| 1/21 | 26 | 28 | 2/2 | 2/4 | 2/9 | 2/11 | 2/16 | 2/18 | 2/23 | 2/25 | 3/2 | 3/4 | 3/9 | 3/11 | 4/6 | 4/8 | 4/13 | 4/15 | 4/20 4/22 |
Next class we'll decide on whether we'll have a class notes system
.
To explore some of these issues we looked at proofs of the Pythagorean Theorem.
We looked at Euclid's statement and proof of
Proposition
47.
In right-angled triangles the square on the side opposite the right angle
equals the sum of the squares on the sides containing the right angle.
Note that Euclid's treatment in its statement or its "proof" never refers
the traditional equation,
a2+b2=c2.
The proof we looked used 4 congruent right triangles and 2 squares and the
the same 4 triangles and the square on the side of the hypotenuse arranged
inside of a square with side "a+b" .
We also considered the shearing proof on Geometer's Sketchpad.
(Based on Euclid's Proof) D. Bennett 10.9.90
1. Shear the squares on the legs by dragging point P, then point Q, to the
line. Shearing does not affect a polygon's area.
2. Shear the square on the hypotenuse by dragging point R to fill the
right angle.
The resulting shapes are congruent. The sum of the squares of the sides equals
the square on the hypotenuse.
In considering the Pythagorean theorem we questioned what kind of assumptions were needed in the proof with the triangles and squares. These included assumptions on how we could identify "equal" objects (congruent figures), how object fit together, and how movements may effect the shapes of objects.
We looked at Euclid's Proposition 1 and Proposition 2. These propositions demonstrate that Euclid did not treat moving a line segment as an essential property worthy of being at the foundations as an axiom, but this was a fundamental tool for all of geometry. In discussing these propositions we also noted that certain points of intersection of circles are presumed to exist without reference to any of the postulates. These presumptions are left implicit for hundreds of years, but are cleared up in the 19th century when careful attention is given again to the axioms as a whole system.
We will look further at the foundations of the proofs of the Pythagorean
Theorem in two ways:
1. Dissections: How are figures cut and pasted together? What
can be achieved using dissections?
2. Transformations: How are figures transformed? What
transformations will leave the "area" and "lengths" of figures invariant
(unchanged)?
Next Class: We'll look at the possiblilities of dissections (like Tangrams) and start using Geometer's Sketchpad in the lab time (2nd hour).
This led to a discussion of the question of whether this necessary condition of equal areas would be sufficient to say that two polygonal regions could be decomposed (cut and pasted) into smaller regions that would be congruent. In a sense this says one could create a set of smaller shapes with which one could make either of the two regions usiing prcisely these smaller shapes. The answer to this question is yes (in fact this is a 20th century result), which was the basis for the remainder of the class lecture.
We considered some of the background results which were known to Euclid: (1) parallelograms results and (2) triangle results. The justifications for these results were reviewed only briefly.
(1) a. Parallelograms between a pair of parallel lines and on the same line
segment are equal (in the sense of being able to decompose one to reconstruct
the other).
Proposition
35.
b.Parallelograms between a pair of parallel lines and on congruent
segments are equal (in the sense of being able to decompose one to reconstruct
the other).
Proposition
36.
(2) a. 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.
b. By rotating the small triangle created by connecting the midpoints of
two sides of a triangle 180 degrees about one of the midpoints, we obtain
a parallogram. (This shows that the triangle's area is the area of
this paralleogram which can be computed by using the length of the base of
the triangle and 1/2 of its altitude- which is the altitude of the
parallelogram.)
The film Equidecomposable Polygons was shown which proved the result:
If two polygonal regions in the plane have the same area, then there is
a decomposition of each into polygons so that these smaller polgons can be
moved idivually between the two polygons by translations or half turns (rotations
by 180 degrees).
Reference was made to the analogous problem in three dimensional geometry where volume equality of polyhedra is a necessary but not sufficient condition for a similar result. This was first demonstrated by Dehn in the 1930's by using another invariant of polyhedra related to the lengths of the edges and the dihedral angles between the faces of the polyhedra.
The next hour was spent in the lab becoming familiar with several of the basic features of Geometer's Sketchpad. Reference was made to definitions in M&I section 1.1. By the end of the class we had constructed a sketch of Euclid's Proposition I in Book I.
After reviewing materials defining, rays, segments, angles, triangles, and
planes, we reviewed eight of the basic Euclidean constructions described
in M&I. several of these constructions rely on some foundations that
assert the existence of points of intersection of circles.
We discussed how constructions also require a justification (proof) that the construction has in fact been achieved. In proving the constructions we used some basic euclidean results, such as the congruence of all corresponding sides in two triangles is sufficient to imply the triangles are congruent (SSS). Also mentioned were SAS and ASA congruence conditions, and well as the result that corrrsponding parts of congruent triangles are congruent (CPCTC).
In considering constructions of tangents to circles we used the characterization
of a tangent line as making a right angle with a radius drawn at the point
it has in common with the circle. (
Book
III Prop. 16.) In our construction, not Euclid's
(Book
III Prop. 17), we also used the result that any angle inscribed in a
semi-circle is a right angle. (
Book
III Prop. 31.)
First Pk where k is an integer was constructed using circles.
After recognizing that we could construct points with fractions using powers
of 2 for the denominator by bisection, we discussed how to contruct
Pk when k is a rational number using the theory of similar triangles.
(With only bisection we could not construct a point for 1/3 although we could
get very close to that point using a binary representation of that common
fraction.) We reviewed the construction of a line though a given point parallel
to a given line.
We then considered M&I's constructions of the same correspondence of
integer and rational points that relied on the ability to construct parallel
lines.
We then considered Euclid's treatment of the side-angle-side congruence [Proposition 4] and how it related to transformations of the plane that preserve lengths and angles. Such a transformation T: plane -> plane, would have T(P)=P', T(Q)=Q' and T(R)=R' with d(P,Q) = d(P',Q') [distance between points are preserved] or m(PQ)=m(P'Q') [measures of line segments are invariant].
After reviewing briefly the outline of Euclid's argument, we noted the key connection between the congruence of figures in the plane and isometeries: Figures F and G are congruent if and only if there is an isometry of the plane T so that T(F) = {P' in the plane where P'= T(P) for some P in F} = G.
We discussed briefly how there were at least four types of isometries of the plane: translation, rotation, reflection and glide reflection.
We watched the video Isometries up to the place where it was
demonstrated that the product of two reflections that have the lines of
reflection intersect at a point O is a rotation with center O through an
angle twice the size of the angle between the two lines of reflection.
We'll watch the remainder of the film and continue the discussion of isometries
next class.
In the second part of class we were in the lab. We worked on constructing a square using Geometer's Sketchpad and then illustrated how GS can do translations, rotations, and reflections. We'll try to look at GS's way to do measurements next time in the lab. This week students should try to reporduce some figure from Book I of Euclid using GS.
In the second part of class we discussed the continuity axiom for a euclidean line: Any nonempty family of nested segments will have at least one point in the intersection of the family. We saw how this leads to the result that any list (possibly infinite) of points in a given segment of a euclidean line will not have every point in that segment on the list. [We also noted that we could make a list of points corresponding to the rational numbers once a unit length had been established. 1/1,1/2,2/1,1/3,2/2,3/1,1/4,2/3,3/2,4/1, ....]
Class concluded with showing more of the isometries video. We stopped after the video had demonstrated that any isometry is the product of at most 3 reflections.
We then watched the remainder of the video on isometries, which classified the product of 3 reflections as either a reflection or a glide reflection. We noted that the four types of isometries can be characterized completely by the properties of orientation preservation/reversal and the existence of fixed points. This is represented in the following table:
| Orientation Preserving |
Orientation Reversing |
|
| Fixed points | Rotations | Reflections |
| No Fixed points | Translations | Glide reflections |
In the second hour we worked on measurements using GSPad.
