ANNOUNCEMENTS
MTH 338 — Spring 2023

6/12/23

As per the HW page, an electronic copy of your final paper should be emailed to me by noon today.

With apologies, the course website was not updated with the due date information in a timely fashion.
If this oversight causes difficulty for you please contact me via email.

I am unexpectedly available from 11 AM–12 PM today via Zoom.

Please include a PDF version to ensure that everything is formatted properly.

6/7/23

I should be on Zoom from roughly 1 PM this afternoon.

I will start a few minutes early if I can, but can not easily stay past 2 PM, and may shut things down early if nobody is present.

6/6/23

I will try to be on Zoom starting shortly after 10 PM tonight.

It may be as late as 10:15 PM before I can connect; keep trying.
I am unlikely to be available after 11 PM...

6/5/23

I will be on Zoom starting at approx 12:15 PM today.

I plan to stay online until at least 1:30 PM, but may not remain after that unless someone is still waiting.

6/4/23

I am still aiming to hold an office hour tomorrow (Monday) starting at approximately 1 PM.

I will start the Zoom session early if I can; check this page for updates, and/or try connecting to the Zoom session.

5/30/23

As discussed in class today, 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!
Further details about these constructions with hyperbolic lunes can be found in MNEG §8.7
(as well as in a paper available here which is about to appear in the College Mathematics Journal).

Here's the classic painting problem we ended class with:

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??

5/27/23

Additional office hours for Week 9:

I plan to be in my office much of the day on T (roughly 10–2), W (10–4), R (10–2).

These times are in addition to my scheduled office hours.

These additional times are not guaranteed unless you make an appointment.

Nonetheless, I'm likely nearby; if I'm not in my office when you look, wait a few minutes and try again.

Late afternoon times are also possible (appointment recommended).

No in-person office hours during Week 10:

I will be away from Corvallis during Week 10...

Class will meet as usual, with a guest lecturer discussing the geometry of special relativity.

I will hold office hours via Zoom at times to be determined.

(Most likely: approx. 1 PM on MW and approx. 10 PM on T)

I expect to be available via email, and may be available for appointments via Zoom at other times.

If you're stuck on where your project is going, come to my office during Week 9!

5/26/23

Here are some reminders about presentation.

• Equations must be grammatically correct parts of sentences, whether displayed or inline.
• All figures must be referred to, and described, in the main text.
• Figure captions can be short, and do not need to be complete sentences.
• References should appear at the end, and must be cited in the text.
• Use the same fonts for inline mathematics as for displayed equations.
  (In $\LaTeX$, use dollar signs around inline mathematics. In other software, you might use italics.)
• The correct spelling is "GeoGebra", with two upper-case letters.

5/25/23

A drawing showing how single elliptic lunes are used to find the area of a triangle can be found here; the double elliptic version can be found here.

Further details about these constructions with lunes, can be found in MNEG §8.6.

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.

5/24/23

I will not be in my office tomorrow morning, but will be available from 10:30–11:30 AM via Zoom.

I will be in my office from 12:30–1:45 PM, and again after class until 4 PM.

5/23/23

TODAY'S CLASS IS CANCELED DUE TO ILLNESS

The activity scheduled for today has been rescheduled for Thursday.

5/22/23

Here are some further suggestions regarding the formatting of your paper. Minor deviations are fine, and more significant deviations may be OK if there is a reasonable justifcation.

• It is difficult to read fonts that are smaller than 12 point.
  (In $\LaTeX$, use something like \documentclass[12pt]{article}.)
• Typical margins are one inch.
  (The default margins in $\LaTeX$ are much too big. One way to fix this issue is by adding \usepackage{fullpage}.)
• Your paper should be about 5 single-spaced pages or the equivalent, not counting figures, lengthy equations, or references.
  Yes, you may double space if you prefer, but single-spaced essays are usually easier to read.
  (A bit longer is fine; much shorter is not. A title page is optional, but does not count toward the total.)
• Your essay should not be handwritten.
  Hand-drawn figures are OK if necessary, but should be drawn very carefully.
• Don't forget about the ground rules for equations, figures, and references, as described here.
• You may use any reasonably standard citation format, such as APA "(Dray, 2021)" [or "Dray (2021)", depending on the context] or numeric "[1]". In either case, full bibliographic details should be given at the end of your paper.

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.

5/18/23

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.")

