Prerequisites: Linear Algebra and Calculus
Meeting Times: Monday Wednesday 2:00-3:40 pm
Professor: C. Sormani
Office: Gillet Hall 200A
Office Hours: Mon/Wed 1:30-2:00 pm & 3:45-4:15 pm
Email: sormanic (at) member.ams.org
Professor's Webpage: https://sites.google.com/site/professorsormani/home
Course Webpage: https://sites.google.com/site/professorsormani/teaching/axiom-geom-s17
Course Description: Geometric theory from an axiomatic viewpoint motivated by Euclidean geometries (plane= E2, solid= E3) and additional non-Euclidean examples (Hyperbolic=H2 and Spherical=S2) . Emphasis on the relationship between proof and intuition. This course is for future math teachers and is not recommended for other math majors.
Text: College Geometry: A discovery approach, by David C. Kay, 2nd Edition. ISBN-13: 978-0321046246 ISBN-10: 0321046242 (HW is not assigned from textbook. Only a few pages are used from this textbook.)
Appendix B has a review of high school geometry.
Appendix C discusses Geometers Sketchpad (we won't cover).
Appendix F has a list of axioms for quick reference (bring to class).
Other Supplies: (not provided, bring your own)
Compasses, Ruler, Protractor, Rubber Bands, Blue Handball, Graph Paper Spiral Notebook (bring to classes)
Euclid's Elements, online
Projects which will be distributed in class
Wikipedia can be a useful resource for mathematics.
Engage NY website with complete info about the NYS Geometry Course
Homework: Complete all reading, review class notes and read projects carefully. Projects will be assigned 1-2 times per week which may be worked on together and submitted in groups. This is a four credit course, so the homework will be at least eight hours a week. No late work will be accepted (email photoed work if you miss class). Some projects may be resubmitted and the new grade will be averaged with the old grade. Always keep a photocopy of a project you have submitted. All proofs must be in 2 columns.
Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.
MAT345 Undergraduate Grading Policy: The 14 projects are worth 5% each, and the exams are worth 15% each. Note that the research report is not required but may be completed for extra credit. To pass the class, the second midterm grade must be at least 85%.
MAT630 Graduate Grading Policy: The 14 projects are worth 5% each, the research report is worth 10%, and the exams are worth 10% each. Note that the research report is required. To pass the class, the second midterm grade must be at least 85%.
Course Objectives:
1. Prove theorems about open sets, unions and intersections in metric spaces (E, F & G)
2. Prove the Euclidean Geometry Theorems for similar, congruent and right triangles (A, E & F)
3. Prove statements about parallelograms, circles, and the coordinate plane (A, B, E, F & G)
4. Identify and describe the main properties of hyperbolic and spherical geometry. (E)
5. Prove theorems about symmetries and transformations. (B, E, F & G)
6. Identify the properties of solid Euclidean Geometry. (E)
Homework assignments and projects available on the course webpage.
Schedule:
Monday 1/30: Pythagorean Theorem (What is a proof?)
including Pythagoras, 3-4-5 triangles-> 12 inches, grading policy
HW: Read Wikipedia entry on the Pythagorean Theorem, email the professor your info.
Wednesday 2/1: Metric Spaces and Set Theory
including balls, subsets, intersections, unions, open sets
Reading: Metric spaces: 78
Project 0: (due Mon 2/6, no resubmission of this project)
A metric space is a space of points with a distance between pairs of points satisfying the metric axioms D1-D3 on page 78 and the triangle inequality. All geometries we study in this course are metric spaces.
A ball, B(p,r), is the set of all points, x, such that d(x,p) is less than r.
(1) Draw balls in the Euclidean plane,E2, and three dimensional Euclidean space, E3.
(2) Prove that B(p,3) is a subset of B(p,5).
EC) If d(p,x)=t then B(x,r) subset B(p,r+t). Hint use triangle inequality.
A set, U, is open if for every point p in U, there is a radius r>0 such that B(p,r) is a subset of U. To prove a set is open: start with any point p in the set and choose the radius depending on that point p.
(3) Prove that a ball, B(q,R), is an open set.
(4) Prove that if U and W are open sets then U intersection W is an open set.
(5) Prove that if U and W are open sets then U union W is an open set.
Monday 2/6: Incidence Axioms of E2, E3, H2 and S2
and disjoint sets, empty set, for all and there exists notation
Reading: Incidence Axioms: 70-73 Hyperbolic Space, H2: 446-447, Spherical Geometry, S2: 547
E2, H2 satisfy axioms I-1, I-2 and I-0.
Axiom I-0: The space contains at least three non-colinear points.
S2 has slightly different axioms. Can you write them down?
E3 satisfies incidence axioms I-1 to I-5.
Homework (due Wednesday 2/8):
Draw triangles in all four geometries (for the sphere bring in a ball with a triangle made of three rubber bands).
Draw two lines (geodesics) in E2,E3, and H2 which are disjoint (have no intersection). What about S2?
Prove: If P and P' are distinct planes in E3 and points x, y, and z are in their intersection, then x,y, and z are colinear.
Start project 1 problems 1-4.
Wednesday 2/8: Betweeness, Segments and Rays
Reading: E2,E3: 79-82.
Project 1: (due Wed 2/15) Be sure to keep track of the axioms you use! You must email me to be marked as attending.
(1) Prove that in a metric space B(p,2) is a subset of B(p,4).
(2) Prove that in a metric space, if d(x,y)=R>5 then B(x,5) and B(y, R-5) are disjoint.
(3) Prove that if p is a point in a metric space and U is the set of points x such that d(p,x) > 5, then U is an open set (use only definition of a metric space and open set).
(4) Prove that if P and P' are planes in E3 and x,y, and z are points in their intersection and x,y,z are not colinear, then P=P' by contradiction (use only incidence axioms).
(5) p88/5 (not a proof) (6) p88/6 (not a proof)
(7) Prove Theorem 2 on page 80, (8) Write p 84 Example 2 as a 2 column proof, skip this if you don't have p84
(9) p89/15 (a 2 column proof), (10) p89/16 (a 2 column proof)
(11) Prove that if P is a plane in E3 and x,y are in the plane, then the ray from x through y is in the plane.
Students may wish to meet on Monday 2/13 to work together on this project. It is a holiday but the building will be open.
Wednesday 2/15: Ruler Axiom and Segment Construction Theorem
and ruler function version of the ruler axiom
Reading: p83-85, Start problems 1-5 of Project 2. You may wish to meet during class time on Mon 2/20 as the building will be open on the holiday.
Wednesday 2/22: Protractor Axioms and Angles
and proving a set is not open, discrete metric spaces, a single point set in E2 is not open,
Reading: p 90-96,
Project 2: (due Mon 2/27)
(1) Write a two column proof of the segment construction theorem.
(2) Use Axioms I-0, I-1 and D-4 to prove that if p is a point in a line L and r is greater than 0 then B(p,r) is not a subset of L.
(3) Prove that a line is not an open set (email me for hint)
(4) Do page 88 problem 6 (not a proof)
(5) Use the ruler axiom to prove that if x and y are points then they have a midpoint, z, such that d(x,z)=d(y,z)=d(x,y)/2.
(6)-(8) Draw an angle in E2, E3 and H2 and shade the interior of the angle.
(9) Prove Theorem 2 on page 96
(10) Do page 99 problem 2 indicating which axioms and theorems you are applying.
Monday 2/27: Perpendicular Lines and Mappings
and images/preimages of sets, the real line as a metric space, the distance function from a point, continuity
Reading: pages 96-99, Try Project 3 problems 1-3 working alone.
Project 1 resubmission due Mon 3/6 (look over right away to ask questions on Wed office hours).
