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.
Prerequisites: Modern Algebra, Linear Algebra and Calculus
Meeting Times: Monday Wednesday 4:00-5:40 pm
Professor: C. Sormani
Office: Gillet Hall 200B
Office Hours: Monday Wednesday 3:30-4:00 pm & 5:40-6:10 pm
Email: sormanic (at) member.ams.org
Webpage: http://comet.lehman.cuny.edu/sormani
Text: College Geometry: A discovery approach, by David C. Kay, Pearson, 2nd Edition. ISBN-10: 0321046242 | ISBN-13: 978-0321046246 |
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).
Supplies: 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
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 grade will be based on proofs completed in projects and in class exams:
To earn an A: the average grade on the projects and exams must be over 90% and the student must complete a three part research report on Hyperbolic or Spherical geometry.
To pass the class: the average grade of the projects and exams must be over 60% and the second midterm grade must be at least 80% demonstrating strong knowledge of Euclidian Geometry, and the final must be over a 60%.
All other grades are determined by taking the average of exams and projects.
MAT630 Graduate Grading Policy: The 15 projects are worth 5% each, the research report is worth 10%, and the exams are worth 5% each. Note that the research report is required. To pass the class, the second midterm grade must be at least 80%.
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. Some projects may be resubmitted and the new grade will be averaged with the old grade. Always keep a copy of a project you have submitted. All proofs must be in 2 columns. Homework assignments and projects available on the course webpage.
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)
Schedule:
Monday January 27: Pythagorean Theorem (What is a proof?)
including Pythagoras, 3-4-5 triangles-> 12 inches, grading policy
HW: Read Wikipedia entry on the Pythagorean Theorem
Wednesday January 29: Metric Spaces and Set Theory
including balls, subsets, intersections, unions, open sets
Reading: Metric spaces: 78
Project 0: (due Mon Feb 3, 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,1) is a subset of B(p,2).
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.
(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.
Samples and hints for Project 0 (photos of notes): Note all proofs are 2 columns: the first is statements and the second is justifications using axioms and definitions only at this point:
Geom1.JPG, Geom2.JPG, Geom3.JPG, Geom4.JPG, Geom5.JPG, Geom6.JPG, Geom7.JPG, Geom8.JPG, Geom9.JPG, Geom10.JPG, Geom11.JPG, Geom12.JPG, Notice the proof is not complete at the end of the last page. Thats for you to try yourselves.
Monday February 3: 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):
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 February 5: Betweeness, Segments and Rays (snow day)
Reading: E2,E3: 79-82.
Project 1: (due Mon Feb 10)
(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,
Monday February 10: Ruler Axiom and Segment Construction Theorem
and ruler function version of the ruler axiom
Reading: p83-85,
Project 2: (due Wed Feb 19)
(1) Write p 84 Example 2 as a 2 column proof,
(2) p89/15 (a 2 column proof),
(3) p89/16 (a 2 column proof)
(4) 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.
(5) Write a two column proof of the segment construction theorem.
(6) 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.
(7) Prove that a line is not an open set.
(8) Do page 88 problem 6 (not a proof)
(9) 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.
Wednesday February 19: Protractor Axioms and Angles
and proving a set is not open, discrete metric spaces, a single point set in E2 is not open,
proof by induction that finite unions are open, example that countable intersections of open sets need not be open.
Reading: p 90-96, Start project 3 problems 1-4.
Thursday February 20: 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.
Project 1 resubmission due Wed Feb 26 (look over right away).
Project 3: (due Mon Feb 24)
(1)-(3) Draw an angle in E2, E3 and H2 and shade the interior of the angle.
(4) Prove Theorem 2 on page 96
(5) Do page 99 problem 2 indicating which axioms and theorems you are applying.
(6) Write the Linear Pair Axiom using notation rather than words: Given A-B-C and a point D not on line AB, then...
(7) Prove the Unique Perpendicular Line Theorem (page 99 Theorem 4)
(8-9) Draw the unique perpendicular line theorem in S2 and H2.
Monday February 24: Convex Sets and Halfplanes
Reading: pages 104-110 Start Project 4
Wednesday February 26: Angle Interiors and the Crossbar Theorem
Reading: pages 108-110, Try Project 4 problems 1-3
Project 2 resubmission due Wednesday March 5, look over right away.
Project 4 due Monday March 3:
(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 March 3: The SAS Hypothesis
Reading pages 120-124
Project 5: (due Monday March 10, no resubmissions of this project)
(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).
