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Week 1

Email Writing: Due Tuesday, Jan 13

1. Self-introduction. Write a paragraph of self-introduction and mail to the email list.
2. Journal/Questions. Write two or three questions or observations about the reading and mail them to the list.

Reading: Read enough by Tuesday to respond in writing and the rest by class.

1. Beyond 3D: Chapters 1 and 3
2. Flatland: Part 1, Sections 1-6, Part 2, Sections 13-20 (you are encouraged to read more)
3. Albers, Banchoff biography

Problems 1: Due at the beginning of class, Thursday, Jan 15

Problems will mostly be written or drawn on paper, but some ask for models. You will turn in the paper problems, so you may wish to make a copy for your own reference. Most models will be observed and checked off in class and returned immediately, but unusual ones may be kept and returned the next week.

1. Slices of the cube. Imagine that a cube passes through Flatland. To A Square the cube appears to be a shape that changes over time; we folks in Spaceland can see this as a succession of slices or cross-sections of the cube by parallel planes.
   • (a) Make a model of the cube and indicate examples of the slices if the planes are parallel to an edge of the cube.
   • (b) Make a model of the cube and indicate examples of the slices if the planes are perpendicular to a diagonal of the cube.
   • (c) Make a model of the cube and indicate examples of the slices for an example when the planes are in "general position", i.e., no two vertices of the cube lie on a single one of the parallel planes.
   • (d) In each case, draw accurate figures on a sheet of paper (graph paper is OK) that show the polygons seen by A Square (include each type of slice that appears).

   Comment 1: One way to do parts (a)-(c) is to make several cube models from cardboard and draw on the models with a pen or pencil. You could also experiment with dipping cubes in water or paint and report on that.

2. Number of Vertices, Edges, etc. of the cube and hypercube.
   • (a) Make a table. Label the rows of the table thus: Point, Segment, Square, Cube, Hypercube. Label the columns of the table thus: Vertices, Edges, Faces, Solids, Hypersolids. The assignment is to fill in numbers in the table. For example, a square has 4 vertices, 4 edges, 1 face, so the numbers will be 4, 4, 1, 0, 0.
   • (b) Explain how you figured out the numbers and why you think they are right.
   • (c) Look for a pattern in the table that will let you figure out the next line for hyperhypercubes (in 5D) by following the pattern rules in the table. Can you give a reason for this pattern?
3. Number of Vertices, Edges, etc. of a "hyperpyramid"

   Description of a "pyramid" figure in 4-space. Consider this recipe for building up "pyramid" shapes. In dimension 0, the shape is a point P_0. In dimension 1, pick a point P_1 distinct from a point P_0 and connect point P0 to P1 by a segment; this gives a dimension-one shape that is a segment P_0 P_1. In dimension 2, start with a segment P_0 P_1; pick a point P_2 not on line P_0 P_1 and connect the points of segment P_0 P_1 to P_2 by segments; this forms a triangle P_0 P_1 P_2. In space, start with a triangle P_0 P_1 P_2 and take a point P_3 not in the plane of the triangle. Connect P3 by segments to each of the points in the triangle. This forms a triangular pyramid or tetrahedron P_0 P_1 P_2 P_3. Now in four-space, start with a tetrahedron P_0 P_1 P_2 P_3 and pick a point P_4 not in the 3-space of the tetrahedron; connect the points of the tetrahedron to P_4 by segments. This forms what we might call a hypertetrahedron, but the usual name is actually a 4-simplex.

   • (a) Make a table as before. Label the rows of the table thus: Point, Segment, Triangle, Tetrahedron, 4-simplex. Label the columns of the table thus: Vertices, Edges, Faces, Solids, Hypersolids. The assignment is to fill in numbers in the table as you did for the hypercube case.
   • (b) Explain how you figured out the numbers and why you think they are right.
   • (c) Look for a pattern in the table that will let you figure out the next line for a 5-simplex (in 5D) by following the pattern rules in the table. Can you give a reason for this pattern?

• Extra Credit Ideas. (a) Carry out an experiment or study on the slicing of some other 3-d shape besides the cube or the tetrahedron and the sphere. (b) Draw and explain a biologically more interesting flatland creature than A Square. In particular, design a method for dealing with food (going in and coming out).

