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Crystal structure and elliptic periodicity. Cubes and dodecahedra.
Solid State Sciences 8 (2006) 730–739 www.elsevier.com/locate/ssscie
Crystal structure and elliptic periodicity. Cubes and dodecahedra. Without π Sten Andersson Sandforsk, Institute of Sandvik, S-38074 Löttorp, Sweden Received 5 January 2006; received in revised form 24 February 2006; accepted 18 April 2006
Dedicated to Professor Hans Georg von Schnering on the occasion of his 75th birthday
Abstract A new periodic and elliptic function, based on the exponential scale, is described. This space contains classic as well as icosahedral crystallography. Crystal structures and crystal shapes of cubic and dodecahedral symmetries are discussed in the spherical space of the new periodic function. © 2006 Elsevier SAS. All rights reserved.
0. Preface I have enjoyed an excellent friendship with Hans Georg von Schnering for almost half a century. We have had great many extraordinary discussions, normally very tough but never unfriendly. Whatever science idea I propose, diffuse and/or theoretical, he understands immediately. He tells me, also immediately, if it is stupid, or not. Normally he is right in his judgement. It has indeed been a remarkable association. Hans Georg is a brilliant man, in science and in friendship. I believe our collaboration has been so fruitful because we have done careful thinking on “What harm a wind too strong at sea might do”, because we are sailors. But it might be more adequate to quote Snorre’s Edda: ett vet jag som aldrig dör, domen över död man.1 Snorre wrote his Edda on Iceland 300 years before The merchant of Venice was written. 1. Introduction The Arabs derived planar groups with the notion of symmetry and applied translation to get structures to describe patterns present on walls or in gardens or parks. Some of these patterns had been around long before the Alhambra. But the Arabs E-mail address: [email protected] (S. Andersson). URL: http://www.sandforsk.se. 1 B.G. Hyde has translated this for me: One thing I know which never dies, the judging of a dead man. Which is much better than what is in the trade. 1293-2558/$ – see front matter © 2006 Elsevier SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2006.04.012
found all the 17 planar groups to describe any planar pattern [1]. Thereby, and with other findings they invented modern mathematics. This was the beginning of many things, among them crystallography. But it took 1000 years for the Western world to rediscover it all. The need was there not only to describe mineral crystals but also to handle the findings of von Laue, and the Braggs. Crystallography had its peak in science during the first half part of last century. Geometry and structure formed the raison d’etre for crystallography during the time up to now with biology and chemistry, the DNA and the protein structures that gave the clue to the understanding of life. And of course, in solid state science where it has been the main tool from the beginning up till now. The geometry we describe in this article is in elliptic space. We may as well say spherical geometry, or just spherical trigonometry for the simpler spaces. The elliptic space allows five-fold symmetry, as well as the ordinary symmetry operations from Euclidean 3D space. The tool to describe it is to stay in Euclidean space and use special functions. We can, for example, describe 20 points regularly spaced on a sphere with a simple function of six terms. This distribution corresponds to a dodecahedron. Another possibility is 20 faces regularly spaced on a sphere, which is the dual, or the icosahedron. Yet another distribution, eight points equally spaced on a sphere, is described with three terms. Earlier we described polyhedra like the great dodecahedron, or the great stellation of the icosahedron in this way using exponential scale methods. Shells were
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formed as a case of spherical geometry, and a great number of points were described from very simple functions [4]. To this we add the radial spherical periodicity in Section 2 below with which we build crystals containing atoms, or bodies. As we use Mathematica we need to establish a few habits: In the Cartesian system of Mathematica, an implicit function of the three spacial coordinates x, y, z will have the cubic symmetry of the coordinate system if the expression is homogeneous in x, y, z. To generate a general position (x, y, z in crystallographic literature) we have to distinguish between the three, and use an expression of a form such as x, 2y, 