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 K & M's treatment of question A. [Two paragraphs are sufficient.]
II. Kinsey and Moore. Show your work, explain your thinking!
1.1: 5-8, 11, 12, [*13 optional]
4.1: 7, 8, 9,10
5.1: 6 (g,h,i)
Classifications by symmetry:
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 one rotational symmetry of order 2 (a half turn).
Group the following letters together in different classes determined by the number and types of symmetries they have as printed on this page. [It is up to you to determine the appropriate classes.]
4.2: 9,10, 12
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
Models of the platonic solids from templates.
1.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?
2. 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.
Cave Metaphor- online.
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.]
|10-23 Higher dimensions.
Make two tori: one from two annuli, one from a single "rectangle."
1. The fourth dimension can be used to visualize and keep track of many things involvimg 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 context the point with coordinates ( 2, 4, 6, 1) might represent a hand with 2 clubs, 4 diamonds, 6 hearts and 1 spade.
Using this convention discuss briefly the following representations of bridge hands: (0, 0, 0, 13), (0, 0, 6, 7), (3, 3, 3, 4).
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 some features of the context.
2. Hypercubes in Higher Dimensions.
The 16 vertices of the 4-dimensional hypercube can be described by the collection of ordered quadruples (a,b,c,d) where the numbers a, b, c, and d are either 0 or 1.
Write a description of the vertices of the 5-dimensional hypercube. How many vertices does the 5-dimensional hypercube have? How many vertices does the 6-dimensional hypercube have? How many vertices does the 10-dimensional hypercube have? What can you say about the vertices for the hypercube of dimension N?
||10-30 Short paper on "Zeno's
Paradox." Look up "Zeno's Paradoxes" in the Encyclopedia (Britannica). Draw a figure that
illustrates the paradox of Achilles and the Tortoise. Describe a common
context 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 (geometric series) is incorrectly compared with
the accumulation of equally sized intervals (arithmetic series).
1. Find and "copy" 3 different world maps. Describe how each map deals with lines of longitude, latitude, and the poles.
Central Projection. [see Figure 7 in A&S.]
2.A On a line mark 11 points that are separated one from the next by one inch. At the middle point draw a circle of radius one inch as in Figure 7.
B. Find the points on the circle that correspond to the points on the line as in Figure 7.
1. Suppose three lines l , m, and k form a triangle. [Draw a large figure to illustrate this situation.] Draw ten points on line l perspective with 10 points on line m with center O. Use these ten points on line m to draw 10 points on line k in perspective with center O'. Draw the lines connecting the corresponding points on line l and k. Describe the figure that these lines suggest.
2. Draw a figure showing a tessellation of the projective plane on one side of the horizon line by parallelograms.
3. Draw a figure illustrating a black and white chess board in perspective.
1. Perspective in Visual Arts.
Find two drawings, paintings, prints, or photographs that have noticable perspective in the composition. Make a sketch or photocopy of the works and locate at least one "infinite" point on the horizon (ideal) line on each of your figures. Find at least three lines in each of your figures that meet at the infinite point.
Coordinates and conics.
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.
2.. Each of the following equations determines a conic curve. Plot 10 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]
3. Draw three separate projective planes including a system of coordinates 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 10 points plotted previously on the standard plane. Connect these points with straight line segments in the projective plane.
||25 No class
||27 No class. Thanksgiving!
With a partner- make a model of a Klein Bottle. [ See Barr Ch. 5 and Appendix pp202-3.
Final Writing Assignment.
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.
|Week||Assnm't Source||Chapter and pages for Reading||Comments, Web Sites to Visit, and other things|
|8-26- to 9-4||Flatland
K & M
|Introduction, Preface, and Part I.
Preface and "To the Reader".
1.1 Measurement pp 1-8
1.1 pp 8-12
1.2 Polygons pp13-16
is available on the web. Perception
Over 30 proofs of the Pythagorean theorem!
Many Java Applets that visualize proofs of the Pythagorean Theorem
Japanese Site with Tangram Puzzles on-line
Here's a web page with many annotated Tangram references
Web references related to scissors congruence- dissections.
|9-4 to 9-11||Flatland
K & M
4.1 Reg. and Semi Regular Tesselations pp 85 - 91
4.2 Irregular Tilings pp94-96
|A wealth of materials can be found by going to this Tesselation
This might be a good time to visit Rug patterns and Mathematics exhibit plus...
|9-11 to 9-23||K & M||6.1 Flatlands pp 180 -184
4.1 Dual Tilings pp91- 93
5.1 Kaleidoscopes (1 mirror)127-130
5.1 (2 mirrors)130 - 134
5.2 Point symmetry 138- 146
|You might want to visit the Geometry Center's Introduction to Tilings as well as the Kali:
Symmetry group page now .
Thursday 9-18 is Tessellation Day: Wear to class clothing that has a tiling pattern on it.
|9-23 to 9-30
||6.1 Flatlands pp 180 -184
4.2 Irregular Tilings pp 97-107
5.3 Frieze Patterns 147-155
[* 4.3 Penrose Tilings ]
[* 5.4 Wallpaper Patterns ]
[*5.5 Islamic Lattice Pattern]
|You might want to look at Penrose tilings by downloading Winlab
The Kali: Symmetry group page is still of interest!
|9-30 to 10-9
||7.1 Pyramids, Prisms, and Anti Prisms pp 208 -215
7.2 The Platonic Solids pp 216-221
7.3 Archimedean Solids pp 224-228
8.1 Symmetries of Polyhedra
|You can look at polyhedra by downloading Wingeom
The Platonic solids is an interesting site with Java viewers for interactive manipulation created by Peter Alfeld of Univ. of Utah.
|10-9 to 10-21
7.2 The Platonic Solids pp 216-221
7.3 Archimedean Solids pp 224-228 AGAIN!
7.4. Polyhedral Transformations pp230-234
Read Plato's Cave Metaphor- online.
||6.2 The Fourth Dimension
Encyclopedia: Zeno's Paradoxes and the infinite.
The Fourth dimension
A Visualization of 4d hypercube (Java applet).
continue to 11-11
10.2 Optical illusions
11.2 Map Projections
pp 1-3, Sections 3,6, 11, 13
|More discussion on Zeno's paradox
A philosophical view on Zeno's Paradox
Try Zeno's Coffeehouse for more logical paradoxes.
Maps and coordinates
| Sections 11, 13
13.3 More on Surfaces
Surfaces in topology
The Moebius strip, The Klein bottle, orientability, and dimension.
Constructing surfaces in general
13.2 -13.4, pp220-222 on Euler's Formula
|Graphs and Surfaces in topology
The Moebius strip, The Klein bottle, orientability, and dimension.
Constructing surfaces in general. The Projective plane.
The Color problems .
|K&M:11.5 looks at surfaces os soap bubbles.
Barr : 9 is an interesting excursion.
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.