## MATH 371 Assignments and Project Spring, 2009

TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE through Moodle.
The Elements by Euclid, 3 volumes, edited by T.L. Heath, Dover (1926)
Proof in Geometry by A.I Fetisov, Mir (1978)
Flatland By E. Abbott, Dover.

 Week Monday Wednesday Reading/Videos for the week. Problems Due on Wednesday of the next week Friday 1 No Class 1-21-09 1.1 Beginnings 1-23 What is Geometry?  Starting to look at Euclid. Prop 1. M&I:1.1, 1.2 E:I Def'ns, etc. p153-5;  Prop. 1-12,22,23,47  A:.Complete in three weeks Due:1-30-09 M&I p5:1-8,11 2 1-26 The Pythagorean Theorem More on Pythagoras and Euclid. 1-30 Prop 2 Begin  disssection theory, M&I 1.2, 1.3  E: I Prop. 16, 27-32, 35-45. Due: 2-4  M&I: p10:1,2,5,10,11-13 Prove:The line segment connecting the midpoints of  two sides of a triangle is parallel to the third side and is congruent to one half of the third side. [ HELP! Proof outline for the midpoint proposition.] 3 2/2Convexity defined. Equidecomposable Polygons Constructions  Isometries  1.1 Def'ns- Objects 2/4 1.2 Constructions 1.4 Continuity 2/6  1.3 Geometry: Constructions and numbers M&I 1.3,1.4  E: III Prop. 1-3, 14-18, 20, 21, 10  Here’s looking at EuclidF. Sect. 11, 25, 31 Watch Due: 2-13 (changed 2-13!) M&I:  p11: 16-19, 24, *27  Problem  Set 1 4 2/9 More on continuity and rational points. 2/11 Start Inversion Orthogonal Circles 2/13More on Cantor Similar triangles Odds and ends. Coordinates. M&I:1.5, 1.6, 2.1  E: V def'ns 1-7;VI: prop 1&2  F. Sect. 32 Due 2-18 M&I: p17: 5, 8-11 p23: 9,10 (analytic proofs) 5 2/16 Odds and ends 2/18 Odds and ends 2/20Start: Transformations - Isometries Transformations - Isometries. M&I: 2.1, 2.2 E: V def'ns 1-7;VI: prop 1&2  E:IV Prop. 3-5 Isometries (Video # 2576 in Library) . Due: 2-25 M&I:1.6:1-12,17,18  Problem Set 2 6 2/23More Isometries: Coordinates and Transformations classification 2/25  More on Isometries 2/27  .... M&I: 2.1,2.2 Due 3-4 Problem Set 3 (Isos Tri)   [3 Points for every distinct correct proof of any of these problems.] 7 3/2Finish Classification of Planar Isometries. Begin Affine Geometry 3/4 Isometries and symmetries 3/6 Proportion and Similarity M&I: 2.2 again;3.1,3.2, 3.5 Quiz #1 on 3/11 See Sample quiz on Moodle. 8 3/9 More on Similarity 3/11 Inversion and Affine Geometry (planar coordinates) the Affine Line. 3/13 Constuctions with Straight edge and compass. M&I:3.6, 3.4,3.7 View video (in Library #4376 ) on "Central similarities" from the Geometry Film Series. (10 minutes) View video (in Library #209 cass.2) on similarity (How big is too big? "scale and form")  "On Size and Shape"  from the For All Practical Purposes Series. (about 30 minutes) 9 Spring break 3/16 No Class 3/18 No Class 3/20 No Class 10 3/23 Visualizing the affine plane. Seeing the infinite.Affine geometry- Homogeneous coordinates and visualizing the affine plane. 3/25 More on Homogeneous coordinates for the plane.A first look at a "Projective plane." Axioms, consistency, completeness and models. 3/27Begin Synthetic Geometry [Finite] Algebraic-projective geometry: Points and lines. Spatial and  Planar Axioms for 7 point geometry. Begin Synthetic Projective Geometry A non-euclidean universe. M&I :3.1, 3.2, 3.5, 3.6, 3.7; 4.1 Due :  4/1  M&I: 3.5: 1,3,4,5,10,11 3.6: 3,7-15  3.7: 1,4,7,10,13 11 3/30 Homogeneous Coordinates with Z2 and Z3 More on Finite Synthetic Geometry and models. Proof of Desargues' Theorem 4/1 -Projective Geometry -Visual/algebraic and Synthetic..Axioms 1-6 Projective Planes. . 4/3. More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry. M&I:4.1, 4.2, 4.3, 2.4 12 4/6 Proofs of some basic projective geometric facts. 4/8 Triangle Coincidences (Perpendicular Bisectors- the circumcenter) Applications of Projective  Geometry Postulates.1-6 4/10 Quiz #2 Desargue's Theorem and Duality Due 4/15 M&I: 4.1:7,15,16;  Prove P6 for RP(2);  4.2: 2,3, Supp:1  4.3: 1-6, Supp:1,5,6 13 4/13 Conic Sections.  More Duality 4/15  More Postulates- Sections (more duality) Complete quadrangles Postulate 9. 4/17 Perspectivities and Projectivities. Projective transformations. Conics Pascal's Theorem ? More on coordinates and transformations. M&I:  4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm) Watch video: "Conics" [VIDEO #628]. Watch video: For All Practical Purposes on conicsFAPP on conics[HSU Video #209- 3. Stand-up conic (Conic sections) (ca. 30 min.)]. WatchVideo: Orthogonal Projection [Video #4223] Due : 4/22Problem Set 4 14 4/20Projectivities. Perspective 4/22Transformations of lines with homogeneous coordinates 4/24 Projectivities in 3 space: More on Projective Line Transformations with Coordinates. Begin Harmonic Sets and Construction of Coordinates. 4.10, 5.4, 2.4  4.11 Due: 4/29 M&I: 4.5:2  ;4.6: 7,8,9;   4.7: 4,7  4.10: 4,5,9,10 [Prove P9 for RP(2),optional] 15 Final exam available  4/30. 4/27Harmonic sets: uniqueness and construction of coordinates for a Projective Line, Plane, Space. 4/29More on Transformations, Coordinates and Harmonic sets. 5/1 5.1,5.4;,5.2, 5.3,5.5, 5.7, 6.1, 6.2 16 Final projects are due for review Tuesday, May 5th. 5/4 Projective generation of conics  Pascal's and Brianchon's Theorem. 5/6 Matrices for familiar Planar Projective Transformations. Inversion angles, circles and lines? 5/8 The Big picture in Summary. Due: M&I: 4.10:1,3,6,7;  5.4:1-8,10;  5.5: 2,3,7 Due:  M&I: 4.10:1,3,6,7;  5.1:5; 5.4:1-8,10;  5.5: 2,3,7 16 Conics revisited. Inversion and the final exam.Quiz #3 . Student Presentations

