February 5

Plan:

  • announcements
      • how we may use the wiki
      • grade based (in part) on regular contributions
      • volunteers for wiki administration?
  • questions? I expect some on Sacchieri and on $\TeX$
  • skip Legendre
  • start reading Lobachevsky

Pre-class Reflection Rob Moray
I found doing the proofs for Euclid's theorem's difficult, especially the parallel postulate. I attempted to read Saccheri's logic, as well as skim through the first chapter. I have become more familiar with $\LaTeX$ and installed the framework on my computer as well as downloaded a suitable editor (TeXshop: http://pages.uoregon.edu/koch/texshop/), though I would still like to develop a better knowledge so I can apply this toward other work. I feel, regarding the choice of my editor, that there is a more suitable editor, and would like one that, if possible, has a way to keep my project updated so I can see my changes without having to continuously compile (a downside of TeXshop). I am frustrated at options for inserting custom graphics, and it would have aided me in doing my homework to have a better understanding of shape creation in $\LaTeX$. I did a number of extensive internet searches but could only find programs that claimed to aid in the process but produced compile-time errors when I imported the code the they generated into my project file. I would like to use this wiki to collaborate and discuss these important concepts, and I have found it helpful to have this kind of structure in MATH 480, which I am also taking this semester.

I also took a look at Joe Cohen's boilerplate .tex file for homework assignments, mostly because I know and respect him and his work in the CS department. I would like to use some of his code as a baseboard for developing my own style for my future homework assignments.


I do not have the book yet, it is on the way, so I can only give a very brief description of today's class.
We basically followed what Prof. Bolker have planned.
We started with the book on page 32 with Lobachevsky, and then on page 33 with his 10 'must known' 'axioms' in order to read his book (I think that is the way to put those 10 axioms, like the 23 definition in Book 1 of Euclid's Elements). We briefly talked about 5 and 7 and one other (I am not sure about the number, and may be two others). Then we talked about 16, 17, 18, 20, and 22, mainly focused on his idea of 'parallel line', which is the first line that does not meet with the existing line. This idea was so confusing to me at the beginning, because I kept thinking 'parallel line' in the way/definition that I have been taught; it took a bit for me to get used to his idea.
That is what I have for today's class, I know it is very brief, so please do add more onto or even change it.
Lin


'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it means just what I choose it to mean — neither more nor less.'
'The question is,' said Alice, 'whether you can make words mean so many different things.'
'The question is,' said Humpty Dumpty, 'which is to be master — that's all.'

Through The Looking Glass By Lewis Carroll.


We also discussed homework:

  1. Homework I is due Thursday, February 7, 2013.
  2. Homework can be collaborative.
  3. It's ok to say you don't know how to do a problem, though it is important to document what steps you've taken (even if you look at an online proof).

We furthermore discussed procedures going forward for the wiki

  1. Contribution is integral to the course, and will make up a significant portion of the final grade.
  2. Students expected to contribute lecture notes
  3. No admin so far

Possible suggestion: Create a separate discussion page, with a similar name format to that being used already, only with "-discussion" appended. For instance, this discussion page would be february-5-discussion.


We briefly examined Sacchieri and his (potentially misguided) attempt to prove Euclid's fifth postulate.
We then moved on to Lobachevsky. Both Sacchieri and Lobachevsky started from the same place - assuming the fifth postulate was not true. But their philosophies were different. Sacchieri was looking for a contradiction, in order to establish the parallel postulate as a theorem. Lobachevsky knew he was creating a new kind of geometry.

We examined some of his definitions, as well as his postulates.

My understanding of Lobachevsky's definition of the word "parallel": Consider a line L and a point p in the plane. Every line through p either meets L or does not meet L. Draw a line segment P from point p to L, perpendicular to L. A line M is parallel to L if it is the first line above, or the first line below, the segment P that does not meet L.

By Lobachevsky's definition of the word parallel, each line can have up to 2 parallels (one on each side of the perpendicular segment P).

Definition:
The boundary lines of one and the other class of those lines will be called parallel to the given line.

We noted another striking difference from Euclid: The introduction of numerical associations. While Euclid did not use the concept of $\pi$ to define a right angle, merely calling it a right angle, Lobachevsky introduces this idea in elucidating the above definition in his text.

Proposition 17:
A straight line maintains the characteristic of parallelism at all its points.
This is an interesting proposition, and something probably taken for granted today. Lobachevsky offers an interesting proof, and the figures from the text are helpful in understanding this proposition.

Proposition 17: A parallel line is parallel everywhere, meaning that wherever we draw the perpendicular segment P, M will be the first non-intersecting line for L.

Proposition 18: If M is parallel to L, then L is parallel to M.

Proposition 19:
The sum of angles in a triangle is less than or equal to $\pi$.

In a rectilineal triangle, the sum of the three angles cannot be greater than two right angles.
This is interesting, because Lobachevsky does not refer to "two right angles" as "$\pi$. This one may be examined last time, but it should be noted that the proof of this proposition has been omitted from the book. Of course, we now accept this proposition as a given.

Proposition 20:
Proposition 20: If there exists one triangle with angles that sum up to $\pi$, than the sum of angles of every triangle is equal to $\pi$.

If in any rectilineal triangle the sum of the three angles is equal to two right angles, so is this also the case for every other triangle.

Professor Bolker has indicated that this will be further examined in the next class session. It is interesting that Lobachevsky attempts to generalize all triangles from one, even though not all triangles are the same (this is defined in Euclid's Elements to some degree). Lobachevsky offers an interesting proof, as illustrated in Figure 1.14 (p. 37) , figure 1.15 and figure 1.16 (p. 38)

Proposition 22:
If two perpendiculars to the same straight line are parallel to each other, then the sum of the three angles in a rectilineal triangle is equal to two right angles
The wording of this seems confusing. It would be interesting to have some more clarity here. Unfortunately, there don't appear to be visual aides accompanying the proof for this proposition. It is interesting to note the use of the capital pi, $\Pi$ as a function such that $\Pi:\mathbb{R}\to\mathbb{R}$.

Rob

Ethan: We can discuss this in class if others want clarity too.

Three interesting articles:


1) Shira's mention of Alice in Wonderland reminded me of this article in NewScientist a few years ago, Alice's adventures in algebra: Wonderland solved

The author's thesis is that some of the characters and scenes that were added to the original story (Cheshire Cat, the trial, the Duchess's baby or the Mad Hatter's tea party) were a criticism about the abstraction that mathematics was going through at the time (1865). You probably know that Lewis Carroll was a pseudonym for Charles Dodgson, who was a mathematician.

More mathematical connections can be found in the Wikipedia entry for Alice's Adventures in Wonderland under 'Symbolism'

After you read this, you'll never look at Alice in Wonderland the same way again.

2) I also ran across an article (which won an award) in the American Mathematical Monthly entitled
'Why Did Lagrange "Prove" the Parallel Postulate'?

Even though we didn't discuss Lagrange (who preceded Legendre), I thought some of you might be interested, as the author tries to capture his motivations and also conveys the state of scientific and mathematical thinking of the time. Perhaps someone might find this useful for their paper.

3) I just noticed this article from a few months ago,
Ancient d20 die emerges from the ashes of time
Its age is estimated from between 300 and 30 BC and was found in Egypt, so maybe Euclid himself rolled it.

===Matt===

In this class we talked mostly about Lobachevskian Geometry. Our text book lists axioms that will help understand his book more clearly, just like Euclid. We looked in the book where the authors skip Lobachevsky's axioms 11 to 15, this means that the author of our book did not thing they were influential to the history of math. But the axioms that are in the text are very similar to that of Euclids. The only thing that i like more about Euclid over Lobachevskian is that Euclid has definitions so you can understand the words he uses and the definitions he uses. It makes understanding the propositions better. Im not sure i can assume that definitions that Eculid uses would be used in the same context in Lobachevskian. I like the way this class is going, it is very interesting and i like this WIKI space. Seeing what others write make the class subjects even more clearer. I know that i am a slower person when it comes to mastering a concept so seeing more than ones person veiw to a topic will help me master the upcoming concepts.

Nicole Nocera


in this class, we talked about Lobachevsky and his geometrical researches on the theory of parallels. Actually I learn some of them in the high school, but now I learned the systematic theory. Lobachevsky talked about the parallels of straight line, but when I was in class I thought about the parallels of non-straight line. I don't know if it's possible that two non-straight line are parallel. for example, if we have two circle with same origin and different radius, suppose one of perimeter is L1 and another one is L2, then L1 never meet L2, could we say L1 and L2 are parallel? also, if we do some changes for 2 functions of sin, the function may never meet, are they parallel? I do not know actually, the ideas just came up when we were talking about the parallels.
Jun


*Introducing Lovachevskian Geometry. And his way of defining parallels. The parallel lines are not just some lines that don't intersect with the given line, but rather, it is the boundaries of this class of lines.
*And the parallel property is kind of symmetric.
*Question: How many lines at most can be parallel to a given line in Lobachevskian space?

Mingzhi


  • Thinking in class is good. Asking is even better. In this case I think you are stretching the word "parallel" much too far. In mathematics you have to pay attention to exact meanings in each particular situation (just like the Humpty Dumpty quote above). For Lobatchevsky a line is "parallel" to another exactly when it's a line that separates crossing from noncrossing lines. It doesn't mean "they don't meet". In this discussion it makes no sense to try to talk about parallel curves, or parallel circles. Also - you can't distinguish betwen straight and non-strait lines. All of Lobachevsky's lines are straight, even though you can't draw pictures that look that way. Ethan

On february 5 we talked about Lobachevsky (1792 - 1856) . Lobachevsky would develop a geometry in which the fifth postulate was not true, but ended up by proving it. What he did was replace the fifth postulate of Euclid by his Parallel Postulate. "There exist two lines parallel to a given line through a given point not on the line." We went through a bunch of his propositions, among them the twentieth that Mr Bolker will attempt to prove at the next meeting. Yawou

  • Lobachevsky did not prove the fifth postulate - is that what you really meant to say? In general, there's no need to write in these notes that "we went through a bunch of propositions". Write things here that will provide useful information to other readers. If you have comments about things you didn't understand, or particularly liked or dislike, put them in the notes page: february-5-discussion.

Gauss in his correspondence speaks about his discoveries on the theory of parallels but there is no scientific publication.
Saccheri found Hyperbolic Geometry and Gauss discovered it.
But we must attribute to Boylai and Lovachevsky the discovery of the Non Euclidean Geometry.
Rachel Bay-Chaparro


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