Wednesday March 5: Congruent Triangles
Reading: Section 3.3, Start Project 5
Project 3 resubmission due Monday March 10, look over right away.
Project 6: (due Wed March 12, no resubmissions of this project)
(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. Warning: X may not be the midpoint.
(10) 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.
Monday March 10: Review of Projects 1, 2, and 4
Bring copies of your projects.
Wednesday March 12: Review of Projects 3, 5 and 6
Study for the midterm exam. Bring sheet of all axioms to the midterm. The midterm is proofs.
Monday March 17: First Midterm, Concurrent Lines
Perpendicular bisectors, angle bisectors, medians
Read and do Project 7 (due Monday March 24):
Extra Credit: Parts III and IV
Research Report Part I (due March 31, no resubmissions, must work alone) :
Choose and complete a project examining the incenter or orthocenter of hyperbolic space,
or the circumcenter, incenter or orthocenter on the sphere
Wedensday March 19: Parallel Postulate:
Parallel Lines, Angle Sum Theorem,
Start Project 8 pages 1-3
Monday March 24: Parallelograms and Bridges:
4.2 Parallelograms, rhombuses and squares
Project 8: (due Wednesday March 26)
Wednesday March 26: Similar triangles:
Similar triangles 4.3, proportions, SOHCAHTOA,
Resubmission of Project 7 due Wednesday April 2
Project 9: (due Monday March 31, no resubmissions of this project)
Monday March 31: Solid Geometry
Read 7.1-7.2 (will be on Euclidean Geometry Exam).
Extra Credit: Solid Geometry: Read 7.1, write out two column proofs of Theorems 1-3, Relate to the NYS Standards Concepts G.G.1-G.G.9,
Resubmission of Project 8 due Wednesday April 9
Research Report Part II: (due Wednesday April 9)
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 April 2: Pythagoras and its Converse:
Project 10: (due Monday April 7, 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 Wed)
Monday April 7: Symmetries and Isometries:
Reflections and Translations 5.2-5.3
Resubmission of Project 9 due Wed April 25.
Project 11: (due Monday April 23, 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. See this link.
Wednesday April 9: Circles, Arcs and Chords:
Circles 3.8 and 4.5, NYS Standards GG49-GG53.
You may resubmit Project 10: Working with Triangles, on Monday 4/23.
Project 12: (due Monday May 5, not on the 2nd exam, no resubmissions)
Circles Project pages 1-3. Page 4 is Extra Credit.
Note the theorems in Project 12 hold on H2 and S2 Part III of your research report may be based on this project if you are working on spherical geometry.
Monday April 23: Review Projects 6 - 10
HW: Study Projects 7-11 for Second Midterm
To pass this course you must score 80% on this exam!
The proofs on this exam are on the level of those on the NYS Geometry Regents Exam.
Must know all NYS Standards Concepts, See sample NYS Regents Exam
Use the Regents Exam to get a sense of the style of questions
but remember we have not yet covered all topics on the exam
like coordinate geometry, or circles or transformations with circles,
or volumes or areas...
You need to know how to do proofs in the plane and state facts on three dimensional space.
For those who do not score 80% and up, you may try again at the end
of the semester to score 80% raw score a Regents Exam style exam
with all topics on the Regents and extra proofs.
Monday April 28: Second Midterm (Euclidean Geometry)
Remember Project 12 is due Monday May 5.
Wednesday April 30: Coordinate Geometry:
4.3, 4.7 Coordinate Plane, Quadrants, Lines through the Origin, Perpendicular Lines
Project 13: (due Monday May 5, no resubmissions)
Monday May 5: Transformations, Areas :
Shift Isometry and Lines, Skews/Shears and Dilations, Skews and Dilations 5.1, 5.4,
Project 14: (due Wednesday May 7, no resubmissions)
Transformations and Rotation Matrices
Wednesday May 7: Area and Volume
Project 15: (due Monday May 12, no resubmissions)
Research Project Part III is due at the Final:
An individual project either about circles on the sphere (Project 12)
or isometries on the sphere or hyperbolic space (Project 13).
Monday May 12: Solid Geometry and Spherical Geometry
Understanding solid geometry using coordinates, and
understanding the sphere as {x2 + y2 + z 2 =1}
Wednesday May 14: 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.
Final TBA
Lehman College, Department of Mathematics and Computer Science,
Professors are not permitted to accept any gifts from students