 M333L Project —SPRING 2005
Goals






To gain experience researching a math topic on your own.
To become more expert in some topic in geometry.
To acquaint you with math topics beyond the standard curriculum.
To gain experience in communicating your ideas to others.
To have a sense of pride in a job well done.
To have fun!
Requirements
You will be required to prepare a written report of your findings. You will also be
required to present your findings in class in the form of a short oral presentation. Projects
will be judged on both the written report and quality of the presentation. You will receive
a presentation grade, which will be 15% of your project grade and a written report grade,
which will be 85% of your project grade.
Time Table/Steps involved in preparing project
1. Select a topic that interests you (very important!) and that you can make
interesting to others. Consult the Topics List and Suggested Sources List below
for possibilities. I will also post a rather long bibliography for topics on the web
page. Begin to find information about the topic. Formulate one or more questions
that you would like answer through your work on this project. The questions
should be mathematical in nature. Make sure that the questions are broad enough
to allow you a chance to do some in-depth exploration. Talk (or send email) to me
at anytime if you are having difficulty with this process.
Due Wednesday, February 9 - Your choice of topic together with the questions
that interest you. Also include a short summary (less than a page) of why you
think it is a good topic and why these questions are of interest to you (enough to
show you have done some research). Finally, give at least two references (more if
references are very short). References to a book must include the title and page
numbers of chapters or sections relevant to the topic. References to journals must
include title of article relevant to the topic. Internet references must include an
author, title, and a current URL.
Oral presentations will be given on April 27, April 29, May 2, May 4, and May 6.
If you have a conflict with any of those days state so now (with a reason) and I
will try to work around that when choosing dates.
2. Continue finding as much information as possible on your topic. Begin making
models and drawings, gathering data, working constructions and examples,
looking for patterns, using Sketchpad or computers if appropriate, organizing
material and whatever else helps you to understand things. Do you have an
answer to your question(s)? Can you make any links to more familiar material or
topics? Can you make any connections to the real world? Does your topic allow
for sufficient mathematical development of an area that is new to you? If you
found that you would like to change your project direction or topic that you
previously submitted, that is fine. However, by March 2 you really should be
committed to a particular project,
Due Wednesday, March 2 - A one-page description of the project area and the
broad limits within which you are working. If possible, give an outline for your
project. Include a description of what you have read or done so far. Include what
progress you have made towards answering your questions. Include a brief
description of the mathematical results pertinent to your topic.
3. Prepare a first draft of your written report. Your report should be in the 8-10 page
range, double-spaced. If you have done some work with Sketchpad or computers,
you would of course include some figures in your report. If you have many
figures, examples, etc. I would list those in an Appendix. See the Evaluation
section below for more specifics on what I will look for in your report.
Due Wednesday, April 13 —First draft of the paper is due. Include a complete
bibliography and an outline of the paper. Once this is returned to you, begin the
process of revising the paper. You should expect to do additional research and a
substantial rewrite of the paper.
4. Start to think about your presentation. The format should be the following. First,
briefly convey the flavor/main ideas of your project. Then focus in on one specific
aspect with the idea of teaching the class something new and interesting. Try to
engage the class. You should allow time for questions, so the actual presentation
should last 7-10 minutes. (This is not very much time!) We’ll have 12 minutes for
each student. See Tips for Presentations below.
Due Wednesday, April 20 —Outline of the presentation is due. Include an
estimate of the time your presentation will take. Describe the medium you will
use (whiteboard, overhead, projector) and any handouts you intend to circulate.
5. Put the finishing touches on your revised report and presentation. Think of good
ways to answer any questions that viewers might ask about your topic.
Due April 27, April 29, May 2, May 4, and May 6— Projects will be presented
on these dates (4 students per day 12 minutes each). All written reports will be
due on May 6, but you can turn in your report early if you wish. Presentation
dates will be chosen semi-randomly (with the special requests submitted on
February 25 taken into consideration). LATE PROJECTS WILL NOT BE
ACCEPTED.
Evaluation
Written Report: I will determine each written report grade. In your report I shall be
looking for the following:








Is the main problem/question(s) clearly stated in the introduction? Has a summary
of how the problem will be approached given? Is the report well organized?
Has the topic has been researched and developed well enough? Have all
reasonable lines of inquiry been investigated and if not, have choices been
explained? Was there linking of familiar topics or connections to real world
examples, if appropriate?
Are the answers well supported? Have examples been given or worked through?
Is there mathematical content and is it correct? Are the definitions stated
correctly? Are the variables defined? Are assumptions underlying formulas
stated? Is each formula derived or provided a reference? Are the arguments
correct?
Does the paper explore an area of geometry that goes beyond the topics covered
in class? Has the student demonstrated a clear understanding of this area?
Is there a creative aspect to the project? Is an original approach used to solve the
problem? Are effective or original models/examples used? Has technology been
used creatively?
Are spelling, punctuation, and grammar correct? Is acknowledgment given where
necessary? Is everything labeled properly?
Were the assignments related to the project due on February 9, March 2, April 13,
and April 20 completed and turned in on time?
Presentation: The class will determine each presentation grade. Each student and I will
assign a score of 0-5 to your presentation and the average will determine your
presentation grade. When assigning a presentation score you should consider the
following:





Did the presentation first present the main theme of the topic and then focus on
one particular aspect?
Did the presentation capture and hold your attention?
Did the presentation engage the audience and get them involved?
Were the visual aids well chosen and appropriate?
Did the presentation facilitate following and understanding?
Attendance is required on presentation days and any absences must be justified.
Sources



Advanced Euclidean Geometry by Alfred Posamentier. If you purchased the
optional text, you will find it a good source of topics with strong mathematical
content.
Geometry by Discovery by David Gay. On reserve in the PMA for M333L. You
can take it out overnight. A GREAT book for potential projects — one could
easily work through a chapter and get a nice project out of it.
Journals in UT Libraries. Most journals have an index in the last issue of each
volume. Some have a five- or ten- year index
MathematicsMagazine (PMA)
TheCollege Mathematics Journal (PMA)
Mathematicsin Schools (PCL)
MathematicsTeacher (PCL)
MathematicsTeaching (PCL)

The QA 445 — 473 shelves of the Undergraduate Library (UGL)

The QA 93 — 95 and QA 440 — 453 shelves of PMA Library (Some of these
will be too advanced, but some will be at the right level.)

Books by Martin Gardner.

Encyclopedias (regular, scientific, and mathematical). There are several
mathematical dictionaries and encyclopedias in the reference shelves of the PMA
library, call numbers beginning QA 5.

The World Wide Web. Try http://forum.swarthmore.edu/math.topics.html or
http://archives.math.utk.edu/index.html
The course web page. I have posted an extensive bibliography compiled by
Martha Smith (UT Math Dept).


Books published by Dover Publications, the National Council of Teachers of
Mathematics (http://www.nctm.org), Dale Seymour Publications
(http://www.cuisenaire-dsp.com), or Key Curriculum Press
(http://www.keypress.com) are often inexpensive and can easily be purchased by
mail order, if you do so early.
Possible Topics
You are free to choose any topic you wish depending on your interests, your reasons for
taking this class or your future career interests. You need to be interested in the topic you
choose — so most anything goes as long as it is related to geometry. Your project must
involve some mathematics (An A or B project will involve substantial amounts of
mathematical reasoning). For example, if you choose the topic "Geometry in African
Art", then the bulk of your paper could involve patterns or tilings and you could use
examples of African art to illustrate the mathematical results in your paper. If you are
interested in teaching/education your project could be from that perspective. I’ve
included some suggestions below, but you aren’t limited to those. Those topics with an *
are especially suited to computer display or work with a computer and others may be as
well. Possibilities here include:

Abstract symmetry in mathematics and physics (Topics ranging from symmetric
conditions in a theorem to symmetric properties of equations to symmetry in
particle physics)

The arbelos ( The region bounded by three suitably placed semicircles — studied
by Archimedes)

Collinearity of points (Simson’s Theorem and others)*

Concurrrency of lines (Ceva’s Theorem and others)*

Conic Sections (They have many interesting properties and applications.)*

Constructing geometric figures from string (Many curved solids actually have lots
of straight lines on them, and so models of them can be constructed of string or
wire.)*
Constructing regular polygons (Lots of history, and connections with number
theory.)


Constructing triangles (Various combinations of data, for example the lengths of
two sides and the altitude to one of these sides, determine a triangle. But how do
you construct the triangle given these data?)*

Constructions with compass only

Curves of constant width

Cycloids and related curves

Dissecting geometric figures (Figures can sometimes be dissected and rearranged
to form other figures. Also, some figures can be dissected into congruent parts
— sometimes even congruent to the original figure.)

Duality theorems (Sometimes theorems about lines and points are true if the roles
of the lines and points are interchanged.)

Duplicating the cube (It is not possible to construct, with just ruler and compass, a
cube having twice the volume of a given cube. Why not? Can this be done by
other means?)

Euclid’s Elements

Euler’s formula (It relates the number of vertices, edges and faces of a
polyhedron.)


Fivefold symmetry (It occurs frequently in flowers, fruits, and molecules, but not
in crystals. However, it recently was discovered in "quasicrystals".)
Fractal geometry*

Geodesic structures

Geometric constructions by paper folding (Many ruler-and-compass type
constructions can be done this way, as well as constructing curves by folding lots
of tangents to them.)