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Week 2

Email Writing: Due Tuesday, Jan 20

1. Journal/Questions. Write two or three questions or observations about the reading and mail them to the list.
2. Outside Reading. Find at least one source (book, article, web site) related to the geometry of higher and lower dimensions. Email the reference with a couple of paragraphs describing something that interested you about this source, questions raised, etc.

Reading: Read enough by Tuesday to respond in writing and the rest by class.

• Banchoff, Flatland intro
• Beyond 3D, Chapters 2, 4

Problems 2: Due at the beginning of class, Thursday, Jan 22

1. Models. Make a good-quality model of something relevant to this weeks reading. Here are some suggestions -- a cardboard model of 3 square-based pyramids that can be assembled into a cube -- a 3D model of a hypercube (perspective drawing) -- an irregular tetrahedron and its mirror image (colored faces) -- some examples of duality of polyhedra (one regular polyhedron inside another (how can you see this?) -- a decomposition of a tetrahedron into rectangular slices to show how to compute the volume.

2. Tetrahedra and Cubes. A regular tetrahedron can be formed by connecting four vertices of the cube. The remaining part of the cube is made of pyramids. What is the combined volume of the pyramids? What is the volume of the tetrahedron in a cube with side of length s?

   What is the volume of a tetrahedron with a side of length t? (Use what the you just showed.)

   Make a model of a regular tetrahedron with side length t and connect the midpoints of the sides with line segments. These form the edges of four smaller "corner" tetrahedra. What is the shape of the figure (the "hole") which results from cutting off these corners? What is the combined volume of the corners? What is the volume of the hole?
3. Volumes of hyperpyramids.

   Explain how a hypercube can be broken up into a number of hyperpyramids with cubical bases. Use this and the Cavalieri principle to give a formula for the volume of hyperpyramids in general.

   The set of points (x,y,z) with nonnegative x, y, z and x+y+z <1 forms a pyramid in 3-space. What are the faces of this pyramid? What is its valume?

   What is the analog of this pyramid on 4-space (i.e., (x, y, z, w) space)? What is its volume? Describe the faces of this hyperpyramid.

4. Extra. (a) Find some additional examples of fractals and explain their fractal dimensions. (b) Compute the volume of the 3-sphere or the 4-volume of the 4-ball.

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Week 3

Class of Thursday, Jan. 22

The class began with a discussion of counting edges, faces, etc., of a hypercube. The small groups considered the question: if a ship in flatlad sails straight East and eventually returns to its starting point, what are possible shapes for Flatland?

Possible answers included a cylinder, a sphere, a polyhedron, a torus. Questions arose about whether the edges would prevent a cube from being a model of Flatland.

The video the Shape of Space was shown, and it most of these same shapes appeared in the video, as well as Mobius strips.

After the video, the class played some games of torus tic-tac-toe. A conclusion was reached that the first player wins.

After some further exploration of Mobius strips made of adding machine paper, the class adjourned to the computer lab for some games of Torus Chess (by J. Weeks).

Email Writing: Due Tuesday, Jan 27

1. Journal/Questions. Write two or three questions or observations about the reading and mail them to the list.
2. Outside Reading: First Step on your Project. Write a paragraph proposing what you plan to do for your project; this of course will be very preliminary, but include some concrete ideas that you have or something interesting that you have learned. (In other words, saying "I am going to do a project on fractals." and that's all, is too weak. Include a couple of citations for references that you have found that are relevant to your project.
   It is definitely possible to change your mind about your project, but you should be doing some prepration each week and giving a progress report.

Reading:

Read enough by Tuesday to respond in writing and the rest by class.

• Handout on gluing.
• Handout on cylinders and cones.

Problems 3:

Due at the beginning of class, Thursday, Jan 29

Part A..

Do the problems 2.3, 2.4, 2.6, 2.9, 2.10 and 2.11 in the shaping space handout.

Part B..

