WHAT ARE THE ATTRIBUTES OF GOD?

The attributes of God can be characterized by a finite list of words and phrases so long as we agree that when a word or phrase is applied to God, the true depth of meaning cannot be fathomed. Here are a few attributes arbitrarily grouped in seven categories. Feel free to add or rearrange.

• Being: Transcendence, Immanence, Infinitude, Eternalness, Immutability
• Wisdom: Exhaustively Definite Foreknowledge, Omniscience, Resourcefulness, Patience, Perseverance, Persistence, Discipline, Prudence, Understanding, Discernment, Stewardship, Courage
• Power: Omnipotence, Humility, Meekness, Gentleness, Obedience/yieldedness, Endurance, Long-suffering/Forbearance, Respect, Responsibility, Dependence, Devotion, Surrender, Commitment
• Holiness: The state of who You are and the acts that You do; Holiness in being and Holiness in action; Absolute purity of heart, will, intellect and action; Perfect love, integrity, righteousness, morality, ethics, character; Perfect being, wisdom, power, justice, goodness, truth
• Justice: Punishment for Sinners; Substitutionary Atonement; Redemption; Forgiveness; Justification; Adoption; Regeneration; Sanctification; Reconciliation; Unification; Glorification
• Goodness: Love, Mercy, Grace, Compassion, Kindness, Benevolence, Charity, Faithfulness, Trustworthiness, Peace, Joy, Fulfillment, Fruitfulness, Faith, Hope, Reconciliation, Healing, Encouragement, Comfort, Strength
• Truth: Logic, Honesty

The infinitude of God is a particularly difficult concept for the human mind. What does it mean to say that God is infinite? How can God be transcendent and immanent at the same time? How many bits of information can be stored by God? Would the capacity of God be strained by comprehending

• The past, present and future status of all space, time, matter and energy in our universe?
• The past, present and future thoughts (intellect, heart and will) of all humans and angels?

Or would these things be no more difficult for God than counting the wheels on a bicycle would be for you?

Consider the meaning of infinity. The smallest infinity is called aleph-null; its cardinality (n) is the number of elements in the set of all positive integers. This infinity is, by definition, "countable." After aleph-null, each element of an infinite set is a subset of elements from its predecessor. The cardinality (number of elements), for each successive infinity, is the number of subsets in the set of all possible subsets constructed from its predecessor. For example, the next infinity after aleph-null is aleph-1 or the continuum; its cardinality ( n' = 2ⁿ ) is the number of subsets in the set of all possible subsets constructed from the positive integers. This happens to be the same as the number of elements in the set of all real numbers. Aleph-1 is not countable. The next infinity is aleph-2; its cardinality ( n'' = 2^n' ) is the number of subsets in the set of all possible subsets constructed from the real numbers. Consider the possibility that God can easily process at least n'''''''^.... bits of information where the number of ' marks is n. What sort of concepts might such a God view as trivial while they confound the most brilliant human minds? Perhaps He could determine the precise array of cosmological parameters that would permit human life to exist in our universe. We exist because:

• For a proton pair, the ratio of electrical to gravitational forces is precisely 10³⁶.
• The fraction of mass converted to energy by the nuclear fusion of hydrogen to helium in a star is precisely 0.007.
• The ratio of gravitational energy to expansion energy was precisely 1.0 in the early universe.
• The ratio of dispersion energy to rest mass energy for a galaxy of stars is precisely 10^-5.
• On a human scale, the number of spatial dimensions is precisely 3.

Significant deviations would preclude our existence. Other concepts which we view as mysterious would likewise be trivially non-mysterious to the mind of an infinite God.

• The location of the seat of consciousness in the human brain is unresolved.
• Holiness cannot be defined separately from God.
• The Golden Ratio and Fibonacci sequence are pervasive throughout creation.

Perhaps all these things are so trivial to God that they occupy an infinitesimally small portion of his thoughts. More consuming issues might be the need for our confession of sin, remorse repentance, faith and obedience.

But wouldn't the sheer size of the known universe -- 100 billion stars in each of 100 billion galaxies -- overwhelm even the most infinite God? To put in perspective the overwhelming sovereignty of God's infinitude, consider the following illustration. Imagine the universe is constructed around a deformable cubic lattice such that the smallest identifiable physical entity is a region called a "voxel" which is bounded by eight vertices, twelve edges and six flat faces. The universe is thereby divided into an ordered (numbered) array of volumes. The volume enclosed by a voxel is on the order of the cube of the local Planck Length (e.g. 1.6 x10^-35 meter). When a voxel is deformed by force, matter or energy, the eight vertices retain their identities but may change their positions, the twelve edges remain straight but each may change its length and each of the six flat faces either remains flat or deforms into a minimal surface with zero mean curvature (e.g. soap film). 13.7 billion years after the beginning, the deformable cubic lattice is no longer perfect but contains line and surface defects analogous to dislocations and stacking faults.

Assume the state of each voxel is 1 or 0. Then the state of the entire universe is defined by a countably infinite sequence of binary digits or, in other words, a single real number. The possible 3-D geometric arrangements of 1's and 0's would be more than sufficient to accommodate the complexity of any "Theory of Everything" that might be devised including those based on Lie Group E8 symmetry.

Now imagine God allows the state of each voxel to change at time intervals equal to the Planck Time (5.4 x 10^-44 seconds). At the end of each time interval, a new real number defines the state of the universe. Nestled among the elements of aleph-3 are particular subsets of aleph-2 that track all possible histories of the universe from the big bang onward. An immanent God would only have to comprehend infinity at a cardinality of n''' to avoid being overwhelmed by this universe. The Christian God understands infinity at least to a cardinality of n'''''''^.... where the number of ' marks is n.

As an interesting aside, if God decided to increase the velocity of light (c) by a factor of say 100, the length of the basic time interval would be smaller by a factor of 100,000. To a transcendent observer, the universe would be changing 100,000 times faster but no entity inside the universe would sense a difference because the other Planck Units below would have changed accordingly.

God would essentially have the ability to "fast forward" and "slow motion" the universe without leaving any evidence detectible by us.

This hypothetical exercise may seem far-fetched given the equations of Quantum Mechanics and General Relativity which treat space and time as continuums. But voxels are merely the three dimensional cellular automata defined by Stephen Wolfram in "A New Kind of Science." We only need a "rule of transformation" for defining what takes place at each time step. Concepts of "state" and "rule of time transformation," more complex than those governing the 256 elementary cellular automata, could be considered. Some concepts might be inspired by certain aspects of Judeo-Christian tradition. For example, one tradition is that the 304,805 character string of concatenated words in the Torah encodes fundamental information about the beginning of the universe and its development over time as well as events in human history.

The remainder of this section contains challenging, theologically connected speculation but no Christian doctrine.

GENETIC CODE FOR THE UNIVERSE?

Perhaps relatively short strings of Hebrew characters from the first chapter of Genesis provide a shorthand for codons undergirding all space/time and matter/energy. Various arrangements of these codons could provide the genetic microstructure and macro-characteristics of magnetism, electricity, normal matter, normal energy, dark matter, dark energy, electromagnetic force, gravitational force, strong force and weak force. However, these codons do not reveal their secrets when read merely as Hebrew character sequences. They must be viewed as mathematical entities.

First invert the character sequence, in the first chapter of Genesis, so the characters are read from left to right. Next connect the first three contiguous characters in the first verse of Genesis to form a string. Now represent each of the Hebrew characters by a base-3 triplet (column vector) according to a rule that starts with Aleph as (000) and ends with Tsadey Final as (222). The three column vectors are (001,201,000). These three base-3 triplets create a 3 by 3 matrix having the elements 0, 1 and 2. Such groups of three Hebrew characters and the associated 3 by 3 matrix could represent fundamental codons for the structure of the universe.

Increasing complexity is created if groups of three contiguous characters are connected to one another. For example, four groups of three contiguous characters could be arranged like tiles on a floor to produce a 6 by 6 matrix. Continuing in this fashion, the individual 3 by 3 matrices from 1, 4, 9, 16, 25, 36, 49 ... groups of three contiguous characters could be arranged to form square matrices of dimension 3, 6, 9, 12, 15, 18, 21 … respectively. Alternately, 3 groups of three characters can produce a Rubik Cube type three dimensional matrix containing 15 individual 3 by 3 matrices.

Any n by n square matrix (M), comprising the integral values 0, 1 and 2, contains much hidden information. It can, for example, be expressed as the product of three, real valued, n by n matrices using the process of Singular Value Decomposition: M = U∑V* where U is a unitary matrix, ∑ is a diagonal matrix and V* is the conjugate transpose of the unitary matrix V. The columns of U and V are eigenvectors of MM* and M*M respectively. The non-zero values on the diagonal of ∑ are the square roots of the non-zero eigenvalues of both MM* and M*M. Each of the matrices U, ∑, and V* can also be subjected to Singular Value Decomposition.

Cosmological information encoded in the Torah would probably not take the form of the fundamental equations of quantum mechanics, general relativity, thermodynamics or any other patterns observed by the human mind. Information would most likely be revealed in a "Bible Code" format. Such a format is so incomprehensible to the human thought process that we are able to find information only if we first know what to look for! The following question serves to illustrate. Can a snapshot of critical parameters, for the early universe, be found within the 28 concatenated characters comprising the first verse of Genesis? One place to look for critical parameters is among the invariant coefficients of matrix characteristic equations. Let the eigenvalues of MM* and M*M -- from a 3 by 3 matrix (M) characterizing a string of three characters -- be denoted by L₁, L₂ and L₃. Now form the 3 by 3 matrix x = {{L₁, 0, 0}, {0, L₂, 0}, {0, 0, L₃}}. The characteristic equation for eigenvalues (L) is L³ + I₁L² + I₂L + I₃ = 0 where the coefficients are given by:

I₁ = - Trace[x]

I₂ = - (1/2)(I₁Trace[x] + Trace[x²])

I₃ = - (1/3)(I₂Trace[x] + I₁Trace[x²] + Trace[x³])

Now do the same for y = x² with II₁, II₂ and II₃ defining the coefficients and z = x³ with III₁, III₂ and III₃ defining the coefficients.

• If M comprises characters 1, 2 and 3 (001,201,000) then the Golden Ratio and its Conjugate are revealed in certain eigenvectors of MM* and M*M. This might explain the presence of that ratio and the Fibonacci sequence throughout nature. The absolute value of III₁ is 144; the reciprocal is 0.00694 which is essentially the mass fraction converted to energy when helium is formed from hydrogen in stars. This number is an indirect measurement of the strong force. Furthermore, the absolute value of II₁ is 28 which is the number of characters in the first verse of Genesis and the fourth magic number (2, 8, 20, 28, 50, 82, 126) of nuclear physics. I₂=4 which could signify the four fundamental forces: strong, weak, electromagnetic and gravitational. II₂=16 which could be the cardinality for the set of parameters that characterize a voxel: x, y and z coordinates, the time step, the rule of transformation, a base-27 number and ten physical property values assigned by that number -- magnetic strength, electrical charge strength, normal matter, normal energy, dark matter, dark energy, electromagnetic force, gravitational force, strong force and weak force. I₃=II₃=III₃=0 which might denote the start time of the Big Bang.
• If M comprises characters 4, 5 and 6 (202,100,210), then the absolute value of II₁ is 138. If that number is the Reciprocal Fine Structure constant just after the Big Bang, then the Fine Structure constant, at that time, was 0.007246 which is within estimated brackets for the early universe. II₂=3⁶ which might connote that the structure of life would be set in place by three persons acting over six "creation days."
• If M comprises characters 10, 11 and 12 (000,102,011), then the absolute value of I₁ is 7 and the absolute value of II₁ is 37. These are the first and eighth terms in the sequence of numbers of spheres (7, 8, 12, 14, 19, 21, 30, 37, 40) that produce solid-like shell structures with stable rings. If M comprises characters 19, 20 and 21 (110,100,212), then the absolute value of I₁ is 12 which is the third term in the sequence of numbers of spheres that produce solid-like shell structures with stable rings.

New patterns emerge if the first 4 groups of 3 contiguous characters are combined to form a 6 by 6 matrix. In that case, we have six eigenvalues represented by L₁, L₂, L₃, L₄, L₅ and L₆ which can be combined to form the matrix x = {{L₁, 0, 0, 0, 0, 0}, {0, L₂, 0, 0, 0, 0}, {0, 0, L₃, 0, 0, 0}, {0, 0, 0, L₄, 0, 0}, {0, 0, 0, 0, L₅, 0}, {0, 0, 0, 0, 0, L₆}}. The characteristic equation for L is L⁶ + I₁L⁵ + I₂L⁴ + I₃L³ + I₄L² + I₅L + I₆ = 0 and the three new coefficients are given by:

I₄ = - (1/4)(I₃Trace[x] + I₂Trace[x²] + I₁Trace[x³] + Trace[x⁴])

I₅ = - (1/5)(I₄Trace[x] + I₃Trace[x²] + I₂Trace[x³] + I₁Trace{x⁴] + Trace[x⁵])

I₆ = - (1/6)(I₅Trace[x] + I₄Trace[x²] + I₃Trace[x³] + I₂Trace[x⁴] + I₁Trace[x⁵] + Trace[x⁶])

with II₄, II₅ and II₆ as well as III₄, III₅ and III₆ denoting the corresponding coefficients for y = x² and z = x³ respectively. The original matrix (MM*) could be used instead of the matrix x having the eigenvalues of MM* on the diagonal. They have the same eigenvalues and characteristic equation. However, the use of "principle" values has a long history in mechanics.

• If M comprises the first twelve characters of the first verse of Genesis, as defined above, then the absolute value of II₁ is 545 and the reciprocal is 0.00183486. Multiplying by 10⁶ yields 1834.86 which is within 0.07% of the current value for the proton to electron rest mass ratio. The reciprocal of III₂ is 5(10^-7) which is equal to the weak force coupling constant. The reciprocal of the absolute value of (II4)(II5)(III2)(III3)(III4)(III5) is 10^-39 which equals the proton/electron gravitational to electrical force ratio.

Now combine the second 4 groups of 3 contiguous characters to form a second 6 by 6 matrix.

• If M comprises characters 13 through 24 then the value of II₂ is 107,611 and the reciprocal is 9.3(10^-6) which is essentially equal to the fraction of rest-mass-energy needed to disperse large cosmic structures such as galaxies and clusters (Q).

Now combine 9 groups of three contiguous characters to form a 9 by 9 matrix. The only unused character in this string is Tsadey Final, the last character in the first verse of Genesis.

• If M comprises characters 1 through 27 in the first verse of Genesis, then the absolute value of (III2)(III3)(III4)(III5)(III6)(II7) is 10⁸⁰ which is the estimated number of elementary particles in the universe.

The numerical values of the dimensionless numbers associated with the strong force, weak force, electromagnetic force (Fine Structure Constant), gravitational force, proton/electron mass ratio, the value of Q and the estimated number of elementary particles are critical to the formation and development of our universe. These numerical values are hidden in the first verse of Genesis and were extracted from the characteristic equations associated with various MM* (and M*M). The corresponding eigenvalues appear in square root form along the diagonal of ∑ in the Singular Value Decomposition of M (i.e. M = U∑V*). But what about the U and V; could any information be hidden in these matrices?

According to the Anthropic Principle, the universe was created as a home for man. Could any primitive information about man be concealed in U? If U is confined to 3 by 3 real matrices with ‖U‖ = ±1 and U^T = U^-1 then U is a rotation matrix. Singular Value Decomposition automatically creates such a matrix. Furthermore, if various U are multiplied together, the products are also rotation matrices. Every rotation matrix in three dimensions defines an axis – a direction left fixed by the rotation – and an angle – the amount of rotation about that axis. Let us examine the angles characterizing various rotation matrices (U) generated by letting M represent the first 27 characters, in groups of three, from the first verse of Genesis and then multiplying each successive U matrix by the ones that precede it.

ROTATION MATRIX	ROTATION ANGLE (deg)
[U123]	85.7
[U123][U456]	16.1
[U123][U456][U789]	16.2
[U123][U456][U789][U101112]	21.6
[U123][U456][U789][U101112][U131415]	77.3
[U123][U456][U789][U101112][U131415][U161718]	43.3
[U123][U456][U789][U101112][U131415][U161718][U192021]	61.3
[U123][U456][U789][U101112][U131415][U161718][U192021][U222324]	32.1
[U123][U456][U789][U101112][U131415][U161718][U192021][U222324][U252627]	38.4

These angles appear to approximate the nominal kinesiology of the human anatomy during walking.

85.7	Nominal knee angle most conducive to joint damage
16.1	Trunk range of angular motion during normal gait
16.2	Nominal value for quadriceps (Q) angle
21.6	Maximum hip angle during normal gait
77.3	Maximum ankle angle during normal gait
43.3	Maximum spinal lumbar rotation and cervical flexion and extension
61.3	Maximum angular velocity of trunk (deg/s) during normal gait; anticipates the definition of "second"
32.1	Minimum ankle angle during normal gait
38.4	Hip angle range of motion during normal gait

Many other things may also hidden in the first chapter of Genesis since this is only a cursory examination. A plethora of mathematical operations can be conducted on n by n, real, square matrices; these include product, Kronecker product, inverse, trace, immanant, various types of decomposition and countless functions with matrix arguments.

A FUNDAMENTAL PARTICLE?

If relatively short strings of Hebrew characters from the first chapter of Genesis provide the codons for all space/time and matter/energy then the universe itself should be constructed based on these codons. In other words, the same 27 characters of the Hebrew alphabet used to define the genetic code would ultimately define the structure of all space, time, matter and energy.

Consider the possibility that the assignment of one of the 27 Hebrew characters -- instead of 1 or 0 -- to a voxel can transform a mere volume of three dimensional space into the most basic building block of matter/energy. Furthermore, the state of the entire universe is still defined by a countably infinite sequence of digits representing a single real number -- now base-27 (with Aleph as zero) instead of base-2. Now connect three contiguous voxels together to generate a string. The three corresponding base-3 triplets generate a 3 by 3 matrix with the elements 0, 1 and 2. Such groups of three voxels and the associated 3 by 3 matrices could represent fundamental codons for the structure of the universe; a string of Hebrew characters now becomes a string of voxels in space.

Increasing complexity can be created by joining together groups of three contiguous voxels (codons) to create various geometric shapes. Such geometric constructions might represent fundamental particles of matter and energy. However, the arrangement of the 3 by 3 matrices does not necessarily coincide with the arrangement of the voxels. For example, four groups of three contiguous voxels could be arranged end-to-end to form a 12 voxel string. But the four 3 by 3 matrices representing these four codons could be arranged like tiles on a floor to create a single matrix of dimension six.

To illustrate a possible connection with Quantum Mechanics, a large number of groups of three contiguous voxels -- e.g. (10¹⁰)² -- could be arranged like tiles to produce a corresponding square matrix of dimension (3)(10¹⁰). This matrix could be used to compute eigenvalues (observables) and eigenvectors (states) for the various particles of physics. To illustrate a possible connection with General Relativity, consider the gravitational field equations G^ij = R^ij – (1/2)g^ijR and G^ij + αP^ij = 0. The Ricci tensor (R^ij) and scalar curvature (R) are derived from the Riemann-Christoffel tensor R^h_irs which is, in turn, connected to the rotation required to pass from one reference frame to another when two nearby points are joined by two different paths in a non-Euclidean space: Ω^h_i = R^h_irsdy^rδy^s . If the Riemannian space was simply a three dimensional space instead of four dimensional space-time, then Ω^h_i could be viewed as a rotation matrix defining both axis and angle.

Recalling that certain dimensionless numbers are associated with every portion of space, is it reasonable to surmise that some primitive particle, comprising these numbers might provide a pervasive and structural influence throughout the entire universe? How about a particle consisting of a string of three groups of three contiguous voxels with the corresponding three 3 by 3 matrices arranged in a Rubik Cube? The Hebrew characters assigned to the first, second and third groups of three voxels could be the first three, second three and third three characters respectively from the first verse of Genesis. The three matrices are therefore defined as:

which denote the three planes parallel to the yz axis. Notice Ayz and Cyz are identical so we really have only two independent matrices with 0, 1 and 2 as elements. These three matrices define all 15 slices. Can a Rubik Cube constructed from these trivial matrices be expected to conceal such numbers as the Fine Structure Constant and the mass ratios of sub-atomic particles? Ignoring possible 3-D matrix algebra, the number of two dimensional 3 by 3 matrices that can be extracted from the cube and arranged like tiles on a floor together with the number of possible arrangements of these tiles is too large to examine. So we will simply make a guess. However, a wild guess would seem unlikely to reveal anything of substance.

Each of fifteen slices through the cube produces 8 matrices by rotation about various axes: dihedral group of order eight (D₄).  In this trial, only one matrix will be selected for each slice. The fifteen selected 3 by 3 matrices can then be permuted in 15! ways. In this trial only one permutation will be chosen. After these assumptions, we still find ourselves short by one 3 by 3 matrix in order to construct a single matrix of dimension 12. So we will add a single 3 by 3 matrix, at the end, to complete the square. That matrix is

which represents the fourth three characters from the first verse of Genesis and is also a Hebrew name for God (Elah). The final 12 by 12 matrix is constructed by the operation

M=ArrayFlatten[{{Ayz,Byz,Cyz,Xyz},{Azx,Bzx,Czx,Xzx},{Axy,Bxy,Cxy,Xxy},{Xxz,Xzy,Xyx,XXX}}]

where Xyz, Xzy, Xzx, Xxz, Xxy and Xyx represent diagonal slices.

But how could anything of substance be obtained from this smorgasbord of assumptions?

Here is the final 12 by 12 matrix, designated as M:

Now define X = MM*, Y = X² and Z = X³ where M* is the transpose of M. X, Y and Z are each 12 by 12 matrices and each will have a characteristic equation with twelve coefficients. Let the coefficients for X, Y and Z be denoted by J₁…J₁₂, JJ₁…JJ₁₂ and JJJ₁…JJJ₁₂ respectively.

• (JJ6)(10^-15) = 0.00704. The fraction of mass converted to energy when helium is formed from hydrogen in stars is currently 0.0071 for a 0.8% deviation.
• Abs{JJ3/JJ2} = 137.705 and Abs{JJ2/JJ3} = 0.007262. The current values for the Reciprocal Fine Structure constant and the Fine Structure Constant are 137.036 and 0.007297 respectively for a 0.5% deviation.
• Abs{(J5)(10^-3)} = 3675.9. The total mass of a proton and a neutron compared to that of an electron is 1836.2 + (1.0014)(1836.2) = 3674.9 for a 0.03% deviation.
• 1/(JJ2) = 4.4(10^-7); JJ10/JJJ10 = 7.3(10^-7); J10/JJ10 = 9.8(10^-7). These are all possible values for the weak force coupling constant which is currently 10^-6 to 10^-7.
• 1/[(JJJ6)(JJJ8)] = 0.47(10^-39). The current value for the proton-electron gravitational to electrical force ratio is 0.435(10^-39) for an 8.0% deviation.
• J11/JJ11 = 8.5(10^-6). This is essentially equal to fraction of rest mass energy needed to disperse large cosmic structures such as galaxies and clusters.
• (JJJ7)²(JJJ9)² = 3(10⁸⁰) which is nominally the number of elementary particles in the universe.
• (JJJ11/JJ11)(2) = 247168. The mass of a Higgs Boson compared to that of an electron is 246037 for a 0.46% deviation.
• 1/(JJ2) = 4.4(10^-7). The mass of a Majorana neutrino compared to that of an electron is 4.7(10^-7) for a 6% deviation.
• (JJJ8)(JJJ9) = 3.43(10⁴⁰). The characteristic length of the universe (velocity of light divided by Hubble's constant) divided by the classical radius of an electron is 4(10⁴⁰) for a 14% deviation.

No more than two coefficients are used for any dimensionless number. The coefficients are combined by simple multiplication and division with an occasional power of 10 or factor of 2. Better estimates may be possible using different combinations of coefficients. Are these remarkable agreements just a coincidence?

The 12 by 12 matrix M, whose elements are 0, 1 and 2, was extracted from the first 12 characters of the first verse of Genesis and appears to conceal fundamental dimensionless numbers, characteristic of the early universe, as combinations of coefficients from the three characteristic equations for X, Y and Z.  If M is, in fact, a ‘magic matrix’ then the same dimensionless numbers may be concealed at various levels of matrix manipulation.  The Reciprocal Fine Structure constant can be used for illustration.  Its current value is 137.036.  We found that |JJ3/JJ2| = 137.705 for a 0.5% deviation.  But what if we form the following 6 by 6 matrix, designated by j₁, from the 36 coefficients of the three characteristic equations?

 

 

             JJJ1    J1        J2          JJJ12    JJJ11     J12

             JJJ2    J6        J3          JJ7         JJ8          JJ9

j₁   =    JJJ3    JJ1      J7          JJ6        J10          JJ10

             JJJ4    JJ2      J4          J8          J5             JJ11

             JJJ5    JJ3      JJ4       JJ5         J9            JJ12

             JJJ6    JJJ7    JJJ8    JJJ9       JJJ10    J11

 

This particular matrix is just one of 36! matrices that can be formed from the 36 coefficients. The trace of this square matrix is equal to the sum of its eigenvalues; any other arrangement of elements having the same trace will have the same eigenvalue sum.  The following connection between the sum of eigenvalues and a dimensionless number might suggest a deeper relationship than one between a simple ratio of coefficients and a dimensionless number.

 

Tr (j₁)/{(2³)(3)(10^-3)(Abs[J11])} = 136.932

 

for a 0.07% deviation from the current value for the Reciprocal Fine Structure constant.  Similarly, the determinant of this square matrix is equal to the product of its eigenvalues and

 

|j₁|/J5 = 1.4(10⁸⁰)

 

This is nominally the number of elementary particles in the universe.

The matrix j₁ comprises a relatively random arrangement of coefficients from the three characteristic equations.  Could a different matrix also be expected to conceal dimensionless numbers?  Consider the following matrix designated as j₂.

 

              JJJ6    JJ3      J4         JJ6          J5          JJ11

              JJJ1    JJJ7    JJ4      J8            J9           JJ12

j₂   =    JJJ2     J1        JJJ8    JJ5         JJJ10    J11

              JJJ3    J6        J2         JJJ9       JJJ11    J12

              JJJ4   JJ1      J3          JJJ12    JJ8         JJ9

              JJJ5   JJ2      J7          JJ7         J10         JJ10

 

The total mass of a proton and a neutron compared to that of an electron is 1836.2 + (1.0014)(1836.2) = 3674.9.  Using j₂ we find

 

Abs[Tr (j₂)/{(3³)(10⁴)(J4)(Abs[J11])^1/2}] = 3672.2

 

for a 0.07% deviation.  Similarly, the current value for the electron-proton electrical to gravitational force ratio is 0.23(10⁴⁰).  Using the determinant of j₂ we find

 

|j₂|/{(JJJ8)³(Abs[J11])} = 0.23(10⁴⁰)

 

for no deviation from the current measured value.  It would appear that predictions of dimensionless numbers, using the sum and product of eigenvalues from j₁, j₂ and undoubtedly other comparable 6 by 6 matrices, may be even closer to the current measured values of these dimensionless numbers than predictions based on simple products and ratios of coefficients from the characteristic equations. 

The dimensionless numbers of physics reflect various properties of space, time, matter and energy.  The universe was created with such precision that any significant variation, in a single, critical dimensionless number, would preclude our existence.  However, simultaneous changes -- in the hundredths of a percent range -- covering several different dimensionless numbers may have occurred over the 13.7 billion years since the beginning.  But if a predicted value deviates by more than a few hundredths of a percent from the current value, then either the dimensionless number is not critical or the best predicted value has not yet been determined.

Let us re-examine the 10 dimensionless numbers previously predicted using simple products and ratios of the 36 coefficients from the characteristic equations.  But this time the eigenvalue sum (trace) and eigenvalue product (determinant) for j₁ and j₂ will be employed.

 

• Tr (j₁)/{(10^-25)(JJ10)(JJJ8)} = 0.0071. The fraction of mass converted to energy when helium is formed from hydrogen in stars is currently 0.0071 for no deviation.

• Tr (j₁)/{(2³)(3)(10^-3)(Abs[J11])} = 136.932 and the reciprocal is 0.007303. The current values for the Reciprocal Fine Structure constant and the Fine Structure Constant are 137.036 and 0.007297 respectively for a 0.07% deviation.

• Abs[Tr (j₂)/{(3³)(10⁴)(J4)(Abs[J11])^1/2}] = 3672.2. The total mass of a proton and a neutron compared to that of an electron is 1836.2 + (1.0014)(1836.2) = 3674.9 for a 0.07% deviation.

• Tr (j₁)/Abs[JJ5] = 8(10^-7) . The weak force coupling constant is currently 10^-6 to 10^-7 for no deviation.

• |j₂|/{(JJJ8)³(Abs[J11])} = 0.23(10⁴⁰). The current value for the electron-proton electrical to gravitational force ratio is 0.23(10⁴⁰) for no deviation.

• JJ10/Abs[Tr (j₂)] = 9.8(10^-6). This is essentially equal to fraction of rest mass energy needed to disperse large cosmic structures such as galaxies and clusters.

• |j₁|/J5 = 1.4(10⁸⁰) which is nominally the number of elementary particles in the universe.

• Abs[Tr (j₂)/{(3²)(10^-8)(J8)(J11)(J5)}] = 245934. The mass of a Higgs Boson compared to that of an electron is 246037 for a 0.04% deviation.

• Tr (j₁)/{(2)(10²)(Abs[J9])(J12)} = 4.7(10^-7) . The mass of a Majorana neutrino compared to that of an electron is 4.7(10^-7) for no deviation.

• {(Abs[JJ1])^1/3(|j₂|)}/{(JJJ8)³(Abs[J11])} = 4(10⁴⁰). The characteristic length of the universe (velocity of light divided by Hubble’s constant) divided by the classical radius of an electron is 4(10⁴⁰) for no deviation.

 

Aside from the sum and product of eigenvalues, only ten J values total were used to compute the ten dimensionless numbers.  These J values are fixed just as the dimensionless numbers are fixed.  The only allowed variation is the functional form of a particular equation.  The equations are all simple products and ratios of numbers from j₁ and j₂ along with powers of 2, 3 and 10.  J11, which is -131100, appears in five equations.  JJJ8, which is 257150395023074017825, appears in three equations.

Probing deeper into this mystery, the dimensionless numbers of physics are not the only type of dimensionless numbers.  The dimensionless numbers of pure mathematics are another type.  Such numbers cannot vary over the life of the universe.  The most pervasive, structural numbers are Pi (π = 3.1415926535…) and Phi or the Golden Ratio (φ = 1.6180339887…).  These numbers are related by

 

π = (5) cos^-1(φ/2)

 

Could either one of these numbers be extracted from our Rubik Cube?  Consider that each individual slice through the cube, not just the 12 by 12 matrix may contain information.  One such slice is the 3 by 3 matrix A_yz representing the first three characters from the first verse of Genesis.  The first eigenvector of A_yzA_yz* -- A_yz times its transpose – is (φ, 0, 1).  The Golden Ratio, and therefore Pi, are concealed in this matrix.

My conjecture is that every fundamental dimensionless constant of physics can be generated using the approach described herein.  Assemble a string of three groups of three contiguous voxels.  Assign, to each of the three groups, a letter sequence from the first verse of Genesis and arrange the corresponding matrices into a Rubik Cube configuration of 27 numbers.  Then create multiple 3 by 3 matrices by slicing through the cube.  Finally, arrange these slices like tiles on a floor to produce multiple 2-D matrices of dimension 3 to 12 or higher.  This may require the use of the Elah matrix to complete the square.  Perhaps these strings of three groups of three contiguous voxels exercise a pervasive, structural influence throughout the universe.

A coincidence is a remarkable concurrence of events or circumstances without apparent causal connection. Given sufficient evidence, however, the bridge between "remarkable concurrence of events" and "causal connection" must be crossed.

EPILOGUE

Could it be that all the mysterious equations of the continuum are merely approximations to the more profound mysteries of the discrete? Could it be that maintaining the universe from one time step to the next requires intelligence? In such a scenario, time would not be the fourth dimension. Time would advance by discrete steps in one direction only. The "rule of time transformation" for each step would reflect the laws of physics as administered by an immanent God. Changes in the length of the basic time step would not be detectable by us.

As before, imagine God allows the state of each voxel to change at time intervals equal to the Planck Time (currently 5.4 x 10^-44 seconds). At the end of each time interval, a new real number defines the state of the universe. Nestled among the elements of aleph-3 are particular subsets of aleph-2 that track all possible histories of the universe from the big bang onward. An immanent God would only have to comprehend infinity at a cardinality of n''' to track all possible histories and futures. The Christian God understands infinity at least to a cardinality of n'''''''^.... where the number of ' marks is n. This is so overwhelming that our human intellect cannot grasp it! The infinitude of God permits Him to instantly track all possible histories and futures of you, me and the universe with no more difficulty than it is for us to count the wheels on a bicycle. Comprehending the cumulative consequences, of all contingent first-cause, human free will decisions on subsequent events, is infinitesimally trivial for God. Couple this concept with the idea that God has inserted Himself into all levels of the "interactive life-game" video. As the life-game designer, God knows all possible paths your life can take from birth to death. God not only knows all contingencies but He has entered the game as your advocate and He wants you to win. He has lovingly given each of us the prevenient grace that brings salvation (Titus 2:11). The requirements of God's law are written on every heart (Rom 2:15). Jesus Christ knocks at the door of every heart (Rev 3:20). The Holy Spirit calls, awakens, draws near, convicts, saves and empowers as He finds response in the heart and disposition to obedience (John 16:8). God's eternal power and divine nature are evident in the world around us (Rom 1:20). He works within each heart, intellect and will. All this and more comprises the infinitude of God!