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Library's review
Indeholder "Preface", "1. Symmetry", "2. eyE s'dniM eht ni yrtemmyS", "3. Never Forget This in the Midst of Your Equations", "4. The Poverty-Stricken Mathematician", "5. The Romantic Mathematician", "6. Groups", "7. Symmetry Rules", "8. Who's the Most Symmetrical of Them All?", "9. Requiem for a
"Preface" er tæt på at være namedropping. Han har snakket med alt og alle for at diskutere symmetri.
"1. Symmetry" handler om symmetri, fx palindromer som Able was I ere I saw Elba eller på ordniveau "Girl, bathing on Bikini, eyeing boy, finds boy eyeing bikini on bathing girl". Også lidt om de forskellige former for symmetri og symmetrigrupperne.
"2. eyE s'dniM eht ni yrtemmyS" handler om dybdeopfattelse, symmetri, psykologi, stereogrammer, definition af en matematisk gruppe. Symmetrigruppe for Isle of Man's logo - tre ben.
"3. Never Forget This in the Midst of Your Equations" handler om ligninger, diofantiske ligninger, Cardano, Tartaglia, Fiore, Ferrari, dal Ferro, Viète, Euler, Bring, Gauss osv.
"4. The Poverty-Stricken Mathematician" handler om Niels Henrik Abel.
"5. The Romantic Mathematician" handler om Evariste Galois.
"6. Groups" handler om gruppebegrebet.
"7. Symmetry Rules" handler om fysik, Einstein, Lie grupper, Ludvig Sylow, Klein, U(1) x UY(2) x SU(3), verden.
"8. Who's the Most Symmetrical of Them All?" handler om klassifikation af de endelige grupper, dvs de 26 sporadiske og de 18 familier.
"9. Requiem for a Romantic Genius" handler om Evariste Galois, Albert Einstein.
"Appendix 1. Card Puzzle" handler om en lille opgave med spillekort.
"Appendix 2. Solving a System of Two Linear Equations" handler om at løse to ligninger med to ubekendte.
"Appendix 3. Diophantus's Solution" handler om en simpel diofantisk ligning og dens løsning.
"Appendix 4. A Diophantine Equation" handler om en simpel diofantisk ligning og dens løsning.
"Appendix 5. Tartaglia's Verses and Formula" handler om huskeremser for specialtilfælde af tredjegradsligningen.
"Appendix 6. Adriaan van Roomen's Challenge" handler om en 45-gradsligning, som er løsbar.
"Appendix 7. Properties of the Roots of Quadratic Equations" handler om andengradsligningen.
"Appendix 8. The Galois Family Tree" viser Galois' stamtræ, hvad man så ellers skal bruge det til.
"Appendix 9. The 14-15 Puzzle" handler om 15-spillet.
"Appendix 10. Solution to the Matches Problem" viser hvordan man anbringer 6 tændstikker i 3 dimensioner, så de danner et tetraeder og dermed 4 ligesidede trekanter.
"Notes" er henvisninger til kilder, hvilket er fint nok, men det virker som skudt lidt over målet, for så videnskabelig er denne her bog jo heller ikke.
"References" er en meget lang liste over meget lange kilder, desværre helt uden indikation af hvilke sidetal, Livio har benyttet sig af eller hvor relevant den enkelte kilde er.
"Index" er et opslagsregister.
Matematisk set er bogen meget tynd i stoffet, for Livio stopper hver gang det er lige ved at blive lidt svært. Ikke min kop te. Til gengæld skitserer han tankegangen i nogle af beviserne, hvilket jeg godt kan lide.
Der er en meget sød overvejelse på side 83, hvor formlen for andengradsligningen med rødder x1 og x2 skrives som:
1/2((x1+x2) +/- sqrt( (x1+x2)^2 - 4 x1x2))
dvs at det her er meget klart at løsningen må være symmetrisk overfor ombytning af x1 og x2.
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Romantic Genius", "Appendix 1. Card Puzzle", "Appendix 2. Solving a System of Two Linear Equations", "Appendix 3. Diophantus's Solution", "Appendix 4. A Diophantine Equation", "Appendix 5. Tartaglia's Verses and Formula", "Appendix 6. Adriaan van Roomen's Challenge", "Appendix 7. Properties of the Roots of Quadratic Equations", "Appendix 8. The Galois Family Tree", "Appendix 9. The 14-15 Puzzle", "Appendix 10. Solution to the Matches Problem", "Notes", "References", "Index"."Preface" er tæt på at være namedropping. Han har snakket med alt og alle for at diskutere symmetri.
"1. Symmetry" handler om symmetri, fx palindromer som Able was I ere I saw Elba eller på ordniveau "Girl, bathing on Bikini, eyeing boy, finds boy eyeing bikini on bathing girl". Også lidt om de forskellige former for symmetri og symmetrigrupperne.