3z. Similarly, in permutations below for a position x, x, x in crystallographic literature we simply use x, y, z. Example: For the permutation of the axes for a general point (x, y, z → z, x, y in the crystallographic literature) we use x, 2y, 3z → 3x, y, 2z. All the 47 Shubnikov polyhedra [2] from the point groups have been described in our book, Group and Structure, published in Sandforsk, Chapter 8 [3]. The one below, pentagonal tristetrahedron, belongs to the space group P23, and the last one from group Pm3m is called hexoctahedron and has 48 identical triangular faces (non-regular), and shown in Fig. 3. This is some spherical periodicity indeed, without π and τ as well! Those polyhera where first produced in the von Federov kaleidoscope. In order to obtain these polyhedra von Federov recommends pouring mercury into his kaleidoscope [2]! The pentagonal tristetrahedron we use as example and calculate its surface. We use the general positions as planes which added together, on the exponential scale, bend over to form the surfaces of the group. This yields the shape of the Shubnikov polyhedron or the mineral crystal. f (x,y,z) = The basic exponential form used is simply e const, where the function f (x, y, z) is the symmetry-distinguishing sum of the components x, y, z, and the sum is taken over all symmetry equivalent permutations in an orbit. The size of the constant is the value of the iso-surface generated, and this value has been determined experimentally for each case shown below. Other values will give surfaces of the same symmetry, but with different curvature, and even different topology. The spacegroup for the pentagonal tristetrahedron is P23 (No. 195) with the general position given by x, y, z; z, x, y; y, z, x; x, −y, −z; z, −x, −y; y, −z, −x; −x, y, −z; −z, x, −y; −y, z, −x; −x, −y, z; −z, −x, y; −y, −z, x; This translates to a set of functions f (x, y, z) x + 2y + 3z; 3x + y + 2z; 2x + 3y + z; x − 2y − 3z; 3x − y − 2z; 2x − 3y − z; −x + 2y − 3z; −3x + y − 2z; −2x + 3y − z; −x − 2y + 3z; −3x − y + 2z; −2x − 3y + z; The general position is 12 j and the first equation is in (1.1). ex+2y+3z + e3x+y+2z + e2x+3y+z + ex−2y−3z + e3x−y−2z + e2x−3y−z
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Fig. 1. The pentagonal tristetrahedron of group P23.
Fig. 2. The final stellation of the icosahedron, equation in Ref. [4].
+ e−x+2y−3z + e−3x+y−2z + e−2x+3y−z + e−x−2y+3z + e−3x−y+2z + e−2x−3y+z = 1016 .
(1.1)
This polyhedron as shown in Fig. 1 is called the pentagonal tristetrahedron by Shubnikov (No. 38) and has 12 faces of identical and non-regular pentagons. So this is our starting example of finite periodicity, which of course is spherical. Of course all polyhedra are examples of periodic spherical geometry. The stellated cases are also excellent cases as reported in Sandforsk [4]. We give one example here, the great stellation of the icosahedron in Fig. 2 with altogether 92 points in a formidable periodic and spherical geometry described in Ref. [4] with only 10 terms. A spherical and periodic geometry without π , but with τ . 2. Background to the radial spherical finite periodicities We continue now with the radial spherical periodicity. A number of GD (Gauss distribution) terms added together with a proper constant give the new periodicity. It was first developed to describe the Larsson cubosomes [5,6]. The key is that each term is peaked. One term consists of two infinite
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a detailed description of the dodecahedral structures here, we leave the icosahedral structures for a forthcoming paper. The cubic symmetry codes x, 0, 0; 0, x, 0; 0, 0, x; −x, 0, 0; 0, −x, 0; 0, 0, −x; x, x, 0; 0, x, x; x, 0, x; x, −x, 0; 0, −x, −x; x, 0, −x; −x, x, 0; 0, x, −x; −x, 0, −x; −x, −x, 0; 0, −x, x; −x, 0, x; x, x, x; x, −x, −x; −x, x, −x; −x, −x, x; x, x, x; x, −x, −x; −x, x, −x; −x, −x, x; −x, −x, −x; −x, x, x; x, −x, x; x, x, −x; The dodecahedral symmetry code x, τ x, 0; 0, x, τ x; τ x, 0, x; x, −τ x, 0; 0, x, −τ x; −τ x, 0, x; Fig. 3. The hexoctahedron of group Pm3m after Shubnikov (No. 47). The corresponding equation is given in Sandforsk [3].
planes in space, two terms give four planes that collaborate to form a cylinder, and with three, there is a sphere. These properties of the GD function in space is the backbone of the 2D periodic properties around the sphere. The radial periodicity in 3D space completes a finite elliptic periodicity, which we find very useful for crystal structure descriptions. The equations in (2.1) and (2.2) define the periodicities. e
−x 2
e
−(x−δ)2
+e
−y 2
−z2
+e
+e
−(y−δ)2
= const, +e
−(z−δ)2
= const.
