A Simulation Hypothesis (Summary)

Programming Planck units from a virtual electron; a Simulation Hypothesis (summary)

The Simulation Hypothesis proposes that all of reality, including the earth and the universe, is in fact an artificial simulation, analogous to a computer simulation, and as such our reality is an illusion. In this essay I describe a method for programming mass, length, time and charge (MLTA) as geometrical objects derived from 2 3 6 2 5 the formula for a virtual electron; fe = 4π r (r = 2 3π αΩ ) where the fine structure constant α = 137.03599... and Ω = 2.00713494... are mathematical constants and the MLTA geometries are; M = (1), T = (2π), L = (2π2Ω2), A = (4πΩ)3/α. As objects they are independent of any set of units and also of any numbering system, terrestrial or alien. As the geometries are interrelated, we can replace designations such as (kg, m, s, A) with a 15 −13 −30 3 rule set; mass = u , length = u , time = u , ampere = u . The formula fe combines these geometries in the following ratio M9T11/L15 and (AL)3/T, as such this electron formula is unit-less (u0). To translate MLTA to their respective SI Planck units requires an additional 2 unit-dependent scalars. We may thereby derive the CODATA 2014 physical constants via 2 (fixed) mathematical constants (α, Ω), 2 (variable) dimensioned scalars and the rule set u. As all constants can be defined geometrically, the least precise constants (G, h, e, me, kB...) can also be solved via the most precise (c, µ0, R∞, α), numerical precision then limited by the precision of the fine structure constant α.

Table 1 Calculated values from (c, µ0, R∞, α) [10] CODATA 2014 Fine structure constant α∗ = 137.035999139 α = 137.035999139(31) Speed of light c∗ = 299792458 u17 c = 299792458 ∗ 7 56 7 Permeability µ0 = 4π/10 u µ0 = 4π/10 ∗ 13 Rydberg constant R∞ = 10973731.568 508 u R∞ = 10973731.568 508(65) Planck constant h∗ = 6.626 069 134 e-34 u19 h = 6.626 070 040(81) e-34 Elementary charge e∗ = 1.602 176 511 30 e-19 u−27 e = 1.602 176 6208(98) e-19 2 ∗ von Klitzing (h/e ) RK = 25812.807 45559 RK = 25812.807 4555(59) ∗ 15 Electron mass me = 9.109 382 312 56 e-31 u me = 9.109 383 56(11) e-31 ∗ −13 Electron wavelength λe = 2.426 310 2366 e-12 u λe = 2.426 310 2367(11) e-12 ∗ 29 Boltzmann’s constant kB = 1.379 510 147 52 e-23 u kB = 1.380 648 52(79) e-23 Gravitation constant G∗ = 6.672 497 192 29 e-11 u6 G = 6.674 08(31) e-11 ∗ −13 Planck length lp = .161 603 660 096 e-34 u lp = .161 6229(38) e-34 ∗ 15 Planck mass mP = .217 672 817 580 e-7 u mP = .217 6470(51) e-7 ∗ −42 Gyromagnetic ratio γe/2π = 28024.953 55 u γe/2π = 28024.951 64(17)e-7

Keywords: virtual electron, mathematical electron, black-hole electron, simulation hypothesis, computer universe, mathematical universe, physical constants, Planck units, sqrt Planck momentum, magnetic monopole, ﬁne structure constant, alpha, Omega;

1 Background mind [3]. The Simulation Hypothesis proposes that all of reality, in- Max Tegmark proposed a Mathematical Universe Hypothe- cluding the earth and the universe, is in fact an artificial sim- sis that states: Our external physical reality is a mathematical ulation, analogous to a computer simulation [2]. structure. That is, the physical universe is mathematics in a In the “Trialogue on the number of fundamental physi- well-defined sense, and in those [worlds] complex enough to cal constants” was debated the number of fundamental di- contain self-aware substructures [they] will subjectively per- mension units required, noting that ”There are two kinds of ceive themselves as existing in a physically ’real’ world” [9]. fundamental constants of Nature: dimensionless (α) and di- Mathematical Platonism is a metaphysical view that there mensionful (c, h, G). To clarify the discussion I suggest to are abstract mathematical objects whose existence is indepen- refer to the former as fundamental parameters and the lat- dent of us [1]. Mathematical realism holds that mathematical ter as fundamental (or basic) units. It is necessary and suf- entities exist independently of the human mind. Thus humans ficient to have three basic units in order to reproduce in an do not invent mathematics, but rather discover it. Triangles, experimentally meaningful way the dimensions of all physi- for example, are real entities, not the creations of the human cal quantities. Theoretical equations describing the physical

