- Email: [email protected]
Particle packing and the mean theory
Physics Letters A 377 (2013) 145–147
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Particle packing and the mean theory Bai-Xiang Xu a , Yang Gao b,∗ , Min-Zhong Wang c,d a
Institute of Materials Science, TU Darmstadt, Petersenstrasse 23, D-64287 Darmstadt, Germany College of Science, China Agricultural University, P.O. Box 74, Beijing 100083, PR China c State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, 100871 Beijing, PR China d Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, 100871 Beijing, PR China b
a r t i c l e
i n f o
Article history: Received 23 July 2012 Received in revised form 28 October 2012 Accepted 9 November 2012 Available online 13 November 2012 Communicated by A.R. Bishop
a b s t r a c t This Letter presents two mean relations between the densities/porosities of random and regular packing modes. The two mean relations work very well for the packing of spherical particles, cubic particles and circular discs. Results confirm the corresponding experimental and computational results. © 2012 Elsevier B.V. All rights reserved.
Keywords: Particle packing Packing density Harmonic–geometric mean Arithmetic–geometric mean
1. Introduction How to pack particles effectively has been a topic of interest since the ancient time. In fact, packing efficiency is not only the interest of fruit vendors in order to lower their transporting cost but also an intriguing topic for scientists and engineers in wide range of disciplines, including e.g. mathematics (discrete geometry), materials science (granular materials, glass, crystalline) and information technology (storage of digital signals). The particle packing problem resembles the structures of granular materials, liquids, glasses, and crystals. For information technology, the particle parking model has even a close relation with the storage of digital signals in a noisy channel. There are a multitude of analytical, computational and experimental investigations on particle parking in the literature [1,2]. As one of the simplest cases, sphere packing is a problem of jamming one large space with hard smooth spheres of the same size. A vivid example used by the experimentalists is packing marbles in a large box. There are four widely known results on sphere packing. The densest packing [3], which can be achieved either by hexagonal √ compact or the face-centered-cubic compact, has a density of π / 18 ≈ 0.74. The density of regular cubic packing reads π /6 ≈ 0.52. Experimental and computational algorithms attained a rather robust density of 0.64 for random packing [4]. This number is conventionally referred as the random close packing density,
*
Corresponding author. E-mail address: [email protected] (Y. Gao).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.11.022
even though there is no strict statistical explanation [5,6]. Correspondingly, there is a random loose packing [7], which has the density of 0.56. The porosity is defined as the unity one subtracted by the corresponding density. Thus for the spherical packing, the porosities for the densest packing, the cubic packing, the random close packing and the random loose packing are 0.26, 0.48, 0.36, 0.44, respectively. A summary of the results is shown in Table 1. Likewise, there are corresponding results for other types of packing, for instance, cubic particle packing and circular disc packing. What interesting is whether there exist certain general average relations between these packing densities, since the random close/loose packing represents certain statistical intermediate state between the two extreme regular packing modes. In this Letter, we present that there are two mean relations which connect the packing densities and porosities of these special packing modes. 2. The two mean relations in particle packing First of all, the definition of the harmonic–geometric mean is introduced. For two positive real values a, b, one can define two series according to the geometric mean and the harmonic mean, respectively. That is,
an =
an−1 bn−1 ,
bn =
(an−1 )−1 + (bn−1 )−1 2
−1 (1)
in which n = 1, 2, . . . , a0 = a, b0 = b. It has been proved that these two sequences converge to the same number, which is called the harmonic–geometric mean of a and b, denoted by HG(a, b). The
146
B.-X. Xu et al. / Physics Letters A 377 (2013) 145–147
Table 1 Packing density/porosity and the mean theory.
Densest packing Random close packing Cubic/square packing Random loose packing Harmonic–geometric mean Arithmetic–geometric mean
Density (sphere)
Porosity (sphere)
Density (square)
0.74 0.64 0.52 0.56 HG(0.74, 0.56) ≈ 0.64 –
0.26 0.36 0.48 0.44 – AG(0.26, 0.48) ≈ 0.36
1.0 0.74 – 0.57 HG(1.0, .57) ≈ 0.74 –
Note: HG(a, b) and AG(a, b) indicate the harmonic–geometric mean and the arithmetic–geometric mean of the variables a and b, respectively. Table 2 The harmonic–geometric mean of the densest packing density 0.74 and the random loose packing 0.56 of spherical packing. n
an
bn
1 2 3 4 5 6
0.74 0.643739078 0.640631268 0.640627517 0.640627517 0.640627517
0.56 0.637538462 0.640623766 0.640627517 0.640627517 0.640627517
Table 3 The arithmetic–geometric mean of the porosity of the densest packing 0.26 and that of the cubic packing 0.48 for spherical packing. n
an
bn
1 2 3 4 5 6
0.26 0.37 0.361635 0.361587 0.361587 0.361587
0.48 0.35327 0.361538 0.361587 0.361587 0.361587
harmonic–geometric mean of the densest packing density 0.74 and the random loose packing density 0.56 for the sphere packing can be calculated. As it is shown in Table 2, HG(0.74, 0.56) = 0.64. It is noticeable that this harmonic–geometric mean happens to equal the random close packing density 0.64. We name this as the first mean relation, i.e. the random close packing density can be approximately determined as the harmonic–geometric mean of the densest packing density and the random loose packing density. A similar law is also found for the porosities of spherical packing. Similar to the harmonic–geometric mean, there is the so-called arithmetic–geometric mean. For two positive real values a and b, two series can be introduced according to
an =
an−1 + bn−1 2
bn =
,
an−1 bn−1
(2)
where n = 1, 2, . . . , a0 = a, b0 = b. It can be proved that these two sequences also converge to the same number, which is called the arithmetic–geometric mean of a and b, denoted by AG(a, b). Gauss discovered the amazing relation between the arithmetic–geometric mean and the length of an ellipse [8]. It is shown that
AG(a, b) =
π
π /2
2 0
1
− 1 dϕ
a2 cos2 ϕ + b2 sin2 ϕ
A comprehensive discussion on the arithmetic–geometric mean, the Gauss formula and their role in the modern computation of the digits can be found in the work by Almvist and Berndt [9]. As it is shown in Table 3, the arithmetic–geometric mean between the porosity of the densest packing 0.26 and that of the cubic packing 0.48 is about 0.36, which nearly equals the porosity of the random close packing [7]. We refer to this as the second mean relation, i.e. the porosity of the random close packing can be nearly
determined as the arithmetic–geometric mean between the porosity of the densest packing and that of the cubic packing. The discovery of the two relations for sphere packing was initially inspired by the Titius–Bode law [10], an empirical rule for semi-major axes of some orbital systems. For solar system, the Titius–Bode law is well satisfied by the planets from Mercury to Saturn, and has correctly predicted the orbits of Ceres and Uranus, even though the eighth orbit predicted by the rule fits roughly that of the dwarf planet Pluto instead of the eighth planet Neptune. Similar rules may be expected in sphere packing systems, since both the planet system and the sphere packing system comprise of spheres under gravitational influence with possible collisions. In fact, the Titius–Bode law can be reformulated in the form of an arithmetic mean or of a geometric mean (see Appendix A). The packing system is of higher complexity due to the large numbers and randomness. This explains the coupled relations of harmonic– geometric or arithmetic–geometric mean. The two relations presented above for sphere packing turn to be also applicable for certain non-sphere packing. For instance, the density relation also holds for the cubic particle packing, where the densest packing density is 1.0, the random close packing density 0.74, the random loose packing 0.57 [11,12]. It can be shown that the random close packing density equals nearly the harmonic– geometric mean of the densest packing density 1.0 and the random loose packing density 0.57. Along with those for the spherical packing, the results are summarized in Table 1. 3. Application of the two mean relations The two relations may be used to predict the random close and the random loose packing densities from two regular densities, with no need of the expensive experimental or computational tests. For example, for the 2D circular disc packing, laying pennies on the top of a large table, it is easy to know the densest packing √ density is the hexagonal packing [13], which has the density of 3π /6 ≈ 0.9069, and the square packing density is π /4 ≈ 0.7854. The corresponding porosities are therefore 0.0931 and 0.2146, respectively. According to the second mean relation on the porosity, the porosity of the random close packing should be nearly equal the arithmetic–geometric mean of the porosity of the densest packing 0.0931 and that of the square packing 0.2146. It can be easily calculated that AG(0.0931, 0.2146) = 0.147533. Thus the porosity of the random close packing of circular disc packing should be around 0.1475, and the corresponding random close packing density is around 0.8525. Furthermore, according to the first mean relation on the density, the random close packing density 0.8525 should equal correspondingly the harmonic–geometric mean of the densest packing density 0.9069 and the random loose packing density. From this relation, one can inversely determine the random loose packing density, which is shown to be 0.8028. To sum up, in use of the two mean relations, one predicts the random close and the random loose packing density, which should be around 0.8525 and 0.8028, respectively. Although these results cannot be strictly verified due to the lack of exact data, they
B.-X. Xu et al. / Physics Letters A 377 (2013) 145–147
indeed lie in the range of 0.80–0.89, which was identified by both experimental and computational tests [4]. This work shows the possibility of predicting densities of random particle packing through simple mathematic mean theories, and sheds lights on the potential mechanism of efficient evaluation methods for the investigation of the packing density of statistically random modes. Note that for packing of particles with more complicated geometry such as ellipsoid and tetrahedron, the presented relations seem to be invalid. As it has been pointed out by Weitz [10], the ellipsoidal particle packing involves not only translational but also rotational motions, for which more complicated models than mean descriptions may be needed.
It can be rewritten as
an =
an+1 + a0
which indicates that the semi-major axis of one planet is the arithmetic mean of the semi-major axis of the Mercury and that of the planet next to it. Eq. (4) can also be rewritten into
an+1 − a0 = 2(an − a0 )
1
The financial support by the Natural Science Foundation of China (No. 11172319) and Chinese Universities Scientific Fund (No. 2011JS046) is acknowledged. The author Xu would also like to thank the German Research Foundation (DFG) for the financial support on the project SFB595.
From Eqs. (6) and (7),
an − a0 =
2
(an − a0 )
(an+1 − a0 )(an−1 − a0 ),
(7)
n = 1, 2, . . .
(8)
which has the form of the geometric mean. It is shown that the axis of the Mercury a0 plays an important role. References
Here it is shown that the Titius–Bode law can also be reformulated in the form of either an arithmetic mean or a geometric mean. In astronomical units, the Titius–Bode law has the form
(3)
where am is the semi-major axis of a planet outward from the Sun, and m = 1, 2, . . . , and a0 = 0.4. The semi-major axis of the planets presents a series of an (n = 0, 1, 2, . . .). For the planets in our solar system, the comparison of the distances calculated from the Titius–Bode law and the corresponding real ones can be found in for example Encyclopedia Britannica [14]. From (3) one can easily verify that
2an − an+1 = a0
(6)
Eq. (6) holds also for any n. Thus
an−1 − a0 =
am = 0.4 + 0.3 · 2m−1
(5)
2
Acknowledgements
Appendix A
147
(4)
[1] J. Zhao, S.X. Li, R.P. Zou, A.B. Yu, Soft Matter 8 (2012) 1003. [2] A.V. Kyrylyuk, M.A. van de Haar, L. Rossi, A. Wouterse, A.P. Philipse, Soft Matter 7 (2011) 1671. [3] T.C. Hales, Discrete Comput. Geom. 36 (2006) 71. [4] J.G. Berryman, Phys. Rev. A 27 (1983) 1053. [5] S. Torquato, T.M. Truskett, P.G. Debenedetti, Phys. Rev. Lett. 84 (2000) 2064. [6] L.V. Woodcock, J. Phys. Chem. B 116 (2012) 3735. [7] S.X. Li, J. Zhao, L.U. Peng, X. Yu, Chinese Sci. Bull. 55 (2010) 114. [8] C.F. Gauss, Nachlass. Arithmetisch geometrisches Mittel, in: Werke, Bd. 3, Koniglichen Gesell Wiss., Gottingen, 1876, p. 361. [9] G. Almvist, B. Berndt, Amer. Math. Monthly 95 (1988) 585. [10] D.A. Weitz, Science 303 (2004) 968. [11] Q. Dong, D.N. Ye, Chinese Sci. Bull. 38 (1993) 54. [12] A.B. Yu, N. Standish, A. Mc Lean, J. Am. Ceram. Soc. 76 (1993) 2813. [13] B. Segre, K. Mahler, Amer. Math. Monthly 51 (1944) 261. [14] Encyclopedia Britannica, http://www.britannica.com.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Particle packing and the mean theory Bai-Xiang Xu a , Yang Gao b,∗ , Min-Zhong Wang c,d a
Institute of Materials Science, TU Darmstadt, Petersenstrasse 23, D-64287 Darmstadt, Germany College of Science, China Agricultural University, P.O. Box 74, Beijing 100083, PR China c State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, 100871 Beijing, PR China d Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, 100871 Beijing, PR China b
a r t i c l e
i n f o
Article history: Received 23 July 2012 Received in revised form 28 October 2012 Accepted 9 November 2012 Available online 13 November 2012 Communicated by A.R. Bishop
a b s t r a c t This Letter presents two mean relations between the densities/porosities of random and regular packing modes. The two mean relations work very well for the packing of spherical particles, cubic particles and circular discs. Results confirm the corresponding experimental and computational results. © 2012 Elsevier B.V. All rights reserved.
Keywords: Particle packing Packing density Harmonic–geometric mean Arithmetic–geometric mean
1. Introduction How to pack particles effectively has been a topic of interest since the ancient time. In fact, packing efficiency is not only the interest of fruit vendors in order to lower their transporting cost but also an intriguing topic for scientists and engineers in wide range of disciplines, including e.g. mathematics (discrete geometry), materials science (granular materials, glass, crystalline) and information technology (storage of digital signals). The particle packing problem resembles the structures of granular materials, liquids, glasses, and crystals. For information technology, the particle parking model has even a close relation with the storage of digital signals in a noisy channel. There are a multitude of analytical, computational and experimental investigations on particle parking in the literature [1,2]. As one of the simplest cases, sphere packing is a problem of jamming one large space with hard smooth spheres of the same size. A vivid example used by the experimentalists is packing marbles in a large box. There are four widely known results on sphere packing. The densest packing [3], which can be achieved either by hexagonal √ compact or the face-centered-cubic compact, has a density of π / 18 ≈ 0.74. The density of regular cubic packing reads π /6 ≈ 0.52. Experimental and computational algorithms attained a rather robust density of 0.64 for random packing [4]. This number is conventionally referred as the random close packing density,
*
Corresponding author. E-mail address: [email protected] (Y. Gao).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.11.022
even though there is no strict statistical explanation [5,6]. Correspondingly, there is a random loose packing [7], which has the density of 0.56. The porosity is defined as the unity one subtracted by the corresponding density. Thus for the spherical packing, the porosities for the densest packing, the cubic packing, the random close packing and the random loose packing are 0.26, 0.48, 0.36, 0.44, respectively. A summary of the results is shown in Table 1. Likewise, there are corresponding results for other types of packing, for instance, cubic particle packing and circular disc packing. What interesting is whether there exist certain general average relations between these packing densities, since the random close/loose packing represents certain statistical intermediate state between the two extreme regular packing modes. In this Letter, we present that there are two mean relations which connect the packing densities and porosities of these special packing modes. 2. The two mean relations in particle packing First of all, the definition of the harmonic–geometric mean is introduced. For two positive real values a, b, one can define two series according to the geometric mean and the harmonic mean, respectively. That is,
an =
an−1 bn−1 ,
bn =
(an−1 )−1 + (bn−1 )−1 2
−1 (1)
in which n = 1, 2, . . . , a0 = a, b0 = b. It has been proved that these two sequences converge to the same number, which is called the harmonic–geometric mean of a and b, denoted by HG(a, b). The
146
B.-X. Xu et al. / Physics Letters A 377 (2013) 145–147
Table 1 Packing density/porosity and the mean theory.
