3

lol, here's a good one

...riding a sphenoid, a sphinx



from 2012-03-25

Good, but it's only a part of it.

Would you have time to hear the whole song?





.

Damn that Rock,Paper,Scissors

In mathematics, the Poincaré conjecture ( /pwɛn.kɑːˈreɪ/ pwen-kar-AY; French: [pwɛ̃kaʁe])[1] is a theorem about the characterization of the three-dimensional sphere (3-sphere), which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

[link to en.wikipedia.org]




Main article: Poincaré conjecture

In November 2002, Perelman posted the first of a series of eprints to the arXiv, in which he claimed to have outlined a proof of the geometrization conjecture, of which the Poincaré conjecture is a particular case.[11][12][13]

Perelman modified Richard Hamilton's program for a proof of the conjecture, in which the central idea is the notion of the Ricci flow. Hamilton's basic idea is to formulate a "dynamical process" in which a given three-manifold is geometrically distorted, such that this distortion process is governed by a differential equation analogous to the heat equation. The heat equation describes the behavior of scalar quantities such as temperature; it ensures that concentrations of elevated temperature will spread out until a uniform temperature is achieved throughout an object. Similarly, the Ricci flow describes the behavior of a tensorial quantity, the Ricci curvature tensor. Hamilton's hope was that under the Ricci flow, concentrations of large curvature will spread out until a uniform curvature is achieved over the entire three-manifold. If so, if one starts with any three-manifold and lets the Ricci flow occur, eventually one should in principle obtain a kind of "normal form". According to William Thurston, this normal form must take one of a small number of possibilities, each having a different kind of geometry, called Thurston model geometries.

This is similar to formulating a dynamical process that gradually "perturbs" a given square matrix, and that is guaranteed to result after a finite time in its rational canonical form.

Hamilton's idea attracted a great deal of attention, but no one could prove that the process would not be impeded by developing "singularities", until Perelman's eprints sketched a program for overcoming these obstacles. According to Perelman, a modification of the standard Ricci flow, called Ricci flow with surgery, can systematically excise singular regions as they develop, in a controlled way.

We know that singularities (including those that, roughly speaking, occur after the flow has continued for an infinite amount of time) must occur in many cases. However, any singularity that develops in a finite time is essentially a "pinching" along certain spheres corresponding to the prime decomposition of the 3-manifold. Furthermore, any "infinite time" singularities result from certain collapsing pieces of the JSJ decomposition. Perelman's work proves this claim and thus proves the geometrization conjecture.


[link to en.wikipedia.org]

I NEVER IMPLEMENTED THIS FEATURE

It reminds me of the skin of a Lion.

It's always plus 1. There's 3 plus 1, that's the way it works.

In mathematics, the Poincaré conjecture ( /pwɛn.kɑːˈreɪ/ pwen-kar-AY; French: [pwɛ̃kaʁe])[1] is a theorem about the characterization of the three-dimensional sphere (3-sphere), which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

[link to en.wikipedia.org]




Main article: Poincaré conjecture

In November 2002, Perelman posted the first of a series of eprints to the arXiv, in which he claimed to have outlined a proof of the geometrization conjecture, of which the Poincaré conjecture is a particular case.[11][12][13]

Perelman modified Richard Hamilton's program for a proof of the conjecture, in which the central idea is the notion of the Ricci flow. Hamilton's basic idea is to formulate a "dynamical process" in which a given three-manifold is geometrically distorted, such that this distortion process is governed by a differential equation analogous to the heat equation. The heat equation describes the behavior of scalar quantities such as temperature; it ensures that concentrations of elevated temperature will spread out until a uniform temperature is achieved throughout an object. Similarly, the Ricci flow describes the behavior of a tensorial quantity, the Ricci curvature tensor. Hamilton's hope was that under the Ricci flow, concentrations of large curvature will spread out until a uniform curvature is achieved over the entire three-manifold. If so, if one starts with any three-manifold and lets the Ricci flow occur, eventually one should in principle obtain a kind of "normal form". According to William Thurston, this normal form must take one of a small number of possibilities, each having a different kind of geometry, called Thurston model geometries.

