ANNOUNCEMENTS
MTH 338 — Winter 2021

3/10/21
Notes from today's class can be found here.
The slides I showed at the end can be found here.
3/9/21
Two mathematicians are talking on the telephone. Both are in the continental United States. One is in a West Coast state, the other is in an East Coast state. They suddenly realize that the correct local time in both locations is the same! How is this possible?
Give up? Some hints can be found here.
3/8/21
Notes from today's class can be found here.
Here are the presentation comments I made at the beginning of class today.
3/5/21
Here are some lightly-edited comments from the end of class about each other's papers.
3/3/21
Notes from today's class can be found here.
3/1/21
Notes from today's class can be found here.
We will do an activity in class on Wednesday.
This activity is done most easily using GeoGebra. If you don't have easy access to GeoGebra during class, you can use pencil and paper – and a ruler.
2/26/21
Several of you have been asking how to format your paper. Here are some guidelines to get you started, but minor deviations are fine, and more significant deviations may be OK if there is a reasonable justifcation.
It's time to be clear about what you actually intend to do.
Exactly which questions are you going to ask (and hopefully answer)?
If you do not yet have a complete list of such questions, I strongly encourage you to see me as soon as possible.
2/24/21
Notes on hyperbolic lunes from today's class can be found here and here.
Discussions of the hyperbolic analog of lunes can be found here and here.
2/23/21
Some additional notes on elliptic lunes can be found here.
Starting next week, I will no longer be available on Monday mornings.
2/22/21
If you submitted a project proposal on Friday, you should have received an email message from me on Saturday with a PDF attachment containing my comments on your proposal.
Several students reported finding this message in their spam folder! If you still don't see it, ask me to send another copy.
2/19/21
The drawing shown in class today in which single elliptic lunes are used to find the area of a triangle can be found here; the double elliptic version can be found here.
As for the formula, the area of a lune with angle $\alpha$ is $\frac{\alpha}{2\pi} (4\pi r^2)=2\alpha r^2$. For a triangle with angles $\alpha$, $\beta$, $\gamma$, the six lunes constructed in class thus have total area $2(2\alpha+2\beta+2\gamma)r^2$. But they cover the sphere (with area $4\pi r^2$) and four extra copies of the triangle. Thus, the area $A_T$ of the triangle satisfies $4(\alpha+\beta+\gamma)r^2=4\pi r^2+4A_T$, so that $A_T=(\alpha+\beta+\gamma-\pi)r^2=Er^2$, where $E$ is the angle excess of the triangle – as expected!
A nice discussion (using obsolete technology) of how to use lunes to determine spherical area can be found here.
Of particular interest is this animation.
2/17/21
There are again two sets of notes today, which can be found here and here.
The missing details from the argument I summarized in class today are in Theorems 6.4.11 and 6.4.12 in RG.
In both cases, the basic idea, as stated in class, is to use the equivalence (same defect) between a triangle and its associated Saccheri triangle, and the fact that the latter depends only on the defect, to show that triangles with the same defect are equivalent, that is, must have the same area.
There is an upper bound to the area of a hyperbolic triangle! The ideal triangle with vertices on the boundary of the Poincaré Disk has all angles equal to zero, and hence area $k\pi$ (in suitable units).
The ideal triangle therefore has finite area but infinite side lengths!
You may want to compare the classic painting problem:
A fence is built along the $x$-axis for $x\ge1$, with height given by $x^{-2/3}$.
What is the area of the fence? How much paint is needed to paint it?
If you build a big bucket as a surface of revolution that just fits the fence, what is its volume?
How much paint is needed to fill it?
How much wood is needed to build the bucket??
2/15/21
There are two sets of overlapping notes today, which can be found here and here.
The applet shown in class today for the construction of the associated Saccheri quadrilateral is available here.
The relevant material can be found in §6.4 of RG.
The fact that ASA and AAS congruence follows from SAS congruence is proved for neutral geometry in §3.3 of RG.
Both results also hold in elliptic geometry, when suitably interpreted.
My apologies for not being available during my office hours this morning.
I will have some free time tomorrow (Tuesday 2/16); contact me to make an appointment.
2/14/21
Several students have asked how long their term paper should be.
A good rule of thumb would be 5–7 pages, single-spaced, not including figures or lengthy equations.
(The WIC requirement is "at least 2000 words.")
2/12/21
Midterm scores have been posted in Gradescope (only).
We will go over the midterm in class today.
IF your grade were determined only by your midterm, it would be:
To estimate your current grade in the class, proceed as follows
Your homework score is the sum of the best 4 of the 5 assignments;
Your exam score is your midterm score multiplied by 5/4;
Your course total at this point is the sum of these two scores (rounded to the nearest integer if necessary).
IF your grade were being assigned now, it would be:
Please be aware that the same procedure will be used to determine your final grade.
Your course grade is not the average of the separate components, but instead determined on a single, combined scale.
(Yes, there will be $\pm$ grades.)
2/10/21
Well, that was exciting! To clarify the rules, you were given 60 minutes from when you started; starting early does not get you extra time. However, given the number of folks who misread the instructions, they were not sufficiently clear.
