Chapter 13: The Five-Dimentional Man

The "Theory of Everything"

This is basically a hope that one day, the whole of physics could become so simplified that we'd have a set of equations small enough to fit on a t-shirt. One of the biggest difficulties is trying to fit gravity into the mix; however gravity has awkward features that make it impossible to unify the final force of nature with the rest of physics.

Of course, this theory may not even be possible at all. So far the mathematical equations (laws of nature) have been pretty successful at explaining our world, but there is absolutely no guarantee that this success will continue. The universe may in fact be a lot less mathematical that physicists ever imagined.

Even though we have mathematical equations that can approximate nature very well, there is no math tahat can capture it exactly. A mixture of "mutually consistant theories" might also provide workable approxiamations valid in different domains. But there may not be one central principle that combines all of those approximations and works in all domains.

Right now we've got a very "mix-and-match" theory, but it is extremely ugly. For example; the huge list of if/then rules: If speeds are small and scales are big use Newtonian Mechanics; if speeds are large and scales are big use special relativity. "If beauty is truth, then mix-and-match must be false. But perhaps the root of the universe is ugly."

Chapter 11: The Clerk From the Patient Office

We know that space is three dimensional; it has three independent directions at right angles to eachother-north, east, and up (like a corner of a building). Although the structure of space at all points is the same, the matter that occupies it may vary. Objects in space can be moved in different ways; they can be rotated, reflected, and translated. But we can also put this into an abstract perspective and say that the transormations being applied are to space itself. This is known as a "change of the frame of reference." The structure of space, as well as the physical laws that express that structure and operate within it, are symmetric under these transformations. Therefore, the laws of physics are the same at all times, and in all locations. When thinking about Earth, we say that the planet is rotating around the sun. But, we can also say that it is the sun that is rotating around the Earth. It all depends on the frame of reference in which we choose. The planets' movement relative to the sun however is much simpler than the planets movement relative to the Earth. The Earth-centered theory is possible but it is extremely complicated.

Chapter 10: The Would-Be Soldier and the Weakly Bookworm

Evariste Galois had a theory of algebraic equations and the key to solving them was symmetry. A man by the name of Marius Sophus Lie wondered if there was a theory for differential equations similar to the theory of algebraic equations. Of course the answer was yes, and once again symmetry was the key. Groups studied by Galois were all finite, however there are many groups that are infinite and include symmetry groups of differential equations.

A common infinite group is the symmetry group of a circle. This group contains transformations that rotate the circle through absolutely any angle. Since the possible angles are infinite, the rotation group of the circle is infinite. This group has the symbol SO(2); the "O" stand for "orthogonal," meaning that the "transformations are rigid motions of the plane"; and the "S" stands for "special," meaning that the "rotations do not flip the plane over." Circles also have an infinite number of axes of reflectional symmetry. A bigge group is formed by adding the reflections, O(2).

Both of these groups are infinite. By specifying the relevant angle, the different rotations can all be determined. When two rotations are composed, you just add the corresponding angles. Lie called this "continuous" and therefore that made the group SO(2) a continuous group. It is also a one dimensional group because only one number is needed to specify the angle. The same applies to the group O(2) because all that is needed is a way to distinguish reflections from rotations which would just consist of changing a plus or minus sign in the algebra.

The left diagram shows that the circle has an infinite numer of rotational symmetries and the right shows that the circle has an infinite number of reflectional symmetries.

Chapter 9: The Drunken Vandal

"William Rowan Hamilton was the greatest mathematician Ireland has ever produced."

Hamilton made a huge breakthrough; an analogy (identity), between mechanics and optics, the science of light. His discovery was amazing. It was a surprise when he discovered that mechanics (the study of moving bodies) and optics (the geometry of light rays) were connected, and an even bigger surprise when he discover that they were the same! This led to the formal setting used by mathematicians and mathematical physicists today. It is used not only in mechanics and optics but in quantum theory too. It is called Hamiltonian systems. What they do is derive the equations of motion of a mechanical system from a single quantity (the total energy). This is now called the Hamiltonian of the system. The equations that result from this involve the position of the parts of the system as well as the speed at which they are moving, also known as the momentum of the system. The equations do not depend on the choice of coordinates which is a great feature.

"Beauty is truth, at least in mathematics. And here the physics is both beautiful and true."

Chapter 7: The Luckless Revolutionary

What is Symmetry?


A symmetry happens when an object is transformed but its structure is still preserved. Symmetry can be seen as more of a process rather than a thing. There are three key concepts to symmetry; transformation, structure, and preserve. We can apply these concepts to an equilateral triangle.

