Talent does what it can, Genius does what it must.
The Theory of Groups is the branch of mathematics in which you do something to something and then compare the result with the result obtained by doing the same thing to something else or something else to the same thing.
The book I recently read, The Equation that couldn't be solved is about these two things chiefly, Genius and Group Theory. I say chiefly because it really does cover a very wide range of topics, but aforementioned themes stand out. The equation in title is the general quintic (which is a just a fancy name for an equation of degree 5, like the quadratic is an equation of degree 2). If nothing else, all our schools taught us the formula to solve the quadratic. Solutions to ax2 + bx + c = 0 are
[Image from Wikipedia]
This formula is effectively known since the time of Babylonians. But what about equation of higher degree? Mathematicians over the years discovered similar (but more complex) formulas for third and fourth degree equations (known as cubic and quartic respectively). It was thought that the trend will continue for equations of higher degrees until two nineteenth century genius, Abel and Galois proved that no such formula (expressed in terms of coefficients, and using only the four arithmetic operations plus extraction of roots) can exist above quartic. But both these genius' led tragic lives and died in their twenties. Consider this
1. Abel proves the insolubility of quintic and sends it to Gauss, the best mathematician of his day. After Gauss' death, Abel's letter is found in his papers, unopened.
2. Mathematical establishment fails to consider even his later work.
3. Abel dies at twenty seven, in his own words 'as poor as a church mouse'.
Irony: Abel prize, which is given in his memory, is worth $992,000 today.
Or about Galois
1. Galois loses his father in political intrigue at a young age.
2. Galois submits his landmark work on the insolubility of quintic (the progenitor of Group Theory) for a prize contest. One of the judges, Fourier takes the work home, but dies after a few days and the work is lost.
3. Galois dies in a duel at twenty, after spending time in prison in the politically turbulent climate.
But despite this not-too-promising start Group Theory survived and has become a keystone of modern science, and the book covers the basics well. The chapter on Galois is a short and very readable mini biography. The history of the solutions of cubic and quartic is worth a separate book in itself. As Group Theory is the official language for describing symmetries of a system (e.g. an equilateral triangle under rotation by 120, 240 or 360 degrees, there is even a bingo ad about this ;-), the book also touches on fields ranging from visual perception to anthropology and music to evolutionary psychology where symmetry pops up. All in all, this book is a masterpiece. Don't miss!!
Enjoy!
-Owen Meredith
The Theory of Groups is the branch of mathematics in which you do something to something and then compare the result with the result obtained by doing the same thing to something else or something else to the same thing.
-James Newman
The book I recently read, The Equation that couldn't be solved is about these two things chiefly, Genius and Group Theory. I say chiefly because it really does cover a very wide range of topics, but aforementioned themes stand out. The equation in title is the general quintic (which is a just a fancy name for an equation of degree 5, like the quadratic is an equation of degree 2). If nothing else, all our schools taught us the formula to solve the quadratic. Solutions to ax2 + bx + c = 0 are
[Image from Wikipedia]This formula is effectively known since the time of Babylonians. But what about equation of higher degree? Mathematicians over the years discovered similar (but more complex) formulas for third and fourth degree equations (known as cubic and quartic respectively). It was thought that the trend will continue for equations of higher degrees until two nineteenth century genius, Abel and Galois proved that no such formula (expressed in terms of coefficients, and using only the four arithmetic operations plus extraction of roots) can exist above quartic. But both these genius' led tragic lives and died in their twenties. Consider this
1. Abel proves the insolubility of quintic and sends it to Gauss, the best mathematician of his day. After Gauss' death, Abel's letter is found in his papers, unopened.
2. Mathematical establishment fails to consider even his later work.
3. Abel dies at twenty seven, in his own words 'as poor as a church mouse'.
Irony: Abel prize, which is given in his memory, is worth $992,000 today.
Or about Galois
1. Galois loses his father in political intrigue at a young age.
2. Galois submits his landmark work on the insolubility of quintic (the progenitor of Group Theory) for a prize contest. One of the judges, Fourier takes the work home, but dies after a few days and the work is lost.
3. Galois dies in a duel at twenty, after spending time in prison in the politically turbulent climate.
But despite this not-too-promising start Group Theory survived and has become a keystone of modern science, and the book covers the basics well. The chapter on Galois is a short and very readable mini biography. The history of the solutions of cubic and quartic is worth a separate book in itself. As Group Theory is the official language for describing symmetries of a system (e.g. an equilateral triangle under rotation by 120, 240 or 360 degrees, there is even a bingo ad about this ;-), the book also touches on fields ranging from visual perception to anthropology and music to evolutionary psychology where symmetry pops up. All in all, this book is a masterpiece. Don't miss!!
Enjoy!
3 comments:
gr8!
good to see you're back on reading..greedily :)
hehe.. thanks.. :)
The quadratic formula. Beautiful.
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