Math 103 Summer, 2003
work in progress!
SUBJECT TO REVISION!
Homework Assignments due:
(* = interesting but optional)
General Reading Assignments (Revised 7-6-2003)
||7/7 Pythagorean Activities
5-8, 11, 12, *13
||7/9 1.2: 1-3,
Web Surfing Activity.
|7/10Work on a portfolio entry.
4.1: 7, 8, 9
||7/15 4.1: 14
(based on 7)
4.1: 10, 24
5.1: 6 (g,h,i)
Design Symmetry Assignment
(now due 7/17)
|7/17 Tesselation Thursday!
Assignment on symmetry.
Start on Lineland paper-due
||7/21 4.2: 9,10
Models of the platonic solids from templates.
Cave Metaphor- online.
Finish 8.1: 6,9
Begin Plato essay due 7-28
|7/24 Dual Tessellations.
||7/ 28 Plato essay
||7/29 See assignment on
Make two tori: one from two annuli, one from a single "rectangle."
assignment on the torus and maps and surfaces
See assignment on Zeno.
|7/31 Symmetry Day.
See assignment on Projection
13.4 : 2-6
||8/4 10.1: 4,5
|8/5 Portfolios and projects due
by 5 pm.
|8/6.See assignment on coordinates
Make a Desargues' Configuration.
See Final paper assignment
||Chapter and pages for Reading
||Comments, Web Sites to Visit, and other things
|7-7 to 7-9
K & M
|Introduction, Preface, and Part I.
1.1 Measurement pp 1-8
1.1 pp 8-12
1.2 Polygons pp13-16
is available on the web. Perception
Over 30 proofs of
the Pythagorean theorem!
Applets that visualize proofs of the Pythagorean Theorem
Japanese Site with Tangram Puzzles
Here's a web page with many
annotated Tangram references
references related to scissors congruence- dissections.
|7-9 to 7-14
K & M
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
This might be a good time to visit Rug
patterns and Mathematics exhibit plus...
Thursday is Tessellation
Day: Wear to class clothing that has a tiling pattern on
|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 .
|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
of the cave. (On line.)
|You might want to look at Penrose tilings by downloading Winlab
You can look at polyhedra by downloading Wingeom
Platonic solids is an interesting site with Java viewers for
interactive manipulation created by Peter Alfeld of Univ. of Utah.
|7-23 to 7-24
||7.4 Polyhedral Transformations
8.1 Symmetries of Polyhedra
||6.2 The Fourth Dimension
10.2 Optical illusions
11.2 Map Projections
Paradoxes and the infinite.
The Fourth dimension
Networks and Euler's formula
Euler's formula, the torus.
Visualization of 4d hypercube (Java applet).
|7-31 to 8-4
|pp 1-3, Sections 3,6, 11, 13
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.
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.)
|8-5 to 8-7
|Sections 11, 13
|Projection and Ideal elements.
Color problems .
||Continuation of Projective geometry
Conics, Euclidean and Non-Euclidean Geometry
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.
Several chapters from the course text will not be covered in class but
can be used for portfolio entries. An entry based on our text should report
on a selection of the included exercises along with the content of the
Use my collection of Visual
Mathematics web sites for surfing visual mathematics and geometry.
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
For All Practical Purposes (COMAP)
Some of the history of mathematics videos from the Open University Series
There are several non-text mathematics books and collections of essays.
Two of these are - K. Devlin's Mathematics: The Science of Patterns
and T. 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
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
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
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.
Due Thursday 8-7.
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.
Due Wednesday 8-6:
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
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
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.
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
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.
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 30. 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.
Tuesday, July 29 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
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?
Wednesday, July 30. 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
Bring in 3 different world maps. Describe
how each map deals with lines of longitude, latitude, and the poles.
Thursday, July 31st, 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.]
Wednesday, July 30. Casting Torus 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?
A. Draw the three regular tilings and one semiregular tiling of
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
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.]
7-21 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
7-16&17-03 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
7-16&17-03Classifications 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