Draw a square and two points A and B in the square (sort of randomly). In each case, draw carefully, 5 line segments connecting A and B (i.e., geodesic segments) given the following gluing rules:

• (a) Glue one pair of sides to get a cylinder (a finite cylinder).
• (b) Glue two pairs of sides to get a torus.
• (c) Glue two pairs of sides to get a Mobius strip.

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Week 4

Class of Thursday, Jan. 29

The class began with reports from each person about the state of their project.

Then the class used Polydrons to build the platonic solids, making a count of faces, edges and vertices.

It was observed that the data for the polyhedra for F, E, V paired up with another polyhedron for V, E, F. This was shown geometrically by duality, which was then explained by a certain amount of handwaving.

A better explanation was given the Platonic Solids video.

After the video, the class adjourned to the computer lab for a Sketchpad experiment with the sum of the exterior angles of a general pentagon. This was an example of total curvature of a closed curve. To be continued.

Email Writing: Due Tuesday, Feb 5

1. Journal/Questions. Write two or three questions or observations about the reading and mail them to the list.
2. Project. Write the equivalent of a full page on your project and email it in by Tuesday. Give at least a couple of precise references.

Reading:

Read enough by Tuesday to respond in writing and the rest by class. Learn the stuff listed below by Thursday.

• Banchoff, Chapter 5
• Learn the names and number of kind of faces, edges, vertices, of the Platonic solids.

Problems 4:

Due at the beginning of class, Thursday, Feb 5

1. Compute the dihedral angles of the regular tetrahedron and the regular octahedron. Explain your method carefully. (Possible methods: construct a cross-section. Use coordinates. Use relations with other polyhedra.)
2. Write a paragraph explaining what the dual of the hypercube is and how to think of it, including numbers of each kind of face, etc.
3. Make a model of a polyhedron that is not a Platonic solid (nicer models get extra credit).

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Week 5

Class of Thursday, 2/5

The class explored angle sums of triangles and other polygons on various "corners" of polyhedra made from Polydrons. These corners turned out to be special cases of cones, and the angle sum is related to the cone angle.

Email Writing: Due Tuesday, Feb 10

1. Journal/Questions. Write two or three questions or observations about the reading and mail them to the list.
2. Project. Write two pages on your project. Have some real content this time. You can email this or hand in hard copy on Thursday. Be prepared to talk about this with the group.

Reading:

Read enough by Tuesday to respond in writing and the rest by class. Learn the stuff listed below by Thursday.

• Handout on geometry of cones

Problems 5:

Due at the beginning of class, Thursday, Feb 12. The first two are just descriptions of what you did in class, presented nicely.

1. Write a nice, clean account of a couple of the examples of angle sums at a polyhedral corner that you did in class 2/5. This should include a net for the corner with a polygon whose angles you can compute exactly.
2. Write down the general rule for the relation between cone angles and polygon angle sums.
3. For what cone angles does a cone have a self-intersecting geodesic?
4. (The Big One). As explained above in the email section. Write two pages of work on your project. This should have some content, not just be a proposal.

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Week 6

Class of Thursday, 2/12

The class explored geodesics on spheres, especially spherical triangles and the relationship between angles and area.

Email Writing: Due Tuesday, Feb 17

Optional this week. See email.

1. Journal/Questions. Write two or three questions or observations about the reading and mail them to the list.
2. Project. Write more on your project. Be ready to report to the group.

Reading:

• Polking on Spheres (Web page)
• Handout on 3-sphere

Problems 6:

Sphere questions on handout.

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Week 7

Class of Thursday, 2/19

Exploration of perspective drawing and the idea of a line at infinity.

Email Writing: Due Tuesday, Feb 17

1. Journal/Questions. Write two or three questions or observations about the Osserman reading and mail them to the list.
2. Project. Final presentation coming up. Either March 5 or March 12.

Reading:

• Handout on Projective Geometry
• Handout on Durer and Perspective Drawing
• Osserman, Poetry of the Universe, Chapters 1, 2, 3.

Problems 7:

Make an egocentric world map for Seattle (see Osserman).

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Week 8

Class of Thursday, 2/26

Tying together loose ends.

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Week 9

Class of Thursday, 3/5

Project Presentations - Group 1.

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Week 10

Class of Thursday, 3/12

Project Presentations - Group 2.