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Volume 3 PROGRESS IN PHYSICS July, 2009
Primes, Geometry and Condensed Matter
Riadh H. Al Rabeh
University of Basra, Basra, Iraq
E-mail: alrabeh rh@yahoo.com
FascinationwithprimesdatesbacktotheGreeksandbefore. Primesarenamedbysome“the elementary particles of arithmetic” as every nonprime integer is made of a uniqueset of primes. In this article we point to new connections between primes, geometry andphysics which show that primes could be called “the elementary particles of physics”too. This study considers the problem of closely packing similar circles
/
spheres in2D
/
3D space. This is in e
ﬀ
ect a discretization process of space and the allowable num-ber in a pack is found to lead to some unexpected cases of prime conﬁgurations whichis independent of the size of the constituents. We next suggest that a non-prime can beconsidered geometrically as a symmetric collection that is separable (factorable) intosimilar parts- six is two threes or three twos for example. A collection that has nosuch symmetry is a prime. As a result, a physical prime aggregate is more di
ﬃ
cult tosplit symmetrically resulting in an inherent stability. This “number
/
physical” stabilityidea applies to bigger collections made from smaller (prime) units leading to larger sta-ble prime structures in a limitless scaling up process. The distribution of primes amongnumbers can be understood better using the packing ideas described here and we furthersuggest that di
ﬀ
ering numbers (and values) of distinct prime factors making a nonprimecollection is an important factor in determining the probability and method of possibleand subsequent disintegration. Disintegration is bound by energy conservation and isclosely related to symmetry by Noether theorems. Thinking of condensed matter as thepacking of identical elements, we examine plots of the masses of chemical elements of the periodic table, and also those of the elementary particles of physics, and show thatprime packing rules seem to play a role in the make up of matter. The plots show con-vincingly that the growth of prime numbers and that of the masses of chemical elementsand of elementary particles do follow the same trend indeed.
1 Introduction
Primes have been a source of fascination for a long time- asfar back as the Greeks and much before. One reason for thisfascination is the fact that every non-prime is the product of aunique set of prime numbers, hence the name
elementary par-ticles of arithmetic
, and that although primes are distributedseemingly randomly among other integers, they do have reg-ular not fully understood patterns (see [1] for example). Theliterature is rich in theories on primes but one could say thatnone-to-datehavemanagedtomakethestrongconnectionbe-tween primes and physics that is intuitively felt by many. Onerecent attempt in this direction is [2], wherein possible con-nections between the atomic structure and the zeros of theZeta function — closely connected to primes — are inves-tigated. We quote from this reference, “Why the periodic-ity of zeros from the Riemann-Zeta function would matchthe spacing of energy levels in high-Z nuclei still remainsa mystery”.In the present work we attempt to relate primes to bothgeometry and physics. We start with the packing of circles ina plane (or balls on a plane)- all of the same size, and pose aquestion;
In a plane, what is the condition for packing an in-tegral number of identical circles to form a larger circle- suchthat both the diameter and circumference of the larger circlecontain an integral numbers of the small circle
? The problemis essentially the same when the 2D circles are replaced withballs on a tray. A surprising result here is the appearance of only two prime numbers 2 and 3 in the answer and
only oneof them is nontrivial- the number 3
. This gives such numbersa fundamental and natural importance in geometry. We mayview this number as a “discretization number of the continu-ous
spaces”. We further study this matter and shed light(using balls to represent integers) on bounds on the growth of primes- namely the well known logarithmic law in the theoryof primes. Still further, we coin the notion that distinct primefactors in the packing of composite collections
/
grouping canhave a profound inﬂuence on the behaviour of such collec-tions and the manner they react with other collections builtof some di
ﬀ
erent or similar prime factors. As many physicsmodels of condensed matter assume identical elements forsimple matter (photons, boson and fermion statistics and theMIT bag model [3,6] are examples) we examine the appli-cability of our packing rules in such case and conclude thatcondensed matter do seem to follow the packing rules dis-cussed here.
2 Theory
Consider the case of close packing of circles on a plane so asto make a bigger circle (Figure 1). The ratio of the radius of thelargecircletothatofthesmallcircleis;
,
54 Riadh H. Al Rabeh. Primes, Geometry and Condensed Matter
July, 2009 PROGRESS IN PHYSICS Volume 3
1 7 13 19
25
31 37 43
49 55
61 67 73 79
85 91
97
...2 8 14 20 26 32 38 44 50 56 62 68 74 80 86 92 98...3 9 15 21 27 33 39 45 51 57 63 69 75 81 87 93 99...4 10 16 22 28 34 40 46 52 58 64 70 76 82 88 94 100...
