Saturday, May 21, 2005

Sylvester's Surfaces

Figure 2. Clebsch's Diagonal Surface: Wonderful.
We are told that "mathematics is that study which knows nothing of observation..." I think no statement could have been more opposite to the undoubted facts of the case; that mathematical analysis is constantly invoking the aid of new principles, new ideas and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world, ...that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention. ...Were it not unbecoming to dilate on one's personal experience, I could tell a story of almost romantic interest about my own latest researches in a field where Geometry, Algebra, and the Theory of Numbers melt in a surprising manner into one another.

I had been looking for the link written by Nigel Hitchin, as this work was important to me, in how Dynkin drawings were demonstrated. Although I have yet to study these, I wanted to find this link and infomration about James Sylvester, because of the way we might see in higher dimensional worlds.His model seem important to me from this perspective.

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

The reason for this post is th ework dfirst demonstrated by Lubos Motl and th etalk he linked by Nigel Hitchin. The B-field, which seems to no longer exist, or maybe I am not seeing it in his posts archived?

In 1849 already, the British mathematicians Salmon ([Sal49]) and Cayley ([Cay49]) published the results of their correspondence on the number of straight lines on a smooth cubic surface. In a letter, Cayley had told Salmon, that their could only exist a finite number - and Salmon answered, that the number should be exactly 27

There had to be a simplification of this process, so in gathering information I hope to complete this, and gain in understanding.

James Joseph Sylvester (September 3, 1814 - March 15, 1897) was an English mathematician and lawyer.

Now as to the reason why this is important comes from the context of geometrical forms, that has intrigued me and held mathematicians minds. Sometimes it is not just the model that is being spoken too, but something about the natural world that needs some way in which to be explained. Again, I have no teachers, so I hope to lead into this in a most appropriate way, and hopefully the likes of those involved, in matrix beginnings, have followed the same process?

The 'Cubics With Double Points' Gallery

f(x,y,z) = x2+y2+z2-42 = 0,

i.e. the set of all complex x,y,z satisfying the equation. What happends at the complex point (x,i*y,i*z) for some real (x,y,z)?

f(i*x,y,z) = x2+(i*y)2+(i*z)2-42
= x2+i2*y2+i2*z2-42
= x2-y2-z2-42.

Has it become possible that you have become lost in this complex scenario? Well what keeps me sane is the fact that this issue(complex surfaces) needs to be sought after in terms of real images in the natural world. Now, I had said, the B-field, and what this is, is the reference to the magnetic field. How we would look at it in it's diverse lines? Since on the surface, in a flat world, this would be very hard to make sense of, when moved to the three coordinates, these have now become six?

fancier way of saying that is that in general, it's okay to model the space around us using the Euclidean metric. But the Euclidean model stops working when gravity becomes strong, as we'll see later.

Now what has happened below, is that what happens in quark to quark distances, somehow in my mind is translated to the values I see, as if in the metric world and moved to recognition of Gaussian curves and such, to decribe this unique perspective of the dynamics of Riemann, lead through geometrical comprehension ad expression. No less then the joining of gravity to Maxwells world.

Like the magnetic field we know, the lines of force represent a dynamcial image, and so too, how we might see this higher dimensional world. Again I don't remember how I got here, so I am trying hard to pave this road to comprehension.

"Of course, if this third dimension were infinite in size, as it is in our world, then the flatlanders would see a 1/r2 force law between the charges rather than the 1/r law that they would predict for electromagnetism confined to a plane. If, on the other hand, the extra third spatial dimension is of finite size, say a circle of radius R, then for distances greater than R the flux lines are unable to spread out any more in the third dimension and the force law tends asymptotically to what a flatlander physicist would expect: 1/r.

However, the initial spreading of the flux lines into the third dimension does have a significant effect: the force appears weaker to a flatlander than is fundamentally the case, just as gravity appears weak to us.

Turning back to gravity, the extra-dimensions model stems from theoretical research into (mem)brane theories, the multidimensional successors to string theories (April 1999 p13). One remarkable property of these models is that they show that it is quite natural and consistent for electromagnetism, the weak force and the inter-quark force to be confined to a brane while gravity acts in a larger number of spatial dimensions."

Now here to again, we are exercising our brane function(I mean brain)in order to move analogies to instill views of the higher dimensional world. The missing energy had to go somewhere and I am looking for it?:) So ideas like "hitting metal sheets with a hammer", or "billiards balls colliding", and more appropriately so, reveal sound as a manifestation of better things to come in our visions?

  • Unity of Mathematics
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