Category Archives: Math

The Three Body Problem and the Appropriation of Images …

Enceladus Geysers, Samuel Nigro Saturn, Samuel NigroThe Sun, Samuel NigroComet Tempel 1 After Projectile Impact, Samuel NigroClouds over Senegal and Mali, Samuel Nigro

At the end of January, I went to the opening of Michael Benson at Hasted Kraeutler, 537 West 24th Street, New York, NY.
The show was a series of large digital prints that he created from the thousands of images sent back to us by probes
we’ve launched into the solar system to answer various scientific questions about the nature
of our planetary neighbors. From the press release, Benson culled the images
taken by “the Cassini Saturn orbiter, the intrepid Mars rovers
Spirit and Opportunity, and the Earth-orbiting
Solar Dynamics Observatory.”

I share Benson’s fascination, and above you’ll find my own culling of his images… You can cull your own set by going to
NASA’s Cassini site, Mars Rover site or the Solar Dynamics Observatory site
or just do a Google Image Search.

I had two over-riding thoughts (beyond the AWE inspired by what lies beyond our own atmosphere)
as I wandered through the show during the opening
and again when I went back for
a second look:

First, I thought of an earlier, and seeming unrelated, lecture I blogged about earlier in Happy Thanksgiving and Gratitude List of 12. Keith Wilson lectured about the collection of Charles Lang Freer’s Buddhist Scultpure from China. He posed the question: At what point does an object of religious devotion become an object of just aesthetic interest? You can ask a similar question about scientific data: at what point, does it become just an object we gaze at for our own aesthetic need?

Second, I have followed the Mars rovers and the Cassini Spacecraft, and I’m not sure how much Benson’s images tell us about what these sophisticated machines have already given us and what their potential is. For example, it is because of the Cassini Spacecraft that we are learning just how incredibly complicated Saturn’s rings are.  There are large masses (as large as and larger than houses and trucks) that move in and out the rings and some of which sweep large portions of the debris and grow and change in size, shape and trajectory.

What the show does do is give us a vision of Michael Benson, and it is heartening to read that he works diligently with scientists as he makes his composite images
so as to be as accurate as possible. So, to Benson’s credit, it feels like he is trying to give us a view as to what it may be like
if we were actually there – floating in front of the Sun, trekking around Mars, or about to sail through Saturn’s rings.
These are things I think about, and find
(in the non-pop-cultural,
non-trivial use
of the term) AWESOME.

Michael Benson and I share the same interest in the line between and the intersection of art and science.

The very next day after the opening, I happened upon the Dynamical Systems Seminar at the Courant Institute by Eugene Gutkin, called The Outer Billiard Map, and I was enthralled by the connection between this and Benson’s show.

WHAT possibly could be the connection between these two experiences!?! you may wonder.
The quick answer is the Three Body Problem.
But, first, I need to explain
what an “Outer Billiard”
problem is.

Imagine billiard balls bouncing around a pool table, hitting each other in a chaotic fashion. The study of Inner Billiards is the study of the physics of this kind of movement. Now, imagine a solid stationary object in the middle of the pool table with billiard balls bouncing around and off of this stationary object. The study of the physics of these kinds of interactions is termed an Outer Billiard problem.

NOW, imagine the stationary object is a large planet or similar galactic object and the billiard balls are satellites or even beams of light… and that is the connection between the gallery show and the seminar. I reveled in the connections.

Billiard Balls flying around a pool table is pretty complicated motion, so to simplify we break it down. The two-body problem (the moon orbiting the earth, for example) is straightforward in terms of classical mechanics and gravity. Add a third body and it gets very complicated, and is, in fact, an old problem that we still only have approximations for. Sir Issac Newton formulated and studied all these issues.

However, we know enough about this motion to not only send spacecraft to various parts of our solar system, but also to infer the existence of other planets (and their sizes!) in remote parts of the galaxy that may even resemble Earth!

This is definitely fun to think about!

Outer Billiard Map_Eugene Gutkin, Samuel Nigro

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Filed under Astronomy, Featured ..., Featured Art Space, Math

my most nostalgic post so far …

Horoballs and Tsvietkova

Horoballs (note: all the spheres are tangent to the z-plane) and Anastasiia Tsvietkova

Anastasiia Tsvietkova – Hyperbolic Structures from Link Diagrams

Anastasiia Tsvietkova – Hyperbolic Structures from Link Diagrams

14 prime knots with 7 crossings

14 prime knots with 7 crossings

I lucky got the chance to go to the last Geometry and Topology Seminar at CUNY Graduate center on December 11th right before the holidays. It was a treat.

Anastasiia Tsvietkova of Louisiana State University presented her dissertation entitled Hyperbolic Structures from Link Diagrams.
Rooted in knot theory and geometric topology, she builds upon W. Thurston’s Hyperbolization Theorem,
which demonstrates that every link in a 3-sphere is

a torus link,
a satellite link
or a hyperbolic link

and these three categories are mutually exclusive. That just SOUNDS satisfying.

Her dissertation lays out an alternative way to compute the hyperbolic link in a 2-DIMENSIONAL PROJECTION.

That deserves a “WOW!”

I enjoyed the talk very much, even though much of the math was over my head.
The fact that the whole lecture only dealt with 3-dimensions made it easier.
I, at least, understood what was at stake and enjoyed following
the structure of the argument.
You can read her paper.

The bottom line:

The lecture got me to open my Knot Theory book and
revisit my drawings of hyperbolic paraboloidal shapes
from Calculus III,

because knot theory is fun and I enjoy calculus.

Hyperbolic Paraboloids from Calc III

Hyperbolic Paraboloids from Calc III

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Filed under Drawing, Featured ..., Featured Thinker, Geometry, Math