1 1 Background world deal with dimensionless quantities and their solutions measurable quantities which we measure using the SI units or depend on dimensionless fundamental parameters. But exper- imperial unit equivalents and assign them to (dimensioned) iments, from which these theories are extracted and by which physical constants that we use as reference, for example all they could be tested, involve measurements, i.e. comparisons velocities may be measured relative to c. These units however with standard dimensionful scales. Without standard dimen- are terrestrial units, although Max Planck proposed a set of sionful units and hence without certain conventions physics natural units, his Planck units are still measured in terrestrial −8 is unthinkable” -Trialogue [5]. terms; Planck mass mP = 2.17647... x10 kg or 4.79825... J. Barrow and J. Webb on the physical constants; ’Some x10−8 lbs. things never change. Physicists call them the constants of Mathematical universe hypotheses presume that our phys- nature. Such quantities as the velocity of light, c, Newton’s ical universe has an underlying mathematical origin. The constant of gravitation, G, and the mass of the electron, me, principal difficulty of such hypotheses lies in the problem of are assumed to be the same at all places and times in the uni- how to construct these physical units, the units that confer verse. They form the scaffolding around which theories of ‘physical-ness’ to our universe, from their respective mathe- physics are erected, and they define the fabric of our uni- matical forms. In the following I describe a system of units verse. Physics has progressed by making ever more accu- that is based on geometrical objects and so is independent rate measurements of their values. And yet, remarkably, no of any particular system of units and also of any numbering one has ever successfully predicted or explained any of the system, yet may be used to reproduce our physical constants constants. Physicists have no idea why they take the special (see table p1) [14]. The model is based on a virtual (unit- numerical values that they do. In SI units, c is 299,792,458; less) electron formula fe from which natural units of mass G is 6.673e-11; and me is 9.10938188e-31 -numbers that fol- M, length L, time T and charge A (ampere) may be derived low no discernible pattern. The only thread running through according to these ratio. the values is that if many of them were even slightly different, 2 6 2 5 3 23 complex atomic structures such as living beings would not be fe = 4π (2 3π αΩ ) = .23895453...x10 (1) possible. The desire to explain the constants has been one of r the driving forces behind efforts to develop a complete uni- (AL)3 L15 units = = = 1 fied description of nature, or ”theory of everything”. Physi- T M9T 11 cists have hoped that such a theory would show that each of the constants of nature could have only one logically possi- The fine structure constant α (4.5.) and Ω (4.7.) are mathe- ble value. It would reveal an underlying order to the seeming matical constants, thus the electron formula fe is also a math- arbitrariness of nature.’ [6]. ematical constant (units = 1). From the above ratio we can extract geometrical objects for ALT (AL as ampere-meter are At present, there is no candidate theory of everything that units for a magnetic monopole) and MLT. The following are is able to calculate the mass of the electron [12]. proposed; Planck units (mP, lp, tp, ampere Ap, TP) are a set of natural units of measurement defined exclusively in terms of five M = (1) (2) universal physical constants, in such a manner that these five T = (2π) (3) constants take on the numerical value of G = ~ = c = 1/4π0 = kB = 1 when expressed in terms of these units. These units L = (2π2Ω2) (4) are also known as natural units because the origin of their def- inition comes only from properties of nature and not from any 26π3Ω3 A = ( ) (5) human construct. Max Planck [7] wrote of these units; ”we α get the possibility to establish units for length, mass, time and temperature which, being independent of specific bodies or 3 Unit u substances, retain their meaning for all times and all cultures, If the mass, space, time and charge units are not independent even non-terrestrial and non-human ones and could therefore of each other then we can then assign a rule set that designates serve as natural units of measurements...”. the relationships between them. u15 (mass) 2 Geometrical objects MLTA u−30 (time) In 1963, Dirac noted regarding the fundamental constants; u−13 (length) ”The physics of the future, of course, cannot have the three u3 (ampere) quantities ~, e, c all as fundamental quantities, only two of them can be fundamental, and the third must be derived We can then construct a table of units, for example; from those two.” [16] Velocity V = length/time = u−13+30=17 Our ‘physical’ universe is defined in terms of fundamental Elementary charge = ampere x time = u3−30=−27