We turned then to Euclid's (Eudoxus') resolution of the issue in
Book
V def'ns 1-5. We looked at these definitions and noted some key
items:
* Ratios exist only between magnitudes of the same type. (Homogeneity)
* For ratios to be equal the magnitudes must be capable of co-measuring.
* Euclid's axioms do not deny the existence of infinitesmals- but will not
discuss equality of ratios that use them.
* It is Archimedes axiom that stays that any two segments either one can
be used to measure the other. [No infinitesmals.] We discussed
briefly how this might be connected to calculus by looking at the change
in the area of a square when the length of a side is changed by an infinitesmal.
We examined the connection between Euclid's definition of proportionality (equal ratios) and real number equality of quotients. We showed for line segments the following two propositions for segments A,B,C,and D with m(A)=a,m(B)=b, m(C)=c, and m(D)=d:
At the end of class we discussed some features of isometries of a line, notation for these using coordinates and ways to visualize these.
We observed that if we let S(Px)=P(x-3) then ST(Px)=S(T(Px))=S(P2x)=P(2x-3) = T*(Px), so ST=T*.
We'll continue the discussion of transformations using plane coordinates next week and beyond.
We turned our attention to right triangles and noted that if an altitude is constructed in a right triangle with the hypotenuse as the base, the figure that results has 3 similar right triangles. Using similarity of these triangles we saw that there is a proportion of the segments of the hypotenuse AD, DB and the altitude CD given by AD:CD::CD:BD.
We then looked at the concept of inverses of points with respect to a given
circle and the construction of inverses. These were connected to the similar
triangles just mentioned showing that the constructions in M&I were correct.
Looking at the issues of doing arithmetic with constructions in geometry,
we noted that this would allow one to construct a point Px' from a point
Px as long as x is not 0 so that x' x = 1.[ Use the circle of radius 1 with
center at P0 to construct the inverse point for Px.]
In the lab we discussed how to create a script in GSP and started to look at the use of coordinates in GSP. Next Wednesday we'll do more with coordinates, along with the use of traces and locus to see some further aspects of coordinate geometry.
We used this fact to construct a circle C2 through a given point B on a circle C1 and a point A inside the circle so that C2 is orthogonal to C1. [ First construct the inverse of A' with respect to C1 and then the tangent to C1 at B and the perpendicular bisector of AA' will meet at the center of the disired circle.] Likewise we discussed constructing an orthogonal circle C2 through two points A and B inside a circle C1, as shown in the sketch below. [still more... this writing is in progress.]
In lab we looked at measurements, calculations, and some visual features such as locus definition, animation, and hide/show buttons.
Discussion turned to five axioms for a committee structure, and then a corresponding geometric structure where committee members are points and committees are lines.
In discussing the issue of whether the 5th axiom could be proven from the other four axioms, we looked at an example of another axiom system, with an axiom N (any pair of lines havng at one point in common) and an axiom P (given a line l and a point P not on that line there is a line m where P is on m and m and l have no common points) which is a version of the parallel postulate (Playfair's - not Euclid's). We gave examples showing that the four axioms and P were possible as well as the four axioms and N were possible. This showed that one can not prove axiom P or N from the other four axioms since P and N are contradictary.
By a similar analysis of the axioms for the seven point geometry we showed that it not possible prove the 5th postulate from the other four. The analysis examines the example of the seven point geometry and notices that by including an 8th and 9th point the resulting geometry would satisfy the other 4 axioms.
We briefly discussed at the end of the first hour how the model we had for affine geometry still satisfies the parallel postulate, since the ideal (infinite) points of affine geometry are not considered as ordinary points of the geometry. However, be removing this distinction between ordinary and ideal points and considering the geometry that results we obtain a geometry in which there are no parallel lines. (A projective plane.) This will be a major focus of discussion for the remainder of the term- especially using the homogeneous coordinates to consider points in this geometry from an analytic/algebraic approach.
