Bully Metric

SI defining constants

Symbol	Defining constant	SI Exact value	Bully Exact value
speed of light in vacuum $c\,$	$1\,$	$1\,$	${\frac {1}{\alpha }}\$	${\frac {1}{\alpha }}\$	$1\,$	$299792458\$	$10^{8}\$
Planck's constant $h\,$	$2\pi \,$	${\frac {2\pi }{\alpha }}\$	$2\pi \,$	$2\pi \,$	${\frac {2\pi }{\alpha }}\$	${\frac {4\times 10^{-18}}{(25812.807)(483597.9)^{2}}}\$	$2\pi \times 10^{-34}\,$
reduced Planck's constant $\hbar ={\frac {h}{2\pi }}$	$1\,$	${\frac {1}{\alpha }}\$	$1\,$	$1\,$	${\frac {1}{\alpha }}\$	${\frac {2\times 10^{-18}}{\pi (25812.807)(483597.9)^{2}}}\$	$10^{-34}\,$
elementary charge $e\,$	${\sqrt {\alpha }}\,$	$1\,$	$1\,$	$1\,$	$1\,$	${\frac {2\times 10^{-9}}{(25812.807)(483597.9)}}\$	$10^{-18}\,$
Josephson constant $K_{J}={\frac {2e}{h}}\,$	${\frac {\sqrt {\alpha }}{\pi }}\,$	${\frac {\alpha }{\pi }}\,$	${\frac {1}{\pi }}\,$	${\frac {1}{\pi }}\,$	${\frac {\alpha }{\pi }}\,$	$483597.9\times 10^{9}\,$	${\frac {10^{16}}{\pi }}\,$
von Klitzing constant $R_{K}={\frac {h}{e^{2}}}\,$	${\frac {2\pi }{\alpha }}\,$	${\frac {2\pi }{\alpha }}\,$	$2\pi \,$	$2\pi \,$	${\frac {2\pi }{\alpha }}\,$	$25812.807\,$	$200\pi \,$
characteristic impedance of vacuum $Z_{0}=2\alpha R_{K}\,$	$4\pi \,$	$4\pi \,$	$4\pi \alpha \,$	$4\pi \alpha \,$	$4\pi \,$	$2\alpha (25812.807)\,$	$400\pi \alpha \,$
electric constant (vacuum permittivity) $\varepsilon _{0}={\frac {1}{Z_{0}c}}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{2\alpha (25812.807)(299792458)}}\$	${\frac {10^{-10}}{4\pi \alpha }}\,$
magnetic constant (vacuum permeability) $\mu _{0}={\frac {Z_{0}}{c}}\,$	$4\pi \,$	$4\pi \,$	$4\pi \alpha ^{2}\,$	$4\pi \alpha ^{2}\,$	$4\pi \,$	${\frac {2\alpha (25812.807)}{299792458}}\$	$4\pi \alpha \times 10^{-6}\,$
Newtonian constant of gravitation $G\,$	$1\,$	$1\,$	$1\,$	$-\,$	$-\,$	$-\,$	$-\,$
electron mass $m_{e}\,$	$-\,$	$-\,$	$-\,$	$1\,$	$1\,$	$-\,$	$-\,$
Hartree energy $E_{h}=\alpha ^{2}m_{e}c^{2}\,$	$-\,$	$-\,$	$-\,$	$1\,$	$\alpha ^{2}\,$	$-\,$	$-\,$
Rydberg constant $R_{\infty }={\frac {E_{h}}{2hc}}\,$	$-\,$	$-\,$	$-\,$	${\frac {\alpha }{4\pi }}\,$	${\frac {\alpha ^{3}}{4\pi }}\,$	$-\,$	$-\,$
caesium ground state hyperfine transition frequency	$-\,$	$-\,$	$-\,$	$-\,$	$-\,$	$9\ 192\ 631\ 770\,$	$2\ 808\ 349\ 000\,$

Six base units are included in the Bully Metric system. Three variants of the apan are defined as space-time units. And three variants of the nat are defined as transformation units.

The time apan (symbol ta) is by definition exactly 30.55 picoseconds. The length apan (or light apan) (symbol la) is by definition the distance light travels in vacuum in 30.55 picoseconds. The gravitational mass apan (symbol ma) is by definition the mass an object would have if its standard gravitational parameter (μ = MG) were equal to the speed of light in vacuum cubed, multiplied by 30.55 picoseconds.

The thermal nat (symbol Tn), has an SI equivalent value of 1.380649×10⁻²³ J/K. The action nat (symbol An) has an SI equivalent value of (1/2pi) × 6.62607015×10⁻³⁴ J⋅s. The rapidity/rotation nat (symbol Rn), does not have an SI equivalent value.

