Differential Geometry without continuity or derivatives

A great deal can be accomplished using chi squared comparison to uniform distributions when we create distributions by projecting all data points, lying within a sphere, perpendicularly onto a plane through the origin of the sphere. We know how many points to expect in each cell if the points are uniformly distributed inside the sphere so

xhiS= sum { (measured - expected )^2/expected }

(over all cells)

measures the deviation from uniformity. It varies as we vary the projection direction. The direction with minimum xhiS serves as a surface normal. The direction of maximum xhiS is that of minimal curvature. Interesting eh?

I read that physicists and mathematicians are pursuing efforts to define space as foamy. It seems to me that the diverse possibilities of this xhiS metric (including the relation to the cell size, including the frangible concepts of "expected" within a sphere) make xhiS a good candidate for handling physical diversities. There could be multiple tangent planes, multiple curvatures, and a spectrum of ideas around the fact that minima and maxim of xhiS can be local rather than global. For example space around a point could be defined as "2D" if the tangent and minimum curvature are global extremes and there are no local extremes to confuse the issue.
Update:
However, space around a subatomic particle or around an atom might have multiple directions that were local minima/maxima. As such the dimension is in question. 
Update 2: Probably this all is nonsense. You would need to already be "in" 3D to measure the proposed dimension.  
Update 3: No, you could be in 3D, making direction dependent measurements there, and still uncover a higher dimension entity within your data.

An actual sphinxmoth - the Pandorus (Grape) Sphinxmoth

Dimensionality of the universe

I cannot imagine why this is a question. There is no physical evidence of three spatial dimensions - no matter how many rulers and clocks you choose to hold in my face as evidence. That would be mistaking your behavior for a physical reality. On the other hand, actions do have dimensions, so there is nothing lost in spite of the criticism.

Update: In terms of the dimension defined by xhiS (in a later post), it is meaningful to ask about things like the dimension of the space around an atom, without having a built-in 3D assumption.

Red shift could be caused by rotation rather than expansion

Anything that causes the shortest path between entities to lengthen will case a red shift. Expansion, where the entities are accelerating away from each other is only one way to lengthen such paths. Another way is for the entities to be on spinning a merry-go-round.

The definition of information, the definition of "life"

This is philosophy of universe week. So to start the ball rolling:

I think a theoretical physics that fails to subsume theoretical biology must be incomplete. The idea that somehow causality is suspended for that brief moment where "life is created" is just not scientific. Nor, for that matter, would a Deus Ex Machina make any more sense for the big bang "universe is created" moment. One thing at a time though.

Grant me the concept of a clock and here is a definition of information: a sequence of increases and decreases of frequency.

Grant me that definition and here is a definition of life: the preservation and perpetuation of information.

So here is that theological moment: the previous two posts on this blog outline a theoretical physics based on information such that the universe is populated entirely with objects interacting based on the information they maintain. According to this post here, that means life never starts, it is simply the stuff of the universe at different levels of complexity. For the benefit of a colleague at work who was using the word "consciousness", let's agree that everything is conscious of itself and its interactions and that collective conscious is possible.

Properties of a universal simulator

This is a long-term discussion with my eldest son about creating C++ objects that simulate or model the behavior of real world objects. The base class is required to have certain properties:

• It can "contain" other simulation objects
  
• It exposes one or more sequences of frequencies
• It maintains a clock with an update() method called once per tick, and sent to each of its contained simulations.
• the contained simulations must run their clocks in synch with the containing object's clock

A goal is to be able to populate a program with objects by declaring/initializing them, and then have the objects freely interact. But what is a "freely interact" supposed to be like when the classes do not know about each other and the programmer (me or you) does not have access to the header file definitions of the classes in use.

So what is a universal language of interaction? The only answer I can come up with is that when exposed frequencies "match" (in the clock of a containing parent class) energy can be exchanged and some kind of communication can occur between interactors. According to yesterday's post, such a "red-shift event"/interaction would need to generate a new class instance (a new "particle") with the frequency difference of the interactors - spun off as an offspring of the harmony.

I guess I don't want you to think I am another crazed mathematician. But I believe there is almost enough here to start creating prototype simulation class definitions, working programs, and some kind of approach to theoretical physics.

A physics of information and red shift

This is not a new topic and I know other people are trying to base a theoretical physics on the notion of information rather than the (tired) ideas of "mass" and "energy". My son and I have been going over it for years. Today I was talking with a colleague at work about how there is still something missing if all you have is information. There is something else needed for the theory. Which reminded me of a different conversation I have been having with my son about resonant frequencies between pairs of particles and how they can exchange information/energy if their frequencies are nearly matched. So a particle with frequency .99 and one of 1.0 could resonate but (according to my son) a 0.01 particle would have to be produced. As the first two particles resonate and experience harmony, they produce an offspring particle. The process is a red-shift event.

This red-shift event defines a spatial component between the two particles and I want to say it should be regarded as happening with a certain probability. One last crazy statement is that probably there still needs to be an observed/observer relation in their to cause probabilities to materialize into facts.

[Another discussion with my son that has been going on for years is about the class definition of a universal simulator.]