SEMESTER PROJECT



• 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 or

• 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 (), Dale Seymour Publications (), or Key Curriculum Press () 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.

 

 

 

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