Millenium prize problem

The Millennium prize problems are seven problems in mathematics where a  prize of $1,000,000 is given to the person who can solve one of them. So far only one problems has been solved: the Poincare conjecture. The Poincare conjecture is one of the most important question in the topology field.

Poincare conjecture:
   Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

 Grigori Perelamn presented a proof of the conjecture in 3 papers. He was offered the prize money and a field medal to which he both declined, stating that it was not only him working on the problem; that people as far back as Hamilton has been working on and contributing to the proof of the conjecture. His proof was honored "Break through of the year".

The remaining Millenium prize problem are

1. P versus NP problem
2. Hodge conjecture
3. Riemann hypothesis
4. Yang–Mills existence and mass gap
5. Navier–Stokes existence and smoothness
6. Birch and Swinnerton-Dyer conjecture

Maybe you'll be the next to solve these famous problems

Unanswered Question:

1. How did the University, that propose these problems, know that these conjectures are true? Are they basing these conjectures on repeated observations?

Evaluation:
Complexity: 5
Interesting: 8
Quality: 7

Final chapters of Greek mathematics

By Ptolemy's time

• the entire eastern Mediterranean was part of the Roman Empire
• the local rulers and romans left the local language and culture intact.

and the great Cicero admitted that the Romans were not that interest in math. 

The last of the Greek mathematians were

1. Nicomachus of Gerasa
2. Diophantus of Alexandria
3. Pappus
4. Hypatia

Not much is know about Nicomachus although it is speculated that he studied in Alexandria. He dealt with perfect primes where he calculated the first four: 6, 28, 496 and 8128. He, however did not present any proof. Besides perfect numbers, Nicomachus also worked with pentagonal, hexagonal and heptagonal numbers.

Pentagonal Numbers

Nicomachus used a proportion in a different sense form Euclid"s. For Nicomachus there was 3 types of proportions:

1. arithmetic proportion- consecutive pair of numbers differ by the same quantity (for example 1,5,9,13 ... etc differ by 4)
2. geometric proportion- "the greatest terms is to the next greatest as that one is to the next" (for example 3,9,27...etc)
3. Harmonic proportion-"greatest term is to the smallest as the difference between the great and mean terms is the the difference between the mean and the smallest terms" (for example: 3,4,6 are harmonic proportion since 6:3 = (6-4): (4-3)

Diophantus worte arithmetica which is divided into 13 books. He introduced symbolic abbreviations and dealt with powers higher then 3, a break frmo the traditional Greek usage.He also presented some intersting problems

1. To divide a give number into two having a given difference
2. To divide a given number into two number such that given fractions (not the same) of each number when added together produced a given number
3. To divide a given square into two squares

Pappus is best know for his Collections,  which are works on various topics in mathematics. Through one of his remarks we learned that women where involved in mathematics in Alexandria.

Hypatia(355-415) was the 1st women mathematician of whom any details are know and her violent death marked the effective end of the Greek mathematician tradition in Alexandria.

Hypatia

Hypatia wa a respected and eminent teach in Alexandria. She was given a thorough education in math and philosophy by her father, Theon of Alexandria. Despite having influential friends, when a rival of her friends spread rumors about her about practicing sorcery a group of citizens gruesomely ended her life.

Heron wrote a detailed work on indirect measurement. For instance, we have two points, A and B, on one inaccessible side of a river (where the observer is on the other side). We want to find the distance between A and B. Heron used similar triangles.

We can measure the distance CA. And the two similar trianlges are DAB and CED.
So we have

CD/AD = CE/AB

So now we can solve for AB

Unanswered questions:

1. How steady was the decline in Greek mathematics? Did it slowly start to fall apart due to political strife and war or was it a long drawn out process?

Evaluations
Interesting: 8
Complexity: 7
Quality:  8

Astronomy and Ptolemy

Ptolemy(100- 178) also made numerous observations of the heavens and wrote several books, one being the Matematical Collections, which consisted of 13 books. The collection was the most influential astronomical work from its time. Islamic scientist bgan calling the book al-magisti or Almagist.

Ptolemy also gave a theorem, which helps derive sum and difference formulas of sine and cosine:


Ptolemy's Theorem:
Given any quadrilateral inscribed in a circle, the product of the diagonals equals the sum of the products of teh opposite sides: AC*BD = AB*DC + AD*BC

The proof of Ptolemy's theorem involves finding 2 pairs of similar triangles,in the diagram above, thus getting 2 different ratios. Then combining the 2 ratios yields the desired equation. With tables of trigonometric ratios the astronomers were able to solve right triangles easily and approach Apollonius' questions.

Unanswered Questions:

1. Did Ptolemy's Theorem have any other practical uses other than find trig identities?

Evaluation:
Interesting: 8
Quality: 7
Complexity:  7

Astronomy

The main reasons that mathematicians studied astronomy was to solve problems dealing with

• the calendar (such as determining the seasons)
• predictions of eclipses
• finding the beginning of the lunar month

In studying astronomy, the mathematics  of Greece created a plane and spherical trigonometry and developed a mathematical model of the universe.

Some terms:
Celestial equator- the axis of which the daily rotation of teh celestial sphere takes place.

vernal and autumnal equinoxes- the points of which the equator and ecliptic intersects one another

The 7 wanderers- They participated in the daily east to west rotation of the celestial sphere but also had their own motions. The 7 wanderers were the sun, the moon, Mercury, Venus, Mars Jupiter, and Saturn.

Great Circle-  A section of a sphere by a lane trough its center. For example the equator is one great circle. The earth has infinitely many great circles.

• The ecliptic, is a great circle that passes through the 12 constellations of the zodiac.

The major contributors to mathematical astronomy were

1. Eudoxus
2. Apollonius
3. Hipparchus
4. Menelaus
5. Ptolemy

Eudoxus was responsible for turning astronomy into mathematical science and is accredited with the invention of the two sphere model. He regarded the models as computational devices.

Hipparchus (190-120) began calculating trigonometric ratios. He also did numerous observations of planetary positions. For his trigonometry, he used chords subtending an arc or central angle in a circle with a fixed radius

Unanswered Question

1. Where there any other attempts to created a spherical model of the Universe? If so, how did they explain the heaven's movement? How did it fail?

Evalueation:
Quality: 8
Complexity: 7
Interesting: 7

Parabolic section

Area(Dark blue triangle) = (1/4)area( light blue triangle)
note: this formula can be applied to any triangle that is drawn in. In general the area of the newer(smaller) triangle will be 1/4th the area of the triangle that was drawn previously.




The formula will be used to find the sum of series in "Quadrature of the parabola" or to approximate the area of a segment of parabola inscribing triangles. Archimedes, using method of exhaustion, constructed more triangles and the area of the triangle, like described above, had 1/4 the area of the triangle drawn in the previous step. The area of the segmet becomes:
                   a +(1/4)a +[(1/4)^2]a+[(1/4)^3]a+...... = k        (any similarities to the geometric series?)

Then k + (1/3)(k-a)=  (4/3)a


Unanswered Questions

1. What were the uses for knowing the area of a segment of paraobla inscribing trianlges  or knowing that a area(one triangle)= (1/4) area(another triangle)?

Evaluation:
Quality: 9
Interesting: 6
complexity: 9

Archimedes and Lemmas

We know that the area of a circle is give by the formula: Area(Circle) = Pi(r^2). However, Archimedes came up with another formula and achieved the same answer: The area of a circle is equal to the area of a right triangle whose legs are the radius and circumference.

One way to see Area(circle) = (1/2)rc is to cut a circle into pieces(or slices) and arrange them as show below. As the slices become thinner (thus making the area more accurate) the rectangle becomes closer to having the sides r and c/2

 

In Archimedes' Measurement of the Circle was two lemmas used to calculate the ratios to the diameter of the perimeters of  regular polygons.

Lemma 1
Suppose OA is the radius of a circle and CA is tangent to the circle at A. Let DO bisect angle(COA) and intersect the tangent at D. Then DA/OA = CA/ (CO+ OA).

The proof of lemma 1 involves extending the line CO to COE where E is a point of the circle and connecting E and A. From there, angle(DOC) is equal to angle(AEO)  which imples that DO is parallel to AE. Using a property involving parallel lines intersecting 2 lines one can derive the desired result.


Lemma 2
Let AB be the diameter of a circle and ACB a right triangle inscribed in the semicircle. Let AD bisect angle(CAB) and meet the circle at D. Connect DB. The (AB/BD)^2 = 1 + [(AB+Ab)/ BC]^2

The proof of Lemma 2 involves manipulating the equation and similar triangles

The first lemma was repeatedly used to develop an algorithm for determining the desired ratio using the circumscribed polygons. Archimedes used the second lemma to get an algorithm for inscribed polygons

Unanswered Question:

1. Is there an advantage to using one formula, for the area of a circle, over the other? Which one appears more in the real world?

Evaluation
Interesting: 8
Quality: 8
complexity: 7 

Levers and Archimdes

Archimedes (287-212 BCE) has more information known about him then any other Greek mathematician. As the son of an astronomer, Phidias, he is credited with the invention, that is used for raising water for irrigation the Archimedian screw. He spent his time solving various problems for Hiero and his successor and it is noted that Archimedes, being very dedicated to his work, would neglect his health and hygiene. For month it was Archimedes military engineering that kept the Roman army  at bay for months during the siege of Syracuse, however eventually the Roman entered the city. During all this chaos Archimedes was working on a math problem and did not obey the Roman solider,  who found him, to come follow him to Marcellus. The solider ended taking his life.

No one before Archimedes had created a mathematical model of the lever by which one could derive a mathematical proof of the law of levers
4 of the 7 Postulate that Archimedes stated in his Planes in Equilibrium

A and B)
Equal weights at equal distances are in equilibrium, and equal weights at unequal distances will have the lever incline towards the weight at a greater distance

C)
When weights are at certain distances are in equilibrium and if more weight is added to one side then the lever will incline towards the side were the weight was added

E)
When weights are at certain distances and weight is taken away from one then the lever will incline towards the side where no weight is taken away.

F)
If magnitudes at certain distances are in equilibrium, other magnitudes equal to them will also be in equilibrium at the same distances

 The Law of the Lever is stated in Proposition 6 and 7:

Two magnitudes, whether commensurable [prop 6] or incommensurable [prop 7], balance at distances inversely proportional to the magnitudes.

Archimedes gave the first proof of the law of the lever and with that law Archimedes found the center of gravity for various figures

Propositions leading to the law of lever

Prop1
Weights which balance at equal distances are equal

Prop2
Unequal weights at equal distances will incline toward the greater weight

Prop3
Suppose A and Bare unequal weights with A> B which balance at point C. Letting AC=a and BC=b then a<b. Conversely, if the weights balance and a<b then A>B

Unanswered Question

1.  What other application does the law of the lever have besides finding the center of gravity?

Evaluation:
complexity: 7
interesting: 7
quality:  8

Construction (with only a ruler and compass)

Surprisingly  there is quite a few things that you can construct with just a ruler and a compass such as

1. A triangle of sides a,b,c
2. A square of side a
3. a perpendicular bisector
4. A square inside a circle
5. A regular hexagon inside a circle
6. An (equilateral) triangle inside a circle

For the first one note that a triangle can not be constructed if a+b>c

•  Starting at one end of the length of a, with the compass stretch out to the lenght of b, mark an arch. do a similar process for the length of a at the other end of a.
• The intersection of the two arcs will be the third point in the abc triangle. 

In drawing a perpendicular bisector to a line AB

• With the compass get the length of AB
• At both ends of AB, using the compass and keeping the length of AB, draw a large enough arc so that there is two intersection
• Align the ruler with the two intersection and draw a line (the resulting line is the perpendicular bisector)

 A square inside a circle
This construction is quite similar to the previous one. In a circle with diameter AB

• draw a perpendicular bisector like before and long enough that the bisector instersects the circle on both end
• Connect the 4 points on the circle which yields a square in the circle

Unanswered Question

1. Was there any practical application, back then,  of constructing with just a ruler and a compass?

Evaluation:
Complexity: 4
Interesting: 5
Quality: 8

Euclid Book 2 propositions

Before diving into some propositions from Euclid's second book some language barriers:

•  The statement "the rectangle contained by 'a' and 'b' " meant a rectangle with a the side length 'a' and ''
• The statement "a together with b" means a+b

Proposition II 4
If a straight line is cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.

In modern terms: This is just squaring a binomial (a+b)(a+b)
note:

• The whole is (a+b) so squaring the whole would yield (a+b)^2
• The segments would be a and b
• rectangle contained by the segments will be ab

Proposition II- 5:
If a straight line is cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

In modern term: (b-x)x+[(b/2)-x]^2 = (b/2)^2
note:

• (b/2)-x  is CD

• Let b (or AB) be a line and divide it into equal and unequal segments
• The equal segment is b/2 or AC and CB
• The unequal segment is AD and DB

Proposition II 11
To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment

In the above diagram the straight line is 'a' where a= x+a-x
In modern term the proposition states that a(a-x)= x^2
note: 
Unanswered Question

1. In book 1, all the propositions lead to the proof of the Pythagorean theorem so do all the proposition, in book 2, lead to any major theorem?

Evaluation:
Complexity: 7
interesting: 8
quality: 7