ANNOUNCEMENTS
MTH 338 — Winter 2022

3/11/22

As per the HW page, an electronic copy of your final paper should be emailed to me by noon on Monday, 14 March 2022.

March 14 is both Pi Day and Einstein's birthday!

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

Although not required, you are encouraged to send me copies of your source flies, such as DOC, TEX, PNG, GGB, etc., perhaps combined into a ZIP file.

If you're looking for last-minute advice, send me your latest draft via email, along with specific questions, and I'll respond as best I can.

3/10/22

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.

3/9/22

Notes from today's class can be found here.

The slides I showed at the end can be found here.

3/7/22

Notes from today's class can be found here.

Here are some comments and reminders about presentations, including the ones I made at the beginning of class today.

• 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.

3/4/22

Notes from the first part of Wednesday's class can be found here.

We will discuss one final geometry next week: Special relativity!

There should be time next Friday for a few students to describe their projects to the rest of the class.

This is an excellent opportunity to get important feedback before finalizing your draft over the weekend.

Please let me know in advance if you would like to present your project.

You will likely have five minutes to describe your project, followed by several minutes of discussion.

This coming week is your last opportunity to get feedback from me about your paper, regarding both content and presentation.

Although I do not plan to hold extra office hours, I will be available much of the week for individual appointments. You are strongly encouraged to meet with me even if you don't think you have any questions!

2/28/22

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 straightedge.

2/26/22

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 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 or lengthy equations.
  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.)
• 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.

2/25/22

As promised, here is a lightly-edited summary of the questions you contributed this morning in class.

I have attempted to remove all reference to a particular geometry, so that the questions apply in many different contexts. Do they apply to your project?

• How does ... affect the shape of ellipses?
• How does adding ... affect taxicab circles?
• Is ... a geometry or a model?
• Can ... be tiled in regular and irregular polygons?
• What is a circle in ...?
• Which triangle congruencies are satisfied in ...?

2/23/22

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.

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??

2/21/22

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.

2/18/22

The course website should be back up.

The mirror site has been taken down.

2/16/22

Rough notes covering the content being discussed this week can be found here.

The relevant material can be found in §6.4 of RG.

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.

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.

2/14/22

MNEG has been updated.

There is new material addressing the content being covered this week.

A slightly improved applet for the Poincaré disk is available here.

Some of the more obscure macros have been removed, which significantly speeds up loading the applet.

2/11/22

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.")

2/10/22

Midterm scores have been posted in Gradescope (only).

We will go over the midterm in class tomorrow.

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

• 64–70: A
• 59–63: AB (too close to call)
• 54–58: B
• 47–53: C
• < 47: 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 multiplied by 10/7;

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:

• 184–200: A
• 172–183: AB (too close to call)
• 151–171: B
• 133–150: C
• 120–132: C−
• < 120: 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.)

2/7/22

New office hours:

This week only, I will hold an extra office hour on Monday (today!), from 3:30–4:30 PM

After this week, I will hold office hours on Wednesdays at 4 PM rather than Sundays, so WF @ 4 PM.

MNEG has been updated.

The applets I demonstrated today are included in the chapter on the Klein disk.

(Some older chapters have also been updated.)

2/6/22

Midterm guidelines
The first question will ask you to certify your agreement to the "House Rules", either by signing and including a copy of the cover sheet, or by writing "I agree to the House Rules" on your answer sheet, then adding your signature.

• The exam is closed book! No books, no notes, whether printed or electronic.
• You may not discuss the exam with anyone other than me.
  I will be available via Zoom to answer questions.
• Your answers and supporting work should be written on paper you provide.
  You do not need to print the exam, or include it with your answers.
  (But see the comments below regarding figures.)
• You do not need to copy the questions.
  (Please clearly label which answer goes with which question.)
• The use of OneNote or similar software that allows you to write electronically and/or annotate PDFs is allowed.
• The use of Word, $\LaTeX$, or similar software that allows you to typeset your answers is acceptable.
  However, doing so can be time consuming... Choose wisely.
• No other software is allowed! In particular, you may not use GeoGebra.
  Please shut down all other software during the exam, such as email, phone, browsers, etc.
• No extra time is available for the use of software to prepare and/or upload your exam.

Some questions involve annotating figures on the exam.

• If you have the ability to annotate PDF files, one option would be to download the exam and write on it electronically, then upload this electronic file when finished.
• If you have a printer, another option would be to print part or all of the exam, write on it as you would an in-class exam, then scan and upload all of your written work when finished.
• You may also redraw figures by hand on your answer sheet.
  You will get full credit so long as your answer is clear (and correct); you do not need to reproduce every detail

2/5/22

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.

Yesterday's xkcd comic strip incorporates the difficulty of mapping the spherical geometry of the globe onto flat maps.

2/4/22

The Klein compass tool has the inputs in the wrong order... (Select the center first, not last.)

This and other limitations of the Klein disk applet are documented here. (Scroll down past the applet.)

My applet with drawing tools for spherical geometry is available here.

Notes filling in some of the details about poles and polars, as discussed in today's class, can be found here.

2/3/22

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.

2/2/22

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.

As announced in class today, the rubric for grading Lab 2 will not be the same as for Lab 1.

For Lab 1, the quality of your example was secondary, with only a minor deduction for using special cases and/or constructing triangles by eye. For Lab 2, the quality of your example will be a primary factor.

(The construction used in class today to duplicate an angle in Euclidean geometry can be found here.)