"2. eyE s'dniM eht ni yrtemmyS" handler om dybdeopfattelse, symmetri, psykologi, stereogrammer, definition af en matematisk gruppe. Symmetrigruppe for Isle of Man's logo - tre ben.
"3. Never Forget This in the Midst of Your Equations" handler om ligninger, diofantiske ligninger, Cardano, Tartaglia, Fiore, Ferrari, dal Ferro, Viète, Euler, Bring, Gauss osv.
"4. The Poverty-Stricken Mathematician" handler om Niels Henrik Abel.
"5. The Romantic Mathematician" handler om Evariste Galois.
"6. Groups" handler om gruppebegrebet.
"7. Symmetry Rules" handler om fysik, Einstein, Lie grupper, Ludvig Sylow, Klein, U(1) x UY(2) x SU(3), verden.
"8. Who's the Most Symmetrical of Them All?" handler om klassifikation af de endelige grupper, dvs de 26 sporadiske og de 18 familier.
"9. Requiem for a Romantic Genius" handler om Evariste Galois, Albert Einstein.
"Appendix 1. Card Puzzle" handler om en lille opgave med spillekort.
"Appendix 2. Solving a System of Two Linear Equations" handler om at løse to ligninger med to ubekendte.
"Appendix 3. Diophantus's Solution" handler om en simpel diofantisk ligning og dens løsning.
"Appendix 4. A Diophantine Equation" handler om en simpel diofantisk ligning og dens løsning.
"Appendix 5. Tartaglia's Verses and Formula" handler om huskeremser for specialtilfælde af tredjegradsligningen.
"Appendix 6. Adriaan van Roomen's Challenge" handler om en 45-gradsligning, som er løsbar.
"Appendix 7. Properties of the Roots of Quadratic Equations" handler om andengradsligningen.
"Appendix 8. The Galois Family Tree" viser Galois' stamtræ, hvad man så ellers skal bruge det til.
"Appendix 9. The 14-15 Puzzle" handler om 15-spillet.
"Appendix 10. Solution to the Matches Problem" viser hvordan man anbringer 6 tændstikker i 3 dimensioner, så de danner et tetraeder og dermed 4 ligesidede trekanter.
"Notes" er henvisninger til kilder, hvilket er fint nok, men det virker som skudt lidt over målet, for så videnskabelig er denne her bog jo heller ikke.
"References" er en meget lang liste over meget lange kilder, desværre helt uden indikation af hvilke sidetal, Livio har benyttet sig af eller hvor relevant den enkelte kilde er.
"Index" er et opslagsregister.
Matematisk set er bogen meget tynd i stoffet, for Livio stopper hver gang det er lige ved at blive lidt svært. Ikke min kop te. Til gengæld skitserer han tankegangen i nogle af beviserne, hvilket jeg godt kan lide.
Der er en meget sød overvejelse på side 83, hvor formlen for andengradsligningen med rødder x1 og x2 skrives som:
1/2((x1+x2) +/- sqrt( (x1+x2)^2 - 4 x1x2))
dvs at det her er meget klart at løsningen må være symmetrisk overfor ombytning af x1 og x2.
Show Less
Genres
Publication
New York : Simon & Schuster, 2005.
Description
Traces the four-thousand-year-old mathematical effort to discover and define the laws of symmetry, citing the achievements of doomed geniuses Niels Henrick Abel and Evariste Galois to solve the quintic equation and give birth to group theory.
User reviews
LibraryThing member nbmars
In Chapter One, Mario Livio promises to open our eyes to the magic of symmetry through the language of mathematics. To do so, he first acquaints us with group theory of modern algebra. A group is any collection of elements (they need not be numbers) that have the properties of (1) closure, (2)
Livio give us a little of the history of algebra, beginning with the ancient Greeks and Hindus, who solved the general quadratic. The story of the solution of the general cubic is a fascinating one involving allegations of cheating and libel among 16th century Italian mathematicians. Moreover, the solution required the invention of imaginary numbers. Once the cubic was solved, the solution or the quadratic quickly followed, but the quintic remained a mystery.
Even Euler and Gauss were stumped by the quintic, and they began to think the problem was insoluble. In fact, the work of two very young mathematicians, Niels Henrik Abel and Evariste Galois, proved that there could be no general solution to the general form of the quintic equation. The solution proved to be a surprise in that it depended on the relations among the coefficients of the variables. Only those quintics with a proper symmetry among the coefficients can be solved by purely algebraic operations. Livio does not actually show why the previous statement is true, probably because it requires real math. Nevertheless, the conclusion is pretty startling even to a math tyro like me.
The book gets a bit bogged down in its biographical sections, devoting more time to Galois’s life than I found interesting. Nevertheless, it is worth reading.
