Date
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What we discussed/How we spent our time
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Jan 14
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Syllabus. Text.
We discussed CE/BCE notation for dating. (No year zero.)
We discussed a coarse timeline of development for Homo sapiens.
We discussed some of the the earliest mathematical objects/writings,
namely:
The Ishango bone:
Plimpton 322:
The Rhind papyrus:
The Moscow papyrus:
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Jan 16
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We discussed an early solution method for quadratic equations.
We gave two proofs of the Pythagorean Theorem:
one attributed to Pythagoras and Euclid's "Bride's Chair" proof.
We defined Pythagorean triples.
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Jan 18
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We gave proofs of the Pythagorean Theorem
attributed to Thabit ibn Qurra, Da Vinci and Garfield.
We started discussing the chord and tangent method for
parametrizing rational points on the circle.
(We established that every rational point
on the circle has the form
((1-t2)/(1+t2),2t/(1+t2))
for some rational t.)
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Jan 23
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We showed how to derive the formula for Pythagorean triples
from the parametrization of rational points on the circle.
Quiz 1 with
98 different solutions.
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Jan 25
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We briefly discussed Hippasus, who is said to have been drowned
for divulging some mathematical secret,
then turned to
commensurable numbers.
We defined segments a and b to be commensurable
if there is a segment c and positive whole numbers m and n
such that |a| = m*|c|, |b| = n*|c|.
We mentioned that a and b are commensurable
iff the ratio of their lengths is a rational number.
We proved √2 is irrational by reductio ad absurdum.
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Jan 28
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We introduced and justified the Euclidean algorithm
for finding the gcd of two positive whole numbers.
We gave a geometric
version of the algorithm and explained how
it could be used to determine the commensurability
of two lengths.
(Construct an a×b rectangle and repeatedly delete
maximal square subregions. The algorithm terminates iff
a and b are commensurable.)
We used this algorithm to show that the Golden Ratio,
φ=(1+√5)/2, is not rational.
Quiz 2.
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Jan 30
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We discussed continued fractions and the Golden Ratio.
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Feb 1
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Classification of polyhedra 1:
We discussed
1. the
Platonic solids
(= convex polyhedra whose
sides are congruent regular polygons and which
have the same number of sides meeting at each vertex),
2. the
prisms,
antiprisms, and
Archimedean solids
(= convex polyhedra not of type 1. whose
sides are regular polygons which meet at each
vertex in the same vertex configuration), and
3. the
Johnson solids (= convex polyhedra not of types
1. or 2.
whose sides are regular polygons which meet
in a strictly convex way at each vertex).
We stated that there are 5 Platonic solids,
infinitely many prisms and antiprisms, either
13 or 15 Archimedean solids, and 92 Johnson solids.
We mentioned that the Platonic solids have been identified with the
classical elements: tetrahedron=fire, cube=earth, octahedron=air,
icosahedron=water, dodecahedron=aether.
We began a proof that there are only 5 Platonic solids.
So far we have introduced Euler's formula: v-e+f=2,
and the relations pf=2e=qv, for a regular polyhedron of type (p,q).
(A regular polyhedron has type (p,q) if its faces are regular p-gons
with q of them meeting at each vertex.)
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Feb 4
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Classification of polyhedra 2:
We showed that a regular polyhedron of type (p,q) -- meaning
all faces are congruent regular p-gons with q meeting at each vertex --
has the following properties:
1. (p,q)=(3,3), (3,4), (4,3), (3,5), or (5,3).
2. (v,e,f) = (4p/(2p+2q-pq),2pq/(2p+2q-pq),4q/(2p+2q-pq))
Quiz 3.
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Feb 6
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Classification of polyhedra 3:
We wrapped up the discussion of the problem of
classifying the regular polyhedra, indicating that
after obtaining the numerical characteristics
of the regular polyhedra one can argue by cases that
only 5 are possible. To show that 5 exist, we must construct them.
We gave ad hoc arguments for the tetrahedron and cube, Pacioli's
argument for the icosahedron, and then obtained the octahedron
and dodecahedron by duality.
We then traced some of the influence on modern mathematics
that developed from this classification problem, namely
the Euler characteristic, Descartes' Theorem about the total
defect of a polyhedron, and the Gauss-Bonnet Theorem.
