Math 103 Summer, 2002
work in progress! (7-24-02 watch for a major revision by 8-1!)

Homework Assignments due: (* = interesting but optional)
1 7/8 Pythagorean Activities
Begin Tangram Activities
7/9 1.1:  5-8, 11, 12, *13 7/10 1.2: 1-3, 8, 9
Web Surfing Activity.
7/11Work on a portfolio entry.
2 7/15   4.1: 7, 8, 9 7/16 4.1: 14 (based on 7)
5.1: 6 (g,h,i)
Tesselation Tuesday!
4.1: 10, 24
5.2: 1,2
Alphabet and Designs
Symmetry Assignment 
7/18 4.2: 9,10
Assignment on symmetry.
Start on Lineland paper-due Monday.
3 7/22 
Lineland paper.
7/23 7.1:  1-4
7.2: 12,13,14
Models of the platonic solids from templates. 
Read Plato handout.
7.2: 20-22
Finish 8.1: 6,9
Dual Tessellations.4.1: 24
Begin Plato essay due 7-29
4 7/ 29 Plato essay  7/30 See assignment on 4th dimension and surfaces.
6.2: 14
See assignment on Zeno.
7/31See assignment on the torus and maps
7/26 Symmetry Day.
See assignment on Projection
13.4 : 2-6
5 8/5 10.1: 2(again),4,5
13.3: 1,2
8/6 Portfolios due by 5 pm.
13.4: 11,12,14
See assignment on Projective Drawing from class.
8/8.See assignment on coordinates and conics
See assignment 
Make a Desargues' Configuration.
Tentative General Reading Assignments (Revised 7-22-2002)
Days Assnm't Source  Chapter and pages for Reading Comments, Web Sites to Visit, and other things
7-8 to 7-10 Flatland 

K & M
Introduction, Preface, and Part I. 

1.1 Measurement pp 1-8 
1.1  pp 8-12 
1.2 Polygons pp13-16 
1.2 pp17-19
 Flatland is available on the web. 

Over 30 proofs of the Pythagorean theorem!
Many Java Applets that visualize proofs of the Pythagorean Theorem

Tangram Introduction
Japanese Site with Tangram Puzzles on-line
Here's a web page with many annotated Tangram references

Web references related to scissors congruence- dissections.
7-10 to 7-15 Flatland 

K & M 
Part II. 

4.1 Reg. and Semi Regular Tesselations pp 85 - 91 
4.1 Dual Tilings pp91- 93 
A wealth of materials can be found by going to this Tesselation Tutorial.
This might be a good time to visit Rug patterns and Mathematics exhibit plus...
Tuesday, July 16th  is Tessellation Day: Wear to class clothing that has a tiling pattern on it. 
7-15 to 
K & M 
K & M 


6.1 Flatlands pp 180 -184 

5.1 Kaleidoscopes (1 mirror)127-130 
5.1 (2 mirrors)130 - 134 
5.1 (3 or 4 mirrors) 134 -135 
5.2 Point symmetry 138- 146 
5.3 Frieze Patterns 147-155 
* 5.4 Wallpaper Patterns 
*5.5 Islamic Lattice Pattern
You might want to visit the Kali: Symmetry group page now .
7-21 to 

4.2 Irregular Tilings pp94-107 
*4.3 Penrose Tilings 

7.1 Pyramids, Prisms, and Anti Prisms pp 208 -215 
7.2  The Platonic solids pp 216-221 
7.3 Archimedean Solids pp 224-228
The metaphor of the cave. (On Handout.)
You might want to look at Penrose tilings by downloading Winlab by Richard Parris. 
You can look at polyhedra by downloading Wingeom by Richard Parris 
The Platonic solids  is an interesting site with Java viewers for interactive manipulation created by Peter Alfeld of Univ. of Utah. 
Polyhedra ! 
7-24 to 7-25 K&M 7.4 Polyhedral Transformations
8.1 Symmetries of Polyhedra

7-25 to 
K&M 6.2 The Fourth Dimension
10.1 Perspective
10.2 Optical illusions
11.2 Map Projections

Encyclopedia: Zeno's Paradoxes and the infinite.
Cartesian coordinates
The Fourth dimension
Networks and Euler's formula
Euler's formula, the torus.
A Visualization of 4d hypercube (Java applet).

8-1 to 8-5 A&S 
pp 1-3, Sections 3,6, 11, 13
10.1 Perspective 
13.2 Surfaces
13.3 More on Surfaces
More on Euler's applications.
Symmetry Day: Bring to class an example of a natural or synthetic physical object that has a non- trivial group of symmetries together with your description of those symmetries. You may bring either the physical object itself or a sketch of the object. 
Projective Geometry
Durer and perspective drawing 

Surfaces in topology
The Moebius strip,  The Klein bottle, orientability, and dimension.
Constructing surfaces in general

The Infinite (Zeno's Paradoxes and the infinite.) 
8-6 to 8-8  A&S

Sections 11, 13

Projection and Ideal elements.

