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Fun w/Digital M Images

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Oren Grad

Active member
Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829

scott kirkpatrick said:
When asked what the complex sculpture represented, the best that sculptor Gustav Vigelund could come up with was "a mind, launched into space."
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Hmm... can't see the back side, but are you sure there isn't something at least vaguely commutative hiding in there somewhere? :)
 
S

scott kirkpatrick

Well-known member
Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829

Oren Grad said:
Hmm... can't see the back side, but are you sure there isn't something at least vaguely commutative hiding in there somewhere? :)
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No the statue is as non-Abelian as you could imagine. If you can imagine Abelianity. Another of his contributions, one which I had never heard of, is that he proved that you cannot solve arbitrary equations of the fifth and higher degree with only the square root at your disposal to express the irrational part of the answer. Inventing group theory gets my vote for a memorial; I just wouldn't have expressed it this way. And since you can't get very far in this without encountering transformations (like rotations) which do not commute, I don't know why they call those "non-Abelian."
 
MGrayson

MGrayson

Subscriber and Workshop Member
Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829

scott kirkpatrick said:
No the statue is as non-Abelian as you could imagine. If you can imagine Abelianity. Another of his contributions, one which I had never heard of, is that he proved that you cannot solve arbitrary equations of the fifth and higher degree with only the square root at your disposal to express the irrational part of the answer. Inventing group theory gets my vote for a memorial; I just wouldn't have expressed it this way. And since you can't get very far in this without encountering transformations (like rotations) which do not commute, I don't know why they call those "non-Abelian."
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:lecture:
Not just square roots, e.g., x^3 - 2 = 0.

I was never clear about the relative contributions of Abel and Galois. Did they know each other's work (the timing of their short productive lives overlap). Mathematicians are very bad at their history.
 
O

Oren Grad

Active member
Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829

MGrayson said:
I was never clear about the relative contributions of Abel and Galois....
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Galois managed to earn himself a vignette in our Algebra 2 textbook... poor Abel had to wait for Modern Introductory Analysis, the following year. (Yes, that was more than 40 years ago!)

Way off on a tangent, but if you're into the history, this is a lot of fun:

https://www.genealogy.math.ndsu.nodak.edu/
 
MGrayson

MGrayson

Subscriber and Workshop Member
Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829

Oren Grad said:
Galois managed to earn himself a vignette in our Algebra 2 textbook... poor Abel had to wait for Modern Introductory Analysis, the following year. (Yes, that was more than 40 years ago!)

Way off on a tangent, but if you're into the history, this is a lot of fun:

https://www.genealogy.math.ndsu.nodak.edu/
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Oh yes. It's great fun. I'm in there, but I never had my own grad students, so my entry is very short. My advisor was quite famous, though.
 
Robert Campbell

Robert Campbell

Well-known member
Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829

Oren Grad said:
Galois managed to earn himself a vignette in our Algebra 2 textbook... poor Abel had to wait for Modern Introductory Analysis, the following year. (Yes, that was more than 40 years ago!)

Way off on a tangent, but if you're into the history, this is a lot of fun:

https://www.genealogy.math.ndsu.nodak.edu/
Click to expand...
If you'd like a daily dose of maths history by email, try this:

http://pballew.blogspot.com/
 
scho

scho

Well-known member
After many days of excessive heat and humidity we were treated to some refreshing early fall air today. No foliage color yet but goldenrod in bloom everywhere. Leica M (262) with 7Artisans 35/2.



 
S

scott kirkpatrick

Well-known member
Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829

MGrayson said:
:lecture:
Not just square roots, e.g., x^3 - 2 = 0.

I was never clear about the relative contributions of Abel and Galois. Did they know each other's work (the timing of their short productive lives overlap). Mathematicians are very bad at their history.
Click to expand...
Thanks for the interesting reading assignment -- I did 2-3 years of solid math as an undergraduate, and you can't learn physics without using group theory, both discrete and continuous, but Abel and Galois were both just exotic names to me. Both were from radical families, living in dangerous times, and for various reasons, very short of money. They both died quite young. Abel came first. The hot problem of their time, more or less the "Goedel's decidability problem" of about 1800 was whether or not all fifth-order and higher polynomial equations could be solved by algebraic means, leading to an answer which at worst involved taking the n-th root of something. The problem attracted crackpot solvers, and both Abel and Galois had their papers ignored or misplaced and lost by the giants of the field, such as Gauss or Legendre.

Abel solved the problem (the answer is no), and found a small math journal in which all of his work appeared. Galois was aware of Abel's proof, but Galois' methods were capable of great extension and application to other problems while Abel's proofs have been subject to small improvements and simplifications down to the present day. Galois invented a permutation group of the basic elements from which a polynomial is created and showed that if the group is non-Abelian, there must be some polynomials with roots not expressible as fractional powers. Analysis of the fundamental groups of quartic and simpler polynomials shows they are Abelian, quintic and higher polynomials are generated by groups which are non-Abelian.

Who knew that this distinction between the simplest irrational numbers and those more irrational has such consequences?

Best reference, but still not the whole story: Wikipedia on "Ruffini-Abel Theorem"
 
scho

scho

Well-known member
Cayuga Inlet near the Ithaca farmers market taken with the Leica M 262 + Biogon 28/2.8 ZM. I posted this image in this thread earlier but this version was run through Topaz A.I. Gigapixel set to increase size to 43 MP. After resizing the image was reduced to 50% of original and exported as a jpeg from LR. Processing is extremely slow but results are excellent.

 
JoelM

JoelM

Well-known member
Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829

scott kirkpatrick said:
Abel invented group theory and solved some amazing problems, but died at 27 in poverty, from TB, a few days before an offer of a professorship at the university in Berlin would have reached him. The statue achieved its prime location in the royal palace park in Oslo largely through extensive artist and sponsor politicking. When asked what the complex sculpture represented, the best that sculptor Gustav Vigelund could come up with was "a mind, launched into space."scott
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Group theory was part of one of the most difficult classes I ever took. The really cool thing about it, though, is that it lets you not only envision, but prove the relevance of chirality and symmetry in the physical world.

Joel
 
scho

scho

Well-known member
A few shots taken with the Leica M 262 + Contax G 45/2 Planar (M mount converted) on a damp woodland walk.





 
scho

scho

Well-known member
Two from a walk in Stewart Park on the south end of Cayuga Lake in Ithaca, NY.

Boathouse - Leica M 262 Contax G 45/2 Planar (M mount converted)



Juxtaposition - Leica M 262 Carl Zeiss 28/2.8 Biogon ZM

 
scho

scho

Well-known member
Some early fall garden color (Fall Crocus and Purple Aster) with the Leica M (262) + Minolta M-Rokkor 40/2.



 
scho

scho

Well-known member
The remnants of Florence passed through last night and dropped off a couple of inches of rain. No major flooding, but high water levels. Leica M (262) with the Zeiss 28/2.8 Biogon.

New tourist signs and maps on the Lake St bridge over Fall Creek.



Ithaca Falls



Old mill remains above trail to Ithaca Falls.



Cayuga Lake

 
scho

scho

Well-known member
Leica M (262) with M-Rokkor 40/2

The municipal "Geese Course"



Rower on Fall Creek



Checking out one of the stages before the weekend music festival

 
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