Math 105 Fall, 2005
Resource List
Final Assignment #9.
[May be submitted until 12-15]
Over the term 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.
Assignment
#8: November 17 - November 29 .
Read Devlin: Pp. 74-79 (Zeno's Paradoxes and the infinite.)
Pp.
115-119 (Conics and cartesian geometry)
Pp. 129 - 133;134-138
(Projective Geometry)
Pp.
106-109 (Euclid's Axioms)
Pp. 122-129
(Non-Euclidean Geometries)
Exercises/Activities: To be collected on Tuesday, November
29
I. Look up "Zeno's paradoxes" in the Encyclopedia (Britannica).
Draw a figure that illustrates the paradox of Achilles and the
Tortoise. Describe an common situation 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.
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.
II. Each of the following equations determines a conic curve.
Plot 8 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]
III. Draw a system of coordinates on the projective plane 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 8 points plotted
previously on the standard plane. Connect these points with straight
line segments in the projective plane.
IV. Discuss the following statements, drawing figures to illustrate
your remarks:
In the projective plane, any conic curve will look like an ellipse. The
main distinction between the curves is the number of ideal points that
lie on the conic.
An ellipse will have no ideal points, a parabola will have one ideal
point, and an hyperbola will have exactly two ideal points.
Assignment
#7: November 3 -15 .
Read Devlin: pp. 138-141 (Dimension)
pp.
179-182.(The Moebius strip, orientability)
pp.
182-186, 187(1st paragraph.) (Surfaces)
Exercises/Activities: To be collected on Tuesday, November
15
I.Cubes.
See page 140 in Devlin.
Sketch the face-on view of the cube and the hypercube.
Label the coordinates of your sketches.
Determine the number of vertices, edges, faces, and cubes in the
hypercube
.
II. Maps.
Find three different types of world maps. Copy each by tracing or photo
or by “cut and paste”. Write a short report (about one page) describing
how both the poles and the global coordinates determined by longitude
and latitude are visualized on each planar map.
III.Surfaces.
A. Describe 5 physical objects that have surfaces that are
topologically equivalent to a (one hole) torus. Bring one example to
class on Thursday.
B. Describe 2 physical objects that have surfaces that are
topologically equivalent to a torus with two or more holes. Bring one
example to class on
Thursday.
IV. Coloring the Torus and the Moebius Strip.
A. Draw a map on the Torus with 5 regions, each having a border with
the
other 4.
B. Draw a map on the Moebius Strip with 5 regions, each having a border
with the other 4.
V. The Fourth Dimension.
The fourth dimension can be used to visualize and keep track of
many things that involve 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 situation the point with
coordinates ( 2,4,6,1) might represent a hand with 2 clubs, 4 diamonds,
6 hearts and 1 spade.
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
the context.
Assignment #6: October 25-November 3
I.Group the following letters, as printed on this page,
together in different classes determined by whether they are
topologically equivalent. [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 (A
recycled problem) 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.
Assignment #5: October 11-October 25
Read Devlin: pp. 174 - 175; 176 - 178
(Topology, Networks, and Euler's formula)
pp. 188-189
(The Four Color Problem)
Exercises/Activities: To be collected on Tuesday, October
25
I. Networks and Euler's Formula.
A. A message has arrived from the planet Yxo describing a road network
connecting 8 major cities on the planet. The message claims that the
network
has exactly 15 edges and divides the planet into 6 separate regions. Based
on Euler's formula discuss the following statement:
Either the planet is not spherical in general shape or the
information is incomplete.
B. A planar graph has been drawn by a designer using exactly 12 edges
and 8 vertices. How many regions are created by this graph?
Explain how you arrived at your count.
Draw a graph that is evidence that such a graph is possible.
II. Networks As Models.
Choose a floor in a building on the Occidental campus.
A. Describe the layout of this floor as a network. Include a sketch of
the network together with a legend explaining the correspondence of
vertices
and edges to the building's features.
B. Suppose there is an electrical blackout at night. Discuss how you
could use your network description to help find your way through the
floor to
leave the building.
C. Compare the situation of the floor layout network to Devlin's London
Underground Network. How are their uses alike? How do they differ?
III. Projections and Cut-up Flattened Semi-Regular Polyhedra
Frameworks.
Draw cut-up flattened frameworks or networks for the following
semi-regular
polyhedra. Iidentify the edges that would be pasted together to
reassemble
the framework in space.
A. The truncated cube. B. The
truncated octahedron.
Assignment #4: September 27 - October 11
Tessellation Thursday, October
6th.
On Thursday,
October 6th, wear to class clothing that has a
tiling (wallpaper) pattern on it.