Transformation: There are alot of things we can do to the triangle; bend it, rotate it, stretch it, etc. However, what we can do is limited by the second word, structure.

Structure: The structure of the triangle consists of its mathematical features, such as the length of a side (therefore no stretching), and the number of sides (no crumpling), also that the sides are straight (no bending).

Preserve: The structure of the transformed object must match the original structure. Therefore the sides must all still be straight and the same length, there must be 3 sides, the location must also be the same therefore sliding it is also out of the picture. Turning it on an angle does preserve some of the structure, but the location may appear different, depending on much of an angle to rotated it.

Chapter 6: The Frustrated Doctor and the Sickly Genius

Many Mathematician had been searching for a solution to a quintic equation. One man named Paolo Ruffini developed a 500 page proof that solving any equation with a dregree higher than 4 was impossible. Mathematicians found this "daunting"; to devote so many hours into proving that something does NOT work. Rumours went around that there were mistakes in his proof, but no one would identify them so he could not defend himself. He knew that his proof was extremely complicated, so he tried again, which did not result any better.

"One reason Ruffini's work was not well received may have been it's novelty.
Like Lagrange, he based his investigations on the concept of a "permutation.""

A permutation is a way to rearrange some ordered list. An example of this would be the results of shuffling a pack of cards. The idea is that you'll end up with a totally random and unpredictable order of the cards. Because the number of permutations possible in a deck of cards, it makes it basically impossible to predict the outcome.



“Permutations arise in the theory of equations because the roots of a given
polynomial can be considered a list. Some very basic features of equations are
directly related to the effect of shuffling that list. The intuition is that the
equation "does not know" the order in which you listed its roots, so permuting
the roots should not make any important difference. In particular, the
coefficients of the equation should be fully symmetric expressions in the
roots-expressions that do not change when the roots are permuted."

This is basically explaining that in a given polynomial, the roots of the equation can be considered a list, so we can incorporate the idea of permutations when trying to solve. "Shuffling" that list is proven to have effects on the equation.


Ruffini also used another of Lagranges ideas. By multipling two permutations together, you would result in getting another permutation. For example, take the symbols a, b, and c. This has 6 permutations: abc, acb, bac, bca, cab, and cba. Now take one of these permutations, cba for example, and we can see this as just an ordered pair from the list. However, it can also be seen as a rule for rearranging the original set. This rule would be, reverse the order. We can apply this to any of the permutions; if we apply it to bca, we'll get acb. Therefore it would make sense that cba times bca would equal abc.



We can also combine the two into one by stacking them on top of eachother. There are two ways to do it. Also, the result depends on which permutation comes first.

Chapter 5: The Cunning Fox

We know that regular polygons can be constructed when they have 3, 4, 5, 6, 8, 10, 12, 15, 16, or 20 sides. We also know that if they have 7, 9, 11, 13, 14, 18, or 19 sides, they cannot be constructed. By examining these numbers, we can see that there is one missing which is the number 17. Until Carl Friedrich Gauss, the construction of a 17-gon (short form) was unknown. However, on March 30th, Guass found a Euclidean construction for a regular polygon with 17 sides. Guass examined two simple properties of the number 17. First, its only divisors are itself anf one; second, it is one greater than a power of two: 17=2^4+1

"If you were a genius like Guass, you could see why those two unassuming
statements implied the existence of a construction, using a straightedge and
compass, of the regular seventeen-sided polygon. If you were any of the other
great mathematicians who had lived between 500 BCE and 1796, you would not even have got a sniff of any connections. We know this because they didn't."

Gauss thought his discovery of the 17-gon was so significant that he asked to have one carved on his tomb stone.

Chapter 4: The Gambling Scholar

The Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55....

This number sequence was created by Leonardo of Pisa (who was later given the name Fibonacci). Every number from 2 onward is the sum of the previous two numbers. This sequence is what made Leonardo famous. It developed through a population growth of rabbits! This sequence has a "remarkable mathematical pattern"; it also plays a very important role in the theory of irrational numbers.

Chapter 3: The Persian Poet

Conic Sections

Discovered around 350 BCE, Greek mathematician Menaechmus discovered "conic sections", special curves that he apparently used to solve the problem of doubling the cube. They were called so because they were done by slicing a cone with a flat plane. One cone is formed by straight line segments extending for a single point and passing through a circle (base). Double cones could also be created by extending the line segments through the point rather than just ending there. A double cone would appear as two ice cream cones that are joined by their pointy ends.