The lab time was spent working on sketches showing ways to understand that result of the CAROMS film about the inscribed triangles of minimum perimeter.
An examination of the problems of transfering an spatial image of a plane
to a second plane using the idea of lines of sight we arrived at an understanding
of how points in the plane would correspond to lines through a point (the
eye).
After the quiz:
An overview of what we've considered so far-
| Euclidean Geometry lines/planes |
Affine geometry lines/planes |
Finite geometry lines |
Projective Geometry lines/planes |
| Axioms Euclid Hilbert |
No Axioms Yet A figure indicating an ideal point or line |
Axioms 7 points/7lines A figure. |
No Axioms Yet A figure indicating all points and lines |
| Parallel lines don't meet | Parallel lines meet at an ideal point. |
All pairs of lines have a common point |
All pairs of lines have a common point |
| Coordinates Analytic/Algebraic |
Ordinary coord's w/ infinite points Homogeneous Coordinates |
Homogeneous Coordinates with cofficients in {0,1}= Z 2 |
Homogeneous Coordinates with real number coefficients |
| Transformations Isometries Siimilarities |
? | ? | ? |
In discussing the 7 point geometry we visualized it using vertices of a cube (besides (0,0,0)) with their ordinary coordinates in standard 3 dimensional coordinate geometry and identified the 7 points . This allowed us to identify "lines" using the homogeneous coordinate concepts and their relation to planes in three dimension through (0,0,0). We identified all but one of the lines easily- the last plane has ordinary equation X + Y + Z = 2... but in this arithmetic for {0,1} we have 1+1=0, so 2=0 and the vertices of that satisfy this equation in ordinary coordinates {(1,1,0), (1,0,1),(0,1,1)} form a line as well.
We then discussed using {0,1,2} for homogeneous coordinates connected to the arithmetic given by the tables
| + | 0 | 1 | 2 | * | 0 | 1 | 2 | |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | |
| 1 | 1 | 2 | 0 | 1 | 0 | 1 | 2 | |
| 2 | 2 | 0 | 1 | 2 | 0 | 2 | 1 |
The homogeneous coordinates for this set identify ordered triples, for example:
<1,0,1>=<2,0,2> and <1,0,2>=<2,0,1>. There
are 27 possible ordered triples, and thus 26 when we exclude (0,0,0), and
these are each paired by the factor 2 with another triple, so there will
be exactly 13 points ( and by the comparable work with lines) and 13 lines
in this geometry.
.... some details still to be reported.
The class ended by watching the Open University video, "A Noneuclidean Universe."
i....related to 2 foci (ellipse and hyperbola) or 1 focus (parabola),
ii...related to a focus and directrix which was also related to eccentricity.
We turned our attention again to projectivities. We showed how the set of projectivities can be considered a group (as did isometries) , i.e. a set together with an operation which satisfies certain nice algebraic properties: closure, associativity, an identity and inverses.
We then considered lines connecting corresponding points in a pencil
of points on a line related by a projectivity (not a perspectivity) and noticed
that the envelope of these lines seemed to be a conic, a line
conic. We briefly discussed the dual figure which would form a more
traditional point conic. [We mentioned how line figures might be related
to solving differential equations e.g. dy/dx=2x-1 with y(0)=3 has a solution
curve determined by the tangent lines deteremined by the derivative: y=x^2-x+3
which is a parabola.]
We turned our consideration to looking at line transformations with homogeneous
coordinates .
These transformations can be identified with matrix multiplications since
A(cv)=cAv for A a matrix, c a scalar, and v a column vector so that pairs
of homogeneous coordinates for one point are transformed to homogeneous
coordinates for a single point.
