Time and Structure

The role of the Inverse Square Law

~Reginald Brooks

©2016, Reginald Brooks, Brooks Design. All rights reserved.

When we look at competing ideas on TIME IN COSMOLOGY, as wonderfully presented at the recent Perimeter Institute for Theoretical Physics conference (organized by: Marina Cortes, Institute for Astronomy;
Lee Smolin, Perimeter Institute; and,
Roberto Mangabeira Unger, Harvard Law School), one is struck by the inevitable conclusions drawn by the elegant Casual Set Theory of Rafael Sorkin (ref: https://www.perimeterinstitute.ca/people/rafael-sorkin):

"order + number = geometry."

And why is that?  
What does it mean to say that the progression of inter-connected number relations forms a geometry --- a geometry of spacetime (ST)?

Take away the overlay of advanced mathematics and look simply at whole integer numbers (WINs) and their relationships to each other. What do you have? Are not numbers simply quantities? So much of this or that? Is not geometry the representation — both in the abstract and physical manifestation state — of those number relationships and an expression of their inter-connectedness? Yes, of course!

But let us step back a moment and frame the concept of time within the cosmology of the universe. What tenants must be satisfied?

  1. Conservation of Energy
  2. Constancy of the Velocity of Light
  3. The ubiquity of the Inverse Square Law (ISL) as the template upon which #1 & #2 depend.

This leads to two additional tenants that are really the inevitable conclusions of combining #1-#3 above:

*If all energy (mater and fields) is conserved;

*If all energy expression (energy, mass, gravity, EM,…) follows the ISL;

*If all energy is an expression of its Space-Time (ST);

*If all energy is expressed in discrete quanta:

Then ALL energy = ALL Space-Time = ALL discrete quanta,
and, follows the ISL in its ultimate dispersion and presentation.

It also follows that if the speed of light is a constant:

That change – a difference — in energy ST quanta is a natural, albeit inevitable, consequence: that the cosmos is in flux (not exactly news!).

But here is the key: the constancy of the speed of light (c), its dependency on its space (wavelength, λ), and time (frequency, ν), means its own ST must not only be in flux – depending on how much energy is onboard – but that the basis for that flux is a consequence of its discrete-quantum nature.
The ST is a pulse-propagated quantum expression of the onboard energy. Both the space the time are discrete quanta – but inter-connected. The greater the energy, the smaller the wavelength, the higher the frequency º all while maintaining a constancy in c.

We can say lightspeed ST = Space-Time = ST.

All ST pulse-propagates into (and out of) existence at c, and c is always constant, because ST is formed of discrete c = λ ν units.

OK, so what does that have to do directly with the ISL and this notion that "order + number = geometry?"

The ISL simply states a numerical relationship – algebraically – for a geometric expression, of how an influence (energy, force,…) becomes diluted as its distance from its source increases.

The ISL is the Universal-Cosmological template upon which the Conservation of Energy = the Conservation of Space-Time is based.

Care to explain?

The ISL can be given a numerical presentation by means of the Brooks (Base) Square Matrix (BBS-ISL Matrix). This simple, but highly inter-connected and ordered whole integer number (WIN) based matrix grid is, like the ISL relationship, infinitely expandable. The BBS-ISL Matrixis a symmetrical, square grid composed of two right-angle isosceles triangles divided by a central diagonal (Prime Diagonal, PD) – the PD being simply the square of the respective Axis numbers (1,2,3,…).

In the Inner Grid, every cell is occupied uniquely by a whole integer number (WIN) that is derived from the PD (TIP: Subtract the two PD numbers to give the Inner Grid cell number). Every Row, Column, and Diagonal within this Inner Grid follows a very simple – but strict – progression pattern out to infinity (TIP: See how the diagonals parallel to the PD have their number sequence increase by +2, +4, +6,…, as one moves away from the PD). The BBS-ISL Matrix is the number-geometry of the ISL. The ISL directive is given order by means of the matrix.

Now here is a crucial point: because of the patterning sequence, every numerical value of every cell from the start at the origin out to infinity is instantaneously known! The “information” of the numbers – the values defining the relationships and inter-connectedness – is instantaneously, and ubiquitously, known for every ST unit quanta born! And it does NOT rely on the velocity of light, c, to get there!

Manifest destiny: when the light-based ST quanta actually get there – one c-step ST pulse at a time – instead.

The birth of ST quanta – forming and stretching to fulfill its c = λ ν units at lightspeed – does so within the cradling template of the ISL (BBS-ISL Matrix). All information – past, present and future – about any path and/or degree of energy expression will be innately onboard. As it pulse-propagates into and out of existence – out past its own local Planck Event Horizon (PEH), as a virtual-like quanta, and back below its PEH to, and through, its own Singularity (S), and into its alternative pool below, and back out again – it does so through its ISL template: “order + number = geometry.”

If this accounts for the "time" in ST, what about the "space?" As ST quanta pulse-propagate through their master ISL template of simple whole numbers (WINs), how can they possibly account for the richness and diversity of the cosmos?

Of course, there are multiple ST quanta interacting. But, built right in the BBS-ISL Matrix are the Pythagorean Triples (PTs). The PTs are so pervasive, so inter-connecting and so thoroughly embedded throughout the matrix that one could nearly start with just the PTs and generate the matrix from there. The connection between the Pythagorean Theorem and the Inverse Square Law – referred to as TPISC – is profound! Here we are talking literally about that connection, not the papers, videos, slideshows, iBooks, etc., that describe it.

Every possible Primitive (parent PPT) and its non-Primitive (child-multiple, nPPT) – and their proofs – are uniquely located within the matrix. (TIP: To find the PTs, simply drop down vertically from any PD value (>4) until it intersects with a squared number on a Row below. That Row contains a PT.)

The PTs, as whole integer number (WIN), non-isosceles Pythagorean right-triangles, introduce key articulation nodes of geometric expression uniquely, but also ubiquitously – throughout the expanding matrix. Effectively, they germinate some asymmetry into the unfolding scaffolding geometric mix.

PTs are non-isosceles, right-triangles and when paired up form non-square rectangles and the non-circular geometries of the oval. To this mix of strictly symmetrical and slightly asymmetrical (or, bilaterally symmetrical) structure, you add the irrationals: Π, √2, and φ(for starters), and here we are! Space has been enriched because ST has been enriched!

ISL (order) + number = discrete geometry = quantum ST = quantum gravity.

Yes, the difference between the past, present and future is an illusion because the flow of time – as ST forming – depends solely on your frame of reference – as pointed out by Sorkin’s article.

So what's next?

Once you realize that ST is discrete, the notion (reality) that that discreteness applies to all energy (thank you Max Planck) and that the velocity of light is constant (thank you Albert Einstein), it becomes apparent that the Planck-Einstein relation holds because lightspeed itself is composed of discrete pulse-propagating, co-dependent space (λ) and time (ν) units acting together as single, discontinuous ST units. These discontinuous ST units are united with the order of the ISL to form an apparent continuum of ST that we, from our humble distance, see and/or measure.

It is but of a small leap to see that the ordering of the ISL, combined with WINs, gives us the geometry of the cosmos, and, because of the Conservation Laws, that there must be a conservation of ST as well, as:

Conservation of Energy = Conservation of lightspeed = Conservation of ST.

Dark what?

And this, I believe, is where the solution to the Dark Matter-Dark Energy problem lies. Dark Matter (+) = Dark Matter (-), inverse of.

Because “order + number = geometry” goes to infinity, so does the information of the ISL contained in the BBS-ISL Matrix.

Every ST unit pulse can, and will, expand out infinitely to the limits of its c = λ ν relationship. At their inflated state, their ST environment so created, is one of highly dilute and widely separated ST, effectively giving a negative gravitational curvature (repulsiveness) of Dark Energy.

Likewise, but on the opposite polemic – and as both are a consequence of the Conservation of SpaceTime – ST is more tightly curved and gravitationally attractive (+ Dark Matter) beyond the measurable mass-energy present, precisely because of the enhanced resonance (interference) of such concentrated ST quanta – both in quantity and intensity. The higher-density ST interference resonances around the masses we see and measure are countered by their opposites the further out the cosmos goes. A changing gravitational constant at hand. We are bobbing comfortably in a somewhat gray surround!

If we could measure ST itself, that lens would show:

ISL order + WIN = geometry = quantum ST = (DM=DE) = quantum gravity = quantum energy

BBS-ISL = Geometry = Quantum Universe


The BBS-ISL Matrix provides the order to the numbers that give rise to the discrete geometry that forms the quanta of Space-Time. Two parallel, but independent, tracks have merged: LightsppedST-MathspeedST and the Causal Set Theory of R. Sorkin (with contributions by F.Dowker, J.Henson, S.Surya, D.Craig, L.Philpott, M.Buck, I.Jubb, and others).

Discrete, discontinuous geometry pulse-propagates into and out of “our” virtual reality, constantly refreshing itself from the “alternative view” via the continuous pool of black hole singularities.

At the moment of birth – post-PEH –each ST quanta pulse is at the origin of its BBS-ISL Matrix – at zero – with zero entropy.

Yet its entire future, as its pulse expands, is numerically ordered and known. The information of the entire matrix is instantaneously and infinitely known. As the pulse expands, both spatially and temporally to fulfill c = λ ν, its lightspeed ST generated follows the strict ordering of the ISL via the BBS-ISL Matrix.

Numbers – as quantities – unfold and grow through ordered, patterned and interconnected sequences (arrays). As Sorkin has hypothesized in “causet” theory, “time” – or what we think we know of “time” – is this process of ordered numerical progression. Running out the matrix, the arrow of time is given direction. Its entropy is increased, at least in the sense that the density of order has decreased.

From the origin of the matrix at zero, with minimal options – options being the number of possible numerical connections – out to infinity, where the number of options – including those re-connecting to its past, laterally to its present sides, and forward to its infinite future – has greatly increased.

The entropy of each ST quanta pulse increases up to the limits of its c = λ ν expansion, collapses back to zero, refreshes and pulses again, and again.

The entropy of 10240, or make that 10^∞ , ST quanta pulse-propagating and resonantly interfering with each other gives rise to the History of the Universe as we know it and the History of the Cosmos as we understand it. Order counts!

      ~~~~~~~~~~Our View continuum~~~~~~~~~~  
  Conservation of ST = (Geometry=3) = Entropy =3  
  Conservation of ST = (Geometry=2) = Entropy =2  
  Conservation of ST = (Geometry=1) = Entropy =1  
  Conservation of ST = (Geometry=0) = Entropy =0  
     ⬆︎              Planck Event Horizon        ⬇︎  


     ⬇︎             Planck Event Horizon         ⬆︎  
  Conservation of ST = (Geometry= 0) = Entropy = 0  
  Conservation of ST = (Geometry=-1) = Entropy =-1  
  Conservation of ST = (Geometry=-2) = Entropy =-2  
  Conservation of ST = (Geometry=-3) = Entropy =-3  
       ~~~~~~Alternative View continuum~~~~~~~

It appears that a bounce is eventually coming!

SIDE NOTE: Imagine--->  
    Picture a tub filled with water. Disregard everything but the water -- it is the singularity pool.

    The water is all the stuff of the cosmos -- all the energy reservoir.  
    It is NOT discrete, but rather continuous.

    Over the top of the water surface you slap down a sieve full of several or  more holes of differing sizes.

    Upon slapping the water surface, water jumps through the holes into the air above the sieve.

    You  have a lot of variables here, but if you time it right and with just enough slapping force, you can repeatedly generate somewhat of the same airborne droplets -- water pulses -- with each slap.

    Change the hole size, number or placement, or change the slapping force, depth, frequency, etc.,and you alter the "virtual reality" of the pulse-propagated water droplets above.

    Multiple sieves, with multiple parameters applied with multiple matrices, will increase the options, the entropy, the presentation we see in our Universe -- the virtual quantum Universe.

    What remains the same throughout: the total amount of stuff -- water -- or energy manifestation, contained within the tub, above and below the sieve.

    The sieve is the Planck Event Horizon and the holes, or pores, are the shape-defining templates of both the rational and irrational numbers. 

    Generally, the smaller the hole (up to a critical size) the smaller and higher the droplet pulses, and vice versa.

    The BBS-ISL Matrix would extend from its origin just out from the PEH.

    The "alternative view" would be a sieve slapping the water reservoir from the bottom, with the pulsing water droplets going down, instead of up. (We are ignoring much of gravity and all the other forces to get our visualization of the formation of discrete ST quanta from a non-discrete--or indiscrete, if you prefer--black body pool of singularities.

    Ordered numbers give rise to discrete ST quanta = geometry (a timed event!)


Rafael Sorkin, “Geometry from order: causal sets” in: Einstein Online Vol. 02 (2006), 1007.

The ideas presented here, while certainly resonating strongly with those of Rafael Sorkin and his 2006 article on “causal sets” – a theory he developed in 1980 – are primarily based on my own work building on and off the “Standard Model” and its many, many creators and architects. A complete list of works are found here. Highlights include: