ANNOUNCEMENTS
MTH 338 — Winter 2020

3/14/20
Change in plans:
3/13/20
Strange but true: The 13th of the month is more likely to be a Friday than any other day of the week!
Give up? Further information is available here.
3/12/20
You do not need to give me hard copy of your essay if the email message that accompanies your electronic submission clearly states that you want me to grade your electronic manuscript.
Be warned that you do so at your own risk, as formatting may appear different at your end and mine. This issue is especially noticeable between different versions of software attempting to read DOC or DOCX files, so you are strongly encouraged to (also) submit as the "official" version a PDF copy whose formatting you have carefully checked.
If you do submit both hard copy and an electronic version, I will treat the hard copy as official unless you indicate otherwise.
I will be in my office from 9 AM–12 PM on Monday.
Feel free to slide your hard copy under my office door if I am not there (or if you want to submit your essay over the weekend). You are encouraged to send me an email message telling me that you have done so.
You have met the deadline if your electronic version reaches me by noon, and your hard copy reaches me reasonably soon thereafter.
All other late submissions require explicit discussion and approval well in advance.
3/9/20
Please bring a printed copy of your current draft to class on Friday.
As decided in class today, we will devote Friday's class to one more opportunity for you to read and critique each other's work.
Having someone else read your writing is an excellent way to get feedback.
2/28/20
As we discussed in class today, 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.
2/27/20
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.
2/26/20
Two versions of the diagram constructed in class today, showing that a hyperbolic triangle together with its 3 exterior lunes combine to make an ideal triangle can be found here and here.
Each of these figures shows both of the two possible sets of 3 exterior lunes.
2/24/20
The drawing shown in class today in which single elliptic lunes are used to find the area of a triangle can be found here.
Discussions of the hyperbolic analog of lunes can be found here and here.
2/20/20
Effective immediately, I will be in my office M 10:45–11:30 AM and WF 9:30–11:30 AM, in addition to my regular MWF afternoon office hour. Most weeks, I can also be available (by appointment) MWF before 9:30 AM and MF after 4 PM.
Please take advantage of these extended hours to discuss your project with me.
2/19/20
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.
2/17/20
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.
2/14/20
On Friday, 2/21/20, we will meet in Kidder 108H (the classroom in the back of the MLC).
Be prepared for an adventure in spherical geometry!
2/10/20
You may bring a straightedge to the midterm. (Yes, a ruler is OK.)
You do not need anything else besides something to write with.
The online "book" (MNEG) has been updated with a description of the Klein Disk.
Take a look here. The circle applet shown in class today is in the second section.
Be warned that both the applet and the description are not yet very polished.
2/9/20
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.
2/8/20
I will hold an extra office hour on Wednesday, 2/12/20, from roughly 9:30–11:30 AM.
You should be able to collect your graded Lab 2 at this time.
This would also be a good opportunity to discuss your choice of topic with me...
2/7/20
One of your classmates suggested that this video about hyperbolic geometry may be of interest to the class.
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 the textbook for this course (although not by me).
2/5/20
The midterm will be held in Bexl 415 on Wednesday, 2/12/20, at 1 PM.
2/4/20
Here is some further information about the midterm:
2/3/20 (updated 2/4/20)
The drawings I showed today in class of the exterior angle theorem on a sphere can be found here.
A GeoGebra applet with the same features can be found here, and will be added shortly to MNEG.
A GeoGebra applet showing stereographic projection can be found here, and will also be added shortly to MNEG.
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 along for each routing.
2/2/20
Here are some additional bugs in my Klein disk applet:
2/1/20
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.
1/31/20
The midterm is confirmed for Wednesday, 12 February 2020 (Week 6).
1/30/20
Here's a fun thing to try in the Poincaré Disk:
Construct an equilateral triangle. (How?) Measure its angles.
1/29/20
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.
1/28/20
The midterm is tentatively scheduled for Wednesday, 12 February 2020 (Week 6).
Please let me know immediately of any conflicts or strong preferences that might affect having the midterm on this date.
1/27/20
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.)
If you want to save your work without resorting to screenshots...
It should now 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.
An online search for GeoGebra applets for hyperbolic geometry will return several alternatives. A particularly good one is this applet, which has a nice array of tools (including a compass).
Be warned that not all of the tools in this applet are labeled! So far as I can tell, the unlabeled tools are, in order:
    Segment, Ray, Line, Circle, ??, Bisect, Midpoint, Perpendicular.
Furthermore, this applet does not appear to provide measurement tools.
1/25/20
How does one duplicate angles?
1/24/20
The online "book" (MNEG) has been updated with the applets shown in class today describing constructions with straightedge and compass, including SAS.
Take a look here.
WARNING: Some URLs have been updated, including elsewhere on this page. You may need to update your bookmarks.
1/22/20
A sample solution to one of last week's homework problems can be found here.
This solution models incorporating the statement of the problem into the narrative.
1/17/20
The online "book" (MNEG) has been updated with the applet used in class today to prove that exterior angles must be larger than nonadjacent interior angles.
Take a look here.
1/21/20
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).
1/20/20
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 computer as a .ggb file.
1/17/20
The online "book" (MNEG) has been updated with applets for taxicab geometry.
Take a look here.
1/15/20
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.
UPDATE 1/25/20: Access to this solution has now been removed.
Some additional resources are listed below:
You may also find these newspaper articles about court decisions involving taxicab geometry to be of interest.
1/13/20
As announced in class, please turn in hard copy for (future) homework assignemnts.
By all means, (also) send me a copy as an attachment (PDF preferred) if you prepared your assignment electronically.
(It's not worth sending me a scanned copy of handwritten work unless there is no alternative.)
1/10/20
I have posted a second excerpt from TG here that should be sufficient for the second homework assignment.
The corresponding sections have not been included, but are not actually necessary to do the problems — and we'll cover that content in class early next week anyway.
Some additional resources are listed below:
1/9/20
My office hours have been posted on the course homepage.
I am often in my office on Wednesday mornings (after 10) and late Monday mornings (after 11).
1/7/20
I have posted an interactive "book" (MNEG) here containing most of the geometric models discussed in class this week.
There are direct links to the two- and three-dimensional GeoGebra drawing interfaces in the introductory sections.
I will attempt to keep MNEG up-to-date as we consider new models.
1/6/20 UPDATE
I have posted an excerpt from TG here that should be sufficient for the first homework assignment (only).
It is not that difficult to find a PDF copy of the entire book online, although I have been unable to verify its legitimacy. Download a copy if you can't wait, but please do obtain an official copy, whether from the bookstore or elsewhere.
(The hard copy is only \$7.95 on Amazon, and the ebook a mere \$4.19...)
1/6/20
Due to an oversight (not mine), the second textbook, Taxicab Geometry (TG), was not ordered on time. My apologies!
The slide I showed at the end of class is available here.
1/2/20
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.
1/1/20
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.