Math 452 Ideas for a Math
452 Project
Spring 2008
Tuesday, Thursday in
Malott 230
2:55 to 4:10 PM
Instructor: Bob Connelly (
connelly@math.cornell.edu)
Office: 433 Malott
Office hours: Monday 3-4, Friday 2-3
Last updated:
March 29, 2008
There are a lot of materials that are available in the Math Library in Malott
Hall, and you can look through them to get ideas. You can see the list here
if you put in the course number, etc. Please come to see me before you start
to work.
1. Copy and scan some of the drawings in some of the art books, or others
you can find, paste it into a drawing program such as Geometer's Sketchpad
and use the algorithms in this course to calculate the viewing distance,
center of vision, etc. How tall was the artist?
2. Read the book by David Hockney that claims that many of the great artists
in the past used mechanical devises to enhance their drawings. Do some calculations,
as in Item 1, here, to see if you agree or disagree with his thesis.
3. Take a picture of the cables of the suspension bridge over Fall Creek
and use the method described in class to show that the nodes lie on a parabola.
4. Show that the Pascal's Theorem implies Desargues' Theorem in any projective
plane.
5. Show that Desargues' property for any projective plane implies that there
is a field (or a skew field) that can be used for coordinates.
6. Make your best guess of a model of Brunelleschi's device that he used
to convince people of the principles of perspective drawing.
7. Show how any Euclidean construction with compass and straightedge can
be done with a compass alone.
8. Show how any Euclidean construction with compass and straightedge can
be done with an arc of a circle in the plane, where the center of the circle
is given, but it cannot be done if the center is not given.
9. Prove Descarte's Theorem that shows the relation between four mutually
tangent circles in the plane, show how this can be used to construct Appolonean
circle packings.
10. Analyse and discuss the "impossible pictures" of the Penroses and Escher,
in particular with respect to perspective.
11. Discuss Grunbaum's criticism of the inaccuracies of the Math Accosication's
dodecahedron logo.
12. Explain the connection of the indefinite metric on 3-space and hyperbolic
space and their connection to special relativity.
13. Discuss and explain the Peaucellier mechanism and how it relates to
inversion, how it relates to Watt's mechanism and give an account of the
history with respect to Phillip Davis's book "The Thread".