Geometry in African art

Geometry in ancient China

Geometry in ancient India

Geometry in ancient Japan

Geometry in archeology

Geometry in architecture

Geometry in astronomy (Shapes of orbits of heavenly bodies; astronomical
calculations)

Geometry in higher dimensions (Four --- and beyond?)

Geometry in machinery (Related to linkages.)

Geometry of animal form and function.

Geometry of billiards*

Geometry of crystals

Geometry of Escher prints (Overlaps with tilings, wallpaper patterns)

Geometry of Galileo

Geometry of genetics (Overlaps with knots.)

Geometry of Honeycombs

Geometry of Islamic art (Islamic Art tends to be very geometric in nature. This
topic overlaps with wallpaper patterns, strip patterns, and tilings.)

Geometry of knots (This has been an important area of research mathematics
in recent years, but many of the basic ideas are easy to understand. Knot
theory has applications to DNA. Knots also appear in artwork, especially Celtic.)

Geometry of molecular structures (Overlaps with crystals.)

Geometry of optics (Reflective and refractive properties of lenses and surfaces
involve a lot of geometry)*

The geometry of perspective drawing*

Geometry of robotics

The geometry of solar energy devices. (Overlaps with geometry of optics, conic
sections)

Geometry of textiles (Geometry occurs both in printed or woven patterns and
in the knitting process. Topic overlaps with knots, patterns.)

Geometry of the kaleidoscope (Related to tilings and optics.) *

The Golden Ratio in geometry

Inversion (A way of transforming the plane so circles and lines go to circles
and lines, but not necessarily in that order. Related to complex numbers
and linkages.)*

Linkages and geometry (Linkages are mechanical devices that do things like
convert straight line motion to circular motion. Related to geometry in
machinery and to inversion.)

Map coloring (How many colors do you need to color a map so that no two
adjacent countries are the same color? What if the map is on a sphere? On a
doughnut? Lots of history and connections with other topics in mathematics.
Overlaps with topological surfaces.)

Minimal surfaces (e.g., soap bubbles)

Networks (Steiner points arise in building a shortest network of rail-lines linking
3 cities. One way of visualizing solid figures in three dimensions is to realize
them as a network in the plane — that’s what you do every time you draw such a
three-dimensional figure in the plane To study Platonic solids such as cubes,
tetrahedra, etc we can study the corresponding networks in the plane. The Euler
characteristic becomes important again here.
Plane geometry. One could take results from Euclidean plane geometry and
develop presentations of them exploiting SketchPad. This is a fascinating topic
because of the richness of results in Euclidean plane geometry and the emphasis
on constructability.


Polyhedra (Properties, symmetries, classification, and constructions)

Projective geometry (Related to perspective drawing)*

Proofs of the Pythagorean theorem (There are lots of them!)*

Proofs without words (A picture is worth a thousand … )


Quadric surfaces (Overlaps with constructing geometric figures from string)
Space tessellations (The same idea as tilings of the plane, but using solid shapes
to fit together to fill up space. Related to crystals.)


Sphere packing
Spherical geometry

Spirals (Overlaps with Golden Section, Fractals)*

Squaring the circle (Can you construct a square whose area is the same as a given
circle? Not by ruler and compass. Are there other means?)

Strip (frieze) patterns (Patterns that repeat in one direction, as in borders and tire
treads)

Taxicab geometry (What if distances could only be measured by going in two
directions, as a taxicab in a city with regularly laid out streets must do?)
Tilings (Also known as plane tessellations. What shapes or combinations of
shapes can fit together, as tiles do, to cover a plane surface uniformly? What
about other surfaces?)*. As we shall learn in class, there are 17 different
categories of plane tilings using ONE color. But it is known that there are 46
different categories using TWO colors.
Topics related to teaching geometry. (Many possibilities here - using Dynamic
Software in the classroom proofs in geometry, etc.)
Topological surfaces




Trisecting the angle. (This is not possible by straightedge and compass, but is
-- to some extent — by other means.)

Wallpaper patterns (Patterns that repeat in two directions)
Tips for Presentations






The oral presentation should communicate effectively the main idea of the project
and its flavor. After, briefly describing the overall project, choose ONE particular
idea that is easy to communicate. Have the goal in mind of teaching your
audience something new and interesting.
You may use the blackboard, overhead projector, and/or computer projector
during your presentation. (If there is some other equipment that you would like,
let me know, and we’ll see if it can be arranged.) There is no requirement that
technology be used in your presentation but would recommend at least using the
overhead projector and prepare a few slides in advance.
Strike a balance between the trivial and complex. For example, don’t belabor the
obvious and don’t subject your audience to a detailed proof. Don’t overwhelm
your audience with jargon, you are an expert on your topic but your audience is
not.
Allow time for questions. Each student will have 12 minutes, so allow 7-10 for
the presentation and the rest for questions.
Your presentation will be more successful if you let the audience participate.
Practice. Time it. Practice again.