The Color problems

Continuation of Projective geometry
Conics, Euclidean and Non-Euclidean Geometry

 Resource List for Portfolio Entries

The following list contains suggestions  for finding resources as well as the names of resources that may be used for one or more portfolio entries. Before reading an article in one of these resources thoroughly it is advisable to scan it quickly to see that it contains something of interest to yourself. Your portfolio entry can report on the content of your reading, illustrate it by examples, and/or follow up on it with your own response and creativity.

The content of the portfolio entry should relate specifically and directly to some visual mathematics. Personal observations , philosophical musings, and aesthetical judgments are not adequate connections to something visual by themselves to qualify as mathematical content.

These articles may also be useful in developing a deper level of understanding on a topic which will suppport your term project. I will add to this list as the term progresses.

Special Assignment Problems and Projects
(These may be assigned - watch for due dates on assignment schedule)

1. We are still trying to describe the cube to a Flatlander, this time using the transformation of the framework of the cube onto the plane by central projections.
Show the image of the cube on the plane under the following projection situations:
A.  One square of the cube is in Flatland and the center of projection is above the cube directly over the center of the cube.
B.  Only one edge of the cube is in Flatland and the center of projection is above the cube directly over the center of the edge in Flatland.
C.  One square of the cube is in Flatland and the center of projection is higher than the cube and not directly over any part of the cube.

2. Parallels on the torus and the sphere. Let's call an arc on a sphere (or the torus) a sline segment if it arises from a cross section of the sphere (or the torus) by a plane that passes through the center of the sphere ( or the torus).
A. Draw a figure showing some sline segments on a sphere with the planes through its center and some sline segments on a torus with the planes through its center.
B. Is the following statement true for any sphere? for any torus?
"Any two sline segments on a sphere (or a torus) can be extended so that they will intersect."

3. Find two drawings, paintings, prints, or photographs that have noticable perspective in the composition. Make a sketch or photocopy of the works and locate at least one "infinite" point on the horizon (ideal) line on each of your figures. Find at least three lines in each of your figures that meet at the infinite point.

4.  During the sessions we have covered many topics in class and through the readings. Choose two topics we have studied for examples in writing a paper (1-3 pages) discussing one of the following statements:
     A. The study of visual mathematics in two dimensions has much in common but also some noticable differences with its study in three dimensions.
     B. The amazing thing about mathematics is how it is able to turn even the simplest things into abstractions and can make the subtlest of concepts clear through a figure.

August 6: Coordinates and conics.
Coordinate geometry is a tool used in intermediate algebra courses to investigate the conic curves. Recall the basic idea is that a point with coordinates (x,y) will lie on a curve in the coordinate plane if and only if the numbers x and y make an equation determining the curve true. For example, a circle with center (0,0) and radius 5 is determined by the equation
X 2 + Y 2  = 25. We can check that the point with coordinates (3,4) is on the circle by verifying that 3 2  + 4 2 = 25.
 1. Each of the following equations determines a conic curve. Plot 10 points for each equation on a standard rectangular coordinate graph. Connect these points with straight line segments to give a polygon that will approximate the curve.
 a. 4X 2  + Y 2 = 25     [an ellipse]       b.   X 2  - Y  2  =  9  [an hyperbola]  c.   X 2  - Y   =  4     [a parabola]

 2. Draw three separate projective planes including a system of coordinates with the horizon line and lines for X=1, 2, 3, 4, and 5, and Y=1, 2, 3, 4, and 5 as well as the X and Y axes. For each of the previous equations, plot 6 points on a projective coordinate plane that correspond to 6 of the 10 points plotted previously on the standard plane. Connect these points with straight line segments in the projective plane.

Projective Drawing:
1. Suppose three lines l , m, and k form a triangle. [Draw a large figure to illustrate this situation.] Draw ten points on line l perspective with 10 points on line m with center O. Use these ten points on line m to draw 10 points on line k in perspective with center O'.  Draw the lines connecting the corresponding points on line l and k. Describe the figure that these lines suggest.