Apan Variants

ta = 30.55 picoseconds (exact)

la = c × 30.55 picoseconds (exact)
   = 9.1586595919 millimeters (exact)

ma = (c3 / G) × 30.55 picoseconds (exact)
   ≈ 12.3330 ronnagrams (approximate)

Nat variants

Tn = 1.380649×10−23 J/K (exact)
An = (1/2pi) × 6.62607015×10−34 J⋅s
Rn = does not have an SI equivalent value.

The above definitions ensure normalization of the speed of light and Newton's gravitational constant when using Bully units:

$c=1.0\,{\frac {la}{ta}}$ (exact)

$G=1.0\,{\frac {la^{3}}{ma\,ta^{2}}}$ (exact)

The Bully system includes units of transformation which are defined by analogy with units of information. These include the nat (n), bit (b), trit(t), and dit or digit (d). For each type of transformation unit, one may convert from nats to bits, trits, or dits, by multiplication with the natural logarithm as shown below:

b = n × loge(2)
t = n × loge(3)
d = n × loge(10)
where loge is the natural logarithm.

The above definitions ensure normalization of Boltzmann's constant and Planck's constant when using Bully units:

kB = 1.0 Tn (exact)
${\textstyle \hbar }$ = 1.0 An (exact)

Planck units and the Bully Metric

The following (table 1) was taken from the Wikipedia Planck units article:

Table 1: Modern values for Planck's original choice of quantities

Name	Expression	Value (SI units)
\frac{Planck time}{time apan}	$t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}$	5.391247(60)×10−44 s
Planck length	$l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}$	1.616255(18)×10−35 m
Planck mass	$m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}$	2.176434(24)×10-8 kg
Planck temperature	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}$	1.416784(16)×1032 K

Since c and G are normalized in the Bully system, this ensures that Bully units should have a simple relationship with Planck's units. As illustrated below, multiplying each SI value from Table 1 by 566.66, results in the corresponding Bully value multiplied by 10^-30:

566.66 × tP = 566.66 × 5.391247(60)×10−44 s
                       = 3055.004(34)×10−44 s
                       = 30.55004(34)×10−30 ps
                       = 1.00001(11)×10−30 ta

566.66 × lP = 566.66 × 1.616255(18)×10−35 m
                       = 915.867(10)×10−35 m
                       = 9.15867(10)×10−30 mm
                       = 1.00001(11)×10−30 la

566.66 × mP = 566.66 × 2.176434(24)×10-8 kg
                       = 1233.298(14)×10−8 kg
                       = 12.33298(14)×10−30 rg
                       = 1.00001(11)×10−30 ma

SI prefixes will have the same meaning and conventions when used with apan variants as they have when used with standard SI units (see table in subsequent section for a list of SI prefixes). The "quecto" (symbol "q") metric prefix means 10^-30. The relationship between Bully units and Planck's units can be summarized as:

566.66 × tP = 1.00001(11) qta
566.66 × lP = 1.00001(11) qla
566.66 × mP = 1.00001(11) qma

Planck units are understood to represent the smallest meaningful size of each quantity. For example, the Planck length is the smallest possible length because looking at small objects requires energy. If one were to build a microscope powerful enough to see objects of Planck length or smaller, the microscope would use so much energy that a black hole would form. In terms of Bully units, the "quecto" of each unit is 566.66 times larger than the absolute minimum size for that unit.

The Planck mass may seem unexpectedly large for a minimum mass value, but keep in mind that in this case the unit is for gravitational mass. There obviously are well defined and detectable masses that are smaller than the Planck mass (for example the electron and proton masses), but the Planck mass represents the boundary between gravitational mass and quantum mass. If an object has a mass larger than the Planck mass then gravitational effects will dominate. If the mass is smaller than the Planck mass then quantum mechanical effects will dominate.

The apan prefix table

Prefix		Base 10	Spacetime Symbols			Transformation Symbols
Name	Symbol		Time	Length	Mass	Thermal	Action	Rapidity
quetta	Q	1030	Qta	Qla	Qma	QTn	QAn	QRn
ronna	R	1027	Rta	Rla	Rma
yotta	Y	1024	Yta	Yla	Yma
zetta	Z	1021	Zta	Zla	Zma
exa	E	1018	Eta	Ela	Ema
peta	P	1015	Pta	Pla	Pma
tera	T	1012	Tta	Tla	Tma
giga	G	109	Gta	Gla	Gma
mega	M	106	Mta	Mla	Mma
kilo	k	103	kta	kla	kma
—	—	100	ta	la	ma
milli	m	10−3	mta	mla	mma
micro	μ	10−6	μta	μla	μma
nano	n	10−9	nta	nla	nma
pico	p	10−12	pta	pla	pma
femto	f	10−15	fta	fla	fma
atto	a	10−18	ata	ala	ama
zepto	z	10−21	zta	zla	zma
yocto	y	10−24	yta	yla	yma
ronto	r	10−27	rta	rla	rma
quecto	q	10−30	qta	qla	qma
minimum value	-	${\frac {10^{-30}}{566.66}}$	${\frac {qta}{566.66}}$	${\frac {qla}{566.66}}$	${\frac {qma}{566.66}}$