Densest packing Random close packing Cubic/square packing Random loose packing Harmonic–geometric mean Arithmetic–geometric mean
Density (sphere)
Porosity (sphere)
Density (square)
0.74 0.64 0.52 0.56 HG(0.74, 0.56) ≈ 0.64 –
0.26 0.36 0.48 0.44 – AG(0.26, 0.48) ≈ 0.36
1.0 0.74 – 0.57 HG(1.0, .57) ≈ 0.74 –
Note: HG(a, b) and AG(a, b) indicate the harmonic–geometric mean and the arithmetic–geometric mean of the variables a and b, respectively. Table 2 The harmonic–geometric mean of the densest packing density 0.74 and the random loose packing 0.56 of spherical packing. n
an
bn
1 2 3 4 5 6
0.74 0.643739078 0.640631268 0.640627517 0.640627517 0.640627517
0.56 0.637538462 0.640623766 0.640627517 0.640627517 0.640627517
Table 3 The arithmetic–geometric mean of the porosity of the densest packing 0.26 and that of the cubic packing 0.48 for spherical packing. n
an
bn
1 2 3 4 5 6
0.26 0.37 0.361635 0.361587 0.361587 0.361587
0.48 0.35327 0.361538 0.361587 0.361587 0.361587
harmonic–geometric mean of the densest packing density 0.74 and the random loose packing density 0.56 for the sphere packing can be calculated. As it is shown in Table 2, HG(0.74, 0.56) = 0.64. It is noticeable that this harmonic–geometric mean happens to equal the random close packing density 0.64. We name this as the first mean relation, i.e. the random close packing density can be approximately determined as the harmonic–geometric mean of the densest packing density and the random loose packing density. A similar law is also found for the porosities of spherical packing. Similar to the harmonic–geometric mean, there is the so-called arithmetic–geometric mean. For two positive real values a and b, two series can be introduced according to
an =
an−1 + bn−1 2
bn =
,
an−1 bn−1
(2)
where n = 1, 2, . . . , a0 = a, b0 = b. It can be proved that these two sequences also converge to the same number, which is called the arithmetic–geometric mean of a and b, denoted by AG(a, b). Gauss discovered the amazing relation between the arithmetic–geometric mean and the length of an ellipse [8]. It is shown that
AG(a, b) =
π
π /2
2 0
1
− 1 dϕ
a2 cos2 ϕ + b2 sin2 ϕ
A comprehensive discussion on the arithmetic–geometric mean, the Gauss formula and their role in the modern computation of the digits can be found in the work by Almvist and Berndt [9]. As it is shown in Table 3, the arithmetic–geometric mean between the porosity of the densest packing 0.26 and that of the cubic packing 0.48 is about 0.36, which nearly equals the porosity of the random close packing [7]. We refer to this as the second mean relation, i.e. the porosity of the random close packing can be nearly
determined as the arithmetic–geometric mean between the porosity of the densest packing and that of the cubic packing. The discovery of the two relations for sphere packing was initially inspired by the Titius–Bode law [10], an empirical rule for semi-major axes of some orbital systems. For solar system, the Titius–Bode law is well satisfied by the planets from Mercury to Saturn, and has correctly predicted the orbits of Ceres and Uranus, even though the eighth orbit predicted by the rule fits roughly that of the dwarf planet Pluto instead of the eighth planet Neptune. Similar rules may be expected in sphere packing systems, since both the planet system and the sphere packing system comprise of spheres under gravitational influence with possible collisions. In fact, the Titius–Bode law can be reformulated in the form of an arithmetic mean or of a geometric mean (see Appendix A). The packing system is of higher complexity due to the large numbers and randomness. This explains the coupled relations of harmonic– geometric or arithmetic–geometric mean. The two relations presented above for sphere packing turn to be also applicable for certain non-sphere packing. For instance, the density relation also holds for the cubic particle packing, where the densest packing density is 1.0, the random close packing density 0.74, the random loose packing 0.57 [11,12]. It can be shown that the random close packing density equals nearly the harmonic– geometric mean of the densest packing density 1.0 and the random loose packing density 0.57. Along with those for the spherical packing, the results are summarized in Table 1. 3. Application of the two mean relations The two relations may be used to predict the random close and the random loose packing densities from two regular densities, with no need of the expensive experimental or computational tests. For example, for the 2D circular disc packing, laying pennies on the top of a large table, it is easy to know the densest packing √ density is the hexagonal packing [13], which has the density of 3π /6 ≈ 0.9069, and the square packing density is π /4 ≈ 0.7854. The corresponding porosities are therefore 0.0931 and 0.2146, respectively. According to the second mean relation on the porosity, the porosity of the random close packing should be nearly equal the arithmetic–geometric mean of the porosity of the densest packing 0.0931 and that of the square packing 0.2146. It can be easily calculated that AG(0.0931, 0.2146) = 0.147533. Thus the porosity of the random close packing of circular disc packing should be around 0.1475, and the corresponding random close packing density is around 0.8525. Furthermore, according to the first mean relation on the density, the random close packing density 0.8525 should equal correspondingly the harmonic–geometric mean of the densest packing density 0.9069 and the random loose packing density. From this relation, one can inversely determine the random loose packing density, which is shown to be 0.8028. To sum up, in use of the two mean relations, one predicts the random close and the random loose packing density, which should be around 0.8525 and 0.8028, respectively. Although these results cannot be strictly verified due to the lack of exact data, they
B.-X. Xu et al. / Physics Letters A 377 (2013) 145–147
indeed lie in the range of 0.80–0.89, which was identified by both experimental and computational tests [4]. This work shows the possibility of predicting densities of random particle packing through simple mathematic mean theories, and sheds lights on the potential mechanism of efficient evaluation methods for the investigation of the packing density of statistically random modes. Note that for packing of particles with more complicated geometry such as ellipsoid and tetrahedron, the presented relations seem to be invalid. As it has been pointed out by Weitz [10], the ellipsoidal particle packing involves not only translational but also rotational motions, for which more complicated models than mean descriptions may be needed.
It can be rewritten as
an =
an+1 + a0
which indicates that the semi-major axis of one planet is the arithmetic mean of the semi-major axis of the Mercury and that of the planet next to it. Eq. (4) can also be rewritten into
an+1 − a0 = 2(an − a0 )
1
The financial support by the Natural Science Foundation of China (No. 11172319) and Chinese Universities Scientific Fund (No. 2011JS046) is acknowledged. The author Xu would also like to thank the German Research Foundation (DFG) for the financial support on the project SFB595.
From Eqs. (6) and (7),
an − a0 =
2
(an − a0 )
(an+1 − a0 )(an−1 − a0 ),
(7)
n = 1, 2, . . .
(8)
which has the form of the geometric mean. It is shown that the axis of the Mercury a0 plays an important role. References
Here it is shown that the Titius–Bode law can also be reformulated in the form of either an arithmetic mean or a geometric mean. In astronomical units, the Titius–Bode law has the form
(3)
where am is the semi-major axis of a planet outward from the Sun, and m = 1, 2, . . . , and a0 = 0.4. The semi-major axis of the planets presents a series of an (n = 0, 1, 2, . . .). For the planets in our solar system, the comparison of the distances calculated from the Titius–Bode law and the corresponding real ones can be found in for example Encyclopedia Britannica [14]. From (3) one can easily verify that
2an − an+1 = a0
(6)
Eq. (6) holds also for any n. Thus
an−1 − a0 =
am = 0.4 + 0.3 · 2m−1
(5)
2
Acknowledgements
Appendix A
147
(4)
[1] J. Zhao, S.X. Li, R.P. Zou, A.B. Yu, Soft Matter 8 (2012) 1003. [2] A.V. Kyrylyuk, M.A. van de Haar, L. Rossi, A. Wouterse, A.P. Philipse, Soft Matter 7 (2011) 1671. [3] T.C. Hales, Discrete Comput. Geom. 36 (2006) 71. [4] J.G. Berryman, Phys. Rev. A 27 (1983) 1053. [5] S. Torquato, T.M. Truskett, P.G. Debenedetti, Phys. Rev. Lett. 84 (2000) 2064. [6] L.V. Woodcock, J. Phys. Chem. B 116 (2012) 3735. [7] S.X. Li, J. Zhao, L.U. Peng, X. Yu, Chinese Sci. Bull. 55 (2010) 114. [8] C.F. Gauss, Nachlass. Arithmetisch geometrisches Mittel, in: Werke, Bd. 3, Koniglichen Gesell Wiss., Gottingen, 1876, p. 361. [9] G. Almvist, B. Berndt, Amer. Math. Monthly 95 (1988) 585. [10] D.A. Weitz, Science 303 (2004) 968. [11] Q. Dong, D.N. Ye, Chinese Sci. Bull. 38 (1993) 54. [12] A.B. Yu, N. Standish, A. Mc Lean, J. Am. Ceram. Soc. 76 (1993) 2813. [13] B. Segre, K. Mahler, Amer. Math. Monthly 51 (1944) 261. [14] Encyclopedia Britannica, http://www.britannica.com.