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Biradial matrix: insights into space-time geometry

Russell Kramer, Syndyne

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Biradial matrix: insights into space-time geometry by Russell Kramer Introduction In the previous bi-radial matrix article a space time network was introduced by extending a geometric analogy to gravitation. This forms two sets of equi-spaced radial lines each of these corresponding to a spherical mass of uniform density; this giving rise to a geometric analogy of two interacting gravitational fields. One of the basic properties of this matrix is that it is “invariant under spatial transformation”. The resulting quantized interference pattern yielded a network of intersection nodes. Applying two connection algorithms a prototypical model of magnetic attraction and repulsion lines were formed. Further analysis revealed inverse square relations over distance embedded in the structure. With respect to quantum reconstruction extending to the macroscopic realm we are deriving a quantum geometry which is scalable from the microscopic to the macroscopic realm. To derive this from a set of fundamental principles. This includes the fundamental relation between space and time, i.e. motion and reevaluating the separate definitions of space and time. From these and other basic principles and the many ways these can be combined arrive at descriptions of more complex phenomenon. The basic properties of time is a subject of serious debate in physics. That space in time are closely interrelated in the form of a space time continuum is integral to general relativity. As in the previous article there are three main sections of code. Section 1 is presented here and the other two are at the end of this article. Biradial Manipulate Begin Package BeginPackage["BiRadialMatrix`", {"GeneralUtilities`"}] Needs[# <> "`", FileNameJoin[DirectoryName[$InputFileName], # <> ".wl" ]]& /@ { "BiradialPlot"} Usage Statements SetUsage[BiradialManipulate, "BiradialManipulate[]"] Begin Private Begin["`Private`"] (* Begin Private Context ) Manipulate BiradialManipulate[] := DynamicModule[  pha = 0, phb = 0, phaseSync = False, phaseInverse = False, showLabels = False, mirrorRayNumbering = False , Manipulate[ BiradialPlot`makeBRMPlot[ d, ra, rb, pha, phb, rule, plotRange, "showPointLabels" -> showLabels, "lineThickness" -> thk, "nodeDiam" -> nodeDiam, "imgWth" -> width, "lineColor" -> lineColor, "pointColor" -> pointColor, "mirrorRayNumbering" -> mirrorRayNumbering ], {rule, "rays", "Field Structure"},  "nodes" -> "Nodes", "rays" -> " Rays ", "attract" -> " Attract ", "repel" -> " Repel ", "hyperbola" -> " Hyperbola " , {{d, 4, Style["D", 14]}, 1, 6, 1}, {{ra, 24, Style[Subscript["R", "a"], 14]}, 4, 60, 1}, {{rb, 24, Style[Subscript["R", "b"], 14]}, 4, 60, 1}, Item[Grid[  Style[Subscript["ϕ", "a"], 14], Dynamic[Manipulator[ Dynamic[pha, (pha = #; If[phaseSync, phb = ra / rb If[phaseInverse, -pha, pha]]) &], {N[-π / ra], N[π / ra]} ]] ,  Style[Subscript["ϕ", "b"], 14], Dynamic[Manipulator[ Dynamic[phb, (phb = #; If[phaseSync, pha = rb / ra If[phaseInverse, -phb, phb]]) &], {N[-π / rb], N[π / rb]} ]]  ], Alignment -> Right], Item[Grid[  "Phase Synch", Checkbox[Dynamic[phaseSync]], "Inverse", Checkbox[Dynamic[phaseInverse]] ,  "Show Labels", Checkbox[Dynamic[showLabels]], "Mirror", Checkbox[Dynamic[mirrorRayNumbering]]  ], Alignment -> Center], {{thk, 0.002, "Line Thickness"}, 0.001, 0.01, 0.001}, {{nodeDiam, 0.01, "Node Diameter"}, 0.005, 0.02, 0.001}, {{lineColor, Purple, "Line Color"}, ColorSlider}, {{pointColor, Black, "Point Color"}, ColorSlider}, {{plotRange, 6, "Plot Range"}, 2, 12, 1}, {{width, 400, "Image Size"}, 400, 800, 100} ], SaveDefinitions -> True( , SynchronousInitialization -> False ) ] Package footer End[] ( End Private Context )EndPackage[] Re-evaluating Space-Time The general view of gravitation from Einstein is that a gravitational field is caused by warping of the space-time in the region of a gravitational mass. Are there specific geometric components of the bi-radial matrix that represent time and space? A general definition of space (s) and time (t) is implied in the definition of motion. The simplest form for an equation describing motion as defined for scientific and engineering purposes is: v=s/t. Equation 1 This indicates that space and time are two reciprocal aspects of that motion. If train A travels twice as fast as train B it is entirely equivalent if we say train A it travels twice as far in the same time as train B or that train A travels the same distance in half the time as train B. This is expressed mathematically in equation 2. 2S 1T = 1S T 2 Equation 2 This reciprocal relation between space and time has a numerical-mathematical and geometric expression using equi-spaced rays and angles within the bi-radial matrix. From equation 1 it is apparent that space and time only co-exist as two reciprocal aspects of motion. Without either space or time there can be no motion: translational motion, rotational motion, oscillatory motion, vibrational motion, orbital motion and so on. While space and time only co-exist as two reciprocal aspects of motion they have distinct geometric properties directly relating to the bi-radial matrix and can be separately defined in a manner that is consistent with equations 1 and 2. From here we can show which parts of the bi-radial matrix correspond to time and to space to more precisely define the space time network. The geometry of the matrix is based on the phenomenon of angular displacement by discrete intervals as shown in diagram 1. Diagram 1 Where do we routinely observe and employ angular displacement indicated by discrete intervals? Time! Diagram2ADiagram2B Time has for millennium been revealed and regarded as angular displacement indicated by discrete quantized angular intervals since the advent of sun dials. Sun dials are based on rotational and orbital components of the Earth Sun system. This involves two reference systems. The bi-radial matrix reflects this. Without two reference systems pole A or pole B are only a “quantum time potentials” where the included angles “a” and “b” represent discrete units of time as shown in diagram 1. Time is relational. Upon the presence of a second time frame or potential, pole “B” the “space” between poles A and B, the “D” segment and the spatial sub-divisions along the rays defined by the nodes occurs. This precisely models the scenario where time is fundamental and space is emergent as proclaimed by Lee Smolin in his book “Time Reborn”. This will be investigated further. Table 1 clearly indicates that time is quantized into seconds, minutes, hours, days, weeks, years and so on. Once we accept the literal truth of this property of time and its reciprocal relationship with space a detailed quantum goemetry emerges which is quite revealing. Table1 Every 360 degrees of the Earth’s rotation around it’s axis is tantamount to saying 24 hours has elapsed. Hence every 15 degrees of the earth’s rotation is tantamount to saying 1 hour has elapsed. Every 360 degrees of the Earths orbital displacement around the sun in tantamount to saying one year has elapsed. We can thus arrive at separate definitions of space and time. Conventional physics recognizes two forms of time: absolute time and clock time. We are here referring to clock time. We have for centuries recognized the rotational nature of time in our time keeping practices and can clearly see from table 1 that time is rotational and cyclical. Once we acknowledge this property of time and take it literally a great deal of things are clarified. General definitions of space and Time Time: discrete angular displacement in any direction. The angle "a" in diagram 1 represents a "moment" or "quanta" of time the result of dividing the 360 degrees around pole A by any counting number. The variable "b" represents a moment or quanta of time the result of dividing the 360 degrees around pole B by any counting number. Note a “quanta” of time is a finite unitized moment of time. A "quantum" of time would be the shortest possible duration of time whose value to my knowledge has yet to be determined. Space: discrete linear displacement in any direction.* Both space and time are forms of displacement, one linear, the other angular. Space is progressive, time is cyclical. These definitions are corollaries of two fundamental postulates forming the basis of a comprehensive physical theory. This explains many important phenomenon. Consider that a wave can generally be defined as the synthesis of linear and rotational displacement. This will be examined further in this series of articles. To further the analysis of space and time, clarify the definition of the space-time network and the field structures introduced in part 1 a bi-radial coordinate system is introduced. Bi-radial coordinate system Diagram3A One of the basic principles relating to the bi-radial coordinate system is that coordinate systems occur in nature and that not all coordinate systems are man made conveniences. Buckminster Fuller known as the inventor of the geodesic dome in his seminal work on geometry “Synergetics” identified a coordinate system in nature called the isotropic vector matrix based on the closest packing of spheres. We assert that nature employs multiple coordinate systems and applying the appropriate system for a given phenomenon is more revealing than applying an arbitrarily constructed system. Each pole represents a separate time frame numbered in opposite directions per their opposing orientation corresponding to opposite time flow. Each number and corresponding ray marks the end of one discrete moment of time and the beginning of another like on clock faces. There is the primary space separating poles "A" and "B", the D segment and the spatial subdivisions along the rays defined by the nodes. Notice that the time intervals are unitized and the spatial intervals are non-unitized; integer vs real. This is a result of the reciprocal relation between space and time. This same reciprocal relation where space and time are inversely related gives rise to another geometric matrix where the spatial intervals are quantized and the time intervals are discrete and non-quantized. This is covered in future articles. With respect to the bi-Radial coordinate system it is far more revealing to use a coordinate system who’s symmetry is similar to the phenomenon being analysed, in this case bi-polar force fields. As referenced above the bi-radial matrix is neither man made or arbitrary. The numbering of the rays is implied in the symmetry of the matrix. In diagram 3A as the poles A and B are opposite each other the rays are numbered in opposite directions. The D segment being the base line has the implied value of zero. Also there are two opposite hemispheres in the bi-radial matrix-the upper and lower hemisphere. Hence the rays in the lower hemisphere below the D segment are numbered in the opposite direction then the rays in the upper hemisphere. From this numbering of the rays in diagram 3A the resulting set of bi-radial coordinates are assigned to each node. In this case the number of rays from each pole is 24. Bear in mind when describing large scale masses the number of rays can be enormous-exceeding the screen resolution and our our optical resolution. The bi-radial equivalency principle allows smaller ray counts to be used and is covered elsewhere. From diagram 3B we can see the quantized angles represent quantized units of time and the line segments along the rays represent discreet non-quantized spatial units. This clarifies the components of the space time network. Also we can examine a specific line segment such as {3,5},{4,5}. From the reference system of pole A this segment traverses a single quantized unit of time from ray 3 to ray 4 while moving slightly further from pole A. From the reference frame of pole B this same line segment traverse a unit of space along ray 5. Hence in the bi-radial matrix the very same phenomenon has different properties depending on the reference system. In any case we define the coordinates {X,Y} where X corresponds to the ray number from pole A and Y corresponds to the ray number from pole B. Each of the coordinates {X,Y} are thus time coordinates. Diagram3B Diagram 3C Diagram 3C isolates the nodes with their corresponding bi-radial coordinates. Note sometimes it is convenient to number the pie sections as they represent the quantized moments of time them selves as opposed to the lines separating them. In this case as shown in diagram 3D the trapezoids become the “coordinates” in the bi-radial matrix. This shows the angular sections numbered and a random sample of the trapezoids with their resulting coordinate pairings. For the present we will use the numbering convention shown in diagram 3A. Diagram 3DAnalyses of Attraction and Repulsion Force Field Structures We can further analyze both attraction and repulsion lines using the bi-radial coordinate system. Diagram 4Diagram 4 shows the associated attraction lines for this bi-radial matrix along with the bi-radial coordinates. From diagram 4 we can observe that the sum of the coordinates along any given attraction line equals a constant. Take for example the 8th attraction line has the coordinates ({1,8}, {2,7}, {3,6},{4,5}, {5,4},{6,3),{7,2) and {8,1). The sums of all these coordinate pairs equal 9. The formula for any given attraction line is thus:X+Y=KEquation 3As will be shown later the set of nodes along any given attraction line fall along the paths of non-concentric circles. The centers of these circles can be calculated and this will be shown in future articles relating the the bi-radial scaling mechanism. Diagram 5 We continue the analysis with the repulsion lines. Diagram 5 shows the repulsion lines with the bi-radial coordinates. We can observe that the difference of the coordinates on any repulsion line equals a constant. Take for example the first repulsion line in the upper right quadrant the coordinates are: {5,4}, {4,3}, {3,2},{2,1}. The formula for any given repulsion line is thus: X-Y=K Equation 4 Among other things we see that the phenomenon of attraction lines in the bi-radial matrix is associated with the operator of addition and the phenomenon of repulsion lines is associated with the operator of subtraction. Also note that while the phenomenon of attraction is considered the “opposite” of repulsion the geometric orientation of the repulsion lines is not diametrically opposed but rather oriented at 90° with respect to the D segment. It will be shown there are other hidden field structures within the bi-radial matrix associated with the operators of division and multiplication. In part one of this series previously published regarding the initial connection algorithms the first one is to find the paths of “shortest distance” from pole A and B passing through the unoccupied nodes. Space is considered an electrical insulator. Since the attraction lines are traversing the shortest possible distance from pole A to pole B through space they are thus traveling the path of least resistance. This corresponds to electricity. The aggregate of this attraction lines correspo0nds to magnetism. The second connection algorithm relates to the paths starting from poles A and B passing within the closest proximity to the vertical axis “V”. Once we recognize time as angular displacement we can describe the various field structures in terms of space and time. The bi-radial coordinate system applies to the gravitational equilibrium zone. We can further the analyses of the gravitational equilibrium zone using the bi-radial coordinate system. Analysis of Gravitational Equilibrium Zone.In diagram 6A we have two equal masses represented by poles A and B each with 36 equi-spaced rays. With two equal masses we would expect the gravitational equilibrium zone to be half way between pole A and B along the D segmentwhere the ratios of the segments d1 and d2 are in the same ration as A and B which is shown in equation 5A. X Y = d 1 d 2 Equation 5That portion of the gravitational equilibrium zone not on the D segment we would expect to be occurring along a vertical axis passing through the mid-point of the D segment. Is there a set of nodes which falls along this path? It turns out that bi-radial coordinates which fall on the this vertical axis V are {1,1}, {2,2}, {3,3}, {4,4}... and the formula for this axis is:X=YEquation 6AIn diagram 6B we represent two unequal masses where Mass A is twice that of mass B. Here the number of rays from pole A is 72 and number of rays from pole B is thus 36. We would expect the gravitational equilibrium zone along the D axis would thus occur at a location where the resulting segments d1 and d2 would be in the same ratio as A and B. That portion of the gravitational equilibrium zone not on the D segment would be biased toward Mass B. Is there a set of nodes which fall along this path? Indeed the set of nodes which fall along this path are: {1,1},{2,2},{3,3},4,4}... and the formula for this curve is :X=YEquation 6BWe thus have the same equation representing two different plots!. This shows that the equation X=Y in the bi-radial coordinate system corresponds to the gravitational equilibrium zone in any bi-radial matrix. To clarify the distinction between equations 6A and 6B we will establish a convention where the number of rays from each pole is specified. Thus equation 6A would be:X=Y36,36Equation 7AEquation 6B would thus appear as:X=Y72,36Equation 7BWhile this system may take getting used to it does appear to precisely correspond to a variety of physical phenomena. With respect to the phenomenon of gravitational and electromagnetic force fields this does appear to be the implied underlying structure. I had visited Mamikon in California while he was at Cal Tech working with the late Tom Apostle as part of Project Mathematics. Mamikon is a trained astrophysicist and I had shown him versions of diagrams 6A and 6B. I asked him if he believed these plots relate to the gravitational equilibrium zone and he had agreed. This was a very informal discussion and many details were not addressed. This relates to a very complex gravitational phenomenon known as the Lagrangian points. There are still many parameters in the bi-radial matrix which need further development to apply it to this area of research which is a personal interest of mine. “...But why does matter have to be introduced explicitly at all? It has been the assumption of traditional physics that even though gravity can be represented in terms of properties of space, other elements of the universe cannot. But in my approach everything just emerges from the same underlying network-or in effect from the structure of space...”NKS page 536-537 Diagram 6A Diagram 6B It is revealing to consider diagrams 6A and 6B in light of Lee Smolin’s description of Space”: “A useful metaphor arising in several approaches to quantum gravity is to imagine that space is not continuous but a lattice of discrete points...”. The bi-radial matrix lends itsewlf to a variety of physical interpretations. This includes electromagnetic and gravitational fields. Could then the nodes represent gravitons or “magnetons”, i.e. particles of magnetic force? There are several other relationships within the bi-radial matrix to be covered in future articles. Summary A geometric definition of time based on established astronomical data and the relationship between space and time are further defined with respect to the fundamental equation of motion V=S/T. From here time and space were shown to have specific geometric components within the bi-radial matrix clarifying the resulting space time network therein. By introducing the bi-radial coordinate system the magnetic attraction and repulsion fields and gravitational fields were further analysed. Fundamental equations were derived to further the analysis. The concept of a quantum time potential was introduced as part of a quantum geometry applied to gravitation and electromagnetism. Further properties including the bi-radial equivalency principle and bi-radial scaling mechanism will be covered in future articles relating gravitation to the microscopic scale. Remaining Sections of program Code Biradial Plot Functions Begin Package BeginPackage["BiradialPlot`", {"GeneralUtilities`"}] Needs[# <> "`", FileNameJoin[DirectoryName[$InputFileName], # <> ".wl" ]]& /@  "VectorGeometry" Usage Statements SetUsage[makeBRMPlot, "makeBRMPlot[d$, ra$, rb$, pa$, pb$, rule$, plotRange$]"] Begin Private Begin["`Private`"] (* Begin Private Context ) makeBRMPlot ( Combined function to generate final plot from set of input parameters )Options[finalizePointLabels] = {"mirrorRayNumbering" -> False}Options[makeBRMPlot] = Join[ Options[finalizePointLabels],  "showPointLabels" -> False, "lineThickness" -> 0.001, "nodeDiam" -> 0.01, "imgWth" -> Automatic, "lineColor" -> Purple, "pointColor" -> Black ] makeBRMPlot[ d_, ra_, rb_, pa_, pb_, rule_, plotRange_, opts:OptionsPattern[]] := Module[  showPointLabels, lineThickness, nodeDiam, imgWth, lineColor, pointColor, pole1, pole2, leftlines, rightlines, leftlines2, rightlines2, posIntPnts, negIntPnts, intPnts, negIntPntsAdj, posAttractPnts, negAttractPnts, attractPnts , ( Retrieve Options )  showPointLabels, lineThickness, nodeDiam, imgWth, lineColor, pointColor  =  OptionValue["showPointLabels"], OptionValue["lineThickness"], OptionValue["nodeDiam"], OptionValue["imgWth"], OptionValue["lineColor"], OptionValue["pointColor"] ; ( Pole positions ) pole1 = {-d / 2, 0}; pole2 = {d / 2, 0}; ( Generate rays ) leftlines = generateRadialLinesLeft[pole1, ra, pa]; rightlines = generateRadialLinesRight[pole2, rb, pb]; ( Split rays point upward from rays pointing downward ) leftlines2 = GroupBy[leftlines, #[[2, 2]] > 0&]; rightlines2 = GroupBy[rightlines, #[[2, 2]] > 0&]; ( Ray intersection points ) posIntPnts = intersectRaySets[leftlines2[True], rightlines2[True]]; negIntPnts = intersectRaySets[leftlines2[False], rightlines2[False] ]; intPnts = Join[posIntPnts, negIntPnts]; negIntPntsAdj = KeyMap[ rb - #[[1]] + If[pb == 0, 0, 1], ra - #[[2]] + If[pa == 0, 0, 1] &, negIntPnts]; ( Apply the Rule ) If[rule != "rays", posAttractPnts = makePoints[posIntPnts, rule]; negAttractPnts = makePoints[If[rule != "hyperbola", negIntPnts, negIntPntsAdj ], rule]; attractPnts = addPoles[posAttractPnts, negAttractPnts, pole1, pole2 , rule]; ]; ( Display the plot *) Show[ Which[ rule == "nodes", makeNodePlot[intPnts, d, plotRange, nodeDiam, imgWth, pointColor ] , rule == "rays", makeBiradialPlot[leftlines, rightlines, intPnts, d, plotRange, lineThickness , nodeDiam, imgWth, lineColor, pointColor] , True, linesPlot[intPnts, attractPnts, d, plotRange, lineThickne