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TEXTS: Fundamentals of Geometry by B. Meserve and J.
Izzo,
A.W. (1969) 
ON LINE at 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  Friday  Reading/Videos for the "week"through Monday.  Problems Due on Wednesday of the next week 

1  1/20 1.1 Beginnings What is Geometry? 
1/22 Start on Euclid Definitions, Postulates, and Prop 1

Watch online: Here’s
looking at Euclid M&I:1.1, 1.2 E:I Def'ns, etc. p1535; Prop. 112,22,23,47 A:.Complete in three weeks 
Due:1/27 M&I p5:18,11 

2  1/25 Euclid Definitions, Postulates,
and Prop 1. cont'd

.1/27 Pythagorean plus. 
1/29 Euclid Postulates/ Pythagoras 
M&I
1.2,
1.3 E: I Prop. 16, 2732, 3545 
Due: 2/3 M&I: p10:1,2,5,10,1113 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 Reading Report due: 2/1  2/1
Euclid early Props/ Pythagoras 
23 Pythagoras and Dissections equidecomposeable polygon . 
25 More on Equidecomposeable polygons Begin Constructions and the real number line M&I's Euclidean Geometry Isometries 1.1 Def'ns Objects 1.2 Constructions 1.3 Geometry: Constructions and numbers 
M&I
1.3,1.4 E: III Prop. 13, 1418, 20, 21, 10 F. Sect. 11, 25, 30, 31 [Optional Sections:110; 12,13,18,20] Watch Equidecomposable Polygons 
Due 2/10 M&I: p17:5, 811 p11: 1619, 24, *27 Problem Set 1 
4Preliminary Project Proposals first review 5 p.m. Wednesday, February 9th.  2/8
More on Details for triangulation of planar polygonal regions and adding parralelograms. 
210 Begin Constructions and the real number line M&I's Euclidean Geometry 
212 Constructions from M&I.  M&I
1.3,1.4 E: III Prop. 13, 1418, 20, 21, 10 F. Sect. 11, 25, 30, 31 [Optional Sections:110; 12,13,18,20] 

5 Reading Report due: 2/15  2/15 1.4 Continuity. Continuity and rational points. Similar triangles  2/17 More on similar triangles and rational
points. Isometries. Start Inversion. Orthogonal Circles 
2/19 Coordinates and Proofs Inversions and orthogonal circles. Odds and ends on continuity axiom More on Cantor 
M&I:1.5,
1.6,
2.1
OPTIONAL[E: V def'ns 17;VI: prop 1&2 ] F. [Sect. 32] pp 5561on Axioms of Continuity Video: Isometries (Video # 2576 in Library) . 

6 
2/22 Inversions and orthogonal circles.  2/24Odds and ends on continuity axiom More on Cantor More Isometries. 
2/26 Begin Transformations  Isometries. coordinates..... 
M&I:
2.1,2.2 E:IV Prop. 35 
Due 2/26 M&I:
p23: 9,10 (analytic proofs) Problem Set 2 
7 Reading Report due: 2/29 
2/29
Finish
Classification
of
Planar
Isometries and coordinates 
3/2 Isometries
and
symmetries More on Similarity 
3/4/  M&I:
1.6, 2.1, 2.2
again Video: "Central similarities" (in Library #4376 ) (10 minutes) "On Size and Shape" (How big is too big? "scale and form")(in Library #209 cass.2) (about 30 minutes) 
Due 3/7 M&I:1.6:112,17,18 Flatland Essay #1 
8 Quiz #1 on Wed. 3/9 
3/7 Begin
Affine
Geometry Proportion and Similarity 
3/9 Quiz #1 Proportion and Similarity in Euclid. 
3/11More on Proportion a la Euclid. 
M&I: 2.1, 2.2
again; Start 3.1,3.2 . 
Due 3/23 Problem Set 3 (Isos Tri) [3 Points for every distinct correct proof of any of these problems.] 
9 Spring break  3/14 No Class  3/16No Class  3/18 No Class  
10 Reading Report due: 3/21 A progress report on the project is due March 23rd 
3/21 Similarity, proportion with numbers vs euclid. The Affine Line. Seeing the infinite. Affine geometry 
3/23  Homogeneous coordinates and visualizing the affine plane. Inversion and 
3/25 Affine
Geometry
(planar
coordinates) A first look at a "Projective plane." 
M&I :3.1, 3.2, 3.4, 3.5, 3.7; 4.1  Due : 4/4! M&I: 3.5: 1,3,4,5,10,11 3.6: 3,715 Problem Set 4 M&I: 3.5: 1,3,4,5,10,11 3.6: 3,715 Submit Outlines of Content for Videos 346 and 628 
11 
3/28 A noneuclidean universe.  3/30Axioms
for
7
point
geometry. Begin Synthetic Projective Geometry 
4/1 The Conics: From cones to equations. 
NonEuclidean Geometry (24 minutes) Open University (History of Math) A NonEuclidean Universe. (24 minutes)(Open University VIDEO346) The Conics (24 minutes) (Open University HSU Libray VIDEO628) 

