## Class Highlights

• Mon, Jan 11th - Intro to Web and Class. HW Read Preface, Section 1.1 and p.12-13 of the textbook
• Wed, Jan 13th Go over HW, inductive and dedective reasoning, and started on logic (negations of statements). HW Convince yourself that the negation of A OR B is NOT A AND NOT B, Wile E. Coyote project (see main web page).
• Fri Conference - work on Wile assignment
• Mon MLKJ Day
• Wed - Intro to Logic via minesweeper.
• Fri - Take 16*16 semi-completed game of minesweeper. Students go around the room presenting proofs (either that a box is a bomb, or that a box is a specific known number) verbally and in order. Things we've learned: proof by contradiction, proof by cases, order or thms matters a lot! (ie you get to use what you've allready proved). HW: review minesweeper, and read section on finite geometries.
• Mon - Intro to geometer's sketchpad and four point geometry.
• Wed-Review of Mon, and 4-line geometry
• Fri-Presentation of Problems
• Mon-Verify Euclid's axioms on geometer's sketchpad, continuation of Presentation of Problems.
• Wed Feb 3rd-Euclid's Axioms. We'll talk about Euclid's Thm1 Book 1 stating that it is possible to construct an equilateral triangle on any given line segment. Within his proof, Euclid needed that 2 specific circles intersected. We showed that A1-A5 are valid on the sphere, in addition to the space consisting of the set of rational points in the plane. In this last space, the specific circles did not intersect, showing that Euclid made false implicit assumptions from his Axioms. In addition, Euclid failed to define numerous terms. Wile E. Revisions due
• Fri- Investigation of Euclid's Axiom 5. Change If then to iff. Intro to Playfair's Postulate replacing Axiom 5. HW Read Section 2.2 and 2.3. Problem Set 1 due at 5.
• Mon- Proof that the sum of the angles in a triangle is 180 degrees. (done on sketchpad, so that proof could easily be changed to be clearer).
• Wed - Negation of Playfair's Postulate to obtain models for Spherical/Hyperbolic Geometry. HW: Read Section about Hilbert's Postulates.
• Fri- Discussion of Hilbert section. HW- negate the statements of the 3 theorems in the Hilbert section, III-5 and IV-1.
• Mon-presentation of negation of Hilbert's Postulates - HW read section on Birkhoff. PS 1 rewrite due at 5pm
• Wed, Feb 17th-discussion of Birkhoff's axioms. HW re-read section.
• Fri, Feb 19th-continued discussion, started SMSP, hw read both of those sections.
• Monday, Visualizing higher dimensions (see geometer's sketchpad page and main page for sketchpad assignment), upper half plane model of hyperbolic space.
• Wednesday, Feb 24th - back to SMSP
To check the web from your appstate e-mail account,
Type "lynx"
The asu web page (text only - no pictures!) will come up.
Type "g"
At the bottom of the asu web page, you will see "URL to open:". Type "www.cs.appstate.edu/~sjg".
My web page will come up. Press space for the next page and or scroll down using the "down arrow key" until the bold 3610 is highlighted.
Use the "right arrow key" to follow the link to our class web page.
Use the "right arrow key" to follow the link to our class web page.
To quit lynx, type "q" and then "y" for yes.
Problem Set 2 due at 5pm
• Fri, Feb 26th - hyperbolic and spherical geometry.
• Spring Break!
• Mon and Wed, March 8th and 10th - oral test
• Fri, March 12th - No class due to conference. Read over section 2.7 and compare to 3.1 and 3.2 (Neutral Geometry). Fill in incomplete proofs in the book.
• Mon, March 15th 3.2 (Neutral Geometry). We compared definitions to spherical and hyperbolic models. We wrote out a complete proof of the first Theorem. Rest of proofs are assigned as a problem set.
• Wed - Why ASA is false on the sphere (using manipulative sphere models), why SAS is false on the sphere.
• Fri, Assumptions for Neutral Geometry do not hold in spherical geometry- how do we fix this?
• Mon, Assumptions for Neutral Geometry revisited.
• Wed started filling in the gaps in the first proof of 3.3.
• Fri, March 26, No class - Assignment - carefully fill in the gaps in the books proofs in section 3.3
• Mon, Students fill in gaps on the board.
• Wed Students prove corollaries on the board.
• Mon 5th Start rectangles.
• Wed 7th. End of Chapter 3. Can you find a rectangle on the sphere or hyperbolic plane? What if we just require the base angles to be 90 degrees, and only require the summit angles to be congruent - can you find these on these spaces? How about if we just require base angles to be equal, and summit angles to be equal, but nothing else-can you find these on these spaces?
• Fri 9th Start Chapter 4
• Mon 12th Back to Geometer's Sketchpad for Euclidean space. In groups, create a script, drawing and text box proof on Geometer's Sketchpad that the sum of the measures of the interior triangle (Euclidean space) is 180 degrees. "Proof check" your proof and then show it to me so that I can "proof check" it.
Homework for Wednesday-be prepared to present a method of approach:
Prove that the following are Equivalent in Euclidean Geometry:
a) Euclid's 5th postulate p.30
b) Playfair's postulate p. 38
c)Converse of the alternate interior angle thm section 3.4
d) Hilbert's parallel postulate Axiom IV-I p. 44
e)SMSG parallel postulate 16
• Wed 14th students present methods of approach from hw from Monday. HW carry out your method to come up with a rigorous proof.
• Fri and Mon 16th 19th - beach ball assignment and reading on hyperbolic geometry (pick this up from my door if you missed class).
• Wed 21 - Start the proof for a formula for the area on a sphere
• Fri 23 - Finish the proof for a formula for the area on a sphere, and use it to prove that the sum of the measures of angles of a spherical triangle is bigger than 180 degrees. Go over the readings on hyperbolic space.
• Mon 26th - On Geometer's Sketchpad, prove that the parallel postulates are all equivalent in Euclidean Space (see above from homework for Wed). Start with a list of the 5 statements and pictures. For the proofs, include pictures in addition to the text comments. DUE Fri.