2 3 Unit u 4 Scalars P9 T 2 π 15 = (2 Ω) 12 (27) 4.1. In order to translate from geometrical objects to a nu- 2πV merical system of units, (dimensioned) scalars are required. I p9/2 T = (2π) , unit = u16∗9/2−17∗6=−30 (28) assign these scalars kltvpa with their corresponding unit u. v6 TV p9/2 k, unit = u15 (mass) (6) L = = (2π2Ω2) , unit = u16∗9/2−17∗5=−13 (29) 2 v5 −30 t, unit = u (time) (7) 8V3 26π3Ω3 v3 A = = ( ) , unit = u17∗3−16∗3=3 (30) p, unit = u16 (sqrt o f momentum) (8) αP3 α p3 v, unit = u17 (velocity) (9) From the Planck units we can solve the physical constants G, h, e, m , k . To maintain integer exponents (for clarity) I −13 e B √ √ l, unit = u (length) (10) replace p with r = p = Ω, unit u16/2=8 a, unit = u3 (ampere) (11) V2L r5 G∗ = = 23π4Ω6 , u34−13−15=8∗5−17∗2=6 (31) The formulas for base units MLTVPA now become; M v2 15 M = (1)k, unit = u (mass) (12) r13 h∗ = 2πMVL = 23π4Ω4 , u15+17−13=8∗13−17∗5=19 (32) − v5 T = (2π)t, unit = u 30 (time) (13) AV 27π3Ω5 v4 P = (Ω)p, unit = u16 (sqrt o f momentum) (14) T ∗ = = , u3+17=17∗4−6∗8=20 (33) P π α r6 2 17 V = (2πΩ )v, unit = u (velocity) (15) 27π4Ω3 r3 e∗ = AT = , u3−30=3∗8−17∗3=−27 (34) L = (2π2Ω2)l, unit = u−13 (length) (16) α v3 6 3 3 10 2 π Ω ∗ πVM α r − ∗ − ∗ A = ( )a, unit = u3 (ampere) (17) k = = , u17+15 3=10 8 17 3=29 (35) α B A 25πΩ v3