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Feb 8
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The Greek construction problems 1:
We discussed straightedge and compass constructions.
We briefly indicated how to construct a line through two points
at opposite ends of the galaxy using ordinary-size straightedge
and compass. We gave recursive definitions of
"constructible point",
"constructible line",
"constructible circle"; all other constructible objects
are defined in terms of these. We showed how to construct:
1. an equilateral triangle,
2. a line through a give point perpendicular to a give line,
3. a line through a given point parallel to a given line,
4. an angle bisector and a segment n-sector,
5. a regular pentagon.
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Feb 11
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The Greek construction problems 2:
We discussed four of the classical Greek construction
problems: squaring the circle, doubling the cube,
trisecting a general angle and contructing a regular
n-gon. Then we began discussing how to
"algebraize" plane geometry by introducing coordinates.
Quiz 4.
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Feb 13
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The Greek construction problems 3:
We continued our discussion of the coordinatization
of affine plane geometries. We noted that the smallest affine plane
has four points {O, I, C, D} and six lines
{{O,I},{O,C},{O,D},{I,C},{I,D},{C,D}}, and that the
associated number system has 2 elements. We noted that
the set of constructible points and lines in the Euclidean
plane is an affine plane, hence has an associated number system.
We labeled it
E0 and called it the field of
of constructible real numbers. We stated that
E0 is the smallest subfield of the real numbers
that is closed under square roots of positive numbers.
We spoke about HW problem #1, and argued that if a regular pentagon
of side length x can be inscribed in a circle of radius 1,
then x must equal
√(2-2cos(72°)). We also argued that
cos(72°)=1/(2φ). Thus, HW problem #1 reduces
to the problem of showing that the side length of the inscribed
pentagon is √(2-1/φ).
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Feb 15
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The Greek construction problems 4:
We gave examples and nonexamples of fields, ordered fields
and Euclidean fields. We described the field
ℚ[√2] that is obtained from
ℚ by adjoining √2.
We began a discussion of why the field
E0
of constructible numbers
is the smallest Euclidean subfield of ℝ.
The most interesting step in showing that
E0 is Euclidean was the step
showing how to construct the square root
of a positive number with straightedge and compass.
We ended the lecture by defining, for any Euclidean subfield E of
ℝ, the set of E-points, E-lines and E-circles
of the plane, and posed the problems of showing that
a line through two E-points is an E-line,
the circle centered at an E-point through another E-point
is an E-circle, and a point of intersection of two E-lines,
two E-circles or an E-line and an E-circle is an E-point.
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Feb 18
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The Greek construction problems 5:
We showed that if E is a Euclidean subfield of ℝ,
then it is closed under straightedge and compass constructions.
As a consequence we obtain that the field of constructible
real numbers is the smallest Euclidean subfield of ℝ.
Quiz 5 without solutions.
Quiz 5 with solutions.
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Feb 20
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The Greek construction problems 6:
Theorem.
r is a real constructible number
if and only if there is a finite chain of subfields
F0⊆…⊆Fk such that
(i) F0=ℚ,
(ii)
Fi+1=
Fi[√p] for some
p∈ Fi+, and
(iii) Fk contains r.
Theorem. If r is a constructible real number, then
r is algebraic and
minr,ℚ(x) has degree that is a power of 2.
Theorem. (Lindemann, 1882) π is transcendental.
Corollary. √π is transcendental. Hence it is impossible
to square a general circle with straightedge and compass.
(Some circles can be squared, but not all. For example,
a unit circle cannot be squared.)
We showed that the problem of constructing a cube with twice
the volume of a given one reduces to the problem of determining if
the cube root of 2 is constructible.
The lecture ended without resolving the problem of
determining whether
x3-2 is the minimal
polynomial of the cube root of 2.
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Feb 22
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The Greek construction problems 7:
We considered the problem of how to determine
the minimal polynomial of an algebraic number. We argued that:
1. Minimal polynomials are irreducible over the rational numbers.
2. A monic polynomial irreducible over the rationals
is the minimal polynomial for any of its roots.
3. The rational root theorem is sufficient to help determine
the irreducibility of a rational cubic.
