In 209b lab: Desargues' Theorem Lab

As I come around, ask me any questions you have on the final research and review presentation

Back in the classroom: projective geometry continued. Slides. sphere. Projective geometry and transformations. Evaluations.

Take questions on the study guide.

Applications of hyperbolic geometry. shape work. "shape of the universe"

tiling octahedrons in shape.

Applications of hyperbolic geometry.

Back in the class: Review the activities #6 on from last week, and today's activities, and discuss the crochet model of hyperbolic space. Review parallels in hyperbolic geometry via Euclid's 5th sometimes but not always holds, and the existence of Playfair's via perpendiculars, but since I-29 doesn't hold we obtain infinitely many parallels to any line through a given point and sum angles proof goes wrong. The Hyperbolic Parallel Axiom states that if m is a hyperbolic line and A is a point not on m, then there exist exactly two noncollinear hyperbolic halflines AB and AC which do not intersect m and such that a third hyperbolic halfline AD intersects m if and only if AD is between AB and AC. How can we make sense of this axiom? axiom. axiom 2. Pythag image of the beginning of Bhaskara's construction in hyperbolic geometry and an image of the attempt at a hyperbolic square. square comic

Comic---guess the punchline, comic 2

Prove that Euclid's 5th Postulate plus Euclid's other axioms before I-31 prove Playfair's.

Discuss different definitions of parallel. Review parallel ideas including same side interior angles being supplementary, alternate interior and corresponding angles being the same, equidistant lines, etc. parallels meeting

Back in the class. Review questions 1-5 from the lab activities and discuss why Euclid's 5th Postulate is vacuously true in spherical geometry. Playfair's Postulate in Euclidean geometry - the existence part by constructing a parallel by using perpendiculars. Which Euclidean propositions are we using? Why are the lines parallel in Euclidean and hyperbolic geometry? Discuss what goes wrong with the existence part of Playfair's on the sphere.

Euclid's 5th Postulate is vacuously true on the sphere so unlike what is listed on the web and in books, the statements are different.

Poincare_Disk_Model_of_Hyperbolic_Geometry.gsp

Discuss Felix Klein and his Erlangen Programm - revolutionizing geometry by understanding a space by its symmetries and transformations.

Review the angle sum on a sphere via clicker questions.

Examine consequences, including AAA on the sphere implying congruence. Consequences for the formula for the area of a spherical triangle - whether the difference between the angle sum and pi is detectable for a 1 mile square area triangle in Kansas. Use the triangles to examine the area of regular polygons on the sphere (n-2 triangles). Discuss Colorado and Wyoming.

Why does it work in Euclidean geometry and what goes wrong on the sphere? Walking and folding an angle sum in Euclidean geometry (use a folding argument to show that parallel implies the sum of the angles in a triangle is 180 degrees), and then complete a Euclidean proof of I-32. Discuss what goes wrong with the proof of I-32 on the sphere.

Discuss Godel's 1930 theorem.

Begin hyperbolic geometry via the Escher worksheet a lab version. number 2

Discuss various ideas of parallel.

Angle sum on a sphere, Sphere activity.

Platonic/Archimedian solid and review the Archimedian solids and their symmetries.

1. Look at this image and recall that we proved (using congruence theorems - SAS) that in Euclidean geometry the circumcenter is the intersection of the perpendicular bisectors of a triangle and that it is equidistant to the vertices.

2. Water Reservoir Problems in Sketchpad

3. If you have finished before we go back to class, work on Project 5.

Back in the class. Go over lab. Measurements with and without metric perspectives. How were circumference, area and volume formulas obtained via axiomatic perspectives and before coordinate geometry and calculus II? Circumference, length and pi. Orange Activity and Archimedes polygonal method.

Angular defect and the connection to the Euler characteristic of a Polyhedra

Metric perspectives in Euclidean and taxicab geometry (a sub-Euclidean geometry). Highlight the possible number of intersections of taxicab circles for different examples. Do 3 non-collinear points determine a circle in taxicab geometry? Squares.

Metric for taxcab geometry and the Euclidean metric. US law is Euclidean. SAS in taxicab geometry. Revisit Euclid's proof of SAS and what goes wrong in taxicab geometry.

Overview of Project 4 and rubric.

Pythagorean theorem water demo

Euclid's proof of the Pythagorean theorem.

A modern version of Euclid's proof of the Pythagorean theorem by Dave Lantz at Colgate

Additional extensions: A review of the Greenwaldian Theorem, as well as the Scarecrow's Theorem.

Polyhedra and platonic solids in Euclidean and spherical geometry

Measurement: Play a few games of taxicab treasure hunt. Introduce taxicab geometry via moving in Tivo and relate to taxicab treasure hunt.

Review the equilateral triangle construction. Replicate the construction in Spherical Easel. Why do the circles intersect? Drag to large circles and compare with the Euclidean proof. Note that for an intersection, you choose the intersection menu and then the objects (the reverse of Sketchpad).

Sibley

Take questions on test 1 and the study guide.

Pythagorean theorem water demo

Euclid's proof of the Pythagorean theorem.

A modern version of Euclid's proof of the Pythagorean theorem by Dave Lantz at Colgate

Additional extensions: A review of the Greenwaldian Theorem, as well as the Scarecrow's Theorem. Burden of Proof. Polyhedra models.

