Math 103I Summer, 2000
work in progress

• Homework Assignments
• Reading Assignments
• Resource List

8-7
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.

7-31 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.
 

7-26 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.

7-25 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.
7-18 1. 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.
7-17 1. 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 Monday.
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 Monday.
 

7-10 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.
3. Finish problems 2-5 on the Torus Activity handout from class.

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.]

II. 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.

• Use Visual Mathematics web sites for surfing visual mathematics and geometry.
• Looking for ideas for your portfolio or a project? Try David Eppstein's Geometry Junkyard. This contains many wonderful and neat resource references!
• Use articles from old Scientific American magazines (located outside my office at Library 48)
• (Older issues) Martin Gardiner's articles are usually short and clear enough to provide material for one or even two even entries.
• (More recent issues) Ian Stewart 's articles are similar and about as playful as the Gardner pieces.
• Some issues  have had articles on special topics that are relevant to our interests. These are usually longer and require a little more effort to digest - though well worth the effort.
• "Topology" by Tucker and Bailey, 1950, pp 8-24.
• A number of liberal arts / mathematics textbooks contain chapters that would be suitable for reporting.
• Mathematics: the Man-made Universe by Sherman Stein.
• Excursions into Mathematics by Beck, Bleicher, and Crowe.
• What is Mathematics? by Courant and Robbins.
• The World of Mathematics by Newman.
• The library has a collection of films and videos that are relavant to our interests.
• For All Practical Purposes (COMAP)
• Some of the history of mathematics videos from the Open University Series (BBC)
• There are several non-text mathematics books and collections of essays similar to the required book by Stewart. Two of these are on the recommended list and are in the book store- Devlin's Mathematics: The Science of Patterns and Banchoff's Beyond the Third Dimension. Martin Gardiner has many books full of puzzles and recreations many of which are relevant.
• The Problems of Mathematics by Ian Stewart.
• The Mathematical Experience by Philip Davis and Reuben Hersh