2.176434(24)×10^-8 kg

$c=1.0\,{\frac {la}{ta}}$ (exact)

$G=1.0\,{\frac {la^{3}}{ma\,ta^{2}}}$ (exact)

Table 2: Planck's units relationship with Bully units

Name	Expression
Planck time	$t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}={\sqrt {\frac {\hbar {\frac {la^{3}}{ma\,ta^{2}}}}{\frac {la^{5}}{ta^{5}}}}}={\sqrt {\frac {\hbar \,ta}{ma\,la^{2}}}}\,ta$
Planck length	$l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}={\sqrt {\frac {\hbar {\frac {la^{3}}{ma\,ta^{2}}}}{\frac {la^{3}}{ta^{3}}}}}={\sqrt {\frac {\hbar \,ta}{ma\,la^{2}}}}\,la$
Planck mass	$m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}=={\sqrt {\frac {\hbar {\frac {la}{ta}}}{\frac {la^{3}}{ma\,ta^{2}}}}}={\sqrt {\frac {\hbar \,ta}{ma\,la^{2}}}}\,ma$
Planck temperature	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}={\sqrt {\frac {\hbar {\frac {la^{5}}{ta^{5}}}}{{\frac {la^{3}}{ma\,ta^{2}}}k_{\text{B}}^{2}}}}$

The 'Bully Metric is an extremely efficient set of time and distance measurement units for representing earth's physical parameters. Bully units can efficiently represent the Earth's sidereal year and tropical year to eight digits; The Bully Metric can also efficiently represent four digit approximations for the Earth's radius (r ≈ 6371), Schwarzschild radius (R), standard gravitational parameter (μ = MG ≈ 3.984e14), and a typical gravitational acceleration on earth's surface (g ≈ 9.813 ).

$${1\,Sidereal\,Year}={1.033\,Eta}={31,558,150\,s}$$

$${1\,Tropical\,Year}=(1.033-0.00004)\,Eta-2\,s=31,556,926\,s$$

$$r_{earth}\approx {\frac {1\,Gla}{\sqrt {2\times 1.033}}}=6371\,km$$

$$R_{earth}\approx {\frac {la}{1.033}}\approx {8.866\,mm}$$

$${\mu }_{earth}\approx {\frac {1}{2\times 1.033}}\,{\frac {la^{3}}{ta^{2}}}\approx {398,400,000,000,000\,{\frac {m^{3}}{s^{2}}}}$$

$$g_{earth}\approx 1{\frac {Ela}{Eta^{2}}}\approx {9.813\,{\frac {m}{s^{2}}}}$$

The Bully Constants

There are a surprising number of physical constants that can be approximated using various algebraic combinations of the following four numbers (click here to learn more):

1.033
30.55
2
0.00004

Sidereal year

The number of seconds in the Earth's sidereal year can be approximated as:

$${1\,Sidereal\,Year}=1.033\times 30.55\,Ms={31,558,150\,s}$$

Tropical year

The number of seconds in the Earth's tropical year can be approximated as:

$${1\,Tropical\,Year}=((1.033-0.00004)\times 30.55\,Ms)-2\,s=31,556,926\,s$$

Great year

The number of tropical years in a Great Year can be approximated as:

$${1\,Great\,Year}\approx ({\frac {1.033}{0.00004}})\,{Tropical\,Years}={25,825\,Tropical\,Years}$$

Earth's radius (r)

An approximate relationship of the speed of light to the Earth's radius (r):

$$r_{earth}\approx {\frac {c\times 30.55\,ms}{\sqrt {2\times 1.033}}}=6371\,km$$

Earth's Schwarzschild radius (R)

An approximate relationship of the speed of light to the Earth's Schwarzschild radius (R):

$$R_{earth}={\frac {2\times GM_{earth}}{c^{2}}}\approx {\frac {c\times 30.55\,ps}{1.033}}\approx {8.866\,mm}$$

Earth's standard gravitational parameter (μ = MG)