−x, −τ x, 0; 0, −x, −τ x; −τ x, 0, −x; −x, τ x, 0; 0, −x, τ x; τ x, 0, −x. We give a number of examples of the new periodicity. 3. Primitive cubic We start with the simplest permutation possible, using the position 6e in Pm3m, with the first symmetry code x, 0, 0; 0, x, 0; 0, 0, x; −x, 0, 0; 0, −x, 0; 0, 0, −x
(2.1)
exponentially it becomes
(2.2)
ex + ey + ez + e−x + e−y + e−z = const
Eq. (2.1) is of the type used to describe the stellations that gave excellent examples of spherical periodicity, as the great stellation of the icosahedron above. With δ this function is now growing radially in space in a true periodic, and also finite, manner. The δs must take the same value in x, y, z to stepwise increase the radius for the spherical periodicity. The motion has 3-fold symmetry. In a cube you have four diagonals of such symmetry and in the dodecahedron, there are ten. The motion now generates the 3D periodicity as said above. The general 3D radial periodicity is elliptic and we will handle this special spherical case of 3D radial periodicity that. The elliptic periodicity is easily developed within group theory, and we hope to do so in a forthcoming article. There are a few simple permutations in space that occur in one form or another in the groups [7], which we call symmetry codes. The symmetry codes on the exponential scale used here make shapes (or polyhedra) which after the crystallographic jargon are: the cube, the rhombic dodecahedron, the diamond, the octahedron, the dodecahedron and the icosahedron. Using GD terms and the new periodicity we have in order as above: pc or primitive cubic structure, bcc or body centred cubic structure, diamond structure, fcc cubic close packing or octahedral structure, the dodecahedron for the dodecahedral structure, and the icosahedron for the icosahedral structure. While we shall give
(3.1)
which is a cube (Fig. 4). We now switch to the GD function, and write periodic functions, which are finite. This makes sense since crystals are of finite periodicity. So we write in Eq. (3.2) a function with only three terms diagonally that give 27 bodies in Fig. 5(a). e−x + e−y + e−z + e−(x−2) + e−(y−2) + e−(z−2) 2
2
2
2
2
+ e−(x+2) + e−(y+2) + e−(z+2) = 2.65. 2
2
2
2
(3.2)
Eq. (3.3) contains seven terms which, due to the properties of true periodicity, gives 343 bodies (73 ) in Fig. 5(b). But the equation itself is a summation and can, equivalently, be written in
Fig. 4.
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Mathematica manners as in Eq. (3.4) below
4. Body centered cubic(bcc) and 3-connectivity(gyroid)
e−x + e−y + e−z + e−(x−2) + e−(y−2) + e−(z−2) 2
2
2
+e
−(x−4)2
+e
−(y−6)2
+e
−(z+2)2
2
+e
−(y−4)2
+e
−(z−6)2
+e
−(x+4)2
2
−(z−4)2
+e
+e
−(x+2)2
+ e−(y+2)
+e
−(y+4)2
−(z+4)2
+e
+ e−(x+6) + e−(y+6) + e−(z+6) 2
2
2
In our next example the symmetry code is 12i, P432, which has the following permutations:
−(x−6)2
x, x, 0; 0, x, x; x, 0, x; x, −x, 0; 0, −x, −x; x, 0, −x;
2
+e
−x, x, 0; 0, x, −x; −x, 0, −x; −x, −x, 0; 0, −x, x; −x, 0, x.
2
= 2.78, 2 2 2 Sum e−(x−n) + e−(y−n) + e−(z−n) , {n, 6, −6, −2} − 2.75 = 0.
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(3.3)
This gives the exponential equation (4.1) and the rhombic dodecahedron in Fig. 6(a).
(3.4)
The Mathematica script used follows as below Sum f, {n, nmin, nmax, di} the sum with n increasing in steps of di. This is of course identical to using .
(a)
(a) (b)
(b) (c) Fig. 5. (a) Three terms in Eq. (3.2) form a small crystal. (b) Bigger crystal from seven terms in Eq. (3.3). Or m = 6 in Eq. (3.4).