(JAB)
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associativity, (3) an identity element, and (4) an inverse operation. The fact that this simple definition leads to a theory that unifies all symmetries amazes even mathematicians. Livio give us a little of the history of algebra, beginning with the ancient Greeks and Hindus, who solved the general quadratic. The story of the solution of the general cubic is a fascinating one involving allegations of cheating and libel among 16th century Italian mathematicians. Moreover, the solution required the invention of imaginary numbers. Once the cubic was solved, the solution or the quadratic quickly followed, but the quintic remained a mystery.
Even Euler and Gauss were stumped by the quintic, and they began to think the problem was insoluble. In fact, the work of two very young mathematicians, Niels Henrik Abel and Evariste Galois, proved that there could be no general solution to the general form of the quintic equation. The solution proved to be a surprise in that it depended on the relations among the coefficients of the variables. Only those quintics with a proper symmetry among the coefficients can be solved by purely algebraic operations. Livio does not actually show why the previous statement is true, probably because it requires real math. Nevertheless, the conclusion is pretty startling even to a math tyro like me.
The book gets a bit bogged down in its biographical sections, devoting more time to Galois’s life than I found interesting. Nevertheless, it is worth reading.
(JAB)
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LibraryThing member jeffreydmoser
I read picked this book because I have since my early algebra days been interested in the quintic (e.g. x^5 + 2x^4 + … + 1). It presented a very good explanation of the history that led up to its ultimate proof that it’s impossible to solve in the general case using standard arithmetic
Although it covered that well, it kind of went off on many tangents to fields that sort of had to do with symmetry. Perhaps I should have got a book focused more — but all around it was interesting.
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operators and extraction of roots. Although it covered that well, it kind of went off on many tangents to fields that sort of had to do with symmetry. Perhaps I should have got a book focused more — but all around it was interesting.
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LibraryThing member shushokan
This book would make a good biography of Abel and Galois but is really a book about maths and not a maths book (if you can see the distinction). We get the intimate details of the two mathematicians' lives but their actual discoveries seem to be an addendum to the book as a whole. If you want a
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popular history and have a basic mathematical knowledge this is for you but I wouldn't recommend it if you want to exit the process knowing something about Galois theory. Show Less
LibraryThing member gregfromgilbert
First 5 chapters give a general history of mathematics centered around the solution to the quintic. It took a proof based on group theory to show it couldn’t be solved using basic operations. In depth focus on Abel and Galois. Chapters 6-8 much more interesting and includes a nice discussion of
Some quotes:
"The theory of groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing." Pg. 180 (actually quoting James R. Newman)
"The unexpected link between permutations and icosahedral rotations allowed Klein to weave a magnificent tapestry in which the quintic equation, rotation groups, and elliptic functions were all interwoven." – pg. 197
"Simple groups are the basic building blocks of group theory in the same sense that prime numbers are the building blocks of all the integer numbers" – pg. 224
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groups and symmetry in quantum physics. Some quotes:
"The theory of groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing." Pg. 180 (actually quoting James R. Newman)
"The unexpected link between permutations and icosahedral rotations allowed Klein to weave a magnificent tapestry in which the quintic equation, rotation groups, and elliptic functions were all interwoven." – pg. 197
"Simple groups are the basic building blocks of group theory in the same sense that prime numbers are the building blocks of all the integer numbers" – pg. 224
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LibraryThing member fpagan
The story of group theory (Abel, Galois, et al) -- another good pop-math book.
LibraryThing member jefware
While the concept of symmetry is fascinating I think that it's application to particle physics may be like applying circles to planetary motions. Nature just isn't symmetric. This book includes a great history of the mathematics of Group Theory.
Subjects
Language
Original language
English
Original publication date
2005
Physical description
x, 353 p.; 25 cm
ISBN
0743258207 / 9780743258203
Local notes
Omslag: Indbundet
Indskannet omslag - N650U - 150 dpi
Side 68: Cardano delights in describing in great and unnecessary detail all the medical problems from which he suffered early in life, including his sexual impotence between the ages of twenty-one and thirty-one.
Side 82: Erland Samuel Bring finder en måde at lave den generelle femtegradsligning om til formen x^5 + px + q = 0.
The equation that couldnt be solved : how mathematical genius discovered the language of symmetry
Indskannet omslag - N650U - 150 dpi
Side 68: Cardano delights in describing in great and unnecessary detail all the medical problems from which he suffered early in life, including his sexual impotence between the ages of twenty-one and thirty-one.
Side 82: Erland Samuel Bring finder en måde at lave den generelle femtegradsligning om til formen x^5 + px + q = 0.
The equation that couldnt be solved : how mathematical genius discovered the language of symmetry
Similar in this library
Pages
x; 353
DDC/MDS
| 512 |