4.2.1. To convert to CODATA 2014 values only 2 of these ∗ M 15 me = , u (36) 6 kltvpa scalars are required to deﬁne the other 4. In this fe example I derive LPVA from MT. The formulas for MT; ∗ −13 λe = 2πL fe, u (37) M = (1)k, unit = u15 (18) 2 ∗ πV M α 7 17∗2+15+13−6=7∗8=56 µ0 = = r , u (38) − αLA2 11 5 4 T = (2π)t, unit = u 30 (19) 2 π Ω ∗−1 α 2 7 34+56=90 9 11 15 = v r , u (39) From the ratio M T = L (eq.1) we extract pvla; 0 29π3 12/15 8π5k4 α k 12/15∗15−2/15∗(−30)=16 ∗ B 29∗4−19∗3−17∗3=8 P = (Ω) , unit = u (20) rσ = ( ) = r, u (40) t2/15 15h3c3 22915π14Ω22 5 2 9/15 ∗ me 1 v 13 2πP 2 k 9/15∗15−4/15∗(−30)=17 R = ( ) = , u (41) V = = (2πΩ ) , unit = u 4πl α2m 22333π11α5Ω17 r9 M t4/15 p P (21) As (α, Ω) have ﬁxed values we need only assign appropriate TV L = = (2π2Ω2) k9/15t11/15, unit = u9/15∗15+11/15∗(−30)=−13 numerical values to 2 of the scalars (i.e.: k, t or r, v) to solve 2 (22) these constants with results as listed in table 1. 3 3 3 ! 4.3. We then note within the electron f ratios M9T 11/L15 8V 64π Ω 1 9/15∗(−15)+6/15∗30=3 e A = = , unit = u 3 αP3 α k3/5t2/5 and (AL) /T, the scalars and units cancel leaving only the (23) unit-less (α, Ω) geometrical objects (eq. 2-5). Consequently 4.2.2. In this example I derive MLTA from PV; these ‘electron’ ratios are independent of any system of units, to quote Max Planck ‘whether terrestrial or alien’. P = (Ω)p, unit = u16 (24) −7 15 k = mP = .21767281758... 10 , u (kg) (42) V = (2πΩ2)v, unit = u17 (25) tp t = = .17158551284...10−43, u−30 (s) (43) MTVA in terms of PV 2π 2 2 2πP p lp M = = (1) , unit = u16∗2−17=15 (26) l = = .20322086948...10−36, u−13 (m) (44) V v 2π2Ω2

3 4 Scalars Apα α 128π4Ω3 a = = .12691858859...1023, u3 (A) (45) α = 2(8π4Ω4)/( )( )2(2πΩ2) = α (60) 64π3Ω3 211π5Ω4 α The scalars ktla and units u cancel; r13 1 v6 1 scalars = . . . = 1 5 7 6 15 15 2 2 15 15 v r r v L lp (2π Ω ) l = = . = 24π19Ω30 (46) 19 9 11 9 11 9 11 9 11 u M T m tp (1) (2π) k t units = = 1 P u56(u−27)2u17 l15 (.203...x10−36)15 u−13∗15 4.6. The Planck units are known with a low numerical preci- = = 1 k9t11 (.217...x10−7)9(.171...x10−43)11 u15∗9u−30∗11 sion, 1 reason why they are not commonly used. Conversely 2 (47) of the CODATA 2014 physical constants have been assigned 3 3 3 3 6 3 3 3 2 2 3 3 3 20 14 15 A L Aplp (2 π Ω ) (2π Ω ) a l 2 π Ω exact numerical values; c and permeability of vacuum µ . . 0 = = 3 = 3 T tp (α) (2π) t α Thus scalars r and v were used as they can be derived directly (48) ∗ ∗ from the formulas for c and µ0 (4.2.2.). a3l3 (.126...x1023)3(.203...x10−36)3 u3∗3u−13∗3 = = 1 (49) c t (.171...x10−43) u−30 v = = 11843707.9..., units = m/s (61) 2πΩ2 In 4.2.2. I defined MLTA in terms of PV. Replacing MLTA with those PV derivations, we find that P and V themselves 211π5Ω4µ kg.m r7 = 0 ; r = .712562514..., units = ( )1/4 cancel leaving only the dimensionless components. We may α s note that throughout this model we find the geometry Ω15 in- (62) dicative of unit-less ratios, a geometrical ‘base 15’. The most precise of the experimentally measured constants is −1 the Rydberg R = 10973731.568508(65) m . Here c, µ0, R are L30 2180π210Ω225P135 218π18P36 2154π154Ω165P99 combined into a unit-less ratio; = / . 18 22 150 18 132 M T V V V ∗ 35 (50) (c ) 2 35 α 9 1 7 30 = (2πΩ ) /( ) .( ) (63) L 2 (µ∗)9(R∗)7 211π5Ω4 22333π11α5Ω17 = (24π19Ω30) (51) 0 M18T 22 (u17)35 A6L6 218V18 236π42Ω45P27 214π14Ω15P9 units = = 1 = . / (52) (u56)9(u13)7 T 2 α6P18 V30 V12 4.7. I have premised a 2nd mathematical constant I denoted A6L6 220π14Ω15 2 Ω. We can define using geometries for (c∗, µ∗, R∗) and then 2 = ( 3 ) (53) 0 T α numerically solve by replacing the geometrical (c∗, µ∗, R∗) 4.4. The electron formula f is both unit-less and non scal- 0 e with the numerical (c, µ0, R) CODATA 2014 values. Rewrit- 0 0 0 0 0 able k t v r u = 1. It is therefore a natural (mathematical) ing eq.63 in terms of Ω; constant, σe has units for a magnetic monopole, σtp a hypo- thetical temperature monopole. (c∗)35 Ω225 = , units = 1 (64) 2295321π157(µ∗)9(R∗)7α26 r9 0 T = (2π) , u−30 (54) v6 Ω = 2.007 134 9496..., units = 1

