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The Unfolding: A Brief Story About How The Rudiments For A Theory Of Everything Took Shape

Ian Beardsley

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I am particular about story telling, especially short stories. Here I present a short story about how the

rudiments for a theory of everything unfolded. I hope by relating this story you can share in the joy of

discovery that came in this adventure which took place in the mind and on paper. It is a journey into the

microcosmos, and out into the vast expanse of the Universe. What took shape here instills a sense of

mystery me in that it formulates the magic that is in the Universe in elegant symbolic language,

mathematics. Let’s begin with a quote from Carl Sagan…

The surface of the Earth is the shore of the cosmic ocean. From it we have learned most of what we know.

Recently, we have waded a little out to sea, enough to dampen our toes, or at most, wet our ankles.The

water seems inviting. The ocean calls. Some part of our being knows this is from where we came. We long

to return. These aspirations are not, I think, irreverent, although they may trouble whatever gods may be.

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List of Constants, Variables, And Data In This Paper

(Proton Mass)

(Proton Radius)

(Planck Constant)

(Light Speed)

(Gravitational Constant)

1/137 (Fine Structure Constant)

(Proton Charge)

(Electron Charge)

(Coulomb Constant)

(The Author’s Solar System Planck-Constant)

(Earth Mass)

(Earth Radius)

(Moon Mass)

(Moon Radius)

(Mass of Sun)

(Sun Radius)

(Earth Orbital Radius)

(Moon Orbital Radius)

Earth day=(24)(60)(60)=86,400 seconds. Using the Moon’s orbital velocity at aphelion, and Earth’s

orbital velocity at perihelion we have:

(Kinetic Energy Moon)

(Kinetic Energy Earth)

: 1.67262E − 27kg

: 0.833E − 15m

h : 6.62607E − 34J ⋅ s

c : 299,792,458m /s

G : 6.67408E − 11N

α :

: 1.6022E − 19C

: 8.988E 9

ℏ

⊙

: 2.8314E 33J ⋅ s

: 5.972E 24kg

: 6.378E6m

: 7.34767309E 22k g

: 1.7374E6m

⊙

: 1.989E 30kg

⊙

: 6.96E 8m

: 1.496E11m = 1AU

: 3.844E 8m

K E

(7.347673E 22k g)(966m /s)

= 3.428E 28J

K E

(5.972E 24kg)(30,290m /s)

= 2.7396E 33J

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The Unfolding Of The Rudiments For A Theory Of Everything

It all started when I was trying to work out a theory for inertia, that property of matter to resist

change in motion: when you push on it, it pushes back. The more of it there is, the more it

pushes back. I had decided to start with the constants, like the gravitational constant, because

I ﬁgured they measured the properties of space and time. I eventually wrote an expression that

to my surprise was equal to 1 second:!

I found that interesting and ﬁgured if the proton was characterized by the second, and the

second came from the ancient Sumerians dividing-up the rotation period of the Earth into 24

hours, each hour into 60 minutes, and each minute into 60 seconds from their base 12 and

base 60 mathematics, that it had to have something to do with the celestial motions and

periods they observed in the sky, that if the second was natural, then it would be in the motions

of the Earth, Moon, Sun, and stars. My ﬁrst guess, which panned out, was that the kinetic

energy of the Moon to the kinetic energy of the Earth times the 24 hour day, should be one

second, or close to it. At ﬁrst I found it was close to it, but then I made an adjustment for the

Earth’s tilt to its orbit of and it came out exact for all practical purposes. I got!

I then thought this was quantum mechanical and that I should make a Planck-type constant for

the Solar System. I found it was in this very equation because it is in joule-seconds which

could be the kinetic energy of the earth times one second in the above equation, so I had:!

I then thought I don’t need to solve the Schrodinger Wave equation of quantum mechanics for

the Solar System, but just look a the equations of Niels Bohr for the Bohr model of the atom,

which he wrote down before the Schrodinger equation existed from suggesting the proton had

discreet orbitals for the electrons and was quantized by , the Planck constant. He didn’t know

why or how it quantized like this by integer multiples of , but he found it worked. He was

inspired to do this by the emission spectra of hydrogen for diﬀerent energies, he suggested

after the electron jumped from one orbital to another by adding energy, that when it fell back in

it would emit light of a particular frequency. So I looked at his equation for the energies of

orbitals and their orbital distances :!

It later turned out for the hydrogen atom the case of Z=1 (1 proton) that the Schrodinger equation had

these as the solutions. That equation in spherical coordinates is:

(

6α

4πh

)

⋅

= 1secon d

θ = 23.5

∘

(Ear th Day)cos(θ ) = 1.0seconds

ℏ

⊙

= (1secon d )KE

ℏ

= −

)

2ℏ

ℏ

Z k

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This equation can be solved for the hydrogen atom giving the results of the the Bohr equations, but that is

for Z=1, for other elements, like Z=2,3,4…The equation becomes too difﬁcult to solve. But Bohr’s

equation works for these other values of Z.

I then looked at Bohr’s equations and changed the Coulomb constant to the gravitational constant ,

which changed charge into mass , put in my Planck-type constant for the Solar System, , did the

dimensional analysis and wrote:

They worked correctly to better than 99% for the Earth/Moon/Sun system, which was the n=3 orbit. I

knew I couldn’t use , which would ridiculously be the number of protons in the Sun, so since this was

gravity not electric ﬁelds, I used the radius of the Sun, . But, since the Moon of the Earth was

mysteriously appearing in my equations, I used it to normalize the radius of the Sun, which gave it a

radius of 400 (400 moon radii). This made it work. I then realized since we were dealing with the Earth,

the planet most optimally suited for life in the Solar System, and thought since this was because the Moon

optimizes the conditions for life on Earth by holding the Earth at its inclination to its orbit around the Sun

allowing for the season’s and preventing temperature extremes, that the Earth perfectly eclipsing the Sun

as seen from the Earth had meaning, and I wrote the the conditions for a perfect eclipse, which are the

radius of the Sun to the radius of the Moon equals the orbital radius of the Earth to the orbital radius of

the Moon. I then thought this might be a condition for habitable planets to be optimally successful and

wrote it:

Later I used this in my solutions for habitable star systems by spectral type of the stars. Finally, I wanted

to formulate the ground state for the solar system and used the equation from the Bohr atom, which is

Thus writing it with the mass of the Moon

−

ℏ

2 m

[

∂

∂r

(

∂

∂r

)

si n θ

∂

∂θ

(

si n θ

∂

∂θ

)

si n

∂

∂ ϕ

]

ψ + V(r)ψ = E ψ

ℏ

⊙

K E

= n

⊙

⋅

2ℏ

⊙

2ℏ

⊙

⋅

⊙

⋅

⊙

planet

moon

star

moon

ℏ

k e

≈ 0.529E − 10 m

ℏ

⊙

(2.8314E 33)

(6.67408E − 11)(7.34763E 22kg)

= 3.0281E 8m

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Noticing this in meters, is about the speed of light in meters per second, if I multiplied by one over , the

speed of light, it would be 1 second:

It was at this point that I was sure I was onto something. Later, playing with the equations, I ended up

with another equation for a second using the proton. It was

Where is the golden ratio (0.618). Equating the left of this with left of my ﬁrst equation the got the ball

rolling

I found it yielded the radius of the proton as

It was at this point that I realized I had the rudiments for a kind of theory of everything that bridged the

microcosms to the macrocosmos. In the next section we will put forth a theory for the mass of a proton.

With what we have so far we can almost apply it to stars of other spectral types (masses, sizes, colors, and

temperatures) for formulating how other stars might structure the mechanics of their habitable planets.

In order to apply this to other star systems, we have to be able to predict the radius of the habitable planet,

presumably in the n=3 orbit. I found the answer to be in the Vedic literature of India. They noticed that the

diameter of the Sun is about 108 times the diameter of the Earth and that the average distance from the

Sun to the Earth is about 108 solar diameters, with 108 being a signiﬁcant number in Yoga. So I wrote the

equivalent:

The surprising result I found was, after applying it to the stars of all spectral types from F through K, with

their different radii and luminosities (the luminosities determine , the distances to the habitable

zones), that the radius of the planet always came out about the same, about the radius of the Earth. This

may suggest optimally habitable planets are not just a function of the distance from the star, which

determines their temperature, but are functions of their size and mass probably because they are good for

life chemistry atmospheric composition, and gravity when they are the size and mass of the Earth.

ℏ

⊙

⋅

= 1secon d

ϕ ⋅

π r

G m

⋅

= 1secon d

(

6 α

4πh

G c

)

⋅

= 1secon d

= ϕ ⋅

planet

= 2

⋆

planet

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In order to get , the distance of the habitable planet from the star, we use the inverse square law for

luminosity of the star. If the Earth is in the habitable zone, and if the star is one hundred times brighter

than the Sun, then by the inverse square law the distance to the habitable zone of the planet is 10 times

that of what the Earth is from the Sun. Thus we have in astronomical units the habitable zone of a star is

given by:

We see our theory has applications to archaeology because the second came to us historically from the

ancient Sumerians because they divided the Earth day (rotation period) into 24 hours, and, because each

hour and minute got further divisions by 60 because their base 60 counting system was inherited by the

ancient Babylonians who were the ultimate source of dividing the hour into minutes and the minutes into

seconds. I have found this system is given by the rotational angular momentum of the Earth in terms the

solar system Planck-type constant, because, as I already pointed out:

This base 60 counting combined with dividing the day into 24 units is mathematically optimal because

the rotational angular momentum incorporates not just the day (rotation period of the Earth) but the mass

and size of the Earth. We can say we are touching on archaeoastronomy. This is because 60/24=2.5 and

the Scottish engineer, Alexander Thom, found ancient megalithic (stone) observatories throughout Europe

may have been based on a unit of length he called the megalithic yard and that the separations between

stones, that align with celestial positions and cycles, are recurrently separated by 2.5 megalithic yards.

Like in Stonehenge.

You will notice the 4/5 in the above equation can be written as 2/5, meaning the equation becomes:

Since , where is the angular velocity of the Earth, we have:

We notice , the ancient Sumerian factors deﬁning the second from the rotation period of the

Earth, We see the rotation, mass and radius (size) of the Earth are optimally formulated with ancient

Sumerian base 12 and base 60 mathematics (they divided the day into 12 hours, and the night into 12

hours).

I have found that the pressure gradient of the protopanetary disc, as a function of radius, that gave birth to

our solar system, is given by:

planet

⋆

⊙

earth

ℏ

⊙

24 = 60

earth

π M

earth

2 π f

= ω

earth

2/5 = 24/60

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This is the solution to:

The protoplanetary disc that evolves into the planets has two forces that balance its pressure, the

centripetal force of the gas disc due to its rotation around the protostar and the inward gravitational

force on the disc from the protostar , and these are related by the density of the gas that makes

up the disc.

I have computed my Planck-type constant, , as such:

Where

this characteristic time of one second is not just in the Solar System, and atom’s proton, but in the basis of

life chemistry, carbon, and the hydrocarbons, the skeletons of life chemistry. I found

is carbon (C)

is hydrogen (H)

Which is to say that six protons, which is carbon, the basis of life as we know it, has a characteristic time

of one second because in the ﬁrst equation above, we have a mass divided by the mass of a proton, times

seconds, giving six protons times a second (6 proton-seconds) which means 6 protons (carbon, the basis

of life) has a characteristic time of one second. This means that 1 proton, hydrogen, has a characteristic

time of six seconds. Hydrogen is the most fundamental element in the periodic table of the elements

which was theoretically created in the so-called big bang that gave birth to the universe, and is the

element from which all of the other heavier elements were made by stars. This six-fold symmetry that is

in hydrocarbons, the skeletons of biological chemistry, is fundamental to deﬁning the periodic table of the

elements because it has been found that the six protons of carbon and their respective charges, interact

P(R ) = P

(

)

−

earth

ℏ

⊙

d P

= − ρ

(

⋆

−

)

⋆

ℏ

⊙

ℏ

⊙

= (hC )K E

hC = 1secon d

C =

⋅

π r

G m

ℏ

⊙

= (hC )K E

earth

= (1.03351s)(2.7396E 33J ) = 2.8314E 33J ⋅ s

6proton s

⋅

4πh

G c

= 1secon d

1proton

⋅

4πh

G c

= 6secon d s

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with its six electrons, their respective charges, to balance to make carbon the most stable element

mathematically in which to describe the rest of the atoms in the periodic table. This is no doubt related to

the regular hexagon, a six-sided polygon which tessellates (tiles a surface without leaving gaps) because it

has its radii equal in length to its sides. This hexagonal tessellating property is used by bees to make their

honeycombs. So we see our theory now goes beyond the atom and the solar system. That it goes to

biological chemistry. But, it does not stop there. It seems to go into cosmology, the study of the origin and

fate of the universe. We see this because my equations link proton properties to 1-second, and protons

were ﬁxed in the universe at 1 second after it, meaning we could be seeing a universal clock that has

inﬂuenced everything since the Big Bang.

The idea is that neutrino decoupling (neutrinos stop interacting with one another) happens when the

reaction rate of weak interactions falls below the Hubble parameter, the expansion rate of the universe

. The reaction rate per particles is given by

is the Fermi constant is about , and is the temperature of the Universe. The

expansion rate of the universe is given by

Where is the Plank mass is about 1.22E19GeV. and have units of inverse time ( ). Neutrino

decoupling happens when

This is when the number of protons in the universe was set in place which, as it would turn out, is close to

one second in rough estimate.

The expansion rate of the Universe is governed by the Friedmann equation

Where is the energy density of the Universe. It is

The Hubble expansion rate is

Since

Γ ≈ G

1.166E − 5G eV

−2

H ≈

−1

8π G

ρ ∝ T

H ∝

≈ 2.4E18G eV

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we have

We said protons and neutrons are set in the universe when it has cooled in its expansion to about 1MeV.

We have

This was done in Planck units where time can be expressed in inverse energy. Since in Planck units

we have

This theory seems, then, to have applications at the core of cosmology, astrobiology (the study of life in

the universe in general), solar system mechanics, particle physics, theories of common structure between

micro-scales and macro-scales, and biology.

To compute the moon’s orbital radius I just use

Where Ag is the molar mass of silver and Au is the molar mass of silver, a connection to the 1.8 that

appears in our Solar System. We use this because we know it works for our Solar System.

This theory has a magical, mystical feel to it in that it involves the condition for the Moon perfectly

eclipsing the Sun as seen from the Earth, the ratio the Solar radius to the lunar orbital radius being in the

ratio of a gold atom mass to a silver atom mass, where gold and silver have been the metals of ceremonial

jewelry on Earth since ancient times, the Sun gold in color, the Moon silver in color, and in that the basis

unit of time in the theory is the second, which came to us from the ancient Sumerians who ﬁrst settled

down from following the herds, and crafting spearpoints from stone to hunt, to invent agriculture,

architecture, mathematics, and writing.

Perhaps one day, humans will travel to other star systems and ﬁnd other worlds whose mathematical

relationships are as beautiful and exciting as they are here in our star system, the Solar System. Perhaps

we will learn of other civilizations on other planets, and learn their ancient histories and how they became

t ∝

2.4E18G eV

(1E − 3G eV )

= 2.4E 24G eV

−1

1G eV

−1

= 5.39E − 25s

t ≈ (2.4E 24)(5.39E − 25)

t ≈ 1.3secon d s

= R

⊙

⋆

(1.8)

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who they are today, what kind of music and art they have, and did they build pyramids or tipis in that

these things may be universal in their geometries.

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Theory For Inertia:

I had two equations that gave the radius of a proton with characteristic times of one second each. I had to

break down the equations in terms of their operational parameters as described by a geometric model.

This is what I came up with, a proton is a 4d hypersphere who's cross-section is a sphere. Of course

occupying the dimension of time (4th dimension in drawing) is the vertical component of the drawing. I

have to draw this 3d cross-section as a circle (we cannot mentally visualize four dimensions). The proton

is moving through time at the speed of light (vertical component in the drawing) it is a bubble in space.

The normal force holding it in 3d space is proportional to the inertia created by the

pliability of space measured by G. So when we push on it (Force applied in drawing) there is a counter

force explaining Newton's action/reaction.

I think you could look at this another way: the cross sectional area of the proton moving against space is

in the opposite direction of the force applied and h is the granularity of space, G still its pliability. That is

to say, the ﬂux of a normal force to a hemisphere is over the area of the cross-section of the sphere.

It is the goal of this opening section to provide a theory for inertia, that quality of a mass to resist change

in motion. We want the the theory to include not just the quantum mechanics constant for energy over

time Planck’s constant, but to include the universal constant of gravitation G, the constant the speed of

light from relativity, and the ﬁne structure constant for theories of electric ﬁelds so as to bring together

the things that have been pitted against one another: quantum mechanics, relativity, classical physics,

electric ﬁelds, and gravitational ﬁelds. Towards these ends we will suggest a proton is a 3D cross-section

of a 4D hypersphere held in place countering its motion through time by a normal force that produces its

inertia (measured in mass in kilograms) much the same way we model a block on an inclined plain

countered by friction from the normal force to its motion. The following is the illustration of such a

proton as a cross-sectional bubble in space:

To get the ball rolling, I had found a wave solution to the Earth/Moon/Sun system where the Earth

orbiting the Sun is like an electron orbiting a proton with a quantum mechanical solution. I found this

solution had a characteristic time of one second. But, I found as well, I could describe the proton as

having a characteristic time of one second, and that this yielded the radius of a proton very close to that

= h /(ct

)

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obtained by modern experiments. So, it is now before me to come up with a theory for the proton in terms

of these characteristic times before I present my theory for a wave solution of the Solar System.

The expressions for the characteristic times of 1-second for the proton that I found, were:

Where is the golden ratio, is the radius of a proton, and is the mass of a proton. We ﬁnd

these produce close to the most recent measurements of the radius of a proton, if you equate the left sides

of each, to one another:

To derive this equation for the radius of a proton from ﬁrst principles I had set out to do it with the Planck

energy, , given by frequency of a particle, and from mass-energy equivalence, :

We take the rest energy of the mass of a proton :

The frequency of a proton is

We see at this point we have to set the expression equal to . So we need to come up with a theory for

inertia that explains it:

The radius of a proton is then

(

6α

4πh

G c

)

⋅

= 1secon d

ϕ ⋅

π r

G m

⋅

= 1secon d

ϕ = 0.618

= ϕ

= 0.816632E − 15m

E = h f

E = m c

E = h f

E = m

⋅

= ϕ =

= ϕ ⋅

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In order to prove our theory for the radius of a proton as incorporating , we will apply our model

outlined involving a normal force, to a 3d cross-section of a 4d hypersphere countering its direction

through time, t. We begin by writing equation 1 as:

Where , the constant of gravitation measures the pliability of space, and the granularity of space, and

c the speed of propagation. measures the inertia endowed in a proton. We write equation 2 as:

We now say that and that the normal force is

This gives us:

Since , we have

This gives

10.

is the cross-sectional area of the proton countering the normal force, , against its motion through

time, this is measured by the constant of gravitation. It is to say that

11.

= ϕ ⋅

6α

4πh

G c

⋅

1secon d

1 =

⋅

π r

G m

⋅

c(1secon d )

⋅

= 1secon d

1 =

⋅

π r

G m

⋅

⋅ F

9α

⋅

(

)

= ϕ

1 =

9α

⋅

3α

⋅

π r

∝

Ar eaCr ossSect ion Pr oton ⋅ F

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And, the coupling constant is

12.

Let us see if this is accurate:

We used the experimental value of a proton . And we have demonstrated that our

model of a proton as a 3D cross-section of a 4D hypersphere countering the normal force against its

motion through time gives its inertia that can counter a force at right angles to its motion through time and

the normal force.

It is thought that the proton does not have an exact radius, but that it is a fuzzy cloud of subatomic

particles. As such depending on what is going on can determine its state, or effective radius. It could be

that the proton radius is as large as

Which it was nearly measured to be before 2010 in two separate experiments. Or as small as

Which is closer to current measurements, which have decreased by 4% since 2010, and could get smaller.

In which case the characteristic time, , could be as large as

Using 2/3 as a ﬁbonacci approximation to . Or, it could be as small as

C =

3α

6.62607E − 34J ⋅ s

(299,792,458m /s)(1s

)

= 2.21022E − 42N

18769

⋅

π (2.21022E − 42N )

6.674E − 11N

(0.833E − 15m) = 1.68E − 27kg

= 0.833E − 15m

⋅

6.62607E − 34

(299,792,458)(1.67262E − 27)

= 0.88094E − 15m

= ϕ ⋅

= 0.816632E − 15m

⋅

π r

G m

⋅

= 1.03351secon d s

ϕ ⋅

π r

G m

⋅

= (0.618)

(352275361)π (0.833E − 15m)

(6.674E − 11)(1.67262E − 27kg)

⋅

6.62607E − 34

299792458

of 16 17

=0.995 seconds

Or perhaps more often it is in the area of:

But, what this tells us is that the unit of a second might be a natural constant. And, since the second comes

from dividing the Earth rotation period into 24 hours, and each hour into 60 minutes, and each minute

into 60 seconds, which ultimately comes to us from the ancient Sumerians who ﬁrst settled down from

hunting, wandering, and gathering and ﬂaking stones into spearpoints to invent agriculture, writing, and

mathematics, that this might be related to the mechanics of our Solar System. We ﬁnd if we take the

second as natural we have a wave mechanics solution to our Solar System with a characteristic time of

one second that is connected to the characteristic time of the proton, thus connecting macro scales (the

solar system) to micro scales (the atom).

6α

h 4π r

G c

= 1.004996352secon d s

of 17 17

The Author