# Math 103I Summer, 2000 work in progress

 Week\Day Mon. Tues. Wed. Thurs. 1 5/29 no class 5/30 5/31 6/1 B&S 4.1.  I: 1,2,3,4,7,*(8,9,10); II: 1,3, *5; III: *2 2 6/5 Tangram puzzle sheet Finish Web Surfing Activity. 6/6Read B&S:  pp219-223 Do: B&S 4.2: I. 2, 3,6,7,4,5 6/7 Do:B&S 4.2: I: 9,10 II: 1, 4, 5 6/8 work on a portfolio entry. 3 6/12 Classify Reflection symmetries of cards. 6/13 Read B&S pp249-252 Symmetry Assignment  Alphabet and Designs 6/14 Do: B&S 4.4. II 2,3,4 Assignment on tilings and symmetry. 6/15 Tesselation Thursday! Do: B&S 4.4 I: 1-6 Project Prelim due by Friday. Start on Lineland paper-due Monday. 4 6/19 Start on making models of the platonic solids for 6/21.  [See B&S 4.5 I:1.] Lineland paper. 6/20 [Work on portfolios.] 6/21 Handout Templates for making polyhedra.  Read Plato handout. 6/22  See assignment for Monday 6/26. 5 6/26 B&S 4.4: I. 7 (photocopy) 4.5. I. 2, 3, 9 II.1 Dual Tessellations. Plato. 6/27  B&S 4.5: I. 10; II:2; III:1 6/28 B&S: p320: 1,2 6/29 B&S: p 365: 2,3 Symmetry Day. 6 7/3 no class :) 7/4 No Class! 7/5: B&S: 5.3: I: 9-11,20; II:1-3, 9, *10 B&S:5.3: I : 14,15; 17 (make one of these with wingeom.) 7 7/10 See assignment on the torus and maps 7/11 B&S: 5.1 (read p 333-335) I; 1-3,5,20 5.2: I :2,3, 13,14 7/12 B&S:5.1: I: 15-17; II:1-3; III:4 5.2: II: 7,9 7/1 B&S  5.1 or 5.2: II.10 4.7: I. 3-5;  II. 3 8 7/17B&S 4.7:II:1 See assignment on 4th dimension and surfaces. 7/18 B&S: 3.1 I:3,4,5,6; II:4 3.2:I:1-3, 8-11,15 See assignment on Zeno. 7/19 Same as 7/18. 7/20 B&S: 3.2: I: 19,20; II:3, 6, 7 III:1 3.3: I: 7 Snowflake sheet (handout) 9 7/24 B&S: 6.1: I : 1-3,5,10 6.3:I:1,8,9,16 7/25 B&S 3.5: I: 1, 3-5 II: 4,5 III:1 See assignment on Projection 7/26 Make a Desargues' Configuration. See assignment on Projective Drawing 7/27 B&S 6.3: 8, 16,20 10 7/31See assignment on coordinates and conics 8/1 8/2 8/3  Projects due for grading by 5 pm. 11 8/7See assignment  Portfolios due by 5 pm. 8/8 8/9 Project Presentations

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.

6-29 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.]
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?

6-26 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

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.]

6-19 Lineland 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

6-14: Describe (and sketch) three distinct tilings of the plane using rectangles that have the long side twice the length of the short side. Discuss the reflection and rotation symmetries of each tiling.

6-13 I. 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.]

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

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.

 Week Assnm't Source Chapter and pages Comments, Web Sites to Visit, and other things 1 Flatland   Stewart  B&S Introduction, Preface, and Part I.  Prefaces and Ch 1 pp.1-7  Welcome  Ch.4.1 (Activity and assignments on Flatland will follow next week.) Flatland is available on the web.  Professor Ian Stewart "The Magical Maze: The Natural World and the Mathematical Mind"  Tangram Information & Software (shareware) by S. T. Han 2 Flatland  Stewart  B&S Part II.  Ch.2 pp8-11; 13-16;16-24;24-26  Ch. 4.2 Bring two congruent equilateral triangles to next class.  This might be a good time to visit Rug patterns and Mathematics exhibit plus... 3 Stewart  B&S ch 7. pp 95-101;109-112  Ch. 4.4 Thursday is Tessellation Day: Wear to class clothing that has a tiling pattern on it.  You might want to visit the Kali: Symmetry group page or this Tesselations site now . 4 B&S Stewart Plato Ch. 4.5 and 4.7 Re-read p16-24 and ch 7. pp 95-101;109-112.  The metaphor of the cave.(On Handout.) Platonic and Archimedean solids, Plato, and Kepler.  The Platonic solids  is an interesting site with Java viewers for interactive manipulation created by Peter Alfeld of Univ. of Utah. 5 Stewart  B&S   Plato ch 11: pp159-166  Ch 4.5  The metaphor of the cave. (On Handout.) (Networks and Euler's formula)  Polyhedra ! 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. 6 B&S Stewart Ch. 5.1 and 5.3 ch 11: review pp159-166 plus 166-169 (Euler's formula, the torus)   More on Euler's applications. 7 B&S Stewart Ch. 5.1, 5.2,  4.7 ch. 10: pp144-155 ch 12: pp 174-179 ch 14: pp 200-208 The Moebius strip,  The Klein bottle, orientability, and dimension. Cartesian coordinates The Fourth dimension. A Visualization of 4d hypercube (Java applet). 8 B&S Stewart ch 3.1, 3.2 and 3.3, 6.1 ch 2 p 13 ch 9 pp 127-139 ch 12 pp  178-187 (optional) Encyclopedia: Zeno's Paradoxes and the infinite. The Infinite (Zeno's Paradoxes and the infinite.)  Constructing surfaces in general 9 B&S A&S ch. 3.5, 6.1 (again) , 6.3, 6.6, (2.6, 2.7) pp 1-3, Sections 3,6, 11, 13 More on the infinite.Fractals Adding the infinite  and limits. Configurations Projective Geometry Durer and perspective drawing  Projection and Ideal elements 10 B&S A&S ch. 4.6 Sections 11, 13 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.
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.
• 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