These conic sections could be used to solve certain cubid equations. There are three main types which can be classified as the ellipse, parabola, and hyperbola. When a plane passes through the cone at an angle it forms a closed oval curve called an ellipse. If the plane slices exactly perpendicular to the cones axis then it forms a circle, which is a special kind of ellipse. When the plane cuts parallel to one of the lines on the surface of the cone then a parabola is formed (a single open curve). Lastly, when a plane cuts through both halves of a double cone. a hyperbola is formes (two symmetrically related open curves). Here we can see how we get asymptotes.

In todays mathematics and applied science, these curves are extremely important. The planets in our solar system orbit in the shape of an elipse.

"This eliptical orbit in one of the observations that led Newton to formulate
his famous "inverse square law" of gravity. This in turn led to the
realization that some aspects of the universe exhibit clear mathematical
patterns. It opened up the whole of astronomy by making planetary
phenomena computable."

Chapter 2: The Household Name

Greek attitude towards Mathematics

These cultures had a very different view than Babylonians and the Egyptians towards mathematics. They saw it in very practical terms such as "aligning shafts through a pyramid so that the ka of the dead pharaoh could be launched in the direction of Sirius." Some Greek mathematicians believed that numbers were the very core of their beliefs rather than just tools that supported mystical beliefs. There was a cult around 550 BCE centered on Pythagoras, that believed that mathematics-especially numbers-were the basis of their creation. We hear of this cult from famous philosophers, Plato and Aristotle.

"They developed mystical ideas about the harmony of the universe, based in
part on the discovery that harmonious notes on a stringed instrument are related
to simple mathematical patterns."

To produce a note of one octave higher than the original, the string must be half the length of what it was when the original note was produced. They investigated many number patterns, especially polygonal numbers. Polygonal patterns were made from arranging different objects into polygonal shapes. For example, the numbers 1, 3, 6, and 10 are known as triangular numbers because when arrange, they form triangles.; whereas 1, 4, 9, and 16, are known as square numbers because they form squares.

Pythagoreanism classified the number 2 to be male, and the number 3 to be female. They had some pretty "nutty numerology". Greek geometry eventually became less mystical however, and they saw it as less of a tool and more of a branch of philosophy.

Chapter 1: The Scribes of Babylon

Positional notation: A numeral system in which each position is related to the next by a common multiplier.

As we are much familiar with, as a digit is moved one place to the left in the decimal system, its numerical value is multiplied by ten. The Babylonians are the earliest known culture to have a similar positional notation, however there was one major difference; instead of being multiplied by ten, everytime a digit was moved one place to the left it was multiplied by 60. This system is known as the sexagesimal (base-60) positional numerical system. They of course did not use the same number symbols as we use today. For example, our numbers 1 to 9 were written with tall, thin wedge symbols grouped together to add up to the number. For numbers greater than 9, a sideways wedge was added which symbolized the number 10. Therefore, two sideways wedges would equal 20.

This particular system stopped at the number 59, for a reason we are still unsure of. Rather than having six sideways wedges equal 60, they babylonians went back to the tall thin wedges. One wedge meant "one times sixty", two wedges meant "two times sixty times sixty", and so on. This caused some problems however because there was no way of telling if a wedge meant "1" or "1 x 60" or "1 x 60 x 60" etc. A symbol for zero was then invented by using a pair of diagonal wedges to show that no number occured in a given space. It is believe that the reason why the sexagesimal system is used rather then the decimal system, is because the number 60 is divisible by a large variety of numbers. This may have come in handy when sharing things among the people (grain, land). Also, it was their method of measuring time. They were known to be excellent astronomers so they would have known that a year was almost precisely 365.25 days, although they found it more convenient to divide a year into 360 days. (360=6 x 60)

Today, we can still see this ancient system being used:

• 360 degrees in a full circle; one degree per babylonian day
• 60 seconds in one minute
• 60 minutes in one hour

Chapter 12: The Quantum Quintet

Eugene Wigner's approach to group theory

Wigner approached this theory in a "simple, classical conext," the vibrations of a drum. A typical musical drum is usually circular, but can be any shape. When the drum is hit, the skin vibrates and a noise is created. Different sounds occur depending on the shape of the drum. The spectrum, or he range of frequencies that the drum can produce, depends in a complex manner on the drums shape. If the drum is symmetrical, it is expected that the symmetry would show up in the spectrum, and it does, but very subtly.

Let's say we have a rectangular drum, typically its vibration patterns would divide the rectangle into a number of smaller rectangles. Here are two examples:

Here are two different virational patterns with two different frenquencies. They are each a snapshot taken at one instant. The dark areas are displaces downwards and the white ones upward.

The symmetries of the drum are connected to the pattern. Any symmetry of the drum can be applied to possible pattern of vibration to produce another possible pattern of vibration, therefore the patterns come in symmetrically related sets.