5 11 17 23 29
35
41 47 53 59
65
71
77
83 89
95
101
...6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102...Table 1: Integers arranged in columns of six.Fig. 1: Close packing of an integral number of circles
/
balls on aplane have one nontrivial solution- 6 balls, plus one at the centre (seealso Figure 2). Here in Fig. 1:
;
;
;
. For integral ratio
,
must be
or
and
.
where
is half the angle between radial lines through the cen-ters of any two adjacent circles. For this number to be aninteger, the quantity
must be an integer and hencethe angle t must be either 30 or 90 degrees. Thus
shouldbe either three or two (see Figure 2b). That is; the diametercan be either two or three circles wide. The number 3 is non-trivial, and gives six circles touching each other, and all inturn tangent to a seventh circle at the centre.Clearly the arrangement of balls on a plane does followexactly the same pattern leading to six balls touching in pairsand surrounding a seventh ball (touching all other six) at thecenter. This result is unique and is independent of the sizeof the balls involved. It is rather remarkable as it gives thenumber 6 a special stature in the physics of our 3D space,parallel to that of the number
in geometry. Such staturemust have been realized in the past by thinkers as far back asthe Babylonian times and the divine stature given to such anumber in the cultures of many early civilizations- six work-ing days in a week and one for rest is one example, the sixprongs of the star of David and the seven days of creation aswell as counting in dozens might have also been inspired bythe same. Before this, the Bees have discovered the same factand started building their six sided honey combs accordingly.Consider now the set of prime numbers. It is known thatevery prime can be written as
, where
is an integer.
Thatis
thenumbersixisageneratorofallprimes. Further, we
Fig. 2: Packing of 2, 3, 4, 7 & 19 (
7
(3
3)
(3
3)) balls in 3D(a,b). The 19 ball case possesses six side and eight side symme-tries (c,d).
note that whereas the number six is divisible into 2 (threes) or3 (twos), an addition of one unit raises the number to seven-a prime and not divisible into any smaller symmetric entities.Put di
ﬀ
erently, an object composed of six elements can easilybreak into smaller symmetrical parts, whereas an object madeof 7 is more stable and not easily breakable into symmetricparts. We know from physics that symmetry in interactionsis demanded by many conservation laws. In fact symme-try and conservation are tightly linked by Noether theorems-such that symmetry can always be translated to a conserva-tion law and vice versa. When we have a group of highlysymmetric identical items, the addition of one at the centre of the collection can make it a prime.Now if we arrange natural numbers in columns of six asshown in Table 1, we see clearly that all primes fall alongtwo lines- top line for the
type and the bottom linefor the case of
type primes (text in bold). If theseare balls arranged physically on discs six each and on top of each other, the two lines will appear diametrically oppositeon a long cylinder. Thus there are two favourite lines alongwhich all primes fall in a clear display of a sign of the closeconnection between primes geometry and physics.We see then that the connection between primes and ge-ometry is an outcome of how the plane and the space lendthemselves to discretization, when we pair such blocks with
Riadh H. Al Rabeh. Primes, Geometry and Condensed Matter 55
Volume 3 PROGRESS IN PHYSICS July, 2009
Fig. 3: (a) Scaling up using small blocks of seven to make largerblocks of seven; (b) Tight packing of circles naturally resulting inhex objects made of hex layers. The number of circles in each layerstrip increases in steps of 6. Note that each hex sector has cannonballs (or conical) packing structure; (c) Easy to construct (square)brick structure to formally replace circles.
the set of positive integers. We may note also that the densityof
and
type primes is the same with respectto the integers. Moreover, if we take the di
ﬀ
erence betweenprime pairs, the distribution of the di
ﬀ
erence peaks at 6 andall multiples of it, but diminishes as the di
ﬀ
erence increases(Figure 4c).In a violent interaction between two prime groups, oneor more of the groups could momentarily loose a memberor more leaving a non-prime group which then become lessstable and divisible into symmetric parts according to the fac-tors making the collection. Clearly in this case, the few noneprimes neighbouring a prime also become important, andwould contribute to the rules of break-up, to the type of prod-ucts and to the energy required in each case.Our packing endeavour can continue beyond 7 to makelarger 3D objects (Figure 2). A stable new arrangement canresult from the addition of 6 balls- 3 on each side (top andbottom) making an object of 13 balls- a new prime ﬁgure.Further 6 balls can be put symmetrically secured on top andbottom to give an object of 19 balls. This last case in additionto being a prime collection has an interesting shape feature.It has six and eight face symmetries and fairly smooth facesas shown in Figures 2(c, d), which could give rise to two dif-ferent groups of 19 ball formations. Further addition of 6’s ispossible, but the resulting object appears less strong. To go adi
ﬀ
erent direction, we can instead consider every 7, 13 or 19ball objects as the new building unit and use it to form furthernew collections of objects of prime grouping. Clearly this canbe continued in an endless scaling up process (Figure 3b).Scaling is a prominent phenomenon in physical structures.Fig. 3b shows that, in a plane, our packing problem and alsothat of the packing of cannon-balls [5] are only subsets of thegeneral densest packing problem and thus it truly is a dis-
Fig. 4: (a) Two overlapping plots of the ﬁrst 104 primes: (1, 2, 3,5,
, 104729) compared to ﬁtting plot (–),
(
serial positions of prime numbers) (
.
); (b) Ratio of a prime (
)to
; (c) Relative number of primes with di
ﬀ
erences of 2, 4,6,
30. Peaks occur at di
ﬀ
erences of 6, 12, 18, 24, 30.
cretization process of space. We note also that circles can bereplaced with squares placed in a brick like structure providedwe only think of the centres of these squares.Intheprocessofaddingnewringsofcirclestoformlargerobjects, both prime and nonprime numbers are met. A primeis formed every time we have highly symmetric combinationwith one to be added or subtracted to it to break the symme-try and produce a prime. If we consider the number of circlesadded in each ring in the case of circular geometry (the sameapplies to hex geometry with small modiﬁcation), the radiusofaringisgivenby
, where
isthenumberoflayersand
is the radius of one small circle set to unity. The numberof circles in each ring is estimated by the integer part of
.For the next ring we substitute
for
in the aboveexpressions and obtain
for a ring. The
relative
in-crease in the number of circles is the di
ﬀ
erence between thesetwo divided by the circumference which gives
. The rel-ative (or probable) number of primes for
-th ring should betaken to come from the contribution of all the items in the ringand this is proportional to
for large
. The actualnumber of primes is an integral of this given by
56 Riadh H. Al Rabeh. Primes, Geometry and Condensed Matter
July, 2009 PROGRESS IN PHYSICS Volume 3
Fig. 5: Relative number of primes in (50000 integer sample): Hexstrips
(*); Circular rings
,
is the numberof rings of circles around the centre (
+
); Interval
,
is theserial number of a prime (
).
since both the radius of a circular strip and the number of circles in that strip are proportional to
. Fig. 5 gives therelative number of primes in one strip and the trend is of theform
, thus conﬁrming the reasoning used above.Figure 4b gives a plot of the ratio
where
is the value of the
-th prime for some 50000 primes samplewhich, forlarge
, equalsthenumberofintegers
/
circlesinthewhole area. Since
for large
, we see thatthis ratio tends to a constant in agreement with the results of the prime number theory (see [1] for example).Further, there are few results from the theory of primesthat can also be interpreted in support of the above argu-ments. For example the well known conjectures suggestingthat there is always a prime between
and
and also be-tween
and
[7] can respectively be taken to cor-respond to the symmetrical duplication of an area and to thering regions between two concentric circles must contain atleast one prime. That is if the srcinal area or sector can pro-duce a prime, then duplicating it symmetrically or adding onemore sector to it will produce at least one prime. The numberof primes in each of the above cases and that of a hex regionare of course more than one and the results from a sample of (1–50000) integers are plotted in Figure 5. The data is gener-ated using a simple Excel-Basic program shown below;
%Open excel > Tools > micro > Basic Editor > paste and runsubroutine prime( )kk=0:% search divisibility up to square rootfor ii=1 to 1e6: z=1: iis=int(sqr(ii+1)+1):% test divisibilityfor jj=2 to iis: if ii-int(ii/jj)*jj=0 then z=0: next jj:% write result in excel sheetif z=1 then kk=kk+1: if z=1 then cells(kk,1)=ii: next ii:end sub:
Fig. 6: Relative number of primes in hex strips (see Figure 3b);primes of the form
(*); primes of the form
,
isthe number of prims around the centre (
+
).
Concentric circles can be drawn on top of the hexagonsshown in Figure 3, and the number of smaller circles tangentto the large circles then occur in a regular and symmetricalway when the number of circular layers is a prime. Some at-tempt was made by one researcher to explain this by formingand solving the associated Diophantine equations. It is notedhere that potential energy and forces are determined by ra-dial distances- that is the radii of the large circles. Also it isknown that the solution of sets of Diophantine equations is agenerator of primes.None prime numbers can be written in a unique set of primes. Thus for any number
we have;
and
where
are integral powers of the prime factors
. Ref. [8] have observed that this relation is equivalentto energy conservation connecting the energy of one large ob- ject to the energy of its constituents- where energy is to beassociated with (
). Further, if the values of
areunity, the group would only have one energy state (structure),and could be the equivalent of fermions in behaviour. Whenthe exponents are not unity (integer
), the group would be-have as bosons and would be able to exist in multiple equiva-lent energy states corresponding to the di
ﬀ
erent combinationvalues of the exponents. Note that
would correspondto the derivative of the prime formula (
) for large
accept for a negative sign.Still in physics, we note that the size of the nucleus of chemical elements is proportional to the number of nucle-ons [3,6] inside it. Since many of the physical and statisti-cal models of the nucleus assume identical constituents, wemay think of testing the possibility of condensed matter fol-
Riadh H. Al Rabeh. Primes, Geometry and Condensed Matter 57
Volume 3 PROGRESS IN PHYSICS July, 2009
I – Elementary particles; II – Particle mass
/
electron mass; III – Nearest primesI-
II- 1 206.7 264.7 274.5 966.7 974.5 1074.5 1506.8 1532.3 1745.5III- 1 211 263 277 967 977 1069 1511 1531 1747I-
II- 1836.2 1838.7 1873.9 1996.1 2183.2 2327.5 2333.6 2343.1 2410.9 2573.2III- 1831 1831 1877 2003 2179 2333 2339 2347 2417 2579I-
D
D
F
D
II- 2585.7 2710.4 3000 3272 3491.2 3649.7 3657.5 3857.1 3933.4 4463.8III- 2591 2713 3001 3271 3491 3643 3671 3863 3931 4463Table 2: Relative masses of well known elementary particles and their nearest primes.Fig. 7: Three normalized plots in ascending order of the relativeatomic weight of 102 elements (
+
); 30 elementary particles (
); theﬁrst 102 prime numbers (*), starting with number 7. Each group isdivided by entry number 25 of the group.
lowing the prime packing patterns as a result. We may alsorepeat the same for the masses of the
elementary particles
of physics which have hitherto deﬁed many e
ﬀ
orts to put a sensein the interpretation of their mass spectrum. To do this weshall arrange the various chemical elements of the periodictable (102 in total) and most of the elementary particles (30in totals) in an ascending order of their masses (disregardingany other chemical property). We shall divide the masses of thechemicalelementsbythemassoftheelementsay, number25, in the list of ascending mass- which is Manganese (mass55 protons) in order to get a relative value picture. The sameis done with the group of elementary particles and these aredivided by the mass of particle number 25 in the list namelythe (Tau) particle (mass 1784 in MeV
/
c
units). Actual unitsdo not matter here as we are only considering ratios. We thencompare these with the list of primes arranged in ascendingorder too. Table 2 contains the data for the case of elementaryparticles. Masses of the chemical elements can be taken fromany periodic table. The nearest prime ﬁgures in the table are
Fig. 8: Absolute-value comparison of the masses of chemical ele-ments and primes. Primes starting from 7 (
+
) and relative masses of the chemical elements of the periodic table in units of Electron massdivided by (137
6) (*).
for information and not used in the plots. In Figure 8 an abso-lute value comparison for the elements is shown. The primesstarts at 7 and the masses of the elements (in electron mass)are divided by 137
6 in order to get the two curves matchingat the two ends.For better ﬁtting, the prime number series had to be start-ed at number 7, not 1 as one might normally do. Comparisonresults are given in Figures 7 and 8. The trends are strikinglysimilar. The type of agreement must be a strong indicationthat the same packing rules are prevailing in all the cases.
3 Concluding remarks
We noticed that primes are closely connected to geometry andphysics and this is dictated by the very properties of discretespace geometry like you can closely pack on a plane onlyseven balls to form a circle. This result and that of the can-non ball packing problem are found to be subsets of the densepacking problem. One clear link between primes and geom-
58 Riadh H. Al Rabeh. Primes, Geometry and Condensed Matter

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