We used these facts to show that x3-2 is the minimal
polynomial of the cube root of 2, and hence it is not possible to
double the unit cube with straightedge and compass.
We then turned to the problem of trisecting a general angle,
and I introduced the polynomial:
x3 - 3x - 2cos(α),
which is a monic polynomial that
has x=2cos(α/3) as a root.
When α=60°, 2cos(α)=2cos(60°)=1,
and the rational root theorem
applies to show that x3 - 3x - 1 is the minimal polynomial
of x=2cos(α/3)=2cos(20°). Since
2cos(60°) is constructible and 2cos(20°) is not,
it is not possible to trisect a 60° angle with
straightedge and compass.
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Feb 25
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The Greek construction problems 8:
We discussed a procedure for finding a monic rational polynomial
(in fact, an integer polynomial) p(x) of degree n such that
x=2cos(α/n) is a root of p(x)-2cos(α)=0.
Quiz 6.
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Feb 27
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Midterm review.
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Mar 1
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Midterm.
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Mar 4
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Section 3.4. We discussed Archimedes' cattle problem,
which reduces to an instance of Pell's equation.
We mentioned how Bhaskara II used Bramhagupta's Identity
to describe an algorithm for solving Pell's equation.
We explained how to use continued fractions to find a fundamental
solution, and then how to use the fundamental solution to generate
all other solutions.
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Mar 6
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Read Chapter 4.
We discussed the method of exhaustion, and used it to
establish the area formula of a circle.
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Mar 8
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Read Chapter 4.
We discussed how to use exhaustion to find a formula
for computing the area of a parabolic sector.
The proof involved exhaustion in two ways: once
to approximate the area by triangles, and the
second time to compute the sum
1+(1/4)+(1/42)+(1/43)+…
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Mar 11
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We completed the last detail of Archimedes'
quadrature of the parabola. Then we worked on a
handout
concerning Pell's equation.
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Mar 13
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Quiz 7.
We discussed the discovery of the cubic formula.
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Mar 15
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We derived the cubic formula.
We discussed the polar form for
complex numbers, and used it to give a geometric
interpretation of multiplication.
We discussed roots of unity,
ω = cos(2π/n)+i*sin(2π/n).
Finally we discussed how cube roots of unity are
used to express all three roots of x3+px+q=0.
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Mar 18
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We discussed Viete's Formulas, which relate
the coefficients of a polynomial to its roots.
Then we discussed how to "depress" a polynomial.
Finally, we started deriving the quartic formula.
Quiz 8.
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Mar 20
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We finished discussing the quartic formula.
We then turned to higher degree equations, noting that
Tschirnhaus developed a way of depressing monic nth degree
polynomials to eliminate the (n-1)rst and (n-2)nd degree
terms, and Bring advanced the procedure for quintics to eliminate
the 4rth, 3rd and 2nd degree terms.
We defined Bring radicals and stated that the roots of
an arbitrary quintic are expressible in terms of
arithmetic operations, rationals, root extraction
and Bring radicals. We discussed
the work of Ruffini, Abel and Galois on the unsolvability
of the general nth degree equation, n≥5.
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Mar 22
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We discussed the problem of determining the number of points
of intersection of two plane curves defined by polynomial
equations
P(x,y)=0 and Q(x,y)=0. We stated Bezout's Theorem,
that the number of intersection points is deg(P)*deg(Q),
provided the number of intersection points is counted "properly".
In particular, we restrict attention to the situation
where the polynomials P and Q have no common factor,
and then we have to consider intersection points with complex
coordinates, we have to count intersection points with the proper multiplicity,
and we have to count "intersection points at infinity".
During the last half of the lecture we discussed how
compute the intersection multiplicity Ip(P,Q).
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Apr 1
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We reviewed Bezout's Theorem and the definition
of intersection multiplicity. We calculated
I(a,b)(P,Q) for some examples.
We then defined the projective plane
as the object obtained from the Euclidean plane
by adding one point at infinity for each parallel
line family and one new line consisting of all points at infinity.
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Apr 3
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We discussed the construction of the projective
plane by adding a line at infinity and also the construction
through homogeneous coordinates. For the first construction,
we showed that the construction also works for other affine
planes, and drew a picture of the Fano plane (which is the
projective plane obtained from the 4-element affine
plane by adding a line at infinity).