I will not penalize anyone for late submission.
A few folks had technical issues with their uploads. I reiterate that it is ultimately your responsibility to ensure that the software you are using does what you think it does, and to know how much time you need to allow for it to do so. If you had a problem this time around, you may want to use different software next time.
I will not penalize anyone for such software-related problems.
My apologies to all for the inevitable anxiety associated with such last-minute confusion – which affected me as well...
2/9/21
The midterm will be available on Gradescope starting at 12:50 PM tomorrow, Corvallis time.
2/8/21
Here is the list of review topics generated in class:
Notes from today's class can be found here.
You may find this video about hyperbolic geometry to be of interest.
A full transcription is available here.
The creator of the video, a mathematician, has written a textbook on hyperbolic geometry.
This book has been used occasionally as a textbook for this course (although not by me).
2/7/21
Yes, the Klein disk applet does have a compass tool, called Elliptic Circle with Center and Radius.
Be careful to select points in the correct order; the first point selected is the center of the new circle.
2/6/21
A sample exam has been posted on Gradescope.
Most of your work and answers should be done on paper you provide, then scanned and uploaded.
You do not need to copy the questions.
Make sure to clearly label which answer goes with which question.
Some questions involve annotating figures on the exam.
Some of these instructions may be different for Wednesday's midterm.
Come to class on Monday prepared to offer suggestions for improvement.
One likely change is that the midterm will not be released until 1 PM (or slightly before) on Wednesday.
2/5/21
Notes from today's class can be found here.
2/4/21
Here is some further information about the midterm:
2/3/21
Notes from today's class can be found here.
The "equator" of the Klein Disk is the bounding circle, which was the equator of the sphere before stereographic projection.
The points "outside" the disk, which were originally in the Southern Hemisphere, are not gone. Rather, they have been identified with their antipodal points in the Northern Hemisphere. So if you try to "leave" the Klein disk, you are wrapped around to the opposite point on the boundary, where you continue into the disk.
As announced in class today, the rubric for grading Lab 3 will not be the same as for Lab 2.
For Lab 2, the quality of your example was secondary, with only a minor deduction for using special cases and/or constructing triangles by eye. For Lab 3, the quality of your example will be a primary factor.
As also announced in class today, my office hour on Friday afternoon, 3/5/21, is canceled.
I should be available most of Friday morning. from 9:30–11 AM on Friday.
(If the Office Hours Zoom session isn't already running, send me an email message requesting an appointment.)
2/2/21
Here are some additional bugs in my Klein disk applet:
Here's an optional challenge in spherical geometry:
Draw a diagram showing the direct route from Portland to Frankfurt, as well as the indirect routes via Reykjavík, New York, and Tenerife. Label each city and determine the total distance for each routing.
2/1/21
Notes from today's class can be found here.
The drawings I showed today in class of the exterior angle theorem on a sphere can be found here.
The annotated versions we discussed in class can be found here, here, and here.
A GeoGebra applet with the same features can be found here.
A GeoGebra applet showing stereographic projection can be found here.
My applet with drawing tools for the Klein Disk model of elliptic geometry is available here.
You should be able to save your work directly from this applet. You may instead wish to download the underlying GeoGebra file, which you can then upload into any standard installation of GeoGebra.
Be warned that there are several known bugs with this home-grown applet.
Make sure to read the notes at the bottom of the page.
Bring to class on Wednesday:
2/1/21
If you are having trouble completing the SAS part of Lab 1, try SSS instead.
Also reread the announcement below regarding SAS.
When trying to duplicate an angle, you may find this website to be helpful
This website is the first hit when searching online for "construct straightedge compass duplicate angle".
1/31/21
The midterm is currently scheduled for Wednesday, 2/10/21 (Week 6).
Please let me know immediately of any conflicts or strong preferences that might affect having the midterm on this date.
The tentative format for the exam will be a traditional, timed, closed-book exam during the regularly-scheduled class period.
I will expect you to sign a statement confirming that the work you submit is your own.
Please let me know as soon as possible of any questions or concerns you have with these ground rules.
1/30/21
A list of potential topics has been posted here.
This would be a good time to reread my advice on how to write mathematical essays.
An older version of this document is available here. Both versions are worth reading.
1/29/21
Notes from today's class can be found here.
A GeoGebra drawing demonstrating the relationship between the angle of parallelism and distance, as constructed at the end of class, can be found here.
When submitting the results of GeoGebra constructions for homework, it is enough to include one or more exported images.
If you prefer, you may send me a copy of the ggb file via email, or post it somewhere online.
1/27/21
Notes from today's class can be found here.
1/25/21
Notes from today's class can be found here.
An applet with drawing tools for the Poincaré Disk model of hyperbolic geometry is available here.
(I believe this applet was downloaded from the GeoGebra website, but am no longer certain.)
It should be possible to save your work directly from this applet. You may instead wish to download the underlying GeoGebra file, which you can then upload into any standard installation of GeoGebra.
Here's a fun thing to try in the Poincaré Disk:
Construct an equilateral triangle. (How?) Measure its angles.