Translation: T(x)=x+3 became
| 1 | 3 | x | x+3 | ||||
| 0 | 1 | 1 | 1 |
.. Reflection R(x)= -x uses the matrix
| -1 | 0 |
| 0 | 1 |
Dilation by a factor of 5 M(x)=5x uses the matrix
| 5 | 0 |
| 0 | 1 |
and inversion I(x)=1/x uses
| 0 | 1 |
| 1 | 0 |
We also considered how composition of these transformations corresponds to
matrix multipication and how these transformations interact with the ideal
point at infinity using homogeneous coordinates.
More to follow.
distinct points on two lines. For a projectivity (the composition of a finite
number of perspectivities) the result is that a projectivity is completely
determined by the correspondence of three points.This result is called
the fundamental theorem of projective geometry and is taken as an axiom by
M&I.
We also watched the film on projective generation of conics which introduced Pascal's theorem and its converse about hexagons inscribed in a conic and showed how to use this result to construct a conic curve passing through any 5 points. This work was also related to projectivities between pencils of lines. We will be considering this further in the course. After a short break we continued using metric ideas to construct an ellipse as a locus on sketchpad and discussed how to do a parabola as well. By next Thursday students should construct examples of the three conics on sketchpad using metric ideas.
We also introduced ths concept of 4 points on a line being related harmonically.
This will be the chief tool we use to introduce (homogeneous) coordinates
into projective geometry.
The idea of a general projective transformation of a projective line using
homogeneous coordinates and invertible 2x2 square matrices was also introduced.
This showed that all the transformations of the line we had previously discussed
were examples of this general type of projective transformation.
These two topic (harmomics and projective transformations) will be a major
topic for the next few weeks.
Turning our attention to the construction on the affine line of the point
P2 from P0,P1, and Pinf, we noticed that the figure used to construct this
point showed that H(P1Pinf,P0 P2). This led to a discussion of constructing
the fourth point, D, on a line given A,B and C already on the line so that
H(AB,CD). We used the construction of P2 to show how to construct the point
D in general.
The issue then became: was the point constructed from the points A, B and
C uniquely determined by the fact that it was in the harmonic relation with
A, B, and C? This is the question of the uniqueness of the point D. We proved
that in fact the point is D. (The proof followed the argument of Meserve
and Izzo. It used Desargues' theorem several times.)
With the existence and uniqueness of the point D established, we turned to
some examples of establishing a coordinate system for a projective line by
choosing three distinct points to be P0, P1, and Pinf. We constructed P2,
P1/2 (in two different ways) and left as an exercise the construction of
P3 and P1/3. We'll continue this discussion next class in showing that that
with the choice of three points on a projective line we can construct points
using harmonics to correspond to all real numbers (as they did in our informal
treatment of the affine line).
The class ended with a review of algebraic projective transformations of
the projective line P(1) which is characterized as a set of pairs of real
numbers <a,b>, not both zero, with <a,b>=<c,d> in the case
there is a nonzero real number t so that c=ta and d=td. These transformations
correspond to 2x2 matrices with non-zero determinant. These form a group
under composition (matrix multiplication) and we looked at one particular
example to see how it transformed points on the projective line with coordinates
to other points on the projective line. We will see how this transformation
is completely determined by the correspondence of three distinct pairs of
points. In some cases the transformation will transform an ordinary point
to an ordinary point, the ideal point to an ordinary point and an ordinary
point to the ideal point, and in some cases the transformation will tranform
the ideal point to the ideal point. A transformation that transforms the
ideal point to the ideal point is called an affine transformation. The
composition of two affine transformations is an affine transformation. The
inverse of an affine transformation is an affine transformation and clearly
the identity transformation is an affine transformation, so the affine
transformations are also a group under the operation of composition (matrix
multiplication). We'll continue this discussion next class to examine more
about the affine group for P(1).
| a | b |
| 0 | 1 |
and therefore T(Px)=Pax+b. Thus an affine transformation of the projective line is a dilation/reflection followed by a translation.
| +/-1 | b |
| 0 | 1 |
or that T(Px)=P+/-x+b. Thus an isometry of RP(1) is a reflection followed by a translation.