Several students also asked about examples of good writing. One such example is linked to my essay on good writing.

WARNING: You may want to avoid reading this paper if you are working on taxicab trigonometry, as it may provide answers to the questions you are investigating.

A more recent example can be found here; this paper won the 2021 WIC Culture of Writing Award in Mathematics at OSU. An expanded version will be published shortly in Pi Mu Epsilon Journal; a PDF version of that article is available here.

Both papers were originally submitted as projects for this course, then later significantly extended for publication.
The published versions of these papers far exceed the standards to get an A in this course.

5/17/23

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.

5/16/23

We will go over the midterm in class today.

IF your grade were determined only by your midterm, it would be:

• 71–80: A
• 70: AB (too close to call)
• 59–69: B
• 45–58: C
• 35–44: D
• < 35: F

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;

Your course total at this point is the sum of these two scores, multiplied by 5/4.

IF your grade were being assigned now, it would be:

• 182–200: A
• 173–182: AB (too close to call)
• 150–173: B
• 125–150: BC (too close to call)
• 100–125: C
• < 100: F

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.)

5/15/23

Additional office hours for the next three weeks (through Week 9):

I will likely be in my office on Tuesdays and Thursdays from 10:30–11:30 AM and 12:30–1:15 PM.

These times are in addition to my scheduled office hours.

These additional times are not guaranteed unless you make an appointment.

Nonetheless, I'm likely nearby; if I'm not in my office when you look, wait a few minutes and try again.

5/12/23

Surprising instances of non-Euclidean geometry. You may find the following links to be of interest.

This video was created by a mathematician who has written a book on hyperbolic geometry that has been used occasionally as a textbook in this course.
A full transcription is available here.

This xkcd comic strip incorporates the difficulty of mapping the spherical geometry of the globe onto flat maps.

5/6/23

I will hold additional office hours this coming week as follows:

In addition to my usual office hours this week, I will also be available as follows:

• Monday (5/8): 7–9 PM via Zoom;
• Tuesday (5/9): 10:30–11:30 AM in Kidder 298A;
• Tuesday (5/9): 7–9 PM via Zoom.

The Zoom link can be found in the first Canvas announcement.

As always, I am available via email at other times; a short consultation via Zoom may also be possible.

5/4/23

Here is some further information about the midterm:

• The midterm will cover taxicab geometry, hyperbolic geometry, and elliptic geometry, as well as the finite geometries discussed during the first few days of class.
• The emphasis will be on qualitative understanding, rather than detailed proofs.
• A basic acquaintance with the structure of the SMSG postulates is recommended
  (i.e. knowing that there are incidence postulates, ruler postulates, etc.).
• Expect true/false questions and short answer questions, as well as computational questions.
• With the possible exception of a single, short essay question, the midterm will be graded for content only, not presentation.
• The midterm is closed book.

5/3/23

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.

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.

5/2/23

The GeoGebra applet I used to illustrate the exterior angle theorem on a sphere can be found here.

The GeoGebra applet I used for stereographic projection can be found here.

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.

• Guess the answers before looking them up.
• There are many websites that will calculate the distance between two locations...
  (Answers from different sources may differ slightly.)
• You can also calculate distance on Google maps! Right click...
• You can check your answers here. (Click on a route to show its length.)

Bring to class on Thursday if you can:

• paper;
• scissors;
• tape.

5/1/23

If you download GeoGebra to run on your local device(s), it is recommended that you download GeoGebra Classic 6, rather than the Calculator Suite.

The macro packages in this course, such as Poincare.ggb, have only been tested with this version.

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.

When trying to duplicate an angle, you may find this website to be helpful

This website is one of the first hits when searching online for "construct straightedge compass duplicate angle".

4/27/23

A further discussion of the angle of parallelism can be found in MNEG.

The last figure demonstrates the relationship between the angle of parallelism and distance.

4/26/23

To verify that SAS congruence implies triangle congruence in this week's lab activity (Lab 0), 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. But you may find this construction useful when working on Lab 1.

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.

4/25/23

The midterm is currently scheduled for Thursday, 5/11 (Week 6), during class.

Please let me know immediately of any conflicts or strong preferences that might affect having the midterm on this date.

How does one duplicate angles?

• By eye. (Not accurate.)
• By measurement. (Not very accurate.)
• Using special angles. (What works? Right angles? Opposite angles? Equilateral triangles?)
• By construction. (See MNEG and RG §4.9.)