2 3 3α AL 7 3 5 r −10 There is a close natural number for Ω that is a sqrt implying σ = = 2 3π αΩ , u (55) 2 2 e π2 v2 that Ω can have a plus and a minus solution; (+Ω) = (−Ω) . 3 7 3 5 3 −10 3 σ (2 3π αΩ ) (u ) r e f = e = , units = = 1 (56) π e −30 Ω = = 2.007 134 9543... (65) T 2π u e(e−1) 3α2T v4 σ = P = 263π2αΩ5 , units = u20 (57) 4.8. We can use the same approach to also numerically solve tp 2π r6 G, h, e, me, kB by ﬁrst rewriting their geometrical formulas in 2 3 2 6 2 5 3 −30 2 20 3 ∗ ∗ ∗ fe = t σ = 4π (2 3π αΩ ) , units = (u ) (u ) = 1 p tp terms of (c , µ0, R ) and then replacing with the CODATA (58) 2014 values for (c, µ0, R, α). Here I solve for Planck’s con- 4.5. The Sommerfeld ﬁne structure constant alpha is a di- stant. mensionless mathematical constant. The following uses a r13 h∗ = 23π4Ω4 , u19 (66) well known formula for alpha (note: for convenience I use v5 the commonly recognized value for alpha as α ∼ 137); 10 ∗ 3 r13 2π (µ ) (h∗)3 = (23π4Ω4 )3, u19∗3 = 0 , unit = u57 2h v5 36(c∗)5α13(R∗)2 α = (59) 2 (67) µ0e c