1. Help/Sample Sketches and Tools/Geometry/Pythagorean Theorem

Go through Behold Pythagoras!, Puzzled Pythagoras, and then Shear Pythagoras. Click on Contents to get to the other Sketches.

2. Pappus and more (Note: if you get to the back side, you could use Help/Sample Sketches and Tools/Custom Tools/Regular Polygons)

3. Go to ASULearn and read my forum posting to you (my comments are in with the class comments)

4. Go through A modern version of Euclid's proof of the Pythagorean theorem by Dave Lantz at Colgate

Back in the classroom. Review lab activities. Take questions on project 2 solutions. Project 2 #9

Go over Project 2 questions 6-9, including proofs of SAS and what goes wrong on the sphere, and AAA and degenerate spherical counterexamples as well as AAA giving congruence rather than similarity for nondegenerate spherical triangles.

1. Turn in hw from the book on similarity, the self reflection from Project 2, and the peer review.

2. Mathematical modeling using geometric similarity. Excel file. Worksheet.

3. If you are finished before we head back to the classroom, then work on

Back in the classroom:

Review geometric similarity in modeling. Discuss what would happen if you thought that a certain amount of body weight is independent of size in adults. Other considerations.

Go over Project 2 questions 1-5.

Meet in 209b lab. Collect hw.

1. Star Similarity

2. Shearing along the x-axis

3. If you are finished before we come back together, you can work on Project 2 or on the Sketchpad problems for Project 3.

Back to the classroom. shearing sheap

linear transformation comic

linear transformations

Applications of similarity:

Read the proof of the trig identity and then fill in the details and reasons using similarity, trig and the Pythagorean theorem, along with some of Euclid's notions. Note that we can show that the Pythagorean theorem is a consequence of similarity as in the next project.

Begin

Sibley

Discuss Wile E Coyote axiom system. Wile - how did they ensure the chase would always begin? That he would continue to see him? How did they ensure Wile would catch the RR when the RR runs faster?

Introduction to "same shape". Fig 8.4 Fig 8.21 Fig 8.32

Count off by 7. Groups prepare short presentations on SSS, SAS, AA, SSA, AAS, ASA, HL (Hypotenuse and leg of a right triangle).

Game 2 of Minesweeper.

Meet in 209b lab. Turn in project 1.

1. Form a group of 2 or 3 people (if you prefer to work alone that should be ok too). Write down your name(s) and your top five problems (unranked) from Project 2.

2. Use the Triangle_Similarity.gsp file (control click and save the file. Then open it from Sketchpad) to complete the Similar Triangles - SSS, SAS, SSA worksheet. Summarize the activities and your responses to them in your notes.

3. Investigate similarity in quadrilaterals. What is the smallest amount of information you need to determine that quadrilaterals are similar? Try this in your own groups either on paper or in Sketchpad - do NOT use an internet search.

4. Work on project 2 once I give your group a problem (and after you have completed 2 and 3). DO use internet searches.

Back in the class discuss folding an angle bisector. Discuss the incenter, circumcenter, orthocenter and centroid. Review Triangle_Similarity.gsp and quadrilateral similarity.

Share where Sketchpad is on campus and take questions on the construction for proposition 1 from the hw.

Where is North? Also discuss 8/08 article

Take questions on project 1. Advice from prior students. Educational goals at ASU

Clicker review of proposition 11 and Sketchpad's built in version of Proposition 11.

Bisecting a line segment. Discuss Proposition 10 in Sketchpad (using Proposition 1 and Proposition 9) and via folding.

Create a slider on a perpendicular bisector--the lines connecting the slider to the corners must be congruent by SAS.

Construct a triangle and construct the perpendicular bisectors. Can we say anything about them? How about as the triangle changes.

Go over an application - a proof that the perpendicular bisectors are concurrent. The circumcenter.

Introduction to Project 2.

Wile E Coyote axiom system.

Read through intro to Geometric Constructions and ask me any questions as I make my way around to help you.

Go through the first side of Euclid's Proposition 1. If you are finished before we come back together, then explore the features of Sketchpad.

Back in the classroom, register the

Go through a proof of Euclid's Proposition 1.

Work on Proposition 11, including paper folding. Reflect on Common Core, transformations and reflections. Sketchpad proposition 11

Where is North? Also discuss 8/08 article

Introduction to the course Form groups of 2 people and discuss how can we tell the earth is round without technology? Mention the related problem on Project 2 [Wallace and West Roads to Geometry 1.1 8].

Inductive versus deductive

Peanut Butter and Jelly Robot: How to make a peanut butter sandwich? Does not compute.

An introduction to minesweeper games as an axiomatic system.

Axiom 1) Each square is a number or a mine.

Axiom 2) A numbered square represents the number of neighboring mines in the blocks immediately above, below, left, right, or diagonally touching.

Examine game 1.

History of Euclid's elements and the societal context of philosophy and debate within Greek society.

Euclid's Postulates

Euclid's first 3

Intro to Geometric Constructions.

Begin Euclid's Proposition 1 by hand, by paper folding

https://www.youtube.com/watch?v=6dA2R8bLb7Q

and by a proof. Comment on the connection between paper folding and reflections and transformations.