Project Proposal Guidelines and Suggestions
The Project. Each student will participate in a course project either  as an individual as a part of a team. Each team will have at most three. These projects will be designed with assistance from myself . The quality of the project will be used for determining letter grades above the C level. Ideas for projects will be discussed during the third week.
Preliminary Project Proposals should be submitted for first review by 5 p.m., February 10th.
A progress report on the project is due March 25th.
Final projects are due for review Tuesday, May 5th. (These will be graded Honors/Cr/NCr.)

A Project Fair will be organized for displays and presentations during the last day of class. Details will be discussed later.

Guidelines for Preliminary Proposals:

• Proposal Format: The proposal should be typed (neatly hand written proposals are acceptable).
• Contents: The content of your proposal should describe, explain or otherwise demonstrate what your project is as you currently envision it. It should also indicate how you will go about completing the project.

• Below are some specific suggestions on features your proposal description might include:
• Title: Include a name( or list of possible names ) for your project.
• Introduction: (Your topic's core idea.) You should explain the idea of your project. Remember that the Introduction is the first place where the reader hears about your idea. You should also explain how the Proposal is organized in the introduction.
• Form and result: Indicate your vision of the final project's form(s), that is, the appearance of the FINAL PRODUCT. What will your project look like in its final ideal form? Note that all forms must include some written explanatory  component.
• Variations: (optional) Since this is a preliminary proposal, indicate some of the possible variations of both substance and form. It might be useful to distinguish the ideal from what may be a minimal project in both substance and form, and perhaps to see the project in stages from minimal to ideal, just in case you run into practical or time problems.
• References and Tools: List references and tools (books, journals, software, people, etc.) that are relevant to your project and that you might use. If you don't have any specific references yet, then indicate the kind of references you might use and where you will find them.
• Methods- Timeline and Task Delegation (for partnerships): Who will do what? When will they do it? If your project has definite parts or subdivisions, then indicate target dates for the completion of each stage.