4 4 Scalars Likewise with the other constants. L5/2 u20(xy2) = = AV, (T ) M3/2T 5/2 P 4π5 √ (e∗)3 = , unit = u−81 (68) 3/2 3 ∗ 4 8 ∗ 27 2 3 M T 3 (c ) α (R∞) u (x y ) = = 1/AT, (1/e) L3/2 5 ∗ 3 √ π (µ ) M5/2 T (k∗ )3 = 0 , unit = u87 (69) 29 2 4 B 3 ∗ 4 5 ∗ u (x y ) = √ = ML/AT, (kB) 3 2(c ) α (R∞) L π3(µ∗) u30(x2y3) = 1/T, (1/t ) (G∗)5 = 0 , unit = u30 (70) p 20 6 11 ∗ 2 2 3 α (R∞) M4T ML u56 x4y7 , µ 10 ∗ ∗ 3 ( ) = 2 = 2 2 ( 0) 16π (R∞)(µ ) L T A (m∗)3 = 0 , unit = u45 (71) e 36(c∗)8α7 5.2. To derive formulas for MLTVA we simply repeat the above, assigning β (unit = u), i (from x) and j (from y). π22(µ∗)9 (l∗ )15 = 0 , unit = (u−13)15 (72) √ √ p 35 24 49 ∗ 35 ∗ 8 8 2 3 α (c ) (R∞) R = P = Ωr, units = u (80) ∗ 225π13(µ )6 2 1/3 1/3 (m∗ )15 = 0 , unit = (u15)15 (73) V 2πR A α P 6 ∗ 5 16 ∗ 2 β = = = ..., unit = u (81) 3 (c ) α (R∞) R2 M 2 gl∗ m∗ 1 p P −13−29=3−30−15=−42 i = , unit = 1 γe/2π = ∗ ∗ , unit = u (74) 15 2kBme 2π(2πΩ) 3 3 ∗ 4 17 8 17∗8 g 3 (c ) r 2 k u 15∗2 −30 (γ /2π)3 = (75) j = = k t = ..., unit = = u u ... = 1 e 8 8 ∗ 3 ∗ 2 v8 r15 u8∗17 2 π α(µ0) (R∞) We can reproduce the (r, v) formulas from 4.2.2. Inserting the above in the alpha formula V 2πΩ2v 8(h∗)3 β = = u (82) α3 = = α3, units = 1 (76) 2 2 ∗ 3 ∗ 6 ∗ 3 R Ωr (µ0) (e ) (c ) 3 6 3 3 3 3 2 2 π Ω v 3 As such, we may numerically solve the least precise physical A = β ( ) = , u (83) α α r6 constants in terms of the 4 most precise. Results are consis- β6 r5 tent with CODATA 2014 (table p1). G j 3π4 6 , u6 = 3 2 ( ) = 2 Ω 2 (84) p 2 π v 5 Unit u as length/mass x time 1 v5 √ L−1 = 4πβ13(i j) = , u13 (85) 5.1. Setting u = L/M.T we construct a table of units (3.). 2π2Ω2 r9 r r4 L √ M = 2πβ15(i j2) = , u15 (86) u, units = = u−13−15+30=2 = u1 (77) v MT P = β16(i j2) = Ωr2, u16 (87) r M9T 11 17 2 2 17 x, units = = u0 = 1 (78) V = β (i j ) = 2πΩ v, u (88) 15 L 13 19 3 4 4 r 19 y, units = M2T = u0 = 1 (79) h = πβ (i j ) = 8π Ω , u (89) v5 This gives us; 23β20 27π3Ω5 v4 T ∗ = (i j2) = , u20 (90) P 6 3/2 πα α r 3 L u A ampere 27 2 3 3 = / / = , ( ) απβ (i j ) α v M3 2T 3 2 e−1 = = , u27 (91) 4 128π4Ω3 r3 6 3 2 u (y) = L /T M, (G) 2 29 2 4 10 απ β (i j ) α r 29 13 kB = = , u (92) u (xy) = 1/L, (1/lp) 4 32πΩ v3 15 2 6 u xy M, m − 1 v ( ) = ( P) T 1 = 2πβ30(i2 j3) = , u30 (93) 2π r9 u17(xy2) = V, (c) π3αβ56 α 19 3 2 µ∗ = (i4 j7) = r7, u56 (94) u (xy ) = ML /T, (h) 0 23 211π5Ω4