2. Draw a figure showing a tessellation of the projective plane on one side of the horizon line by parallelograms.

3. Draw a figure illustrating a black and white chess board in perspective.

August 1.Central Projection. [see Figure 7 in A&S.]
On a line mark 11 points that are separated one from the next by one inch. At the middle point draw a circle of radius one inch as in Figure 7.  Find the points on the circle that correspond to the points on the line as in Figure 7.  Describe the relation of a point on the circle to the corresponding point on the line with regard to the point O where the circle and the line touch.
July 301. Look up "Zeno's paradoxes" in the Encyclopedia (Britannica). Draw a figure that illustrates the paradox of Achilles and the Tortoise. Describe a common context today to which Zeno's argument about Achilles and the Tortoise could be applied. Using your situation, discuss where the accumulation of small and infinitely divisible intervals is incorrectly compared with the accumulation of equally sized intervals.
July 301. The fourth dimension can be used to visualize and keep track of many things involvimg four distinct qualities that can be measured in some fashion.
A. For example, a 13 card bridge hand can be thought of as a point in four dimensions where the coordinates represent the number of cards of each suit present in the hand. In this context the point with coordinates ( 2, 4, 6, 1) might represent a hand with 2 clubs, 4 diamonds, 6 hearts and 1 spade.
Using this convention discuss briefly the following representations of bridge hands: (0, 0, 0, 13),  (0, 0, 6, 7),  (3, 3, 3, 4).
Suppose a bridge hand is represented by the point with coordinates (x, y, z, w).
Explain why x + y + z + w = 13.
B. Describe another context where four dimensions can be used in representing some features of the context.
2. Hypercubes in Higher Dimensions.
The 16 vertices of the 4-dimensional hypercube can be described by the collection of ordered quadruples (a,b,c,d) where the numbers  a, b, c, and d are either 0 or 1.
Write a similar description of the 8 vertices of the 3-dimensional cube.
Write a description of the vertices of the 5-dimensional hypercube. How many vertices does the 5-dimensional hypercube have? How many vertices does the 6-dimensional hypercube have?  How many vertices does the 10-dimensional hypercube have? What can you say about the vertices for the hypercube of dimension N?
3. Surfaces.
A. Describe 5 physical objects that have surfaces that are topologically equivalent to a (one hole) torus. Bring one example to class on Tuesday.
B. Describe 5 physical objects that have surfaces that are topologically equivalent to a torus with two or more holes. Bring one example to class on Tuesday.

July 31 1. Make two tori: one from two annuli, one from a single "rectangle."
2. Bring in 3 different world maps. Describe how each map deals with lines of longitude, latitude, and the poles.

1. This Thursday is Symmetry Day: Bring to class an example of a natural or synthetic physical object that has a non-trivial group of symmetries together with your written description of those symmetries. [You may bring either the physical object itself or a sketch of the object.]
2. Casting Shadows on Flatland.
The sphere is still trying to explain some of the features of the torus to a Flatlander. This time the sphere has decided to show the Flatlander different shadows that are cast by the projection of the torus onto Flatland.
A. Draw three different shadows that the torus could cast.
B. Do you think it is possible to make a torus that would cast a shadow on Flatland that completely covers a circle and its interior? If so, describe some of the features of such a torus. If not, give some reasons for your belief. In other words, is it possible that a Flatlander might mistake a torus for a sphere based on the shadow it casts?

Due Wednesday, 7-24 Dual Tessellations.
A. Draw the three regular tilings and one semiregular tiling of the plane.
B. Use a red pencil to mark the center of each of the polygons. Join any two centers of polygons that share a common side. This should give new tilings of the plane in red.
C. Describe the new tilings you obtain in part B. These are called the duals of the original tilings.
D. For each of the 8 tilings (from parts A and B) make a list of its symmetries.
E. Compare the symmetries of each tiling with the symmetries of its dual tiling. Explain any connections you notice between these symmetries

Due Monday 7-29 Plato and Shadows: The Greek philosopher Plato describes a situation where a person lives in a cave and can only perceive what happens outside the cave by observing the shadows that are cast on the walls of the cave from the outside.

Write a brief essay discussing a situation in the contemporary world where indirect experiences are used to make observations. How are the observations made? How are they connected to the actual situation? Do you think the inferences made from the observations are always accurate? [3 or 4 paragraphs should be adequate.]

Due Monday , 7-22Lineland Paper: Imagine you are a Flatlander talking to a Linelander.
Write an explanation of symmetry to a Linelander from the point of view of a Flatlander. Discuss and illustrate the kinds of symmetry that are possible in Lineland. Which Flatland symmetries (if any) would you associate with Lineland symmetries? Explain the association briefly.
Here are some terms you might use in your discussion: Reflection  Rotation  Translation  Orientation

Due  Thursday 7-18 Find (or create) three graphic designs (in advertisements, logos, or icons) that have  (i) reflection symmetry only, (ii) rotational symmetry only, and (iii) reflection and rotational symmetry.

Due: 7-17 Classifications by symmetry:
   It is often useful to classify visual objects by their symmetries. For example, the letter "T" as it appears on this page has only a reflection symmetry determined by a vertical line, whereas the letter "I" has two reflection symmetries and one rotational symmetry of order 2 (a half turn).
Group the following letters together in different classes determined by the number and types of symmetries they have as printed on this page. [It is up to you to determine the appropriate classes.]