Dr. Sarah's Math 3510 Class Highlights
### Dr. Sarah's Math 3510 Class Highlights Spring 2002 Page

The following is not hw unless you miss class.
See the Main Class Web Page
for due dates.

**Mon May 6**
Maple Demo on
Rotations and the Space Shuttle
If time remains, then begin
Maple Demo on
Quaternions

**Wed May 8**
Finish Maple Demo on
Quaternions
and review Rotations and the Space Shuttle.
Apollo 13
and gimbal lock.
WebCT test
on Geometry and Numbers and Rotations and the Space Shuttle.

**Mon Apr 29**
Geometry and Number Theory via
summary of Wiles and Fermat's Last Theorem and
the article in Geometry at Work.

**Wed May 1**
Finish up Geometry and Numbers via article in Geometry at Work.
WebCT test 4 on
Computer Learning to Diagnose and Categorize and Geometry and Probability.
Work on draft 2 of your worksheet.

**Fri May 3** No class - instead, arrange to swap
worksheets with your partner so that you can complete their worksheet
and give suggestions for improvement (due Monday and you will be graded
on this process).

**Mon Apr 22**
Discuss worksheet guidelines
and goals for the drum and sum of the angles in a spherical triangle
worksheets.
Computer Learning to Diagnose and Categorize.
In the process, review the dot product, proof that the dot product of
2 vectors is equal to the product of their norms times the cosine of the
angle between them, and the equations of planes in higher dimensions via
vectors. Look at data set on heart disease from Netscape:
Heart Disease Data main page,
Heart Disease Data description,
the actual data shortened to the
14 attributes used from the Cleveland Data

**Wed Apr 24**
Review the actual data shortened to the
14 attributes used from the Cleveland Data
and discuss how to place it into Excel via saving it into word
and replacing , with ^t, and then pasting it into Excel.
WebCT test 3.
Using Netscape, pick a topic other than Heart Disease from the
UCI Machine Learning Repository Content Summary
and investigate the data.
Begin geometry and probability:
To open a .gsp sketchpad link,
from NETSCAPE, click on the link. You will see some garbage text symbols. Under file, click on Save as... Double click on the public folder/Save
files here folder which is located on the desktop. When you are inside that folder you can save the file. Then open up Sketchpad (GSP 4.00) and under
File, open, find your file and open it.
Use this to open
Buffon's Needle
(adapted from Paul Kunkel's Sketchpad)
, open it
with Sketchpad 4, and then go through the pages in order to begin
geometry and probability.

**Fri Apr 26**
Review Buffon's Needle
(adapted from Paul Kunkel's Sketchpad)
and the
proof of Buffon's Needle.
Attempts to
use
this to estimate Pi.
As a practical matter,
Buffon's needle experiment is not a very efficient method of approximating pi.
According to Richard
Durrett, to estimate pi to four decimal places with L = 1 / 2 would
require about 100 million tosses!
If time remains, conclude geometry and probability via spinners.

**Mon Apr 15**
Finish up hearing the shape of a drum worksheet and turn this in -
geometry and biology continued -
the structure of viruses.

**Wed Apr 17**
Review Dr. Holly Hirst's
Writing Mathematics.
Copy and paste the text from the
beachball activity into Word.
Use Equation Editor to change the math into nice looking math symbols.
Save often.
Copy and paste the pictures into Word in the relevant places.
Work on formatting issues to turn this into a professional looking worksheet.
As part of this process,
write an introduction to the worksheet that places it in
the context of spherical geometry so that it can be a self contained
activity.
(Recall that I was trying to limit the sheet to 1 double sided page,
and that I wanted to place it on the web as text, and so I sacrificed
professional writing guidelines in order to do so.)

**Fri Apr 19**
Finish up geometry and biology via
DNA and Knot Theory,
Voronoi diagrams in Biology,

**Mon Apr 8**
Divide up into groups of two and
continue reading The Twin Paradox in a Closed Universe, MAA Monthly Vol 108 #7,
Aug-Sept 2001, p. 585-590,
by Jeffrey R. Weeks.
Come back together and talk about group dynamics and go over the article.

**Wed Apr 10**
Respond to any remaining questions on mapping the brain and
physics and geometry.
Begin the relationship of geometry and biology by searching the web
for this relationship or the relationship of geometry to
the structures of DNA, crystals, or viruses.
Each person must find at least one item to briefly present on Friday.
Review course syllabus and goals.
Discuss final classroom worksheet.
Review the original and this (4th) version of the
Beachball activity.
If time remains then
begin the classroom worksheet on Hearing the Shape of a Drum.

**Fri Apr 12**
Begin biology via presentations (see Wed).
Continue worksheet on Hearing the Shape of a Drum.

**Wed Apr 3**
Geometry at Work p. 78-80 Mathematics to the Aid of Surgeons.
Conclude Mapping the Brain.
Begin the relationship of geometry and physics by searching the
web. Each person must find at least one item to briefly present on
Friday.

**Fri Apr 5**
Begin physics via
presentations (see Wed). Divide up into groups of two and
begin reading The Twin Paradox in a Closed Universe, MAA Monthly Vol 108 #7,
Aug-Sept 2001, p. 585-590,
by Jeffrey R. Weeks. Reading within a group of two is very different
than reading alone (the brain article).

**Mon Mar 25**
Download class data and open in Excel.
Discuss Does height predict armspan via a
linear regression line
in Excel and the r^2 value
(From Excel click on A, hold down the apple key and click on B.
Then under Chart, release on chart, click on
XY (Scatter) and click on Finish. Click on the white part of the graph.
Under Chart, release on Add Trendline... Click on Options and
check the bottom two boxes - Display equation on
chart and Display R-squared value on chart.)
Discuss the golden mean and look at armlength/handlength.
Discuss applications of the 4th dimension to business,
data, medicine, art
and an intro to Geometry in Learning from Geometry at work
(we will continue this at a later date after all the references that
we ordered on Friday are in).

**Wed Mar 27**
Road Map for the Mind
Read and discuss.
Use ideas in article to do additional searching on
mapping the brain.
Find
MONICA K. HURDAL's pages.

**Fri Mar 29**
Geometry at Work p. 76-78 Mathematics to the Aid of Surgeons
Highlight the differences between reading an article, and the usual
way that material is taught. Each person reads the article themselves
and tries to understand each step of the article.

**Mon Mar 18**
Finish exploring the Shape of Space by Jeff Weeks p. 96-102
Shape of space video part 2 and interview with Jeff Weeks.
Review
Shape of the Universe -- Mind Bending Ideas!,
Henderson's 3-Manifolds Shape of Space,
Shape of Space latest news,
Cosmology
News
Excerpts from Week's paper on Topological Lensing in Spherical Spaces
page 1,
page 12.

**Wed Mar 20** WebCT test 2 on the shape of the universe.
If time remains, then
Torus and Klein Bottle Games - students play each other.

**Fri Mar 22** Discuss the assignment due today -
how everyone tracked down the first page of their article,
and the depth of resources at ASU.
Hand out a different
reference finding assignment and students search for their
reference and order it online.
Conclude geometry of the earth and universe via a discussion
of remaining questions.
Measure
height, armspan, handlength, and armlength.
If time remains,
enter data into Excel.
Discuss how the 4 variables can be thought of as the 4th dimension.
Discuss Music 1
and Music 2.

**Mon Mar 4** Discuss Mathscinet searches.
The shape of the universe continued.
Review possible 2-d Euclidean universes.
Discuss 2-d spherical and hyperbolic universes via
the classification of complete, connected surfaces which locally look
like the sphere or hyperbolic space.
In the process, discuss the isometry group structures
of Euclidean, spherical and hyperbolic spaces as topological
groups and as algebraic groups (semi-direct product)

**Wed Mar 6** MathSciNet Searches and bibliographic entries.

Ramin Shahidi, *Mathematics to the Aid of Surgeons,*
Geometry at Work: Papers in Applied Geometry (C. Gorini, ed.),
MAA Notes 53, Mathematical Association of America, Washington, DC,
2000, pp. 76-80.