• For partnerships:This project is a collective effort and should reflect the work and effort of all. Indicate when and where you will meet outside of class and how often. When possible, estimate the number of hours you are allocating to each task.
• Record keeping: Indicate how you will keep track of the progress of your project and the time spent by each individual participant on the project's work.

Results of Brainstorming and other suggestions from previous courses :)

 Tiling patterns - tesselation 3d tiling MC Escher perspective Curves: conics, etc. optical illusions knots fractals Origami KaleidescopeSymmetry The coloring problemPatterns in dance and other performance arts Flatland sequel (4d) MapsJugglingstructural Rigiditydimension Polyhedra bridgemaking (architecture) Models (3d puzzles) paper mache or claymobiles sculptureA play - movie build three dimensional shapes power point performance website

Problem Set 1

DEFINITIONS: A figure C is called convex if for any two points in the figure, the line segment determined by those two points is also contained in the figure.
That is, if A is a point of C and B is a point of C then the line segment AB is a subset of C.

If F and G are figures then F int G  is { X : X in F and X  in G }.
F int G is called the intersection of F and G.
If A is a family of figures (possibly infinite), then  int A = { X : for every figure F in the family A, X  is in F }.
int A is called the intersection of the family A.
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1. Prove: If F and G are convex figures , then F int G is a convex figure.

2. Give a counterexample for the converse of problem 1.

3. Prove: If A is a family of convex figures, then int A is a convex figure.

4. Prove: The line segment RS is convex. [ Refer to M & I pg.2.]

Problem Set 2

1. Suppose n is a natural number. Given P0 and P1 , prove by induction that you can construct with straight edge and compass (SEC) a point P sqrt(n)   which will correspond to the number  sqrt(n) on a Euclidean line.

2. Suppose we are given P0, P1, and P a where P a corresponds to the real number a>0. Give a construction with SEC of a point Psqrt(a) which will correspond to the number  sqrt(a) on a Euclidean line.

3. Given points P0, P1, Px, and Py on a Euclidean line corresponding to the real numbers x>0 and y>0, give constructions with SEC for the following points.

 a) P x + y b) P x - y c) P x *y d) P 1/x
4. Construct with SEC on a Euclidean line:   sqrt(5)/sqrt(3)  + sqrt(sqrt(6)) .

5. Suppose that d(A,B) = d(A',B') and that l is the perpendicular bisector of the line segment  AA'. Let B'' be the reflection of B across l, i.e., B''= Tl(B). Prove that if  B' is not equal to B''  then A' lies on the perpendicular bisector of the line segment B''B'.

Problem Set 3

1.  Prove: Two of the medians of an isosceles triangle are congruent.

2.  Prove: If two of the medians of a triangle are congruent then the triangle is isosceles.

3.  Prove: The angle bisectors of congruent angles of an isosceles triangle are congruent.

4.  Prove:  If two of the angle bisectors of a triangle are congruent then the triangle is isosceles.

Problem Set  4

1. Use an affine line with P0 , P1 , and Pinf given. Show a construction for P1/2 and P2/3.

2. Use an affine line with P0 , P1 , and Pinf   given. Suppose x > 1.
Show a construction for  Px2 and Px3 when Px is known.

3. D is a circle with center N tangent to a line l at the point O and C is a circle that passes through the N and is tangent to l at O as well.
Suppose P is on l and PN intersect C = {Q}; Q' is on C so that Q'Q is parallel to ON; and {P'} = NQ'  intersect l.

Prove: a) P and Q are inverses with respect to the circle D.
b) P' and Q' are inverses with respect to the circle D.
c) P and P' are inverses with respect to the circle with center at O and radius ON.

4. Suppose C is a circle with center O and D is a circle with O  an element of D.
Let I be the inversion transformation with respect to C.

Prove: There is a line l, where I(P) is an element of  l for all P  that are elements of  D -{O}.