|1||7/8 Pythagorean Activities
Begin Tangram Activities
|7/9 1.1: 5-8, 11, 12, *13||7/10 1.2: 1-3,
Web Surfing Activity.
|7/11Work on a portfolio entry.|
|2||7/15 4.1: 7, 8, 9||7/16 4.1: 14
(based on 7)
5.1: 6 (g,h,i)
4.1: 10, 24
Alphabet and Designs
|7/18 4.2: 9,10
Assignment on symmetry.
Start on Lineland paper-due Monday.
Models of the platonic solids from templates.
Read Plato handout.
Finish 8.1: 6,9
Dual Tessellations.4.1: 24
Begin Plato essay due 7-29
|4||7/ 29 Plato essay||7/30 See assignment on 4th dimension
See assignment on Zeno.
on the torus and maps
|7/26 Symmetry Day.
See assignment on Projection
13.4 : 2-6
|5||8/5 10.1: 2(again),4,5
|8/6 Portfolios due by 5 pm.
See assignment on Projective Drawing from class.
|8/8.See assignment on coordinates
Make a Desargues' Configuration.
|Days||Assnm't Source||Chapter and pages for Reading||Comments, Web Sites to Visit, and other things|
|7-8 to 7-10||Flatland
K & M
|Introduction, Preface, and Part I.
1.1 Measurement pp 1-8
1.1 pp 8-12
1.2 Polygons pp13-16
is available on the web.
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.
|7-10 to 7-15||Flatland
K & M
4.1 Reg. and Semi Regular Tesselations pp 85 - 91
4.1 Dual Tilings pp91- 93
|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...
Tuesday, July 16th is Tessellation Day: Wear to class clothing that has a tiling pattern on it.
|K & M
K & M
|6.1 Flatlands pp 180 -184
5.1 Kaleidoscopes (1 mirror)127-130
5.1 (2 mirrors)130 - 134
5.1 (3 or 4 mirrors) 134 -135
5.2 Point symmetry 138- 146
5.3 Frieze Patterns 147-155
* 5.4 Wallpaper Patterns
*5.5 Islamic Lattice Pattern
|You might want to visit the Kali: Symmetry group page now .|
|4.2 Irregular Tilings pp94-107
*4.3 Penrose Tilings
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
The metaphor of the cave. (On Handout.)
|You might want to look at Penrose tilings by downloading Winlab
You can look at polyhedra by downloading Wingeom by Richard Parris
The Platonic solids is an interesting site with Java viewers for interactive manipulation created by Peter Alfeld of Univ. of Utah.
|7-24 to 7-25||K&M||7.4 Polyhedral Transformations
8.1 Symmetries of Polyhedra
|K&M||6.2 The Fourth Dimension
10.2 Optical illusions
11.2 Map Projections
Encyclopedia: Zeno's Paradoxes and the infinite.
The Fourth dimension
Networks and Euler's formula
Euler's formula, the torus.
A Visualization of 4d hypercube (Java applet).
|8-1 to 8-5||A&S
|pp 1-3, Sections 3,6, 11, 13
13.3 More on Surfaces
|More on Euler's applications.
Symmetry Day: Bring to class an example of a natural or synthetic physical object that has a non- trivial group of symmetries together with your description of those symmetries. You may bring either the physical object itself or a sketch of the object.
Durer and perspective drawing
Surfaces in topology
The Moebius strip, The Klein bottle, orientability, and dimension.
Constructing surfaces in general
The Infinite (Zeno's Paradoxes and the infinite.)
|8-6 to 8-8|| A&S
|Sections 11, 13
|Projection and Ideal elements.
The Color problems .
||Continuation of Projective geometry
Conics, Euclidean and Non-Euclidean Geometry
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.
2. Parallels on the torus and the sphere. Let's call an arc on a sphere
(or the torus) a sline segment
if it arises from a cross section of the sphere
(or the torus) by a plane that passes through the center of the sphere
( or the torus).
A. Draw a figure showing some sline segments on a sphere with the planes through its center and some sline segments on a torus with the planes through its center.
B. Is the following statement true for any sphere? for any torus?
"Any two sline segments on a sphere (or a torus) can be extended so that they will intersect."
3. 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.
4. During the sessions 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.
August 6: Coordinates
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.
1. 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]
2. 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.
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.
Projection. [see Figure 7 in A&S.]
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. Find the points on the circle that correspond to the points on the line as in Figure 7. Describe the relation of a point on the circle to the corresponding point on the line with regard to the point O where the circle and the line touch.
July 301. 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 is incorrectly compared with the accumulation of equally sized intervals.
July 301. 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 similar description of the 8 vertices of the 3-dimensional cube.
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?
A. Describe 5 physical objects that have surfaces that are topologically equivalent to a (one hole) torus. Bring one example to class on Tuesday.
B. Describe 5 physical objects that have surfaces that are topologically equivalent to a torus with two or more holes. Bring one example to class on Tuesday.
July 31 1. Make
two tori: one from two annuli, one from a single "rectangle."
2. Bring in 3 different world maps. Describe how each map deals with lines of longitude, latitude, and the poles.
Due Monday 7-29 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.]