Hyper gourd theory-- solving simultaneously the mysteries in particle physics, biology, oncology, neurology, economics,and cosmology

ORIGINAL ARTICLE

Hyper-gourd theory: solving simultaneously the mysteriesin particle physics, biology, oncology, neurology, economics,and cosmology

Ken Naitoh

Received: 26 March 2012 / Accepted: 23 August 2012 / Published online: 21 September 2012

� ISAROB 2012

Abstract The inevitability of various particle masses for

hadrons, quarks, leptons, atoms, biological molecules,

liquid droplets of fossil fuel and water, living cells

including microorganisms and cancers, multi-cellar sys-

tems such as organs, neural systems, and the brain, stars,

galaxies, and the cosmos is synthetically revealed. This is

possible because each flexible particle is commonly gen-

erated by a mode in which a larger particle breaks up into

two smaller ones through a gourd shape with two lumps.

These masses, sizes, frequencies, and diversity dominated

by super-magic numbers including the silver ratio, in

fractal nature can be derived by the fusion of the quasi-

stability principle defined between absolute instability and

neutral stability, the indeterminacy principle extended for

quantum, statistical, and continuum mechanics, and the

spherical Lie group theory. The analyses also result in a

new mathematical definition of living beings and non-

living systems and further explain the standard network

patterns of various particles and also the relation between

information, structure, and function, because the proposed

theory based on gourds posits a new hyper-interdisciplinary

physics that explains a very wide range of scales, while the

Newton, Schrodinger, and Boltzmann equations describe

only a narrow range of scales.

Keywords Biological � Cosmic � Interdisciplinary �Subatomic � Quark

1 Introduction

There are still so many mysteries about non-living and

living systems from subatomic to cosmic scales, which are

related to the masses of subatomic particles, quantum

entanglement, the magic numbers, morphogenetic pro-

cesses, the standard circuit pattern of the neural network in

the brain, overall mechanism of cancer, fusion of left–right

symmetric and asymmetric organs, the definition of life,

side effects, functions of introns and junk, economic sys-

tems generated by the human brain, dark matter, and super-

cosmos outside our universe.

The reason why so many mysteries have not been

explained is related to the fact that the traditional Newton,

Schrodinger, and Boltzmann equations [1–5] can reveal only

a narrow range of scales, although each equation describes

several classes of conservation of mass, momentum,

moment, and energy. Super-interdisciplinary and supermulti-

scale physics, which synthesizes the whole span from the

pico- to the peta-scale, is necessary to go further.

Our previous researches [6, 7] offered some hypotheses

and predictions regarding these mysteries. One concerns

the function of introns and junk. Another concerns the left–

right asymmetric part inside the brain system, which brings

feelings of comfort. Later, some experimental researches

done by other people verified our hypotheses and predic-

tions [8, 9].

Here, we will reveal the mysteries above more intensively

and extensively. First, we will classify the natural processes

into three modes: breakup–collision, expansion–compression,

and excitatory–inhibitory process (acceleration–depression)

This work was presented in part at the 17th International Symposium

on Artificial Life and Robotics, Oita, Japan, January 19–21, 2012.

K. Naitoh (&)

Faculty of Science and Engineering, Waseda University,

3-4-1 Ookubo, Shinjuku, Tokyo 169-8555, Japan

e-mail: [email protected]

URL: http://www.k-naito.mech.waseda.ac.jp/;

https://www.wnp7.waseda.jp/Rdb/app/ip/ipi0211.html?lang_kbn=0

&kensaku_no=4181

123

Artif Life Robotics (2012) 17:275–286

DOI 10.1007/s10015-012-0056-y

(Fig. 1). We name the modes of breakup–collision, expan-

sion–compression, and excitatory–inhibitory process (accel-

eration–depression), as gourds I, II, and III, respectively. This

is because each one of the three modes has shape of gourd in

Fig. 2 or occurs after gourd shape. Breakup–collision mode

clearly shows a gourd shape having two kinks, while excit-

atory–inhibitory process that in chemical reaction apparently

has a gourd in connection pattern of two molecular particles at

collision timing. The mode of expansion–compression in

Fig. 1 also has a gourd shape of two kinks connected at later

stage of expansion as shown in the later section, although

earlier stage often shows only one kink. Nature is dominated

by gourds at rate-determining timing, rather than by strings or

spheres at equilibrium state. In short, our previous theoretical

models [6, 7, 10–13] are extended here and also we propose

here a physical theory that explains the fractal nature in the

gourds of the three models, although the fractal concept has

only been used in mathematics so far.

2 Methodology

2.1 Quasi-stability principle [7, 10–13]

Natural systems are essentially discontinuous in three-

dimensional space but relatively continuous in time,

because from the subatomic to the cosmic scale they con-

sist of particles such as quarks, hadrons, atoms, molecules,

fluid particles, cells, and stars. Accordingly, various natural

processes for non-living and living systems can approxi-

mately be described by the following momentum equation

systems:

dðnÞ

dtðnÞyi tð Þ ¼

X

j

fij yi; yj

� �þ ui or

oðnÞ

otðnÞyi t; xkð Þ ¼

X

j

fij yi; yj

� �þ ui;

ð1Þ

where yi, xk, t, n, fij, and ui denote physical quantities of

particles such as velocity, particle deformation rate, and

pressure, spatial coordinates, time, order of derivative,

function of yi and yj, and random disturbance related to

indeterminacy coming from the small number of particles,

respectively [7, 10–13]. Quasi-stability is defined as a

principle in which one part (one term) on the right-hand

side of Eq. 1 is zero, when disturbance of ui enters the

system. When at least one of the various terms of i or j for

fij is zero, the system is also quasi-stable against the dis-

turbance. Quasi-stability as it is used here lies between

neutral stability and an absolutely unstable condition. The

quasi-stability is different from meta-stability denoted in

thermo-physics, because this quasi-stability for momentum

is weaker than the meta-stability for energy conservation.

(One example of evidence is that the elimination of only

one term among various numerical error ones in a finite

difference equation derived with the Taylor series yields an

approximate solution for a physical phenomenon such as

the transition to turbulence [14, 15].)

It is emphasized here that traditional stability analyses

based on a mathematical variable transformation for matrix

diagonalization are meaningless for revealing the nature of

living beings, because life is not in mathematical space.

2.2 Indeterminacy level

As shown above, various natural phenomena consist of

particles, i.e., discontinuity in space. This discontinuity of

particles leads to indeterminacy (stochasticity) for several

Fig. 1 Three modes

Fig. 2 Gourd shapes as two kinks connected

276 Artif Life Robotics (2012) 17:275–286

123

stages of phenomena, such as electron particles described by

the Schrodinger equation. An important point is that the level

of indeterminacy, i.e., degree of variation, varies according to

window scales for averaging (stochastic determinism win-

dow) and the number of particles. As an example, when the

system being analyzed consists of a smaller number of parti-

cles, the level of indeterminacy increases (Fig. 3).

Statistical mechanics based on the Liouville and Boltz-

mann equations [1] tells us that a very large window for

averaging the aggregation of a huge number of particles

brings deterministic continuum mechanics, whereas a small

window for a small number of particles leads to a sto-

chastic differential equation [15–18]. When a small win-

dow for averaging is used, physical quantities such as

mass, size, and velocity are defined with indeterminacy,

i.e., vagueness [10, 11, 15, 17, 18].

The baryons and mesons are constructed of only three and

two quarks, respectively. There are only two electrons inside

the smallest orbit around an atom. The number of carbon,

oxygen, and nitrogen particles inside a nitrogenous base is not

enough for continuum, because of the order of ten. These

small numbers of particles lead indeterminacy in the gov-

erning equations. On the other hand, a biological cell or a

liquid droplet of over 1 mm in size includes a lot of molecular

particles that result in deterministic governing equations.

The most important point is that a system of only one particle

is also deterministic, although such systems are of an infinitely

small number (see Appendix on the details of window of

averaging, indeterminacy level, and boundary conditions).

3 Breakup phenomena (Gourd Ia)

3.1 Size and weight

The breakup phenomena on various spatial scales com-

monly show the shape of a gourd having two lumps at the

time of breakup (Fig. 2). Thus, we can model the gourd

having two lumps using two flexible spheroids connected

as an approximation [7, 10–13, 19–22]. Here, we define a

parcel as a flexible spheroid (lump) having two long and

short radii a(t) and b(t) dependent on time t, for a quark or

lepton, the aggregation of some quarks, leptons, and hadrons

generated by a high energy experiment, the aggregation of

neutrons and protons in each child atom resulting from the

fission of a large atom such as a uranium 235, a nitrogenous

base in biological base pairs of nucleic acids hydrated with

a lot of water molecules, an amino acids hydrated, a bio-

logical cell just before division, a liquid droplet at breakup,

and a star (or dark matter) at breakup in the cosmos. The

parcel becomes a sphere of the radius rd (=[ab2]1/3) under

an equilibrium condition. The deformation rate c(t) is

defined as a(t)/b(t), while a sphere without deformation

corresponds to c = 1. Next, we derive a theory for

describing the deformations and motions of the two con-

nected spheroid parcels having two radii of rd1 and rd2

under equilibrium conditions and two deformation rates of

ck [k = 1, 2], while the size ratio of the two parcels is

defined by e = rd1/rd2.

We model the relative motion between the two parcels,

nonlinear convections inside the parcels, and the interfacial

force at the parcel surface. The interfacial force is evalu-

ated in the form of r/rm where m and r are constants and

r is the curvature of the parcel surface. Several types of

forces such as nuclear force, van der Waals force, surface

tension, Coulomb force, and the force of gravity can be

explained by varying m. The relation m = 1 implies the

surface tension of liquid. Mean density of the parcels is qL.

We assume that the convection flow inside a flexible

parcel is irrotational, i.e., potentially one. This potential

flow is applicable, because fluctuations entering the parcels

such as bases, cells, and atoms will be those of thermal

fluctuations with very high speeds, which are less dissi-

pative (fluctuation dissipation theorem). The potential

assumption is also valid for star breakups, because of their

large size and high speed. Breakup of subatomic particles

occurs with energy input of extremely high speeds at the

level of light.

Moreover, we must consider that a parcel is not often a

continuum, because the number of nucleons and water

molecules inside the parcels like atom and nitrogenous

base will be fewer than 1,000. Thus, this leads to a weak

indeterminacy of physical quantities such as deformation

rate and density.

Here, we derive the relation between the dimensionless

deformation rate ck (:ak/bk [k = 1, 2]) of each parcel

dependent on dimensionless time �t and the size ratio of two

parcels of e = rd1/rd2.

The stochastic governing equation having indeterminacy

can be written for momentum as

Fig. 3 Window for averaging

Artif Life Robotics (2012) 17:275–286 277

123

with, mci, mcj, msi, msj, Det, B0k, C0k, and E0k [for k =

1, 2], which are defined in Refs. [7, 10–13, 20–22], with

the parameter dst denotes random fluctuation. Equation 2 is

derived only by the above assumptions and also purely

mathematical transformation. The long derivation of Eq. 2

is in Refs. [7, 10] confirmed by the referees.

Next, we also define yi = ci – 1 as deviation from a

sphere. The momentum equation is

which is approximated by the first order of the Taylor

series [7, 10–13, 19–22].

It should be stressed that the indeterminacy caused by

the small number implies a slightly indeterminant (vague)

shape of parcel, which is weakly different from spheroid.

Thus, the present theory based on Eqs. 2 and 3 also

explains breakup processes of various shapes of parcels.

For each m, the quasi-stable size ratios of 1.0 and about

1.44 appear in Eq. 3, i.e., in the 1st-order term of the

Taylor series. It is stressed that a deformation disturbance

entering a particle, i.e., that for yi, leads to an asymmetric

size ratio around e = 1.44, while a disturbance of defor-

mation speed, dyi=d�ti, brings a symmetric ratio of 1.0.

The higher-order of the Taylor series for parcel 1 results in

d2yi

d�t2i

¼X1

k¼1

bk e; mð Þyk�1i

!dyi

d�ti

� �2

þX1

k¼1

ak e; mð Þyki þ f;

ð4Þ

where the last term on the right-hand side includes influ-

ences from parcel 2. Concrete functions for ak and bk can

be found using a computer program MATHEMATICA,

because of very long terms.

The higher order of terms of the Taylor series obtained

with the statistical fluid dynamics model (Eq. 4) also

reveals the various super-magic numbers for the size ratios

of (Table 1). It is stressed that, although there are various

ratios in Table 1, the ratios are desultory or discontinuous.

We should recall that various values of m bring the same

quasi-stable ratios of 1.0 and about 1.44 for the first order

of the Taylor series, whereas the higher-order terms lead to

various other ratios. (The fractional Taylor series also show

quasi-stable ratios including about 1.44 and 1.0 in Table 1.)

The size and weight ratios of purines and pyrimidines in

DNA are often around 1.45, which is close to 1.44 related to

surface force in Table 1, while the size ratios of identical bases

in RNA is 1.0 related to convection in Table 1. Heavy chains

in immune globulins (IgX) have size ratios of about 1.5, while

the frequency ratio of small and large types of IgXs is also 1.5:

two types of large immune globulins (IgM and IgE) and three

small ones (IgG, IgD, and IgA) [7, 11–13, 16].

The largest amino acid is about three times larger than

the smallest one, which corresponds to 3.58 in the higher

order of the Taylor series for m = 1 in Table 1. It is well

known that there are several types of hydrogen-bond con-

nections [33, 34]. Thus, the several orders of the Taylor

series will also be related to the variety of hydrogen-bond

connections. (Liquid sprays such as water and fuel also

show about threefold variation of droplet sizes [7].)

The ratios of chromosomes in human beings are about

six times at maximum, which may be seen as 4.54 in the

higher order of the Taylor series for m = 1, 6.11 in the

higher order of the Taylor series for m = -1, or the ratios

over 4.0 for m [ 1. (This will be because of chromosomes

are related to covalent bond connection [20–22].)

These ratios of 1:1 and about 1:1.5 also correspond to those

of child atoms generated by the breakup of uranium 235. This

means that the probabilities of the size ratios of 1:1 and about

2:3 are relatively high, because uranium will have the shape of

a gourd with two kinks just before its breakup. (The effect of

special theory of relativity between mass and energy will not

influence for the size ratio of child atoms obtained after the

fission process of uranium 235 very much, because the effect

work on both two child atoms at an identical rate. Moreover,

impact of only one neutron to uranium will not produce fast

deformations of uranium at the later stage of fission.)

The halo structures such as H10 and M32 have the

number ratios of neutrons and protons over 2:3 [10, 20].

d2

d�t2i

ci ¼ mcid

d�tici

� �2

þmcjd

d�tjcj

� �2

þmsi c53�2

3m

i þ msj c53�2

3m

j

( ),Detþ dst for i ¼ 1; 2 j ¼ 1; 2 i 6¼ j½ � ð2Þ

d2yi

d�t2i

¼ � 2

3ð3� e3 � 2e2þmÞ dyi

d�ti

� �2

þ3ð3� e3Þmyi � 4e1þm dyj

d�tj

� �2

þ12e1þmmyj

" #,½3ðe3 þ 1Þ� þ d0st; ð3Þ


123

These number ratios will also be explained by the present

theory of the higher-order of accuracy.

Table 1 also reveals the magic numbers appearing in the

weight ratios in various atoms generated by cold fusion:

about 3.6, 2.1, and 1.8 [20, 21].

The constant m will often have values larger than 1.0 or

1.0 for subatomic forces such as those of nuclear force.

When m has a value between 1.3 and 3.0, the ratios of e are

between 1.0 and about 105, which correspond to the mass

ratios in very small particles including quarks and leptons

or those in the hadron classes of baryons and mesons. (It is

stressed that a weight ratio of about 2:3 is also observed in

dark matter [20, 21].)

If we redefine a parcel as an electron cloud, Eqs. 2–4

and Table 1 may also explain the mysterious electron

orbits such as 4f more unstable than 5p and 6s, because

quasi-stable ratios do not increase monotonously according

to increasing of the orders of the Taylor series, while those

are related to the energy levels of orbits. For examples, the

3rd and 8th orders of terms bring the relatively large quasi-

stable size ratios of about 3.58 and 4.54, whereas the 4th

and 5th orders lead to the relatively small ones of about

2.47 and 2.10. It will also be stressed that both the numbers

of electron orbits and the orders of the Taylor series

inducing various quasi-stable size ratios are about seven or

eight.

3.2 Permanent and tentative components inside parcels

Parcels can essentially include two types of components,

which are permanent and tentative ones. The permanent part

exists both before and after the breakup process, i.e., during

the whole processes including the breakup timing, while the

tentative part appears and shows important features only at

the time of breakup. In several cases of natural phenomena,

the permanent part is a one-dimensional string or ring, having

a mass proportional to the parcel size e. Parcels including

both tentative and permanent parts have a mass proportional

to e3. Some natural phenomena have no tentative part, in

which case only the permanent part is proportional to e3.

Let us examine the following examples. In biological

systems, these magic numbers of e explain those for bio-

logical molecules of one-dimensional strings and rings

such as nitrogenous bases, amino acids, and proteins,

whereas the values of e3 correspond to biological molecules

hydrated by water molecules. In subatomic processes, the

aggregation of baryons as permanent part is one-dimen-

sional, while the total mass including baryons, gluons, and

the quark condensation effect as tentative parts is three-

dimensional. The tentative part is similar to the immersed

mass in fluid dynamics. Pure droplets of water or fuel and

stars are three-dimensional because they only have the

permanent part.

Table 1 Super-magic numbers

Values outside parentheses

mean the quasi-stable size ratios

of parcels, whereas values in

parentheses for m = 1 imply the

orders of the Taylor series

m = -3 m = -1 m = 1 m = 1.04 m = 1.3 m = 1.5 m = 2 m = 3

Convection

1 1 1 (1) 1 1 1 1 1

1.23 1.34 1.27 (2) 1.27 1.26 1.25 1.24 1.21

1.39 1.4 1.35 (3) 1.35 1.34 1.34 1.32 1.29

3.51 1.42 1.39 (4) 1.27 9 1030 10,321 256 16.1 4.36

1.42 1.43 1.4 (5) 1.38 1.38 1.37 1.36 1.33

4.22 1.41 (6) 9.08 9 1022 1,151 68.7 8.33 3.07

1.43 1.42 (7) 1.4 1.4 1.4 1.39 1.36

4.44 1.43 (8) 4.06 9 1022 1,033 64.4 8.02 2.86

1.44 1.43 (9) 1.41 1.41 1.41 1.4 1.38

4.5 2.82 9 1025 2,472 108.7 10.4 2.99

4.52 1.42 1.42 1.42 1.41 1.39

4.53 1.95 9 1034 37,325 553.6 23.5 4.25

1.43 1.43 1.43 1.42 1.4

Surface force

1.44 1.44 1.44 (1–9) 1.44 1.44 1.44 1.44 1.44

1.8 3.58 (3) 4.57 3.14

1.7 2.47 (4) 1.99

1.63 2.1 (5) 1.84

1.59 1.79 (7) 1.74

6.11 1.71 (8) 4.42

4.54 (8) 1.68

2.97


123

The group theory shows that one- and three-dimensional

spheres are in a group. This may support the idea that e and

e3 repeatedly appear in various natural particles (parcels),

while nature is relatively obviative to two-dimensional, e2.

(Many types of particles in nature take an axis such as that

of spheroid parcel, which is one-dimensional characteristic.

Thus, it is relatively difficult that aggregations of the par-

cels are two-dimensional.)

3.3 Discontinuity around m = 1

When the curvature is 1/r, we define the surface force as

one proportional to 1/rm. We can employ the simple form

of 1/rm, because we consider only the breakup timing of the

gourd having two lumps.

Let us think about the reason why discontinuous jumps

of size ratios in Table 1 can appear around m = 1. Taking

the r-integral of 1/rm as the potential, the functions dis-

continuously change for m increasing around 1.0. This may

be the reason why values of m a little larger than 1.0 dis-

continuously induce extremely large size ratios.

There are extremely large size ratios over 1030 in

Table 1 for 1.0 \ m \ 1.1. These very large ratios might

possibly correspond to the size ratios of vapor molecules

and water droplets produced at breakup and also to those of

subatomic particles and stars, or the very large or infinite

ratios for 1.0 \ m \ 1.1 may explain interaction between

particles at a infinite distance, i.e., the quantum

entanglement.

The potential flow assumption, irrotational flow one

without viscosity, is applicable even for very small sub-

atomic and molecular systems, because very high energy,

i.e., very high speed (in a very short period), is put for the

particle. Actually, energy at the level of sound speed is put

into the molecular system, while energy at the level of light

speed is put into the subatomic experiment, although the

period is very short.

3.4 Number of particle types and frequencies

The size asymmetry of around 3:2 of the main rings in

purines and pyrimidines naturally leads us to an asym-

metric number of types, i.e., ‘‘two’’ types of purines and

‘‘three’’ types of pyrimidines [7, 10–13]. The multiplicative

inverse of the asymmetric number of types is the size

asymmetry. This can be easily understood from the mass

conservation law, i.e., from the fact that the main rings of

purines have ‘‘nine’’ molecules of carbon and nitrogen,

while ‘‘six’’ molecules of carbon and nitrogen form the

main rings of pyrimidines. Accordingly, the number of

base types is proportional to the frequency of bases inside

RNA [7, 10–13]. In qualitative terms, the sizes and

molecular weights of the twenty types of amino acids are

also inversely proportional to the frequencies.

There are mesons with ‘‘two’’ quarks and baryons with

‘‘three’’. Therefore, an analysis based on Eqs. 2, 3, and 4

may also clarify the ratios in the elemental particles such as

quarks.

3.5 Relation between indeterminacy and quasi-stability

Our previous reports [7, 10–13, 19–22] support that the

quasi-stability weaker than the neutral stability controls the

natural particle sizes. Here, we will show another evidence.

The neutral stability demands that all of the terms of

momentum and deformation rate (position) in Eqs. 3 and 4

become zero, which leads to deterministic behavior,

whereas the quasi-stability permits that one term is not

zero, which brings indeterminate behavior. This quasi-

stability corresponds to the well-known indeterminacy

principle for quantum mechanics. Thus, the essential

indeterminacy principle in quantum mechanics [5] also

gives an evidence for the inevitability of the quasi-stability.

The hypothesis that both terms of momentum and position

can be deterministic is not true, because various natural

systems from subatomic to cosmic are in the quasi-

stability.

The reason why very small particles such as six quarks,

leptons, and gauge bosons cannot be taken out will also be

related to the indeterminacy, i.e., the quasi-stability.

4 Strings and rings produced by coalescence

(Gourd Ib)

4.1 Clover structure

An extremely large frequency ratio for purines and

pyrimdines in tRNA, say, far larger than 1.5, cannot produce

the stem in tRNA, because purine and pyrimidine pairs do

not easily form in the presence of only one type of base

[10–13, 16]. It is also known that, as purines and pyrimi-

dines in DNA have the same density, they form a pair at

each locus due to hydrogen bonding. Thus, this frequency

ratio of 1.0 for purines and pyrimidines cannot generate

loops in tRNA [10–13, 16]. This is the reason why a fre-

quency ratio between 1.0 and 1.5 for purines and pyrimi-

dines promotes clover structures having stems and loops

(Fig. 4). More complex structures such as rRNA can also

be explained by the above-mentioned dynamical mecha-

nism [10–13, 16]. Concavity and convexity, like thumbs in

RNAs, play an important function for grasping objects,

including nucleic acids and proteins. The ‘‘information’’

such as asymmetric frequency ratios between 1.0 and 1.5

induce ‘‘structures’’ such as convexoconcave shapes of


123

RNAs, which lead to ‘‘functions’’ such as grasping. (Dur-

ing differentiation and proliferation of the morphogenetic

process, repeats of symmetrical and asymmetrical cell sizes

also induce the structures of concavity and convexity in

multi-cellar systems, thereby leading to functional parts

such as arms and legs [11–13, 23, 24].)

Chromosomes in multi-cellar systems also have three-

dimensional repeat structures of concavity and convexity. It is

known that introns and junk are concave in shape, while exons

having the clear function of producing proteins are convex in

shape. (Several sequence data in the world-wide databases

may show a tendency of purines richer than pyrimidines in

RNA, which may be contradict to the above principle that

smaller bases are more. Non-coding RNA reported recently

may solve this contradiction. There may also be other

unknown RNAs which are rich with pyrimidines.)

4.2 Comfort

Human brain feels the golden and silver ratios of around 2:3 to

be comfortable. Thus, our previous reports predicted that there

will be about 2:3 ratios in the aggregations of neural cells of

the brain [7]. Recently, some researchers found the left–right

asymmetric part inside the habenular nucleus of the zebrafish

brain, which has size ratios between 1:1 and 1:2 [9]. It is

known that the habenular nucleus is related to fear, which is

the opposite feeling of comfort. Thus, the asymmetric size

ratios of neurons and networks inside the brain, which may be

induced by some different neurons such as GnRH neurons and

glia cells, also have sympathetic vibration with the asym-

metric ratios in picture images entering from the outside into

the eyes. The sympathetic vibration brings comfort.

Next, let us think about the musical scale inducing

comfortable music. Musical scale Perfect 5 uses the ratio of

2:3 for the sound frequencies, while perfect 4 employs 3:4.

Music also uses the magic numbers shown above. Then,

perfect 5 consists of ‘‘seven’’ half tones, while perfect 4 is

with ‘‘five’’ half tones. It is also stressed that the magic

numbers derived by Eq. 4 result in the ratio of 7:5 close to

the silver ratio, which is the rhythm used for Japanese

poems such as ‘‘haiku’’ and ‘‘tanka’’.

5 Aggregations generated by more coalescences

(Gourd Ic)

5.1 Inner asymmetry in cells and morphogenesis

Let us think about the morphogenetic process of the human

beings further. The starting point is colony of microor-

ganisms. Symmetric and asymmetric size ratios are also

observed at the cell level of microorganisms [10–13].

There are terminal cells and basal cells of different sizes

in the morphogenetic processes. This difference in cell size

also shows asymmetry [10–13]. Embryo stem (ES) cells

also show asymmetric cell divisions (differentiation) such

as a division to glial cells and neurons.

The left–right symmetric distribution of arms and legs is

observed in outward appearance, although the inner body,

including the heart and liver, is asymmetric. Outer cells

close to the surface move relatively easily in relation to the

absolute origin on the earth, because one part of the cell is

free without any connection to other cells. However, inner

cells receive forces from many directions due to the pres-

ence of other cells in a homogeneous field, making it dif-

ficult for them to move relative to the origin on the earth.

Therefore, inner cells deform relatively easily without any

translational motion of the gravity center.

Equations 3 and 4 explain this important characteristic

of the asymmetric division of inner cells and the symmetric

division of outer cells. This is because the asymmetric size

ratio of cells (the size ratio of about 1.45) is relatively

quasi-stable against the disturbance of deformation that

easily affects inner cells and also because the second term

on the right-hand side of Eqs. 3 and 4 implies cell defor-

mation. Outer cells divide into identical sizes of cells,

because the first term corresponds to the translational

motion of a cell [10–13].

5.2 Protons and neutrons in atoms

It is also well known that several atoms in nature have

number ratios of protons and neutrons between 1:1 and 2:3.

Here, let us examine the reason why larger atoms have

larger number ratios close to 2:3.

As shown above, the inner and outer parcels of baryons

determine whether the number ratio of neutrons and pro-

tons is asymmetric or symmetric, respectively [10, 11].

Larger aggregations of parcels such as thorium (Th),

which contains more baryons than helium (He), have more

Fig. 4 Schematic diagram of symmetric and asymmetric density

ratios in DNA and one leaf of tRNA


123

inner baryons, because the surface/volume ratio of the

aggregation becomes smaller as the size increases. More

inner baryons for larger atoms bring more asymmetric

number ratios of protons and neutrons [10, 11]. [Mysteri-

ously, the masses of stable protons and neutrons are almost

the same, while child atoms generated by fission of ura-

nium 235 and nitrogenous bases (pyrimidines and purines)

have a different weight ratio around 2:3. Some reasons are

shown in our previous reports [10–12]. There is the other

analogical evidence supporting the inevitability of the ratio

close to 1:1 for proton and neutron, that the Watson–Crick

pair of nitrogenous bases also has the asymmetric weight

ratio of about 2:3, whereas the weight ratio of the pair

including sugars inside DNA and RNA is close to 1:1

because of addition of sugar for each base.]

6 Repeats of breakup and coalescence (Gourd Id)

As shown above, one particle alone is deterministic without

indeterminacy, because of the Newton’s momentum

equation. Of course, systems having a number of particles

from three to several hundred can also be indeterminant,

when macroscopic modeling is done. Systems having more

particles may be approximated as a continuum, i.e.,

deterministic. Aggregation of a lot of small particles results

in a large particle at the next scale, which is close to a

sphere or a spheroid. The new large particle is determin-

istic again, because it is only one. (Two flexible particle

systems shown in Eq. 2 become deterministic when there

are no particles colliding to the two particle system as the

disturbance. If there are disturbance particles, the two

particle system is indeterminant.)

As an example, let us look at the biological system.

Only one DNA existing inside a cell is deterministic,

because DNA must accurately determine the structures and

functions of living beings. Two male and female DNAs for

mating filled with water can produce a weak diversity with

indeterminacy by crossing over. There are extremely large

numbers of genes, proteins, and water molecules inside a

cell, which leads to a continuum, i.e., the next deterministic

behaviors and sustainability during the cell’s life time.

Compartment of the large number of molecules due to cell

membrane leads to the determinacy as sustainability. The

loss of many molecules during the aging process results in

indeterminacy, instability, and death. (It is stressed that a

few electron orbits around atomic cores are indeterminant.

Stars are nearly deterministic, because weak gravity forces

lead to individual motion for each star.)

Therefore, further thought experiments based on the

indeterminacy level mentioned above may reveal the total

numbers of macroscopic groups of bio-molecules inside

human beings, including presently unknown ones. The

death rate of children will be between 10 and 0.1 %,

although the rate depends on economical situation of each

country. This may imply that the total number of macro-

scopic molecular groups inside healthy human beings is

between 10 and 1,000. (We find that at least six macro-

scopic molecular groups are necessary for living beings

[10–12, 25–29].)

Moreover, if the total numbers and types of macroscopic

molecular groups are revealed between 10 and 1,000 in detail,

the present indeterminacy level analysis may bring a new

insight for medicines such as those after becoming cancer.

7 Expansion and compression (Gourd II)

Other natural phenomena such as the morphogenetic and

aging process are in the topological mode of expansion and

compression. This mode is like a gourd expanding, which

has two spherical parts and a duct for suction.

An example of the gourd expanding is unborn baby

inside mother, in which two spherical parts correspond to

head and body. Our unsteady three-dimensional flow sim-

ulations obtained by solving the stochastic Navier–Stokes

equation qualitatively revealed the three-dimensional

structure of the morphogenetic processes of human beings,

including organs and the brain [30]. The morphogenetic

process of the main blood vessels inside the brain was also

simulated [31]. The computational simulations demonstrate

left–right asymmetric organs in the inner region of the

body, while symmetric organs and parts are relatively

outside the body. The simulation results also support the

principle of inner asymmetry and outer symmetry obtained

in the foregoing sections by the quasi-stability principle

(Eq. 3). Moreover, the simulation results also show another

principle of early symmetry and later asymmetry [27].

(Compression with shrinking will correspond to death,

because compression process is very unstable as seen in

turbulence increase during the compression stage of piston

engine.)

Our universe may also be expanding. If there are no

super-universes outside our universe, our one is symmetric.

8 Excitatory and inhibitory mode (Gourd III)

8.1 Self-replication

Our previous reports clarified the minimum excitatory

network of chemical reactions necessary for biological

self-replication [12]. The minimum system has four types

of macroscopic molecular groups: two information groups

x11 and x12 and two functional molecular groups x21 and

x22 (Fig. 5) [12]. This four-stroke system in Fig. 5 works


123

as a closed loop. Microorganisms including bacteria and

archaea and cancer cells basically employ this network of

the four molecular groups, because of a monotonic increase

without any depression. Emphasis is placed on the fact that

Fig. 5 shows a new, concrete, mathematical definition of

life. We can topologically see the symmetric and asym-

metric circles of reaction networks in Fig. 5, which also

have gourd shapes connected with symmetric and asym-

metric lumps. These gourds may commonly appear in both

parcels in a base-pair and those in the network, because

both are related to interaction of particles at the rate-

determining stage. Fusion of asymmetric and symmetric

size ratios of molecules (bipolarity of sizes of 1:1 and about

2:3) will naturally result in the fusion of asymmetrical and

symmetrical network patterns (bipolarity of topology).

Complementary pairs of RNA such as double-strand

RNA (dsRNA), i.e., only one molecular group, may form

the simplest excitatory cycle. However, information and a

catalytic function are undetached in this type of system,

because each strand of dsRNA has both of them. This leads

to the fact that the dsRNA and DNA suitable for stabilizing

information is not conducive to the production of various

functions for inducing multi-cellar systems having com-

plex geometries. Thus, living beings select the detachment

of information and function.

8.2 Morphogenetic and economic processes

The main temporal mystery is the standard clock, i.e., the basic

molecular instrument regulating the biological rhythm com-

mon to the cell cycle, proliferation and differentiation induced

by the stem cell cycle, neural pulse, neural network, and cir-

cadian clock. In order to generate the standard clock, at least

two more inhibitory molecules (molecular groups) of infor-

mation and function should be added to depress a monotonous

increase in DNA [10, 26, 27, 29].

Here, we define x13 and x23 as the other molecular groups

for inhibitory factors repressing reactions. These two groups

are incorporated in the four groups of x11, x12, x21, and x22,

because today’s cells, the morphogenetic processes of multi-

cellar systems, and neural systems use negative controllers

such as Oct-4 and SOX2 for producing tissues and organs [32].

This leads to a macroscopic model having six types of

molecular groups, or in other words, a six-stroke engine

(Fig. 6a). We can describe the densities of the six molecular

groups at generation N after the mother cell generation in the

morphogenetic process, by the following equations.

xNþ11i � xN

1i ¼ ai1xN1i � xN

21; i ¼ 1; 2; 3ð ÞxNþ1

2i � xN2i ¼ ai2d xN

1i � nixN23

� �� xN

22; i ¼ 1; 2; 3ð Þ;ð5Þ

where xij � xkm denotes the smaller value among xij and

xkm and also where d(x) denotes the larger value of x or 0,

i.e., max (x, 0) [25–27]. Statistical mechanics inevitably

leads to the mathematical form on the right-hand side in

Eq. 5, because of collision probability.

Numerical solutions for Eq. 5 show about a sevenfold beat

cycleofdensities formolecular groupson average,whilevarying

the parameters in Eq. 5 results in four- to tenfold beat cycles.

An important point is that the actual morphogenetic pro-

cesses show about seven-beat cycles of molecular densities

[25–29]. Another example may be that, in the morphogenetic

processes, both human beings and giraffes have ‘‘seven’’ neck

bones. These facts provide substantiating evidence for Eq. 5.

8.3 Neural network

Equation 5 will also reveal the standard topology of the cor-

tical neural circuit (network), the integration mechanism of

brain functions, the neural system for muscle control, and the

chemical reaction network inside a neuron [28].

Figure 6b shows the standard pattern for neural net-

works, which includes inputs and outputs. For the network,

six variables of xij (i = 1–2, j = 1–3) are redefined as the

activation level of neurons, related to the density of mol-

ecules and amount of total energy inside the neurons.

Fig. 5 Four-stroke molecular engine


123

There are two sides in Fig. 6b, one for inputs and the other

for outputs, which correspond to information and functional

molecules in Eq. 5 and Fig. 6a. The upside-down topology of

inputs and outputs in Fig. 6b will also be possible.

The most important point is that the present equation

(Eq. 5) and Fig. 6 describe the essential physics underlying

the network of neural cells and the molecular network

inside a neuron, whereas the Hodgkin–Huxley (H–H)

model describes only the outer electrical quantities such as

electron flow and voltage for a single neuron. (It should be

added that some variations modified from the network

pattern in Fig. 6 and Eq. 5 are also possible, by varying the

arbitrary constants and also by adding more molecular

types except for x13 and x23.)

It is stressed that the equation generated by the six groups

for describing the seven-beat cycle on average (Eq. 5) also

shows a quasi-stable feature for healthy conditions, whereas

an unstable condition will be for sickness [10].

Circadian clock of about 24–25 h are also seven times the

fundamental temperature oscillation of about 3.5 h [28, 29].

Equation 5 is an ordinary differential one that eliminates

spatial variations of quantities, because the spatial diffusion

of molecules and cells is relatively fast in comparison with

temporal oscillations and also because a lot of molecules

move between cells.

Moreover, emphasis is placed on the fact that the cycles

of boom and bust appearing in economic and social sys-

tems are also the seven-beat on average, because economic

systems are produced by human brains. Flux and reflux of

companies and capital can also be clarified by the present

analysis. (For standard neural network systems, the initial

conditions or inputs may be inputs from outside networks.

The initial inputs can be included on the right-hand side of

Eqs. 5 and 6 by the Delta function.)

8.4 Total energy limit

The model (Eq. 5) in the previous sections was derived

under the assumption of an infinite energy supply. How-

ever, energy supplied for molecular networks, cell colo-

nies, organs, neural networks, or economic systems will be

limited, because the surface-to-volume ratio of each system

decreases according to an increase in the number of mol-

ecules, cells, neurons, or populations, leading to a condi-

tion of insufficient energy. Thus, a new energy restriction

term should be added to Eq. 5, which results in Eq. 6 [28].

xNþ1ij� xN

ij¼ aijðxN

1j � bijxN23Þ � xN

2i � eij ½xNij �

q;

xij� 0; xN1j � bij xN

23� 0 ði ¼ 1� 2; j ¼ 1;�3Þ; ð6Þ

where q [ 2 is set in case that the symbol � is defined as

product, whereas q [ 1 if the symbol � means smaller

value among two.

Let us solve the time-dependent process including

morphogenesis and aging processes by using Eq. 6.

Numerical solutions for the equation extended with total

energy limit show a transition to sick situation such as

cancer in the aging process of the human beings including

the brain, i.e., a mysterious transition from chaotic oscil-

lation at 2nd stage to periodic one at 3rd stage, while the

vibration amplitude keeps a constant level (Fig. 7).

This limitation on the total mass and energy is also

evident in today’s economic systems, because a huge

amount of information travels at the very high speed of

light through the worldwide internet, whereas the speed of

cargo shipment is still at the sonic level. This unbalance

between information and objects may induce very complex

oscillations and economic crisis, which do not have the

sevenfold beat mentioned above.

8.5 Higher pressure for evolution and self-organization

The scenario described above by Eqs. 5 and 6 for a universal

network pattern in fractal nature can reveal the reason why

living beings from the pre-biotic process to multi-cellar sys-

tems can be induced stably in a self-organizing manner.

Random processes cannot generate the present living system

having complex chemical reaction processes.

Fig. 6 Six-stroke engines for morphogenetic processes and neural

systems


123

9 Conclusion

This approach solves mysterious problems concerning the

origin of life, evolution, molecular biology, microbiology,

system biology, morphogenesis, economics, medicine,

brain science, subatomic theories, high energy physics, and

cosmology.

The proposed theory based on gourds at rate-determin-

ing stage posits a new hyper-interdisciplinary physics that

explains a very wide range of scales, while the Newton,

Schrodinger, and Boltzmann equations describe only a

narrow range of scales. Analysis based on gourd shapes at

rate-determining timing shows a new paradigm, whereas

superstring theories based strings or spheres will mainly

reveal at equilibrium state (Fig. 8). Quasi-stability princi-

ple is extremely important for various phenomena and

stages.

Acknowledgments This article is part of the outcome of research

performed under a Waseda university Grant for special research

project (2009B-206). The author thanks Mr. Hiromi Inoue and Mr.

Kenji Hashimoto of Waseda University for their help on this study.

Appendix

Various window sizes for spatial averaging, which are

smaller than those for continuum approximations such as

the Boltzmann and Navier–Stokes equations (deterministic

equations) describing inner analytical domains vary the

indeterminacy levels (degrees of vagueness for physical

quantities such as parcel shape, density, pressure, and

temperature). We should show a way, which determines

the window size for averaging. This can be done using

boundary and initial conditions, because these conditions

are also indeterminant due to existing in the outer unknown

region. Basically, the indeterminacy level of physical

quantities in inner region is set to be identical to that of

initial and boundary conditions. Several examples on the

indeterminacy level are reported in our previous reports

[10, 11, 15–21, 28].

References

1. Hirschfelder JO, Curtiss CF, Bird RB (1964) Molecular theory of

gases and liquids. Willey, New York

2. Aris R (1989) Vectors, tensors, and the basic equations of fluid

mechanics. Dover, New York

3. Tatsumi T (1982) Fluid dynamics. Baifukan, Tokyo

4. Landau ED, Lifshitz EM (2004) Fluid mechanics, 2nd edn.

Butterworth-Heinemann Elsevier, Oxford

5. Landau ED, Lifshitz EM (1981) Quantum mechanics, 3rd edn.

Butterworth-Heinemann Elsevier, Oxford

6. Naitoh K (1998) Introns for accelerating quasi-macroevolution.

JSME Int J Ser C 41(3):398–405

7. Naitoh K (2001) Cyto-fluid dynamic theory. Jpn J Ind Appl Math

18-1:75–105 (also in Naitoh K (1999) Oil Gas Sci Technol 54-2)

8. Hirotsune S, Yoshida N, Chen A, Garrett L, Sugiyama F,

Takahashi S, Yagami K, Wynshaw-Boris A, Yoshiki A (2003)

An expressed pseudogene regulates the messenger-RNA stability

of its homologous coding gene. Nature 423:91–96

9. Aizawa H, Bianco IH, Hamaoka T, Miyashita T, Uemura O,

Concha ML, Russell C, Wilson SW, Okamoto H et al (2005)

Laterotopic representation of left-right information onto the

dorso-ventral axis of a zebrafish midbrain target nucleus. Curr

Biol 15(8):238–243

10. Naitoh K (2012) Spatiotemporal structure: common to subatomic

systems, biological processes, and economic cycles. J Phys Conf

Ser 344 (also in Naitoh K (2011) A force theory. RIMS Kok-

yuroku 1724:176–185)

11. Naitoh K (2010) Onto-biology. Artif Life Robot 15:117–127

12. Naitoh K (2008) Stochastic determinism. artificial life and robotics

13:10–17 (also in Naitoh K (2008) Onto-biology: inevitability

of five bases and twenty amino-acids. In: Proceedings of 13th

international conference on biomedial engineering (ICBME),

Singapore. Springer, Berlin)

13. Naitoh K (2006) Gene engine and machine engine. Springer,

Tokyo

14. Naitoh K, Kuwahara K (1992) Large eddy simulation and direct

simulation of compressible turbulence and combusting flows

in engines based on the BI-SCALES method. Fluid Dyn Res 10:

299–325

Fig. 7 Transition to sick condition as the 3rd stage in the aging

process of the human being including the brain

Newton eq.

Boltzmann eq.

Shroedinger eq.

Subatomic Theories

Hyper-gourd theory: [for rate-determining stages based on quasi -stability and statistic fluid mechanics]

[Mainly analyses for equilibrium states]

Fig. 8 Hyper-gourd theory as warp and traditional equations as weft


123

15. Naitoh K, Shimiya H (2011) Stochastic determinism capturing

the transition point from laminar flow to turbulence. Jpn J Ind

Appl Math 28(1):3–14 (also in Naitoh K, Nakagawa Y, Shimiya

H (2008) Stochastic determinism approach for simulating the

transition points in internal flows with various inlet disturbances.

In: Proceedings of 5th international conference on computational

fluid dynamics, (ICCFD5) and computational fluid dynamics.

Springer, and in proceedings of 6th international symposium on

turbulence and shear flow phenomena (TSFP6), 2009)

16. Naitoh K (2010) Onto-biology: a design diagram of life, rather

than its birthplace in the cosmos. J Cosmol 5:999–1007

17. Naitoh K (2011) Stochastic determinism: revealing the critical

Reynolds number in pipe and fast phase transition. In: Proceed-

ings of 10th international symposium on experimental and

computational aerothermodynamics of internal flows (ISAIF),

Paper no. ISAIF10-045, 2011

18. Naitoh K, Noda A, Kimura S, Shimiya H, Maeguchi H (2010)

Transition to turbulence and laminarization clarified by stochastic

determinism. In: Proceedings of 8th international symposium on

engineering turbulence modeling and measurements (ETMM8),

vol 775, Marseille

19. Naitoh K (2012) Hyper-gourd theory. In: Proceedings of 17th

international symposium on artificial life and robotics (AROB17)

20. Naitoh K (2011) Quasi-stability theory: explaining the inevita-

bility of the Magic numbers at various stages from subatomic to

biological. In: Proceedings of 15th Nordic and Baltic conference

on biomedical engineering and biophysics, IFMBE proceedings,

vol 34, Denmark, pp 211–214 (also in Naitoh K (2011) Quasi-

stability theory: explaining the inevitability of the Magic numbers

at various stages. Proceedings of ISB2011, Brussels)

21. Naitoh K (2011) The inevitability of biological molecules con-

nected by covalent and hydrogen bonds. In: Proceedings of

European IFMBE conference, IFMBE proceedings, vol 37,

pp 251–254

22. Naitoh K, Hashimoto K, Inoue H (2011) The inevitability of the

biological molecules. Artif Life Robot 16

23. Naitoh K (2011) The engine: inducing the ontogenesis. In: Pro-

ceedings of 15th Nordic and Baltic conference on biomedical

engineering and biophysics, IFMBE proceedings, vol 34, Den-

mark, pp 215–218

24. Naitoh K (2008) Physics underling topobiology: space–time

structure underlying the morphogenetic process. In: Proceedings

of 13th international conference on biomedial engineering,

(ICBME). Springer, Singapore (2008)

25. Naitoh K (2011) Morphogenic economics. Jpn J Ind Appl Math 28:1

26. Naitoh K (2011) The universal bio-circuit: underlying normal and

abnormal reaction networks including cancer. In: Proceedings of

European IFMBE conference, IFMBE proceedings, vol 37,

Budapest, pp 247–250

27. Naitoh K (2011) Onto-oncology: a mathematical physics under-

lying the proliferation, differentiation, apoptosis, and homeostasis

in normal and abnormal morphogenesis and neural system. In:

Proceedings of 15th Nordic and Baltic conference on biomedical

engineering and biophysics, IFMBE proceedings, vol 34,

Denmark. pp 29–32

28. Naitoh K (2011) Onto-neurology. In: Proceedings of JSST2011,

international conference modeling and simulation technology,

Tokyo, pp 322–327

29. Naitoh K (2009) Onto-biology: clarifying also the standard clock

for pre-biotic process, stem-cell, organs, brain, and societies. Proc

CBEE, Singapore

30. Naitoh K (2008) Engine for cerebral development, artificial life

robotics, vol 18. Springer, Berlin

31. Naitoh K, Kawanobe H (2012) Engine for brain development.

Artif Life Robot 16:482–485

32. Takahashi K, Yamanaka S (2006) Induction of pluripotent stem

cells from mouse embryonic and adult fibroblast cultures by

defined factors. Cell 126:663–676

33. Nishio M (2004) Weak hydrogen bonds. Encyclopedia of

supramolecular chemistry. Marcel Dekker, USA, pp 1576–1585

34. Umezawa Y, Nishio M (2005) The CH/Pai hydrogenbond. Seitai

no Kagaku 56(6):632–638


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