| 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. p153-5; Prop. 1-12,22,23,47 A:.Complete in three weeks |
Due:1/27 M&I p5:1-8,11 |
|
| 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, 27-32, 35-45 |
Due: 2/3 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 Reading Report due: 2/1 | 2/1
Euclid early Props/ Pythagoras |
2-3 Pythagoras and Dissections- equidecomposeable polygon . |
2-5 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. 1-3, 14-18, 20, 21, 10 F. Sect. 11, 25, 30, 31 [Optional Sections:1-10; 12,13,18,20] Watch Equidecomposable Polygons |
Due 2/10 M&I: p17:5, 8-11 p11: 16-19, 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. |
2-10 Begin Constructions and the real number line M&I's Euclidean Geometry |
2-12 Constructions from M&I. | M&I
1.3,1.4 E: III Prop. 1-3, 14-18, 20, 21, 10 F. Sect. 11, 25, 30, 31 [Optional Sections:1-10; 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 1-7;VI: prop 1&2 ] F. [Sect. 32] pp 55-61on 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. 3-5 |
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:1.6:1-12,17,18 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: 3,7-15 Problem Set 4 M&I: 3.5: 1,3,4,5,10,11 3.6: 3,7-15 Submit Outlines of Content for Videos 346 and 628 |
| 11 |
3/28 A non-euclidean universe. | 3/30Axioms
for
7
point
geometry. Begin Synthetic Projective Geometry |
4/1 The Conics: From cones to equations. |
Non-Euclidean Geometry (24 minutes) Open University (History of Math) A Non-Euclidean 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
Z2 and Z3 More on Finite Synthetic Geometry and models. Proof of Desargues' Theorem |
4/8Projective Geometry -Visual/algebraic and Synthetic.. Synthetic Projective Geometry Algebraic-projective 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: 1-6, 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(p94-97).4.7, 4.10, p105-108 (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: 1-6, Supp:1,5,6 Problem Set 5 Reading Report on Monday 4-25. |
| 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 8:15-10:00 and by appointment or chance |
Student Presentations will be made Wednesday 5-11 10:20-12:10 | Final Exam DUE Friday, May 13, before 5 P.M. |
||
1. 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 `cap` C = {Q}; Q' is on C so that Q'Q is parallel to ON; and {P'} = NQ' `cap` `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.
2. 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}`.
Problem Set 4
1. Use an affine line with P0 , P1 , and `P_{oo}` given. Show a construction for P1/2 and P2/3.
2. Use an affine line with P0 , P1 , and `P_{oo}`
given. Suppose x > 1.
Show a construction for Px2 and Px3 when
Px is known.
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.
Flatland Essay #1
Problem Set 2
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.]
Project Proposal Guidelines and Suggestions
Preliminary Project Proposals should be submitted for first review by 5 p.m. Monday , February 8th.
A progress report on the project is due March 23rd.
Final projects are due for review Friday, April 29th. (These will be graded Honors/Cr/NCr.)
A Project Fair will be organized for displays and presentations during the final exam period. 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.
- 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.
- 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.
Below are some specific suggestions on features your proposal description might include:
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.
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 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 |