## MATH 371 Assignments and Project Spring, 2007

TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE with  HSU ONCORE through Moodle.
The Elements by Euclid, 3 volumes, edited by T.L. Heath, Dover (1926)
Proof in Geometry by A.I Fetisov, Mir (1978)
Here's Looking at Euclid..., by J.Petit, Kaufmann (1985).
Flatland By E. Abbott, Dover.

Due: 4/25
M&I: 4.10:1,3,6,7;
5.4:1-8,10;  5.5: 2,3,7
 Week Monday Wednesday Reading/Videos for the week. Problems Due on Wednesday of the next week Friday Math 480 Lab Assignments 1 1/15 No Class 1/17 1.1 Beginnings 1/19 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:1/24 M&I p5:1-8,11 2 1/22 The Pythagorean Theorem  Lab: Intro to Geometer's Sketchpad/ Wingeom 1/24 Convexity defined. The Pythagorean Theorem 1.2 1/26 Guest Lecture  on Euclid Here’s looking at Euclid M&I 1.2, 1.3  E: I Prop. 16, 27-32, 35-45. Due: 1/31 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.] Lab Exercises 1: Due: 1/29 Construct a sketch with technology of  1. Euclid's Proposition 1 in Book I.  2. Euclid's Proposition 2 in Book I.  3. One "proof" of the Pythagorean Theorem. 3 1/29 Equidecomposable Polygons Constructions  Isometries  1.1 Def'ns- Objects  1.2 Constructions 1.3 Geometry: Constructions and numbers 1/31 1.4 Continuity 2/2   Breather: Start: Transformations - Isometries M&I 1.3,1.4  E: III Prop. 1-3, 14-18, 20, 21, 10  F. Sect. 11, 25, 31 Watch Due: 2/7 [extended to2/14] M&I: p17:5, 8-11  p11: 16-19, 24, *27  Problem  Set 1 Lab Exercises 2: Due by : 2/5 Do Construction 3, 4, 6, 7, and 8 from Meserve and Izzo Section 1.2. BONUS:Show how to "add" two arbitrary triangles to create a single square. 4 2/5 More on continuity and rational points. Similar triangles 2/7 More on Cantor Start Inversion. 2/9Orthogonal Circles Odds and ends. Transformations - Isometries. Coordinates. M&I:1.5, 1.6, 2.1  E: V def'ns 1-7;VI: prop 1&2  F. Sect. 32 Due 2/19 [Changed  as of  2-9] M&I: p23: 9,10 (analytic proofs) M&I:1.6:1-12,17,18  Problem Set 2 Lab Exercises 3: Due 2/12 1. Construct a scalene triangle using Wingeometry. Illustrate how to do i) a translation by a given "vector", ii) a rotation by a given angle measure, and iii) reflections across a given line.. 2. Create a sketch that shows that the product of two reflections is either a translation or a rotation 5 2/12 More Isometries: Coordinates and Transformations 2/14 Isometries / 2/16 coordinates/ 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) . Lab Exercises 4: Due 2/19 1. Draw a figure showing the product of three planar reflections as a glide reflection. 2. Draw a figure illustrating the effects of a central similarity on a triangle using magnification or dilation that is a) positive number >1, b) a positive number <1, and c) a negative number. 6 2/19 classification 2/21  More on Isometries 2/23  .... Finish Classification of Planar Isometries. M&I: 2.1,2.2 Due 2/28 Extended to 3/7! Problem Set 3 (Isos Tri)   [1 Point for every distinct correct proof of any of these problems.] Lab Exercises 5: Due  2/26. 1. Construct the inverse of a point with respect to a circle a) when the point is inside the circle; b) when the point is outside the circle. 2.  Given a circle O and two interior points A and B, construct an orthogonal circle O' through A and B.  3. Draw two intersecting circles O and O' and measure the angle between them. 7 Quiz #1 on Monday in class. 2/26Isometries and symmetries Begin Affine Geometry Proportion and Similarity 2/28 More on Similarity 3/2 Inversion and Affine Geometry (planar coordinates) the Affine Line. M&I: 2.2 again;3.1,3.2, 3.5 Lab Exercise 6:  Due 3/5 1.Draw sketches for each of the following triangle coincidences: 1. Medians. 2. Angle Bisectors. 3. Altitudes. 4. Perpendicular Bisectors 8 3/5 Visualizing the affine plane. Seeing the infinite. 3/7 Affine geometry- Homogeneous coordinates and visualizing the affine plane. 3/9 More on Homogeneous coordinates for the plane. 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) Lab Exercise 7: Due  3/19.  1. Inversion: Investigate and sketch the result of inversion on lines and circles in the plane with a given circle for inversion.  When does a line invert to a line? When does a line invert to a circle? When does a circle invert to a line? when does a circle invert to a circle?  Show sketches where each case occurs. [ Remember the inverse of the inverse is the original figure.]  2. Use inversion with respect to the circle OP to invert

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 13th.
A progress report on the project is due March 26th.
Final projects are due for review Tuesday, May 1st. (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 (revised 2-7-07)

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.

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}.