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Monday November 6, 2006 5:30 am Lethbridge Sunrise 7:28 Sunset 17:02 Hours of daylight: 9:34
A. Morning Musings
5:30 am It is +7 C at the moment with a high of +17 C forecast.
Now to sit back and enjoy the coffee. The first sip is very nice. Yesterday had a feeling of accomplishment as I finished reading "Marie Antoinette" and I completed nailing down the mainline track for my model train layout.
The biography of Marie Antoinette was superb. I am beginning to feel that I have a sense of French history, having read a novel (in 4 volumes) based on the life of Napoleon, now one on Marie Antoinette, and am part way through "Citizens", a history of the French Revolution. I am definitely motivated to finish reading "Citizens". One question does occur to me: why do I enjoy the biographies yet do not feel it necessary to make any notes, whereas while reading "Citizens" I want to make notes after every chapter? I am also feeling that I want to begin reading a two volume series on Canadian history entitled "Origins".
I must return to my mathematics. It has been over a week since I last looked at "Fearless Symmetry". I still don't know what quadratic reciprocity is all about.
B. Plan
Immediate Health Walk & exercise 1 hr Model Trains Begin assembly of 6 quad hopper cars 1 hr Mathematics Read "Fearless Symmetry" chap 7: quadratic reciprocity 1 hr Continue reading "The Equation that Couldn't be Solved" 2 hr History Continue reading "Citizens" 1 hr Literature Continue reading "The Breakfast of Champions" by Kurt Vonnegut 1 hr Later Chores Investigate water softeners for home Technology Read manual for cell phone Make notes for chap. 4 of "Switching to the Mac" Begin reading "iPhoto" digital photography - learn about using the various manual settings
Philosophy Read "The Art of Living" by Epictetus Mathematics Larson "Calculus" Read "The Computational Beauty of Nature" Chap 3 Gardner "The Colossal Book of Short Puzzles" History Watson "Ideas" Model Trains Build oil refinery diorama: add ground cover Assemble second oil platform kit Assembly of CN 5930, an SD40-2 with a NAFTA logo Puzzles The Orange Puzzle Cube: puzzle #9
C. Actual/Notes
8:00 PM
I have finished reading "The Equation that Couldn't be Solved". Stunning! This is an amazing book. The variety of examples of how symmetry is a part of our life reads like a renaissance document. The detailed biography of Galois' life is fantastic. As a motivator to learn more about group theory the book is without peer. However the book is deliberately light on the actual mathematics underlying both symmetry and group theory. This is where "Fearless Symmetry" shines.
Now that I have finished reading the book, the question arises, "Should I make some notes about this, and if so, how?"
I think I will begin by noting some of the yellow highlighted sentences.
The Equation That Couldn't Be Solved (2005). Mario Livio
Chapter 1 Symmetry [p. 1 - 28]
- "I hope that the story as a whole will depict both the humanistic side of mathematics and, even more importantly, the human side of mathematicians." [p. 2]
- "Yet group theory, the mathematical language that describes the essence of symmetries and explores their properties, did not emerge form the study of symmetries at all." [p. 2 - 3]
- "Bilateral symmetry is so prevalent in animals that it can hardly be due to chance." [p. 6]
- "... all directions on the surface of the Earth are not created equal. A clear distinction between up and down ... is introduced by the Earth's gravity." [p. 6]
- "Having all the sensory organs ... in the front clearly helps the animal in deciding where to go and how best to get there." [p. 6]
- "There is nothing major in the sea, on the ground, or in the air, to distinquish between left and right." [p. 7]
- "... many multicellular animals have an early embryonic body that lacks bilateral symmetry. The driving force behind the modification of the 'original plan' as the embryo grows may indeed be mobility. ... Life forms that are anchored in one place and are unable to move voluntaraily, such as plants and sessile animals, do have very different tops and bottoms, but no distinquishable front and back or left and right." [p. 7]
- "Any civilization sufficiently evolved to engage in interstellar travel has likely long passed the merger of an intelligent species with its far superior computational-technology-based creatures. A computer-based super-intelligence is most likely to be microscopic in size." [p. 8]
- "Have nothing in your houses which you do not know to be useful or believe to be beautiful." [p. 17]
- "The arts and sciences are chock-full of fascinating examples of symmetry under the operations of translation, rotation, reflection, and glide reflection ..." [p. 22]
- "An interesting transformation that is not geometrical in nature involves permutations - the different rearrangement of objects, numbers, or concepts." [p. 22]
- a = b is symmetric under the interchange of a and b, but a < b is not symmetric under such an interchange. [p. 23]
- roulette is an example of a game that is symmetric with respect to the players - everyone has the same chance of winning. Blackjack is an example of a game that is not symmetric with respect to the players - good strategy is important. [p. 24 - 25]
Chapter 2 The Mind's Eye in Symmetry [p. 29 - 50]
- "To the Gestalt psychologists, therefore, symmetry was one of the key elements to contributing significantly to the 'goodness' of the figure." [p. 34]
- "Two other important elements in the Gestalt principles of organization are proximity and similarity." [p. 35]
- "Symmetry plays an important role in the recognition of similarity because it represents a true invariant - an immunity to change." [p. 35]
- "Humans have been 'practicing' perception for generations, and through their endless number of perceptual encounters they have learned what to expect." [p. 37]
- "With every step toward the revolutions of relativity and quantum mechanics, the role of symmetry in the laws of nature has become increasingly appreciated." [p. 43]
- "A group ... is a set that has to obey certain rules with respect to some operation. ... The properties that define a group are"
- closure
- associativity
- identity element
- inverse.