This is similar to formulating a dynamical process that gradually "perturbs" a given square matrix, and that is guaranteed to result after a finite time in its rational canonical form.

Hamilton's idea attracted a great deal of attention, but no one could prove that the process would not be impeded by developing "singularities", until Perelman's eprints sketched a program for overcoming these obstacles. According to Perelman, a modification of the standard Ricci flow, called Ricci flow with surgery, can systematically excise singular regions as they develop, in a controlled way.

We know that singularities (including those that, roughly speaking, occur after the flow has continued for an infinite amount of time) must occur in many cases. However, any singularity that develops in a finite time is essentially a "pinching" along certain spheres corresponding to the prime decomposition of the 3-manifold. Furthermore, any "infinite time" singularities result from certain collapsing pieces of the JSJ decomposition. Perelman's work proves this claim and thus proves the geometrization conjecture.


[link to en.wikipedia.org]

The sphere and toroid as symbol of the head and halo or to some the crown of thorns.

The sphere as echoing dreamer with buoyant rings (toroids) radiating forth with electromagnetic interplay.

The distillation of the dream circulating, concressing and returning to echoing sphere.

[link to www.laetusinpraesens.org]

HAH!

This is exactly what I was thinking...well, so far after only reading a few paragraphs.

Basically it was an idea I had about forming governance using the blueprint as to how non-material manifests into material, or how toroids are the stability of aether.

In effect the "thorns" are the various pieces of a global system which lack appropriate connectivity, resulting in unsustainability and instability -- and potentially chaos and collapse.

HAH!

This is exactly what I was thinking...well, so far after only reading a few paragraphs.

Basically it was an idea I had about forming governance using the blueprint as to how non-material manifests into material, or how toroids are the stability of aether.

What an incredible website!

hmmm:

"The subtle combination of creativity, irrationality, surprise and "cognitive catastrophe" are associated with the non-linearity of the Knight's move. Change is to be understood and achieved by the possibility of "getting off a train" of linear thought -- otherwise expressed as "out-of-the-box"."

he 'knight's move' is perpendicular.


and:

think entropy. being able to describe it's 'shape' still doesn't come close to what or why.

just sayin'

how do you explain a machine who`s parts transform into other parts that transform into other parts that...........

...........for ever

a transient machine

feels like eternal

whoa!

our new philosophy feels tangible

Time-invariant system


[link to en.wikipedia.org]

then we discovered our environment (universe) utilizes time invariant system (timeless) that manifest transient dynamical systems

a layer of a layer

What an incredible website!

Ok, I just started this thread.

Is this the theory as to why things smooth out naturally? Accretion within dust disks, aftermath of celestial events, etc. Basically it is an organizational quality embedded in the aether...I'm thinking.

hmmm:

"The subtle combination of creativity, irrationality, surprise and "cognitive catastrophe" are associated with the non-linearity of the Knight's move. Change is to be understood and achieved by the possibility of "getting off a train" of linear thought -- otherwise expressed as "out-of-the-box"."

he 'knight's move' is perpendicular.


and:

think entropy. being able to describe it's 'shape' still doesn't come close to what or why.

just sayin'

A Knight's move is spiral in form as well.

time for u to 'square' the spiral, SS?

We are the Hollow men:

[link to english.pravda.ru]

An interesting article.

The scientist was working on his dissertation under the direction of academician Aleksandrov. "The subject was not hard: "Saddle surfaces in Euclidean geometry." Can you imagine equal-sized and irregularly spaced surfaces in infinity? We have to measure the cavities between them," the mathematician said.

I suppose so.

No worries, it's just a thrown peddle which starts the rippling cascade. And then again you don't even have to throw it.

Just peck on a keyboard and walk away.

Horatio Alger and his bootstrap mechanism just rolled in their graves.

Back to the abyss!