Wednesday 3/1: Convex Sets and Halfplanes
Reading: pages 104-110
Project 3 due Mon 3/6 (must have new partners you have never worked with before)
(1) Write the Linear Pair Axiom using notation rather than words: Given A-B-C and a point D not on line AB, then...
(2) Prove the Unique Perpendicular Line Theorem (page 99 Theorem 4)
(3) Draw the unique perpendicular line theorem in S2 and H2.
(4) Prove the Vertical Pair Theorem (page 99 Theorem 5)
(5) Prove that that if K and K' are convex, then their intersection is convex.
(6) What about the union of convex sets K and K'?
(7)-(10) Problems 1-4 on pages 111-112 (not proofs)
Monday 3/6: Angle Interiors and the Crossbar Theorem
Reading: pages 108-110, Try Project 4 problems 1-3
Project 2 resubmission due Mon 3/13, look over right away to ask questions on Wed office hours.
Wednesday 3/8: The SAS Hypothesis
Reading pages 120-124
Project 4: (due Monday 3/13, no resubmissions of this project) (resubmission is now allowed Monday 3/27)
(1) Prove Pasch's Theorem (page 107 Thm 2)
(2) Prove the interior of an angle is a convex set.
(3) Write a proof of the Crossbar Theorem in two columns.
(4)-(10) Do problems 3-9 on pages 126-7 (no proofs, but explain clearly which sides and angles match)
The sphere and hyperbolic half plane also satisy SAS. This is easy to see on the sphere. It doesn't look true on Hyperbolic space but that is because the distances between points have a complicated definition (see page 450-451).
Monday 3/13: Congruent Triangles
Reading: Section 3.3, Start Project 5
Project 3 resubmission due Monday March 20, look over right away.
Project 5: (due Monday 3/20, no resubmissions of this project) (This is now due Friday 3/25 before midnight)
(1) Write up the proof of SSS. (Read 123-124 and 139-142 on SAS Postulate/Hypothesis and ASA Theorem, Ex 1-3, Isosceles Triangle defn and Theorem first. Also read the construction of the perpendicular bisector.)
(2-3) On graph paper using compass and straight edge, construct the perpendicular bisector of segment AB, where A=(1,3) and B=(5,7), then where A=(-2,4) and B=(-6, 2).
(4) Suppose A,B, and C are noncolinear, prove there exists a unique line perpendicular to line BC passing through A.
(5-6) Given a line L and a point P, the distance from L to P is defined as the length of the unique perpendicular line segment from P to L. If R= dist(P,L) then B(p,r) intersected with L is empty iff r is less than or equal to R. Prove this (two directions).
(7) Prove that a half plane is an open set.
(8) Prove that the interior of an angle is an open set.
(9) Angle bisector construction: Given angle ABC with AB=BC, show that if a point X has AX=CX, then ray BX is the angle bisector. You may use Theorem SSS to do this proof.
(10) SKIP THIS ONE Write up proof of SSA as stated on page 175 (or in handout). (Read proof of ASA in handout).
Note we pospone the proof of AAS which can be proven quickly once we know the sum of the angles of a triangle is 180 degrees. That cannot be proven until we introduce Euclid's Parallel Postulate. Right now everything we've proven is true for more general geometries like the sphere and hyperbolic space. For a proof of AAS in these more general spaces, you may read 3.6 if you wish.
Wednesday 3/15: Concurrent Lines, Perpendicular bisectors, angle bisectors, medians
Read and do Project 6 (due Wed 3/29 after the midterm): Resubmitting Project 4 is now allowed on Monday 3/27.
Extra Credit: Parts III and IV
Monday 3/20: Review of Projects 1-5
Study for the midterm exam. Bring sheet of all axioms to the midterm. The midterm is proofs.
