groups have?
• Who is Koloman Moser?
• What is meant by glide-reflection symmetry?
• What is a Kaleidocycle?
3. Process documents: Document your process and progress, showing sketches of various different attempt at tessellated shapes. Create 3 roughs. Use color pencil or markers to test out various different color schemes. This is due during week 10.
4. Create an Escher-like tesselation design based on one of your rough on 10x10 bristol with a 12x12 mat. Color your design. Be very precise with you measurements and make sure everything is square, borders are even. This design should be "portfolio-worthy".

Part2: 3D Tessalation

1. Due week 10: Construct and assemble one of each of the 5 platonic solids. See http://www.korthalsaltes.com/ for templates.
2. Create an Escher-like tesselations mapped onto the 2 Platonic solids or 1 Platonic solid and a kaleidocycle. See http://ccins.camosun.bc.ca/~jbritton/jbpolydodeca.htm and http://www.cs.berkeley.edu/~j-yen/cs285/as1.html for examples
3. Create a zerox copy (or tracing paper copy) of your 2 design and leave uncut and unfolder. These copies will not be returned.

Esher was once asked:

Q. ...Your pictures do not involve only simple shapes-- you tessellate things like lizards and birds, how do you achieve that?

A. Interesting you should ask--I developed a technique for that myself. To begin with you take a shape that on its own will tessellate. Proceed by ‘cutting’ out pieces of the shape and ‘putting’ them around the outside in a specific way. The easiest is in a square or rectangle where you can put the piece on the same place on the other side of the shape, but, for shapes that involve rotations to tessellate, it is much more complex. In a hexagon for example you must place the piece removed on the opposite part of an adjacent side. If you ‘cut’ a piece out of the side that you just ‘put’ a piece on, you must move that piece to the side that the other piece came from. The artistic part in this is to make that pieces you ‘cut’ out and ‘put’ on look like something. Below is an example (actually drawn by Escher!!) of how a complex tesselation starting from a hexagon might come out.

see: http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/escher.html for complete interview and Escher's own diagram demonstrating the technique.

Due dates:

• Week 10: Thursday, Dec 12:
  • Research Report: minimum 1 page plus one traced image
  • Tessaltion Worksheet handouts
  • Thumbnails and roughs for Tessaltion project
  • Construct and assemble all five platonic solids

• Week 11: Thursday, Dec 19:
  • Tessaltion Worksheet handouts
  • Thumbnails and roughs for Tessaltion project
  • Construct and assemble all five platonic solids

   

References:

• M.C. Escher et al, M.C. Escher: His Life and Complete Graphic Work, New York: Abrams, Inc., 1992.
• Jinny Beyer, Designing Tessellations: The Secrets of Interlocking Patterns, Chicago, IL: Contemporary Books, 1999.
• "Tilings and Patterns", by Branko Grunbaum and G.C. Shephard (W.H. Freeman, 1987). This is a most complete treatment of the topic I know of, with formal mathematical language, but also with lots of pictures. If you want all of the chapters, including those on non-periodic tilings, get the hardback version.
• "Introduction to Tessellations", by Dale Seymour and Jill Britton (Dale Seymour Publications, 1989). This is written for teachers, and as the title suggests, is an introduction to the topic.
• "Islamic Art in Context", by Robert Irwin (Harry N. Abrahms, 1997). Covers the history, religious and secular architecture, literature, and more of art, in which a highly-developed geometry plays a central role.
• "M.C. Escher, His Life and Complete Graphic Work", by F.H. Bool, J.R. Kist, J.L. Locher, and F. Wierda (Harry N. Abrams, 1982). A very complete source on the life and work of Escher.
• "Visions of Symmetry", by Doris Schattschneider (W.H. Freeman, 1990). A wonderful book, which reproduces Escher's notebooks in which he works out tessellation systems.
• "The Magic of M.C. Escher", by Eric The, Joost Elffers, and Andreas Landshoff (Harry N. Abrahms, 2000). A new book which interweaves well-known prints with numerous unpublished drawings, the artist's eloquent words, and observations by Escher expert J.L. Locher. A dramatic and rather cinematic presentation of Escher's art, with a number of large gate folds.
• "Escher on Escher - Exploring the Infinite", by M.C. Escher (Harry N. Abrahms, 1989). The artist speaking in his own words about his art, with 120 accompanying illustrations. Includes the complete text of a series of lectures Escher was scheduled to give in the United States, but which were cancelled due to illness.
• "Dissections: Plane & Fancy", by Greg Frederickson (Cambridge University Press, 1997). The best book I know of on geometric dissections - makes for very entertaining reading for math lovers.
• "Polyominoes: Puzzles, Patterns, Problems and Packing", by Solomon W. Golomb (Princeton Science Library, 1996). The definitive treatment of the topic, originally published many years ago.
• "M.C. Escher Kaleidocycles", by Doris Schattscneider and Wallace Walker (Pomegranate, 1987). A fascinating and fun book in which Escher's tessellations are adapted to innovative three-dimensional paper models.