2/1/22

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.
(We will discuss this model tomorrow in class.)

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.

1/31/22

Notes from today's class can be found here.

Drawings illustrating the exterior angle theorem on a sphere can be found here.

The GeoGebra applet I used in class for the same purpose can be found here.

A GeoGebra applet showing 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 Wednesday if you can:

• paper;
• scissors;
• tape.

1/30/22

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.

1/28/22

Supplementary notes outlining the proofs of the parallelism properties discussed at the end of class can be found here.

A GeoGebra drawing demonstrating the relationship between the angle of parallelism and distance 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.

1/27/22

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 second hit when searching online for "construct straightedge compass duplicate angle".

1/26/22

Notes from today's class can be found here.

The midterm is currently scheduled for Wednesday, 2/9/22 (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.

Such an exam would be delivered electronically, and would not be proctored.

• I would be available over Zoom to answer questions.
• I would expect you to sign a statement confirming that the work you submit is your own.
• If your schedule requires you to be on campus during the scheduled time, you can of course use the classroom to take the exam, but you would need to be able to access Gradescope unless you make other arrangements with me in advance.

1/24/22

Partial 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.

1/22/22

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.

1/22/22

My apologies to students who were unable to join my office hours yesterday afternoon.

(An earlier meeting ran longer than expected...)

I will be available during my usual office hour on Sunday at 4 PM.

If you'd rather not wait that long, I am available most of the day both today and tomorrow (Saturday and Sunday, 1/22 and 1/23). Send me an email message to propose a time.

1/21/22

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.
• Images of the taxicab constructions made in class today can be found here and here.

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 RG §4.9...)

1/19/22

An annotated version of the 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.

A simple GeoGebra applet for taxicab geometry can be found here (at the bottom of the page).

If you would like a copy of this applet for yourself (and can't figure out how to download the underlying ggb file), ask me.

I was asked at the end of class whether parallel is a transitive property.

The answer is no! As should be clear from the hyperbolic parallel postulate, if there are two lines through a given point parallel to a given line, these two lines are obviously not parallel, as they intersect at the given point!

1/14/22

Notes from today's class can be found here.

The bug in the GeoGebra applets in MNEG has been fixed.

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).

1/13/22

HW #1 has been graded. Your numerical scores will show up on Canvas, but you will need to go to Gradescope to see and/or download your corrected assignment.

• If you received a lower score than expected, do not panic!
• Roughly speaking, 20–25 (out of 25) represents excellent work, and 15–20 represents good work.
• 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.

1/12/22

Some information about (taxicab) ellipses and hyperbolas can be found here and here.

An illustration of "sliding circles" to construct ellipses can be found 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.

I have posted an interactive book (MNEG) here containing most of the geometric models discussed so far.

There are direct links to the two- and three-dimensional GeoGebra drawing interfaces in the introductory sections.

See especially the chapter on taxicab geometry, which includes the GeoGebra applets shown in class today.

This resource will be updated as we add more models.

1/10/22

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:

• A detailed introduction to GeoGebra can be found here.
• An introduction to taxicab geometry using GeoGebra can be found here.
• GeoGebra applets for some standard taxicab geometry constructions can be found here.

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

1/7/22

A summary of today's class can be found here.

I will hold office hours via Zoom on Fridays and Sundays at 4 PM.

I may close the Zoom session early if nobody is present, so let me know in advance if you plan to arrive late.
I am also available by appointment, with Wednesdays at 4 PM being a particularly good choice.

1/6/22

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.

A sheet of taxicab graph paper is available here.

1/5/22

Office Hours:

• I will hold an office hour on Friday (1/7/22) at 4 PM.
• I will hold a second office hour on Sunday at either 4 PM or 7 PM. Please let me know in class which you prefer.
• Going forward, please let me know in class which of W/F/Su at 4 and/or 7 PM work best for you, or if none of them do.
• Appointments at (some) other times are also possible.
• All office hours will be held via Zoom; the link is posted on Canvas.

1/3/22

The slides shown at the beginning and end of class are available online:

• Models;
• Overview.

With apologies, I failed to record this morning's class...

Fortunately, I recorded a very similar lecture last year, which I have made available on Canvas (in the Media Gallery).

1/1/22

Below is some information about how this course will be run.

Overview:

• Class meetings will be held in person at the scheduled time.
• A remote option will also be available, via Zoom.
• Expect a combination of lecture, discussion, and both individual and group problem solving.
• Some "reading" assignments may involve watching short videos of me explaining a particular concept.
  Watch these videos before class, via Canvas.
• All class meetings will be recorded and available afterward to watch online via Canvas.
• All assignments will be submitted via Gradescope.

Details:

• General information about getting started with Zoom is available here.
• General information about submitting assignments via Gradescope can be found here.
• Further information can be found on my own information pages for Gradescope and Zoom.
• Each assignment exists in 3 places: on this website, in Gradescope, and on Canvas:
  • The assignment itself can be found (only) on the homework page.
  • Each assignment has a name, such as "Use Gradescope" or "HW 1".
  • When you have completed the assignment, export or scan it to PDF.
    Please do not take photographs of your work except as a last resort.
  • Upload your PDF to Gradescope, following the instructions here.
  • After grading, your corrected assignment will be available on Gradescope.
  • After grading, your score will be available on Canvas.

Let me know if you have difficulties with any of these steps.

These instructions are likely to evolve...

12/31/21

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 $\LaTeX$.

• 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.

10/20/21

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.