1/24/21
If you did not achieve the presentation score you were hoping for, you are encouraged to take advantage of the writing resources listed below in an earlier announcement, some of which are repeated here.
1/23/21
To verify that SAS congruence implies triangle congruence in this week's lab activity, it is enough to construct by any means a second triangle so that SAS congruence holds, then measure the remaining side and angles. However, the gold standard would be to construct the second triangle using only straightedge and compass.
If you successfully accomplish this task using GeoGebra, the second triangle should remain congruent to the first when you alter the initial triangle.
Duplicating an arbitrary angle requires several steps...
If you're stuck, try constructing a right triangle.
If you're still stuck, try constructing an equilateral triangle.
You might want to reread RG §2.2, which contains both Euclid's construction of an equilateral triangle, and Euclid's demonstration that one can copy a given line segment to a new starting point.
Try the "Compass" tool in GeoGebra.
Again, there's nothing to turn in for this activity.
However, you can save your work if desired, either by creating a GeoGebra account when prompted, or by declining to login, then saving to your local device as a .ggb file.
1/22/21
Notes from today's class can be found here.
The applets used in class today describing constructions with straightedge and compass can be found here.
How does one duplicate angles?
1/20/21
Notes from today's class can be found here.
The applet used in class to show that exterior angles must be larger than nonadjacent interior angles can be found here.
Classroom video has now been posted in Canvas both in the Media Gallery and as ungraded assignments.
1/18/21
The shortened URL originally advertised for this website no longer works.
Please update your bookmarks to use the official URL, namely http://math.oregonstate.edu/~tevian/onid/MTH338.
Update: As of 1/19, the shortened URLs (without "people") are working again...
1/17/21
As you will have seen from the corrected versions of your definitions, I will make comments directly on your writing assignments in PDF format.
You do not need to submit PDF versions of writing assignments yourself unless you prefer to do so, but be warned that format conversions can occasionally fail, especially for figures, and occasionally for special symbols, such as equations.
However, your final paper does need to be submitted as a single PDF.
1/15/21
Notes from today's class can be found here.
From the (old) notes for this course at UC Denver:
Non-Euclidean Geometry is not not Euclidean Geometry. The term is usually applied only to the special geometries that are obtained by negating the parallel postulate but keeping the other axioms of Euclidean Geometry (in a complete system such as Hilbert's).
1/13/21
Some information about (taxicab) hyperbolas can be found here.
A screenshot of our intial attempt to draw taxicab hyperbolas can be found here.
I have posted an interactive "book" (MNEG) here containing most of the geometric models discussed in this class.
There are direct links to the two- and three-dimensional GeoGebra drawing interfaces in the introductory sections.
See especially the chapter on taxicab geometry.
1/11/21
Notes from today's class can be found here.
Screenshots of the images drawn in class are available here, here, and here.
I have posted a sample solution to the first homework assignment here.
This solution models a good mix of description and figures, but no equations. An arguably better presentation would be to incorporate the statement of the problem into the narrative, but that choice depends on the audience.
Reminder: When submitting assignments to Gradescope, please follow the instructions here.
In particular, please make sure that the filename includes your name, and that you avoid uploading photos of handwritten work if possible – use a scanning app instead (preferably to PDF, not JPG).
1/10/21
This would be a good time to try using GeoGebra, which we will use in future activities.
GeoGebra can be run online in a browser, or downloaded to most computers, tablets and smartphones.
You shouldn't have to create an account in order to save files locally.
Some additional resources are listed below:
You may also find these newspaper articles about court decisions involving taxicab geometry to be of interest.
1/8/21
Partial class notes from today's class can be found here.
1/6/21
It has been brought to my atention that the OSU book store has run out of copies of TG.
Please let me know immediately if you have not yet been able to obtain a copy of this textbook.
1/5/21
A sheet of taxicab graph paper is available here.
1/4/21
The slide I tried to show at the end of class today is available here.
The two slides shown near the beginning of class today are available here.
1/1/21
Welcome to remote teaching! Below is some information about how this course will be run.
Overview:
Details:
Let me know if you have difficulties with any of these steps.
These instructions are likely to evolve...
12/23/20
Please explore the course website, noting in particular the criteria I will use to evaluate written work.
Please also read this document with some comments on wordprocessing formats.
Nothing else is as good as $\LaTeX$ at typesetting mathematics. Especially if you are planning to become a mathematician, you are strongly encouraged to learn $\LaTeX$. I am happy to help with $\LaTeX$ coding questions, but not with installation or editor-specific problems.
You may use any wordprocessing software you wish, so long as I can read the equations.
Finally, you may find some of the writing resources listed below to be helpful.
12/15/20
To the best of my knowledge, the (older, hard cover) 3rd edition of Roads to Geometry (RG), from Pearson, is identical to the (newer, paperback) 3rd edition, from Waveland.
If you're buying a new copy, the paperback is significantly cheaper. If you're buying used, you may only find the former — but do make sure it's the 3rd edition. Either should work fine for this course.
10/9/20
The main text (RG) is also available as an eTextbook from Amazon, either for purchase or for rent.
We will also make frequent use of my own notes (MNEG) on non-Euclidean geometry.