




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: pp219223
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 pp249252
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: 16 Project Prelim due by Friday. Start on Lineland paperdue 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: 911,20; II:13, 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 333335) I; 13,5,20
5.2: I :2,3, 13,14 
7/12 B&S:5.1:
I: 1517; II:13; III:4 5.2: II: 7,9 
7/1 B&S 5.1 or 5.2: II.10
4.7: I. 35; 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:13, 811,15 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 : 13,5,10
6.3:I:1,8,9,16 
7/25 B&S 3.5: I: 1, 35
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 
87
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 (13 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.
731 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.
726 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.
725 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.
718 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.
717 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 4dimensional 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 3dimensional
cube.
Write a description of the vertices of the 5dimensional hypercube.
How many vertices does the 5dimensional hypercube have? How many vertices
does the 6dimensional hypercube have? How many vertices does the
10dimensional 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.
710 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 25 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.]
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.17 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 pp811; 1316;1624;2426 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 95101;109112
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
Reread p1624 and ch 7. pp 95101;109112. 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: pp159166 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 pp159166 plus 166169 
(Euler's formula,
the torus)
More on Euler's applications. 
7  B&S
Stewart 
Ch. 5.1, 5.2, 4.7
ch. 10: pp144155 ch 12: pp 174179 ch 14: pp 200208 
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 127139 ch 12 pp 178187 (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 13, 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 NonEuclidean Geometry 