S. Greenwald,
*Diameters of spherical Alexandrov spaces and curvature one orbifolds,*
Indiana Univ. Math. J. **49** (2000), no. 4, 1449-1479.
MR 2002c:53052

See 36 and 37 for biblio format of articles on the web that haven't
been published elsewhere.

Finish up hyperbolic classification from Monday.

**Fri Mar 8** The shape of the universe.
Review classification from last time and highlight the closed, finite
2-d universes.
Discuss the 4th dimension and questions from
last Wed.
Open up sketchpad and open
Sketchpad/Samples/Sketches/Geometry/HigherDimensions/hypercube.gsp
to view a creation of the hypercube. Discuss why the universe is not
thought to be a hypercube.
Gauss' attempts at measuring the universe.
Shape of the Universe -- Mind
Bending Ideas!
Begin
exploring the Shape of Space by Jeff Weeks p. 96-102.

**Mon Feb 25** 3D Homer continued

**Wed Feb 27**
The Fourth Dimension,
Is Space Finite?,
go through
Davide Cervone's talk on The Cube and the Hypercube: Rotations and Slices
by clicking on the image that looks like a triangle filled in and pointing
to the right. If controls appear below a picture that means that it is
a movie. Play each movie by clicking on the image (second from the right)
that looks like a filled in triangle with a greater than sign on its right.

Questions to think about:

How is a hypercube formed from a cube? (Hint: Use an analogy similar to Professor Frink's description of how a cube is formed from a square.)

How many "faces" (3-d boxes) does a hypercube have?

What might one layer of Homer's skin look like in 4-d if he were to change from 3-d to 4-d? (Hint: In 3-d, to the naked eye, a layer of skin looks like a 2-d piece of paper with holes or pores in it - think about what this would change into if it gained a dimension.)

Look at the main web page for hw.
**Fri Mar 1**
Hand out Mathematics into Type p. 70-75 on bibliographic styles
The shape of space continued.
Shape of space video part 1 - 2 dimensions.
Euclidean 2-d closed finite universes (torus and Klein bottle)
via the classification of complete, connected surfaces which locally look
like the plane (cylinder Mobius Band, torus, Klein bottle).
Torus
and Klein Bottle Games.

**Mon Feb 18** Projective geometry continued. Models and
axioms.

**Wed Feb 20** Projective geometry completed
with sketchpad activities.
Skim readings and perform activities listed.
To open a .gsp sketchpad link, from NETSCAPE, click on the link.
You will see some garbage text symbols. Under file, click on
Save as... Double click on the public folder/Save files here
folder which is located on the desktop. When you are inside that folder
you can save the file. Then open up Sketchpad (GSP 4.00) and under
File, open, find your file and open it.
CD6_1_3.gsp,
solution - creating a perspective view
of the triangular floor pattern,
CD6_1_7.gsp,
CD6_1_8.gsp

**Fri Feb 22**
3D Homer

**Mon Feb 11** Hyperbolic models continued.
Escher's Sun and Moon and Heaven and Hell and the relationship to
different geometries. History of geometry continued.

**Wed Feb 13** WebCT test, perform a web search and find
the website on Euclid's Elements that contains the proofs that Dr.
Sarah handed out. Write down the words that resulted in your successful
search and the address of the website. Then explore the website.

**Fri Feb 15** Projective geometry continued -
the history of projective geometry and
the
relationship to the art of perspective

**Mon Feb 4** Spherical Pythagorean theorem via coordinate
geometry,
a cone of 450 degrees and the fact that shortest is not always straight
(on a sphere (2 straight paths between any 2 points only one of which is
shortest) and on the 450 degree cone (symmetry)).
Discuss theorem that on a smooth surface, locally a straight line
is the shortest distance
between 2 points, and compare with the sphere to see that
globally this is not the case.