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Apr 5
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We discussed the equations for polynomial curves
in the projective plane. We observed that the homogenizations
of y-x2 and of xy-1 are the same up to a permutation
of variables, suggesting that parabolas and hyperbolas are
look the same in the projective plane. We gave a geometric argument
explaining why all irreducible conics look like the circle in the
projective plane. We found the points at infinity on
y=x2 and xy=1. We ended by describing
how projective transformations may be represented by matrices,
and how the columns of the matrix may be found by looking
at the image of the triangle with vertices
[1:0:0], [0:1:0], [0:0:1].
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Apr 8
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We started the classification of quadratic curves
in the real projective plane. The bulk of this lecture
was spent in explaining why invertible 3×3-matrices represent
collineations. We stated the classification theorem:
a subset of the real projective plane that is defined by
a homogeneous quadratic polynomial is projectively equivalent
to a circle, a pair of lines (possibly equal),
a single point, or the empty set.
(The last 2 possibilities do not occur in the complex
projective plane.)
Quiz 9.
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Apr 10
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We showed that any homogeneous quadratic polynomial
whose zero set does not lie entirely on a line
defines a set that is projectively equivalent to
a circle or to a pair of distinct lines.
I reset the HW due date from this Friday to next Monday.
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Apr 12
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We proved Pascal's Theorem.
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Apr 15
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We worked on this collaborative project.
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Apr 19
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We discussed the discovery of hyperbolic geometry
and started describing the Poincare model of
the hyperbolic plane.
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Apr 22
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We discussed reflection through hyperbolic
lines in the Poincare model (via circular inversion)
and showed that
any point in the Poincare model can be moved
to any other point using 1 or 2 reflections.
Quiz 11.
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Apr 24
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We explained why the first four of Euclid's axioms hold
in the Poincare model while the fifth one fails.
We mentioned that hyperbolic geometry is a rich and strange
geometry in which angle sums of triangles are always
less than 180 degrees and where the Pythagorean Theorem
takes an unusual form.
We discussed the idea of "bi-interpretable" structures,
and gave as an example the real number field
and the complex number field equipped with complex conjugation.
The Poincare model is an interpretation of hyperbolic
plane geometry into Euclidean plane geometry. I mentioned
that there is an interpretation in the reverse direction, so that in fact
Euclidean geometry and hyperbolic geometry are bi-interpretable.
This means that a fact about one of the geometries is equivalent
to a fact about the other
(possibly expressed in a different way). In particular,
the two geometries are equiconsistent.
We then turned to area, noting that (whether in the Euclidean plane
or the hyperbolic plane) area ought to be a function
α defined at least on unions of triangles such that
(i) α(R)>0 if R has nonempty interior, (ii)
α(T1)=α(T2) if
T1 and T2 are
congruent triangles, and
(iii) α(P∪Q)=α(P)+α(Q) if
the intersection of P and Q has
empty interior. We mentioned that this determines area
on unions of triangles
up to a normalizing factor: in the Euclidean plane
it determines an area function whose values on rectangles
are given by base×height, while in the hyperbolic plane
it determines an area function whose vlaues on triangles
are give by defect.
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Apr 26
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We discussed equidecomposability of polygons
in the plane. We started the proof of
the Wallace-Bolyai-Gerwien Theorem, and completed
all except the proof that any rectangle is equidecomposable
with a square of the same area.
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Apr 29
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We completed the proof of the W-B-G Theorem about equidecomposability
in the plane.
Quiz 12.
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May 1
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We discussed the first three of Hilbert's problems,
and explained to what degree they have been resolved:
the Continuum Hypothesis, the problem
of determining the consistency of arithmetic, and
the equidecomposability of polyhedra.
(With regard to the third problem, we described
the Banach-Tarski paradox.) The bulk of the lecture
was devoted to the third problem,
in particular to defining the Dehn invariant
and sketching the proof that if P and Q are equidecomposable
polyhedra, then P and Q must have the same volume and the same Dehn invariant.
We calculated the Dehn invariant of a cube to be zero.
We identified two tetrahedra with equal bases and heights
but different Dehn invariants.
I distributed a
review sheet for the final.
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May 3
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We reviewed for the final.
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