... this simple definition can lead to a theory that embraces and unifies all the symmetries of our world." [p. 46]Chapter 3 Never Forget This in the Midst of Your Equations [p. 51 - 89]
- "Concern for man himself and his fate must always constitute the chief objective of all technological endeavors ... in order that the creations of our mind shall be a blessing and not a curse to mankind. Never forget this in the midst of your diagrams and equations." (Albert Einstein) [p. 51]
- "Diophantus is best known today for a special class of equations that bears his name - Diophantine equations." (these are sets of equations that have integral solutions) [p. 59]
- "Muhammad ibn Musa al-Khwarizmi (ca. 780 - 850) ... was the first to expose in a systematic way the solutions of quadratic equations." [p. 61]
- "... the solution to the general cubic (and quartic) equation defied mathematicians until the sixteenth century." [p. 63]
- "Lagrange made the important discovery that the properties of equations and their solubility depend on certain symmetries of the solution under permutations." [p. 84]
- "Gauss gave his first proof of what has become known as the fundamental theorem of algebra - the statement that every equation of degree n has precisely n solutions. ... The fundamental theorem demonstrated unambiguously that the general quintic equation must have five solutions. But could these be found by a formula?" [p. 85]
- "This was the setting into which two young men, perhaps the most tragic figures in the history of science, appeared. The Norwegian Niels Henrik Abel and the Frenchman Evariste Galois were about to change the course of algebra forever." [p. 89]
Chapter 4 The Poverty-Stricken Mathematician [p. 90 - 111]
- an excellent biography of Abel.
Chapter 5 The Romantic Mathematician [p. 112 - 157]
- a superb biography of Galois.
Chapter 6 Groups [p. 158 - 197]
- In addition to the general concept of a permutation, are the related ideas of even and odd permutations (depending on the number of reversals of the elements). [p. 161]
- Another important idea is the idea of a cyclic permutation [p. 163]
- "Galois ... discovered an ingenious way to determine whether an equation is solvable from an examination of the symmetry properties of permutations of its solutions." [p. 163]
- "The identification of permutations as crucial mathematical objects worthy of study thus set Galois on the road to formulating group theory." [p. 164]
- "Permutations and groups are intimately related." [p. 164]
- "... two groups that have the same structure or the same 'multiplication table', such as the group of permutations of three objects and the group of symmetries of the equilateral triangle, are called isomorphic." [p. 167]
- "... certain subsets of the members of a group may by themselves satisfy all the four requirements of being a group (closure, associativity, identity, inverse). In that case the subset is said to form a subgroup." [p. 168]
- "If we divide the order (number of members) of the parent group ... by the order of the subgroup ... we obtain the composition factor." [p. 168]
- "An important theorem due to Lagrange ... The order of a finite subgroup always evenly divides the order of its finite parent group." [p. 168]
- "Galois started by showing that every equation has its own 'symmetry profile' - a group of permutations (now called the Galois group) that represents the symmetry properties of the equation. ... Before Galois, equations were always classified only by their degree: quadratic, cubic, quintic, and so on. Galois discovered that symmetry was a more important characteristic." [p. 170]
- "Galois was able to prove that for any degree n, one can always find equations for which the Galois group is actually the full permutation group. In other words, he showed that for any degree, there are equations that possess the maximum symmetry possible." [p. 170]
- "Galois then defined a normal subgroup. If any member of a subgroup satisfies the property that multiplying it from the left by a member of the parent group and from the right by the inverse gives a member of the subgroup, then the subgroup is called a normal subgroup." [p. 171]
- "Galois called a group solvable if every single one of the composition factors generated by its descendent maximal subgroups was a prime number. ... the condition for an equation to be solvable by a formula is that its Galois group should be solvable." [p. 171]
The remainder of the book returns to the theme of the first three chapters where a large variety of examples of symmetry in various scientific, artistic and psychological endeavors.
The concept of a normal subgroup seems to appear out of thin air. Where did this particular combination come from?
I still do not "fully" understand Galois' proof, but I do feel that I have a general sense of what he acomplished and definitely realize that group theory is an incredibly important branch of mathematics.
I can hardly wait until tomorrow when I hope to return to "Fearless Symmetry".
10:00 PM
D. Reflection