Wednesday 3/22: First Midterm on Topics in Projects 1-5
Look at the Engage NY website: this has all the material needed to prepare for Exam II
Research Report Part I (due April 7, no resubmissions, must work alone) :
Choose and complete a project examining the incenter or orthocenter on hyperbolic space,
or the circumcenter, incenter or orthocenter on the sphere
Monday 3/27: Parallel Postulate:
Parallel Lines, Angle Sum Theorem,
Start Project 7 alone pages 1-3 See also the Engage NY website for this and all other topics in Euclidean Geometry.
Wednesday 3/29: Parallelograms and Bridges:
4.2 Parallelograms, rhombuses and squares
Project 7: (due Wed 4/5 everyone submit their own project)
Monday 4/3: Similar triangles:
Similar triangles 4.3, proportions,
Project 8: (due Friday 4/21 no resubmissions of this project)
Wednesday 4/5: SOHCAHTOA
Resubmission of Project 6 due Fri 4/28 (reliable video online)
Research Report Part II: (due Mon 4/24)
Write 2 pages with diagrams explaining why parallel postulate theorem, sum of angles is 180 degrees and one other theorem from the past two weeks are all false on either hyperbolic space or the sphere.
Wednesday 4/19: Pythagorean Theorem and its Converse:
Project 9: (due Monday 4/24, no resubmissions of this project)
Working with Triangles Project, problems 1-10 on the third page must be handed in on graph paper using the first two pages as reference. Students wishing to complete a research paper who have not yet started may still do so. If you did not do part I as assigned above, you may instead write up something about solid geometry (7.1-7.2). You must hand in Part II of the research paper (handwritten is fine) by next Mon 4/24)
Thursday 4/20 the class has an extra meeting and students should work together on the project in the classroom if they can make it on a Thursday. I will be here to answer questions.
Monday 4/24: Symmetries and Isometries:
Reflections and Translations 5.2-5.3
Project 10: (due Monday 5/1, no resubmissions of this project)
Note the theorems in Project 10 hold on H2 and S2. Part III of your research report may be based on this project. In fact one may use reflections across lines to see what congruent triangles in the hyperbolic plane look like.
Wednesday 4/26: Start Circles, Arcs and Chords.
Monday 5/1: Circles, Arcs and Chords (the professor will be at a meeting at the GC so just work in groups, there is a makeup lesson at the end)
Circles 3.8 and 4.5, NYS Standards GG49-GG53.
Project 11: (is due Wed 5/11, no resubmissions)
Circles Project pages 1-3. Page 4 is Extra Credit.
Note the theorems in Project 11 hold on H2 and S2
Part III of your research report may be based on this project if you are working on spherical geometry
Wednesday 5/3: Coordinate Geometry and Transformations
Project 12: (due Monday 5/8, no resubmissions)
.
Monday 5/8: Transformations, Areas :
Shift Isometry and Lines, Skews/Shears and Dilations, Skews and Dilations (taught on last Wed)
Resubmission of Project 7 and Project 9 due Friday 5/12 going over these in class today.
Project 13: (due Friday 5/12, no resubmissions) (went over last Wed)
Transformations and Rotation Matrices
Wednesday 5/10: Area and Volume (Project 11 on circles is due today, resubmit Project 7 inc EC on Fri May 12)
Project 14: (due Monday 5/22, no resubmissions)
Prepare for the Second Midterm which is a NYS Regents Exam (projects 6-14) To pass this course you must score 85% on this exam!
The proofs on this exam are on the level of those on the NYS Geometry Regents Exam.
See sample NYS Regents Exams.
Monday 5/15: Second Exam on Euclidean Geometry. This is based on the NYS Regents Material (see the Engage NY website)
Research Project Part III is due at the Final: An individual project either about circles on the sphere (Project 11)
or isometries on the sphere or hyperbolic space (Project 10).
Wednesday 5/17: Review and Hyperbolic Space
We will go through the theorems of the semester and see which work on hyperbolic space and which fail.
The final will cover material from the entire course. No notes permitted on the final.
Make Up Lesson: Review Class during Finals Week: Mon May 22 2-3:30 pm
and Final during Finals Week: Final Wed May 24 1:30-3:30 pm