**Wed Feb 6**
Geometer's Sketchpad continued - poincare.gsp -
Is Euclid's 5th postulate ever, always or never true in hyperbolic space?
Is the Pythagorean theorem ever, always or never true in hyperbolic space?
Review differential geometry thm that smooth surface implies that
an intrinsically straight line is always the shortest path between
nearby points. Discuss the fact that if the surface is complete then
any two points can be joined by a shortest distance
path. Compare with a plane with a hole removed and
related to Euclid's 2nd and 3rd postulates.
Reading on
Spheres and GPS - How it Works

**Fri, Feb 8** Euclid's 5th postulate and Playfair's
postulate - review this via spherical and hyperbolic geometry and
discuss the history of the 5th postulate and non-Euclidean geometry.
Elliptic geometry via the projective plane (S^2/<-id>)
obtained by the quotient of S^2 by the Z_2 group (intro to groups).
Discuss which of Euclid's axioms hold in projective geometry and
discuss the relationship to art.
Mercator Map, models of the hyperbolic plane
(from paper annuli, crochet, and polyhedral constructions).

**Mon, Jan 28** Euclid's proof of SAS in Book 1 of the
Elements (prop 4) and a more rigorous version of it using
reflections. Examined where the proof failed on
the sphere. Discussion of the fact that while
postulate 1 as is seems to be true on the sphere, really
uniqueness was a part of this postulate even though it was not
stated as such. Hence Postulate 1 (with uniqueness) is false on
the sphere.
Examined Euclid's proof of the sum of the angles in a triangle
being equal to 180 degrees (two right angles) (prop 32)
and examined where this fails on the sphere.
Examined the statement of the phythagorean theorem
(prop47) and the difference between a "square on the side"
and a "square of the side".
Examined a modern proof of the pythagorean theorem
and where the proof fails on the sphere.

**Wed Jan 30**
Intro to constructions and geometer's sketchpad via
Euclid's Elements Book 1 Proposition 1.
Highlights the difference between a sketchpad construction and a proof.
Students work on
Proposition 3.
Intro to to hyperbolic space via
investigations within the Poincare disk model (
Sketchpad/Samples/Sketches/Investigations/Poincare Disk.gsp).
Given a hyperbolic line and a point off of the line,
how many parallels can be formed?
What is the sum of the angles in a hyperbolic triangle?
How large can the sum of the angles get? How small can the sum
of the angles get?
Create a right triangle. Ask
whether the Pythagorean theorem can ever hold in the hyperbolic disk.
We will answer this next Wed.
Highlight the discrete nature of sketchpad.

**Fri, Feb 1**
Dr. Foley's Geometric Constructions
description.
Beachball activity on the sum of
the angles of a spherical triangle

**Mon, Jan 21** MLKJ Holiday

**Wed, Jan 23** Revisit geometry of the earth questions
with advanced techniques (questions 7, 6 , 5).
Hand out definitions, postulates, common notions, and propositions
from Book I of Euclid's Elements.
Discuss the ideas and history behind intuition, inductive and deductive
reasoning, including Thales, Pythagoreans, and Euclid.
Notice that Euclid's axiom 1 discusses
a straight line.
Conclude with "What is straight?" on a plane and a sphere. Discuss the
intuition behind the idea of a zero covariant derivative for
a longitude on a sphere, and a non-zero covariant derivative for
a non-equator latitude.

**Friday, Jan 25**
The Proof -
a Nova video about the solution of Fermat's Last Theorem,
paying special attention to:

influences, support and barriers to becoming a mathematician
and conducting research

how mathematical research is described

how people get the flashes of insight needed to do research

whether the mathematicians often collaborate or instead mostly
work by themselves

the use of geometry in attempts to solve a number theory problem
(algebraic geometry)

**Monday, January 14** Fill out Index Sheet, go over office hours,
rough overview of course, begin
geometry of the earth project.

**Wed, Jan 16** Go through the course web pages and then
post onto the WebCT bulletin board.
Work on the geometry of the Earth project.
**Fri, Jan 18** Collect reports. Presentations.
Go over intuition and lower level responses.