The applet I used in class today with drawing tools for the Poincaré Disk model of hyperbolic geometry is available here.

Use the "disk" menu (furthest to the right of the geometric icons) for hyperbolic constructions.

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.

4/24/23

Students who have completed five 300-level mathematics courses and have at least a 3.0 math GPA should consider applying for membership in Pi Mu Epsilon, the national mathematics honor society.

Further details are available here.

4/21/23

I have posted (anonymously) the definitions that resulted from yesterday's group discussion here.

Several students have asked about my grading scheme.

• As stated on the course home page, your grade will be based equally on the term paper, the midterm, and the other assignments.
• Some factors that determine your course grade are described here.
• I do not use a rigid grading scale of the form 90%=A, etc. Rather, each component of your grade will be compared with the rest of the class, with the cutoffs being determined by class performance.
• Based on past experience, I expect that anyone who completes all of the assignments will earn a grade of C or better, and anyone who completes all of the assignments reasonably well will earn a grade of B or better.

Finally, it has been brought to my attention that the link on the course home page to MNEG only works from within Canvas if you open the link in a new tab or window.

I believe this behavior is due to Canvas rejecting the software used in the book, which is beyond my control.

4/20/23

I have posted a sample solution to the last problem on the second assignment here.

This solution incorporates the problem statement into the narrative.

The discussion of rectangles I presented in class can be found here.

You may want to compare this presentation with the one in RG §3.6.

As you saw during class, GeoGebra can graph absolute values and inequalities, but not both together.

Desmos appears to be able to graph this combination, although it can not handle double inequalities involving two variables, nor can it (to my knowledge) be programmed to draw shapes such as taxicab circles.

4/18/23

The applet used in class to show that exterior angles must be larger than nonadjacent interior angles can be found here.

Let me know if you would like a copy of any of my GeoGebra applets for yourself (and can't figure out how to download the underlying ggb file).

4/13/23

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).

Regarding HW #1:

• If you received a lower score than expected, do not panic!
• Lower scores suggest a mismatch between your understanding and my expectations, which you are strongly encouraged to discuss with me during office hours.
• I will drop the lowest homework score when computing your final grade.

4/11/23

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.

4/7/23

The phone booth problem posed at the end of yesterday's class is TG §2:10.

Spoiler: The solution is given on the last page of the introductory notes below.

You may find these newspaper articles about court decisions involving taxicab geometry to be of interest.

Some notes on the topics from yesterday's lecture are available online:

• Taxicab intro;
• Models;
• Overview;
• Graph paper.

4/6/23

You can find the applets I used in class today here and here.

There is an applet on the latter page that includes tools for drawing taxicab circles and for measuring taxicab distance.

Please explore the course website, noting in particular the criteria I will use to evaluate written work.

Most Writing Assignments will be ungraded, but feedback will still be provided using similar criteria.

Please also read this document with some comments on wordprocessing formats.

Nothing else is as good as $\LaTeX$ at typesetting mathematics. If you are planning to become a mathematician, you are strongly encouraged to learn $\LaTeX$.

• A good if exhaustive introduction to $\LaTeX$ is available online here.
• $\LaTeX$ is available in the MLC computer lab.
• $\LaTeX$ can also be used online, for instance at Overleaf.

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.

• My advice on writing a mathematical essay can be found here.
• OSU has a WIC Survival Guide, which can be found here.
• A list of further resources can be found here, including a link to OSU's Writing Center.

4/5/23

My in-person office hours occur before and after class. If I am not in my office at those times, I might be in the classroom.

I am often in my office before and after the posted times. Feel free to check!

4/4/23

The Canvas course page has been restored and published.

The main source of information about the course is this website, which can be reached from Canvas, but is hosted elsewhere.
(Links to external sites from within Canvas may cache an old version. You can check by opening the link in another tab.)

However, links for remote office hours via Zoom and for my live, in-class notes have been posted as a Canvas announcement.

I believe I have corrected the minor errors and omissions on these pages that were pointed out to me during class.

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.

Finally, you can find the applets I used in class today here.

Feel free to browse through the other chapters of my online book of models, which is available here. However, you are strongly encouraged to work through problems yourself before reading too far ahead or playing with applets that might inadvertently reveal information that you would be better off discovering on your own. (Working with others is fine, so long as everyone contributes.)

3/19/23

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.

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.