Math 325

Spring 04

Group Take-Home Test 1

 

The Rules:

 

            1. You may only work with people in your group.  You may come to me for help/clarification/hints (no guarantee that you’ll get exactly what you want…).

            2. You may use your textbook and other references, including the internet; however, you are strongly discouraged from using the internet to search for exact solutions to problems on the test.

            3. When your group is ready, schedule a time for your oral exam.  Exams should take place outside of class and before March 23.

            4. At the oral exam, for each problem I will randomly select one student to present.  After this student is done, others may elaborate, disagree, etc.  Thus, everyone in your group should be ready to present all problems (and to answer questions about the problems).

            5. You may bring visuals, models, etc. to your oral exam.  Visuals can include overhead transparencies, drawings, posters, web pages, etc.

            6. You will get your grade at the end of your oral exam.  Everyone in the group will get the same grade.  Grades will be determined as follows:

            A  Excellent work.  Shows insight into all problems, with very strong work on at least three problems and good work on the fourth.

            B  Very good work.  Shows insight into all problems, with very strong work on at least two problems and good work on most of the rest.

            Incomplete  If you are not up to a B at the first meeting, I will have you reschedule.  You may also choose to get an incomplete instead of a B.  Lower grades will be discussed later, if necessary.

            7.  Enjoy!  This exam is an opportunity for you to work on some interesting and challenging problems.  It is intended to be a learning experience, not just an assessment experience.

 

1.  Descartes’ Theorem

 

            The angle deficit at a vertex of a polyhedron is 360º minus the sum of the measures of the angles of the faces meeting at the vertex.  For example, the deficit at a vertex of a cube is 360º-(3 x 90º) = 90º.  The deficit at a vertex where a regular pentagon, a square, and two equilateral triangles meet is 360º-(108º+90º+60º+60º) = 42º.  In this problem, we work in Euclidean Geometry.

 

a.  The total deficit of a polyhedron is the sum of the angle deficits at all vertices.  Find the total deficit of ten different polyhedra that we’ve studied in class.  They should all be the same number.  Descartes’ theorem states that the total deficit of any convex polyhedron is this number.

 

            The next five parts of the problem outline a method to prove Descartes’ Theorem by adding up all the angles in all the polygons comprising the polyhedron in two different ways.  If you wish, you are free to ignore parts b-f and find your own way to prove the theorem.

 

b.  Find an expression for the sum of the angles in a polygon with e edges.

 

c.  Use the above to express the result of adding up all angles in a polyhedron in terms of E, the number edges in the polyhedron, and F, the number of faces.

 

d.  Now express the angle sum at a vertex in terms of the deficit, d, at that vertex.

 

e.  Use part d. to express the total angle sum in terms of D, the total deficit of the polyhedron, and V, the number of vertices in the polyhedron.

 

f.  Equate the results in parts b and e, and then use Euler’s formula to prove Descartes’ Theorem.

 

g.  Use Descartes’ Theorem to explain why there is no semi-regular polyhedron (Archimedian Solid), with septagons (seven sided regular polygons) as faces.

 

2.  Buckyballs

 

            A buckyball is a polyhedron whose faces are all regular pentagons or regular hexagons, with exactly three polygons meeting at each vertex.  We built buckyballs out of origami units.  In this exercise, you will prove, in two different ways, the remarkable fact that any buckyball must have exactly 12 pentagonal faces.

 

            Let H = the number of hexagonal faces and let P = the number of pentagonal faces in a buckyball.  Let V, E, and F, as usual, represent the total number of vertices, edges, and faces, respectively, in the buckyball.

 

a.  Find an expression for F in terms of P and H.

 

b.  Find an expression for E in terms of P and H.

 

c.  Find an expression for V in terms of P and H.

 

d.  Substitute the three expressions above into Euler’s Formula, and show that P=12.

 

e.  Now, use Descartes’ Theorem to prove that P=12.   You might look at the possible vertices and how much the pentagons and hexagons contribute to the angular deficits.

 

f.  Can you build a buckyball with twelve pentagons and one hexagon?  Do it, or explain why it’s impossible.

 

 

3.  Perspective Drawing in Sketchpad

 

Begin by going to http://mathforum.org/workshops/sum98/participants/sanders/ and try out all the mathematical links in the tutorial.

 

a.  Make a sketchpad drawing of a castle in two-point perspective.  Make it elaborate, and prepare to demonstrate how you constructed the castle and how you can adjust the vanishing line, as in the online applet.

 

b.  Make a sketchpad drawing of something in one-point perspective.  Choose something that has columns, railroad tracks, tiles, or something else that makes one point perspective especially impressive.  Once again, prepare to discuss your construction and to adjust the vanishing point and discuss what happens to the picture.

 

4.  Cylinders and Cones

 

            Explore all aspects of problem 4.1 in Henderson. 

 

Copyright 2005, Debra K. Borkovitz.  You may copy or edit this material for non-profit, educational use only.

 

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