Read Devlin:
pp. 112-115 (Euclid- inscribed angles in semicircles,Platonic
solids, Plato, and Kepler)
pp 129-132 (Introduction to the geometry of projection)
Plato's Cave Metaphor-
online.
Exercises/Activities: To be
collected on Tuesday October 11.
I.Creating new
tessellations by modifications.
Create a tessellation of curved figures following the ideas from the
activity of September 29th applied to the regular tiling of the plane
by (1) hexagons and (2) equilateral triangles.
II. The Sphere has brought the Cube to visit
his new friend in Flatland. Describe two possible successions of
different planar shapes the Cube might appear as while passing
through Flatland.
Draw the sequences as they would be seen in
Flatland.
III. Projections: 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?
IV. 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.
V.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.]
Assignment #3:
September 20 - September 29
Read Devlin: pp. 153-157 (lattices and sphere packing);
pp. 163-164 (wallpaper patterns); pp. 144-150 (Symmetry Groups again!); pp.165-170 (tiling again!).
Exercises/Activities:
To be collected on Thursday September 29.
I. 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
II. Dual tessellations and symmetry.
A. Draw the tessellation of
the plane with equilateral triangles. Place a point in the
center of each triangle and connect these points with a line segment
when they are
in adjacent triangles. Describe the symmetries of the resulting tiling
of
the plane.
B. Draw the
semiregular tessellation of the plane that uses squares and
octagons.[See Devlin p.166.] Place a point in the center of
square and octagon and connect these points with a line segment when
they are in adjacent figures. Describe the symmetries of the resulting
tiling of the plane.
III. [This problem may be done with a partner]. Symmetry group
of a square.
There are 8 symmetries for any square.
A. Provide a sketch of the square and illustrate the
8 symmetries as in the triangle example. [See the class notes.]
B. Determine the "multiplication table" for the
group of symmetries of the square. [See Devlin page 150]
Assignment #2: September 8 - September 20.
Read Flatland:
Part II.
Read Devlin: pp.
144-150 (Symmetry Groups); pp. 165-169 (Tiling).
Exercises/Activities: To be collected
on Tuesday September 20.
I. Using the seven pieces of the tangram puzzle create
A. A rectangle and B.
A right triangle.
II. Suppose that the square made using the seven tangram pieces has a side of length 4.
A. Find the length of the sides of each of the seven pieces.
B. Find the area of each of the seven pieces.
III. Classifications by reflection
symmetries:
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 "J"
has no reflection symmetry.
Group the following letters, as printed on this page,
together in different classes determined by the number of reflection
symmetries they have. [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
IV. Assume the following pattern is extended indefinitely on the plane.
This pattern has many lines of reflectional symmetry. Draw in all
the lines of symmetry using colored pencils so that any two lines with
the same color are parallel and any two lines that are parallel are
colored with
the same color.
V. Find (or create) a graphic design (in advertisements, logos, or
icons) that
(i) has exactly one reflection symmetry, (ii)
has exactly two reflection symmetries, and (iii) has exactly three
reflection symmetries.
Indicate and discuss the symmetry line(s) in
each separate design.
Assignment
#1: September 1 - September 8.
Read Flatland. Introduction, Preface, and Part I.
(Activity and assignments on Flatland will follow next week.)
Read Devlin:
Prologue, pp. 1-7.
Chapter 1,
Greek Mathematics, pp. 14 - 18, 21, and 31.
Exercises/Activities: To be collected on Thursday, September
8.
I. Ask your adviser or an
instructor you know in your major the following two questions:
A. How do you define mathematics? (or What
is mathematics?)
B. Can you give one example of how mathematics is
used in your discipline? Ask for an explanation of the response.
Report the responses to these two
questions and relate them to Devlin's treatment of question A.
[Two paragraphs are sufficient.]
II. [See Devlin, p.21]
Draw a rectangular figure arranging 30 discs into 5 rows.
A. Draw a straight line through the rectangle that
divides the discs into two groups each with 15 discs.
B. According to the formula on Devlin page 21, using
n = 5,
1 + 2 + 3 + 4 + 5 = 5 (5+1) / 2.
Explain the relation of this equation to the arrangement of the 30
balls in part A.
C. Describe an analogous figure to explain why
1 + 2 + 3 + ⋯ + 98 + 99 + 100 = 10,100/2 = 5,050
Other resources: 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
Web
references related to scissors congruence- dissections.
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.
- 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 chapter.
- Use my collection of Visual
Mathematics web sites for surfing visual mathematics and geometry.
- Use articles from old Scientific American magazines
(available
on-line?)
- (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.
- There are several non-text mathematics books and collections
of essays.
- Mathematics: The Science of Patterns by K.
Devlin
- Beyond the Third Dimension by T. Banchoff.
- 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