p 5 5 Unit u as length/mass x time 3 90 ∗− π αβ α A charged rotating black hole is a black hole that pos- 1 = (i6 j11) = v2r7, u90 (95) 0 23 29π3 sesses angular momentum and charge. In particular, it ro- 5.3. We require 3 units to cancel both u and the 2 scalars. tates about one of its axes of symmetry. In physics, there is With 2 units we may cancel u but retain our numerical SI a speculative notion that if there were a black hole with the values, i, j suggest a limit to the values the SI constants can same mass and charge as an electron, it would share many of take, the boundary imposed between 10−59 to 1060. the properties of the electron including the magnetic moment and Compton wavelength. This idea is substantiated within a r17 k17/4 series of papers published by Albert Einstein between 1927 = k2t = = ... = .812997...x10−59, units = 1 (96) v8 v15/4 and 1949. In them, he showed that if elementary particles were treated as singularities in spacetime, it was unnecessary In SI terms β has this value; to postulate geodesic motion as part of general relativity [11]. √ The Dirac Kerr–Newman black-hole electron was intro- v 1 v a1/3 = = = √ ... = 23326079.1...; unit = u duced by Burinskii using geometrical arguments. The Dirac 2 2/15 1/5 r t k k wave function plays the role of an order parameter that sig- (97) nals a broken symmetry and the electron acquires an extended space-time structure. Although speculative, this idea was cor- In summary I have described a programmable approach [15] roborated by a detailed analysis and calculation [8]. using universal geometrical objects based on a formula for a virtual electron (mathematical constant) from which we may References derive the CODATA 2014 values with associated units via; 1. Linnebo, Øystein, ”Platonism in the Philosophy of - 2 (fixed) mathematical constants (α, Ω), Mathematics”, The Stanford Encyclopedia of Philos- - 2 (variable) unit-dependent scalars ophy (Summer 2017 Edition), Edward N. Zalta (ed.), - a unit u rule-set. plato.stanford.edu/archives/sum2017/entries/platonism- mathematics In the “Trialogue on the number of fundamental physical constants” was debated the number, from 0 to 3, of dimen- 2. Nick Bostrom, Philosophical Quarterly 53 (211):243–255 sionful units required [5]. Here the answer is both 0 and 1; (2003) 0 in that the electron, being a virtual particle, has no units, 3. https://en.wikipedia.org/wiki/Philosophy-of-mathematics yet it can unfold to form the Planck units and these can be (22, Oct 2017) de-constructed in terms of the unit u, and so in terms of the 4. Leonardo Hsu, Jong-Ping Hsu; The physical basis of nat- physical universe, being a dimensioned universe (a universe ural units; Eur. Phys. J. Plus (2012) 127:11 of measurable units) the answer is 1. 5. Michael J. Duff et al JHEP03(2002)023 Trialogue on the number of fundamental constants 6 Additional notes on the physical constants 6. J. Barrow, J. Webb, Inconstant Constants, Scientific Amer- In the article ”Surprises in numerical expressions of physical ican 292, 56 - 63 (2005) constants”, Amir et al write ... In science, as in life, ‘sur- 7. M. Planck, Uber irreversible Strahlungsforgange. Ann. d. prises’ can be adequately appreciated only in the presence Phys. (4), (1900) 1, S. 69-122. of a null model, what we expect a priori. In physics, theories sometimes express the values of dimensionless physical 8. A. Burinskii, Gravitation and Cosmology, 2008, Vol. 14, constants as combinations of mathematical constants like pi No. 2, pp. 109–122; Pleiades Publishing, 2008. or e. The inverse problem also arises, whereby the measured DOI: 10.1134/S0202289308020011 value of a physical constant admits a ‘surprisingly’ simple ap- 9. Tegmark, Max (February 2008). ”The Mathematical Uni- proximation in terms of well-known mathematical constants. verse”. Foundations of Physics. 38 (2): 101–150. Can we estimate the probability for this to be a mere coinci- 10. planckmomentum.com/SUHessayformulas.zip (down- dence? [13] load article formulas) ”The fundamental constants divide into two categories, 11. Burinskii, A. (2005). ”The Dirac–Kerr electron”. units independent and units dependent, because only the con- arXiv:hep-th/0507109 stants in the former category have values that are not deter- 12. https://en.wikipedia.org/wiki/Theory-of-everything mined by the human convention of units and so are true fun- (02/2016) damental constants in the sense that they are inherent properties of our universe. In comparison, constants in the latter 13. Ariel Amir, Mikhail Lemeshko, Tadashi Tokieda; category are not fundamental constants in the sense that their 26/02/2016 particular values are determined by the human convention of Surprises in numerical expressions of physical constants units” -L. and J. Hsu [4]. arXiv:1603.00299 [physics.pop-ph]

6 6 Additional notes on the physical constants 14. Macleod, Malcolm J. ”Programming Planck units from a virtual electron; a Simulation Hypothesis” Eur. Phys. J. Plus (2018) 133: 278 15. Macleod, Malcolm J. ”Plato’s Cave; Source Code of the Gods”, the philosophy and physics of a Virtual Universe, online edition (2017) http://platoscode.com/ 16. Dirac, Paul, The Evolution of the Physicist’s Picture of Nature, June 25, 2010 https://blogs.scientiﬁcamerican.com/guest-blog/the- evolution-of-the-physicists-picture-of-nature/

7 6 Additional notes on the physical constants