12Reading Report due: 4/4 
4/4 The affine plane and homogeneous coordinates for points. Lines and homogeneous linear equations. 
4/6 Homogeneous
Coordinates
with
Z_{2 }and Z_{3} More on Finite Synthetic Geometry and models. Proof of Desargues' Theorem 
4/8Projective Geometry Visual/algebraic and Synthetic.. Synthetic Projective Geometry Algebraicprojective geometry: Axioms, consistency, completeness and models. Points and lines. Spatial and Planar 
Orthogonal Projection [HSU Library videorecording] VIDEO 4223 (11 minutes) Central Perspectivities [HSU Library videorecording] VIDEO 4206 (14 minutes) Central Similarities  [HSU Video 4376] YouTube Video for central similarities10:36 https://www.youtube.com/watch?v=0stCtGH4lxY 
Submit Outlines of Content for Videos 4223, 4206, and 4376. M&I:3.7: 1,4,7,10,13 4.1:7,15,16; Prove P6 for RP(2); 4.2: 2,3, Supp:1 4.3: 16, Supp:1,5,6 
13 
4/11 
4/13 Quiz #2? 
4/15 

14 
4/18 Proofs Using PlaneProjective Geometry Postulates 
4/20 Duality. Intro to Transformations of RP(1) and matrices. 
4/22 The complete quadrangle. Perspectivities and projectivities. 
M&I: 4.5,4.6(p9497).4.7, 4.10, p105108 (Desargues' Thrm) 5.4 Projective Generation of Conics Video 2574
Math
History 8 of our work on the projective plane. 
Submit Outline of Content for Video 2574 4.1:7,15,16; Prove P6 for RP(2); 4.2: 2,3, Supp:1 4.3: 16, Supp:1,5,6 Problem Set 5 Reading Report on Monday 425. 
15 Last Reading report due 4/25 Final Exam Distributed 4/29 
4/25 
4/27 
4/29 

16 
5/2 
5/4 Quiz # 3 
5/6 

17 Final Exam Week. 
Office Hours for Exam Week MTWRF 8:1510:00 and by appointment or chance 
Student Presentations will be made Wednesday 511 10:2012:10  Final Exam DUE Friday, May 13, before 5 P.M. 
1. Use an affine line with P_{0} , P_{1} , and `P_{oo}` given. Show a construction for P_{1/2} and P_{2/3}.
2. Use an affine line with P_{0} , P_{1} , and `P_{oo}`_{ }
given. Suppose x > 1.
Show a construction for Px^{2} and Px^{3} when
Px is known.
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.
1. Suppose `n` is a natural number. Given `P_0` and `P_1` , 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)` [square root of `n`] on a Euclidean line.
2. Suppose we are given `P_0, P_1`, and `P_a` where `P_a` corresponds to the real number `a>0`. Give a construction with SEC of a point `P_{sqrt(a)}`which will correspond to the number `sqrt(a)` [square root of `a`] on a Euclidean line.
3. Given points `P_0, P_1, P_x`, and `P_y` 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}` 
5. Suppose that `d(A,B) = d(A',B')` and that `l` is the perpendicular bisector of the line segment `A A'`. Let `B''` be the reflection of `B` across `l`, i.e., `B''= T_l(B)`. Prove that if `B'` is not equal to `B''` then `A' ` lies on the perpendicular bisector of the line segment.
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 ∩ G is { X : X ε F and
X ε G }.
F ∩ G is called the intersection of F and G.
If A is a family of figures (possibly infinite), then ∩A =
{ X : for every figure F in the family A, X ε F
}.
∩ A is called the intersection of the family A.

1. Prove: If F and G are convex figures , then F ∩ 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 ∩ A is a convex figure.
4. Prove: The line segment RS is convex. [ Refer to M & I pg.2.]
A Project Fair will be organized for displays and presentations during the final exam period. Details will be discussed later.
Guidelines for Preliminary Proposals:
Tiling
patterns 
tesselation 3d tiling MC Escher perspective Curves: conics, etc. optical illusions knots fractals Origami Kaleidescope Symmetry The coloring problem Patterns in dance and other performance arts Flatland sequel (4d)  Maps Juggling structural Rigidity dimension Polyhedra bridgemaking (architecture) Models (3d puzzles) paper mache or clay mobiles sculpture A play  movie build three dimensional shapes power point performance website 