Directions:  You may work together, but please write your answers in your own words.  Please write (neatly) on one side of the paper, cut off any messy edges, and staple the papers together.

1. Answers vary.  Draw any 2 of the following.
1. A leopard with 6 cubs (see p. 220, Fig. 4.42).
2. Trunks of the kajana tree, starting with triangles with 6 dots on each side instead of 5 (see p. 209, Fig. 4.8b).
3. A 5x5 version of the lion's stomach design (see p. 214, Fig 4.22)
4. A resizing of the spider in its cobweb design that's different from the ones shown in p. 215, Fig. 4.24 or Fig 4.25.
5. A flying duck design with 4 ducks that is monolinear and symmetrical.
6. A 4x4 or 5x5 Celtic knot that's similar to the ones on this handout but has a different grid (i.e. a different set of ``mirrors'').  You can read more about knot construction here.
2. The following questions refer to the article ``Knots in chemistry and physics'' on p. 240-245 in your course packet.  There is a lot of mathematical detail here--just try to get a sense of the main ideas.
1. What is the main goal of knot theory?  The goal is to classify all knots.  If two knots are equivalent (see #2) they are considered ``the same'' in the classification.  Knots are classified by the number and order of their crossings.
2. When are two knots equivalent to each other?  (Think about the exercises we did in class, too.)  Knots are equivalent if one can be made into the other without cutting the loop of string.  (This is called a ``continuous deformation''--but all that means is that you can't cut the string.)  In class, we showed that the left and right trefoil knots are not equivalent to each other, because you can't make one into the other without cutting the string.
3. What is the unknot?  The unknot is a loop of string with no crossings.  If a knot can be ``untangled'' to form a simple loop, it is equivalent to the unknot.
4. What is the ``world's smallest knot?'' (as of 1993)  The world's smallest knot is a trefoil knot made of molecular threads by Christine Dietrich-Buchecker and Jean-Pierre Sauvage in 1988.
5. What is one surprising fact you learned from this article?  Answers vary.
6. Vaughan Jones is the chief mathematician associated with knot theory.  On the Internet or elsewhere, find an interesting fact about Vaughan Jones.  Be sure to cite your source.  Answers vary.
3. Take Barry Dayton's interactive strip pattern tutorial.  When you are done, email me the code.  Answers vary.
4. Read about sipatsi on p. 248-252.
1. Classify the three strips on p. 250, Fig. 1.1, the strip in Fig. 1.3, and the strip in p. 251, Fig. 1.5, using the pq system.
1. Fig 1.1:  pqpq
2. Fig 1.1:  oooo
3. Fig 1.1:  cccc
4. Fig 1.3:  pdbq
5. Fig 1.5:  pdpd
2. What is the period of the strips in Fig. 1.1?  The period is the number of strips needed to make one copy of the motif (which is the smallest design that is used to build the entire design).
1. 6
2. 6
3. 6
3. How are common multiples important in sipatsi weaving?
1. Each strip design must have a whole number of motifs.  Therefore, the length of the gipatsi must be a common multiple of the length of the motif of each individual pattern.  For technical reasons, the length must also be a multiple of 4.  Therefore, the total length is a common multiple of 4 and all the different motifs.
4. What does Fig. 1.9 on p. 252 represent? (I had to laugh!)  It represents patterns in tire treads.
5. Read about titja baskets on p. 253-257.  Look at the five photographs on p. 256-7.  What types of symmetries do the baskets have?  Which have reflections?  If they have rotational symmetry, find the degree of the rotation.  All have rotational symmetry.  In addition, A, B, C, and D have reflections.  Specifically,
1. Rotation of 90 degrees.  Reflection horizontally, vertically, and on 45 degree diagonals.
2. Rotation of 360/5 = 72 degrees.  Reflections on axes spaced by 72/2 = 36 degrees.
3. Rotation of 360/6 = 60 degrees.  Reflections on axes spaced by 60/2 = 30 degrees.
4. Same as #3.
5. Rotation of 360/3 = 120 degrees.  No reflections.
6. The article on p. 259-273 is the text of a lecture by the famous Dutch artist M.C. Escher.  You may recognize his designs from T-shirts or posters.  On p. 269 he discusses some of the designs from various cultures that inspired him, pictured in "Eight copies of space-filling designs" above.  Of these eight designs,
1. Which have reflectional symmetry?  Is it horizontal, vertical, or along a diagonal?  1 (45 degree diagonal), 3 (45 degree diagonal), 4 (horizontal and vertical), 5 (horizontal only), 6 (vertical, 60 degree diagonals), 7 (horizontal and vertical)
2. Which have rotational symmetry of 180 degrees?  1, 2, 3, 4, 7 Of 90 degrees?  1, 2, 3 Of 120 degrees?  6, 8
3. What type of symmetry does Escher's "Swans" on p. 261 illustrate?  glide reflections.
7. Extra Credit.  Use the Wallpaper Drawer to make one of the designs shown in the Escher article.  Use Print Screen to print your work.

Here's a possible answer: