# Quintessence (Planck unit)

Quintessence = sqrt(velocity/mass) as a 5th and principal Planck unit

The MKSA system of units is a physical system of measurement that uses the meter, kilogram, second and ampere (MKSA) as base units and forms the base of the SI International System of Units. As Planck units these would constitute Planck mass, Planck length, Planck time, Planck ampere. Quintessence as a Planck unit would be characterized by encompassing the 4 MKSA Planck units (i.e.: being that unit from which the 4 MKSA units may be derived), thus qualifying as a 5th Planck unit in the classical elements sense.

## Quintessence historical

### Quintessence

According to ancient and medieval science, quintessence (quinta essentia or fifth element), also called aether, æther, aither, or ether, is the material that fills the region above the terrestrial sphere. Believing that the movements of the heavenly bodies are continuous, natural and circular, and that the natural movements of the four terrestrial elements (water, earth, fire, air) are rectilinear and discontinuous, Aristotle concluded that the heavenly bodies must be composed of a fifth element, aither [sic] [1]

### Classical elements

Classical elements typically refer to water, earth, fire, air, and (later) aether, which were proposed to explain the nature and complexity of all matter in terms of simpler substances. [2][3] Ancient cultures in Greece, w:Ancient Egypt, Persia, w:Babylonia, Japan, Tibet, and India had all similar lists, sometimes referring in local languages to "air" as "wind" and the fifth element as "void". The Chinese Wu Xing system lists Wood ( ), Fire ( huǒ), Earth ( ), Metal ( jīn), and Water ( shuǐ), though these are described more as energies or transitions rather than as types of material. These five elements have been associated since Plato's Timaeus with the five w:platonic solids.

#### Platonic solids

In Timaeus, Plato talks about the five, and only five, possible regular solids – those with equivalent faces and with all lines and angles, formed by those faces, equal. They are the four-sided tetrahedron (fire), the six-sided hexahedron or cube (earth), the eight-sided octahedron (air), the twelve-sided dodecahedron (quintessence), and the twenty-sided icosahedron (water).

 w:Tetrahedron w:Cube w:Octahedron Dodecahedron Icosahedron Four faces Six faces Eight faces Twelve faces Twenty faces

#### Godai (Japanese philosophy)

Godai, (五大?, lit. "five – great, large, physical, form") five elements philosophy in Japan is derived from Buddhist w:dharma and traditional Chinese medical doctrine that traveled from China throughout east Asia to Japan.[4][5]

The Japanese Buddhist Godai is attributed to esoteric Japanese Buddhism during the tenth century CE under the name of gorin (the "five wheels" or the "five rings"). Godai and gorin is also seen within the practice of w:ninjutsu, where these principles became an essential aspect of the esoteric ninja teachings (the ninpo-mikkyo) whereas the theory of gogyo moved into the functional theory of traditional Japanese medicine and exoteric Buddhism.[6]

The godai is a static or inert philosophical understanding of the traditional Japanese elements and study, similar to the Greek w:classical elements. The four main elements or building blocks are Earth, Water, Fire, Wind, and Void is non substantial.[7]

### Fifth element

In Plato's Timaeus (58d) speaking about air, Plato mentions that "there is the most translucent kind which is called by the name of aether (αἰθήρ)"[9] but otherwise he adopted the classical system of four elements. w:Aristotle, who had been Plato's student at the Academy, agreed on this point with his former mentor, emphasizing additionally that fire has sometimes been mistaken for aether. However, in his Book w:On the Heavens he introduced a new "first" element to the system of the w:classical elements of Ionian philosophy. He noted that the four terrestrial classical elements were subject to change and naturally moved linearly. The first element however, located in the celestial regions and heavenly bodies, moved circularly and had none of the qualities the terrestrial classical elements had. It was neither hot nor cold, neither wet nor dry. With this addition the system of elements was extended to five and later commentators started referring to the new first one as the fifth and also called it aether, a word that Aristotle had not used.[10]

Aether differed from the four terrestrial elements; it was incapable of motion of quality or motion of quantity. Aether was only capable of local motion. Aether naturally moved in circles, and had no contrary, or unnatural, motion. Aristotle also noted that w:celestial spheres made of aether held the stars and planets. The idea of aethereal spheres moving with natural circular motion led to Aristotle's explanation of the observed orbits of stars and planets in perfectly circular motion.

### Quintessence (Physics)

In w:physics, quintessence is a hypothetical form of w:dark energy, more precisely a w:scalar field, postulated as an explanation of the observation of an accelerating rate of expansion of the universe. It has been proposed by some physicists to be a fifth fundamental force.[11][12][13][14]

## Quintessence as a Planck unit

Assigning geometrical objects to the Planck units via 2 dimensionless physical constants, the fine structure constant α and Omega Ω and by setting a mathematical relationship un between them, we can construct this table.

Geometrical units
Attribute Geometrical object Relationship
Quintessence ${\displaystyle Q=2\pi \Omega }$ ${\displaystyle unit=u^{1}=u}$
mass ${\displaystyle M=1}$ ${\displaystyle unit=u^{15}}$
time ${\displaystyle T=2\pi }$ ${\displaystyle unit=u^{-30}}$
length ${\displaystyle L=2\pi ^{2}\Omega ^{2}}$ ${\displaystyle unit=u^{-13}}$
speed of light ${\displaystyle V=2\pi \Omega ^{2}}$ ${\displaystyle unit=u^{17}}$
ampere ${\displaystyle A={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}}$ ${\displaystyle unit=u^{3}}$

From these relationships, we note that certain ratios of units cancel and become unit-less

${\displaystyle {\frac {u^{3*3}u^{-13*3}}{u^{-30}}}={\frac {u^{-13*15}}{u^{15*9}u^{-30*11}}}\;...\;=1}$

### SI units (derivations)

Setting 2 unit-less ratios (x, y) in terms of MLT:

${\displaystyle u,\;units={\sqrt {\frac {L}{MT}}}}$
${\displaystyle x,\;units={\sqrt {\frac {M^{9}T^{11}}{L^{15}}}}=u^{0}=1}$
${\displaystyle y,\;units=M^{2}T=u^{0}=1}$

Gives the units for the dimensioned constants;

${\displaystyle u^{3}={\frac {L^{3/2}}{M^{3/2}T^{3/2}}}=A}$ (Ampere)
${\displaystyle u^{6}(y)=L^{3}/T^{2}M,\;(G)}$ Gravitation constant
${\displaystyle u^{13}(xy)=1/L,\;(1/l_{p})}$ Planck length
${\displaystyle u^{15}(xy^{2})=M,\;(m_{P})}$ Planck mass
${\displaystyle u^{17}(xy^{2})=V,\;(c)}$ speed of light
${\displaystyle u^{19}(xy^{3})=ML^{2}/T,\;(h)}$ Planck constant
${\displaystyle u^{27}(x^{2}y^{3})={\frac {M^{3/2}{\sqrt {T}}}{L^{3/2}}}=1/AT,\;(e)}$ elementary charge
${\displaystyle u^{29}(x^{2}y^{4})={\frac {M^{5/2}{\sqrt {T}}}{\sqrt {L}}}=ML/AT,\;(k_{B})}$ Boltzmann's constant
${\displaystyle u^{30}(x^{2}y^{3})=1/T,\;(1/t_{p})}$ Planck time

### SI constants (derivations)

To convert to the SI unit values we need 2 scalars, here are used r and v, numerically

${\displaystyle v=11843707.9...}$
${\displaystyle r=.712562514...}$

Assigning (with i as the numerical x and j as the numerical y from above)

${\displaystyle Q=2\pi \Omega {\frac {v}{r^{2}}},\;unit=u}$
${\displaystyle i={\frac {1}{2\pi {(2\pi \Omega )}^{15}}},\;unit=1}$
${\displaystyle j={\frac {r^{17}}{v^{8}}},\;unit=1}$

Gives

${\displaystyle A=Q^{3}({\frac {2^{3}}{\alpha }})={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}{\frac {v^{3}}{r^{6}}},\;u^{3}}$
${\displaystyle G={\frac {Q^{6}}{2^{3}\pi ^{2}}}(j)=2^{3}\pi ^{4}\Omega ^{6}{\frac {r^{5}}{v^{2}}},\;u^{6}}$
${\displaystyle L^{-1}=4\pi Q^{13}(ij)={\frac {1}{2\pi ^{2}\Omega ^{2}}}{\frac {v^{5}}{r^{9}}},\;u^{13}}$
${\displaystyle M=2\pi Q^{15}(ij^{2})={\frac {r^{4}}{v}},\;u^{15}}$
${\displaystyle P=Q^{16}(ij^{2})=\Omega r^{2},\;u^{16}}$
${\displaystyle V=Q^{17}(ij^{2})=2\pi \Omega ^{2}v,\;u^{17}}$
${\displaystyle h=\pi Q^{19}(ij^{3})=8\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}},\;u^{19}}$
${\displaystyle e^{-1}={\frac {\alpha \pi Q^{27}(i^{2}j^{3})}{4}}={\frac {\alpha }{128\pi ^{4}\Omega ^{3}}}{\frac {v^{3}}{r^{3}}},\;u^{27}}$
${\displaystyle k_{B}={\frac {\alpha \pi ^{2}Q^{29}(i^{2}j^{4})}{4}}={\frac {\alpha }{32\pi \Omega }}{\frac {r^{10}}{v^{3}}},\;u^{29}}$
${\displaystyle T^{-1}=2\pi Q^{30}(i^{2}j^{3})={\frac {1}{2\pi }}{\frac {v^{6}}{r^{9}}},\;u^{30}}$

### Solutions

Solving for these constants using α, Ω, r, v

Physical constants; geometrical vs experimental (CODATA)
Constant In Planck units Geometrical object Calculated (r, v, Ω, α*) SI CODATA 2014 [15]
Speed of light V ${\displaystyle c^{*}=(2\pi \Omega ^{2})v,\;u^{17}}$ c* = 299 792 458, unit = u17 c = 299 792 458 (exact)
Fine structure constant α* = 137.035 999 139 (mean) α = 137.035 999 139(31)
Rydberg constant ${\displaystyle R^{*}=({\frac {m_{e}}{4\pi L\alpha ^{2}M}})}$ ${\displaystyle R^{*}={\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}}{\frac {v^{5}}{r^{9}}},\;u^{13}}$ R* = 10 973 731.568 508, unit = u13 R = 10 973 731.568 508(65)
Vacuum permeability ${\displaystyle \mu _{0}^{*}={\frac {\pi V^{2}M}{\alpha LA^{2}}}}$ ${\displaystyle \mu _{0}^{*}={\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}}r^{7},\;u^{17*2+15+13-6=7*8=56}}$ μ0* = 4π/10^7, unit = u56 μ0 = 4π/10^7 (exact)
Planck constant ${\displaystyle h^{*}=2\pi MVL}$ ${\displaystyle h^{*}=2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}},\;u^{15+17-13=8*13-17*5=19}}$ h* = 6.626 069 134 e-34, unit = u19 h = 6.626 070 040(81) e-34
Gravitational constant ${\displaystyle G^{*}={\frac {V^{2}L}{M}}}$ ${\displaystyle G^{*}=2^{3}\pi ^{4}\Omega ^{6}{\frac {r^{5}}{v^{2}}},\;u^{34-13-15=8*5-17*2=6}}$ G* = 6.672 497 192 29 e11, unit = u6 G = 6.674 08(31) e-11
Elementary charge ${\displaystyle e^{*}=AT}$ ${\displaystyle e^{*}={\frac {2^{7}\pi ^{4}\Omega ^{3}}{\alpha }}{\frac {r^{3}}{v^{3}}},\;u^{3-30=3*8-17*3=-27}}$ e* = 1.602 176 511 30 e-19, unit = u-19 e = 1.602 176 620 8(98) e-19
Boltzmann constant ${\displaystyle k_{B}^{*}={\frac {\pi VM}{A}}}$ ${\displaystyle k_{B}^{*}={\frac {\alpha }{2^{5}\pi \Omega }}{\frac {r^{10}}{v^{3}}},\;u^{17+15-3=10*8-17*3=29}}$ kB* = 1.379 510 147 52 e-23, unit = u29 kB = 1.380 648 52(79) e-23
Electron mass ${\displaystyle m_{e}^{*}={\frac {M}{f_{e}}},\;u^{15}}$ me* = 9.109 382 312 56 e-31, unit = u15 me = 9.109 383 56(11) e-31
Classical electron radius ${\displaystyle \lambda _{e}^{*}=2\pi Lf_{e},\;u^{-13}}$ λe* = 2.426 310 2366 e-12, unit = u-13 λe = 2.426 310 236 7(11) e-12
Planck mass M ${\displaystyle m_{P}^{*}=(1){\frac {r^{4}}{v}},\;u^{15}}$ mP* = .217 672 817 580 e-7, unit = u15 mP = .217 647 0(51) e-7
Planck length L ${\displaystyle l_{p}^{*}=(2\pi ^{2}\Omega ^{2}){\frac {r^{9}}{v^{5}}},\;u^{-13}}$ lp* = .161 603 660 096 e-34, unit = u-13 lp = .161 622 9(38) e-34
Planck time T ${\displaystyle t_{p}^{*}=(2\pi ){\frac {r^{9}}{v^{6}}},\;u^{-30}}$ tp* = 5.390 517 866 e-44, unit = u-30 tp = 5.391 247(60) e-44
Ampere A ${\displaystyle A^{*}={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}{\frac {v^{3}}{r^{6}}},\;u^{3}}$ A^* = 0.148 610 6299 e25, unit = u3

### Conclusion

In the Trialogue on the number of fundamental constants [16] was debated the number of fundamental constants required by the universe. In terms of dimensioned constants (alpha and Omega are dimensionless) it appears that 1 (Quintessence) is required as from this unit the 4 mksa units can be derived.