An approximate relationship of the speed of light to the Earth's standard gravitational parameter (μ = MG):

$${\mu }_{earth}=GM_{earth}\approx {\frac {c^{3}\times 30.55\,ps}{2\times 1.033}}\approx {398,400,000,000,000\,{\frac {m^{3}}{s^{2}}}}$$

Earth's gravity (g)

A typical gravitational acceleration on earth's surface can be approximated as:

$$g_{earth}={\frac {{\mu }_{earth}}{{r_{earth}}^{2}}}\approx {\frac {c}{30.55Ms}}\approx {9.813\,{\frac {m}{s^{2}}}}$$

Definition of the apan

The above approximations for earth's physical parameters can be further simplified by introducing new measurement units. The apan will be defined with two variants. The time apan (symbol ta) is by definition exactly 30.55 picoseconds. The length apan (or light apan) (symbol la) is by definition the distance light travels in vacuum in exactly 30.55 picoseconds.

SI prefixes will have the same meaning and conventions when used with the apan as the have when used with standard SI units (see table in subsequent section). For example:

One million ta = 1,000,000 ta = 1 mega time apan = 1 Mta = 30.55 μs
One million la = 1,000,000 la = 1 mega light apan = 1 Mla = 9.1586595919 km

Approximations for earth's physical parameters can be written in terms of the apan as follows:

Sidereal year

The number of seconds in the Earth's sidereal year can be approximated as:

$${1\,Sidereal\,Year}={1.033\,Eta}={31,558,150\,s}$$

Tropical year

The number of seconds in the Earth's tropical year can be approximated as:

$${1\,Tropical\,Year}=(1.033-0.00004)\,Eta-2\,s=31,556,926\,s$$

Earth's radius (r)

An apan based approximation of Earth's radius (r):

$$r_{earth}\approx {\frac {1\,Gla}{\sqrt {2\times 1.033}}}=6371\,km$$

Earth's Schwarzschild radius (R)

An apan based approximation of the Earth's Schwarzschild radius (R):

$$R_{earth}\approx {\frac {la}{1.033}}\approx {8.866\,mm}$$

Earth's standard gravitational parameter (μ = MG)

An apan based approximation of the Earth's standard gravitational parameter (μ = MG):

$${\mu }_{earth}\approx {\frac {1\,{\frac {la^{3}}{ta^{2}}}}{2\times 1.033}}\approx {398,400,000,000,000\,{\frac {m^{3}}{s^{2}}}}$$

Earth's gravity (g)

A typical gravitational acceleration on earth's surface can be approximated as:

$$g_{earth}={\frac {{\mu }_{earth}}{{r_{earth}}^{2}}}\approx 1{\frac {Ela}{Eta^{2}}}\approx {9.813\,{\frac {m}{s^{2}}}}$$

The apan prefix table

Prefix		Base 10	Bully Metric		SI Equivalent
Name	Symbol		Time	Length	Time	Length
quetta	Q	1030	Qta	Qla	30.55 Es	9.1586595919 Rm
ronna	R	1027	Rta	Rla	30.55 Ps	9.1586595919 Ym
yotta	Y	1024	Yta	Yla	30.55 Ts	9.1586595919 Zm
zetta	Z	1021	Zta	Zla	30.55 Gs	9.1586595919 Em
exa	E	1018	Eta	Ela	30.55 Ms	9.1586595919 Pm
peta	P	1015	Pta	Pla	30.55 ks	9.1586595919 Tm
tera	T	1012	Tta	Tla	30.55 s	9.1586595919 Gm
giga	G	109	Gta	Gla	30.55 ms	9.1586595919 Mm
mega	M	106	Mta	Mla	30.55 μs	9.1586595919 km
kilo	k	103	kta	kla	30.55 ns	9.1586595919 m
—	—	100	ta	la	30.55 ps	9.1586595919 mm
milli	m	10−3	mta	mla	30.55 fs	9.1586595919 μm
micro	μ	10−6	μta	μla	30.55 as	9.1586595919 nm
nano	n	10−9	nta	nla	30.55 zs	9.1586595919 pm
pico	p	10−12	pta	pla	30.55 ys	9.1586595919 fm
femto	f	10−15	fta	fla	30.55 rs	9.1586595919 am
atto	a	10−18	ata	ala	30.55 qs	9.1586595919 zm
zepto	z	10−21	zta	zla	30.55e-3 qs	9.1586595919 ym
yocto	y	10−24	yta	yla	30.55e-6 qs	9.1586595919 rm
ronto	r	10−27	rta	rla	30.55e-9 qs	9.1586595919 qm
quecto	q	10−30	qta	qla	30.55e-12 qs	9.1586595919e-3 qm