Fig. 6. (a) Rhombic dodecahedron. (b) m = 4. (c) m = 4.5.
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e(y+z) + e(x+z) + e(x+y) + e−(y+z) + e−(x+z) + e−(x+y) + e(y−z) + e(x−z) + e(x−y) + e−(y−z) + e−(x−z) + e−(x−y) = 1013 .
(4.1)
We give structure and periodicity directly in the summation below: 2 2 2 2 Sum e−(x+y−n) + e−(x−y−n) + e−(y+z−n) + e−(y−z−n) 2 2 + e−(z+x−n) + e−(z−x−n) , {n, m, −m, −2} − 5.25 = 0.
(4.2)
(a)
(b) Fig. 8.
Eq. (4.2) gives a small crystal of bcc, with the outer shape of a rhombic dodecahedron in Fig. 6(b). A phase shift (m = 4.5) gives a new structure as shown in Fig. 6(c). This structure has a 3-connected net originally derived by Wells [8] and this is a very beautiful structure indeed. This has been discussed several times in our books in connection with the γ -Si and SrSi2 structures. Each body is surrounded in a planar way by three other bodies. If the plot is made to include finite borders, the crystal shape is again that of a rhombic dodecahedron. The group for bcc is Im3m, and for Wells’ net the group is I 41 32 with bodies in 8(a). 5. Diamond structure This structure has the Wykoff position 4e as it occurs in the cubic groups with the positions
(a)
x, x, x; x, −x, −x − x, x, −x; −x, −x, x. The points are used to describe the planes forming a tetrahedron (Fig. 7) with the equation ex+y+z + ex−y−z + e−x+y−z + e−x−y+z = 108 .
(5.1)
Due to the non-centric structure of the tetrahedron we have to make the summations as in Eq. (5.2) producing a small diamond, or the adamantane structure in Fig. 8(a) and a bigger piece of diamond in Eq. (5.3), and shown in Fig. 8(b) 2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, 1.5, −0.5, −2} = 3.7, (5.2)
(b) Fig. 9. (a) Octahedral diamond. (b) Tetrahedral diamond.
2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, 1.5, −2.5, −2} = 3.7.
(5.3)
We make a still bigger piece of diamond after Eq. (5.4) and show it in Fig. 9(a) where we clearly see that the shape is octahedral, and will remain so whatever size we build.
Fig. 7.
2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, 3.5, −4.5, −2} = 3.7.
(5.4)
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735
(a)
Fig. 10. Inner part of a diamond crystal sized 3 microns.
Fig. 11. Octahedron.
Only using positive phases as in Eq. (5.5) we obtain a shift in crystal shape as in Fig. 9(b). 2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, −0.5, −8.5, −2} = 3.75. (5.5) The summation concept makes it easy to extend to any crystal size, and in Eq. (5.6) below the crystal size is about three microns. We have with these boundaries for the crystal calculated a small part in the interior of the crystal as is obvious in Fig. 10. The calculation is carried out under rather low resolution—still it took almost an hour on my Mac. A normal high resolution picture like the one in Fig. 9(b) takes 35 seconds. 2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, −0.5, −104 , −2} = 3.83. (5.6) 6. Octahedral code or ccp And finally for the octahedron we take the 8g in Pm3m, which has the following permutations: x, x, x; x, −x, −x; −x, x, −x; −x, −x, x; −x, −x, −x; −x, x, x; x, −x, x; x, x, −x. And from the octahedral planes we obtain the octahedron via Eq. (6.1) in Fig. 11.
(b) Fig. 12.
ex+y+z + ex−y−z + e−x+y−z + e−x−y+z + e−(x+y+z) + e−(x−y−z) + e−(−x+y−z) + e−(−x−y+z) = 108 .
(6.1)
As we use a symmetric summation we may use the tetrahedral summations shown in Eq. (6.2). Bodies meet to form the ccp packing in Fig. 12(a) with a constant of 2.45 and it is clear that each body has 12 next neighbours. 2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, 6, −6, −3} = const. (6.2) And for a constant of 2.97 we see the complete finite periodicity for this symmetry in the form of a cubic close packed arrangement of 75 bodies in an outer octahedral shape in Fig. 12(b). 7. The dodecahedral code, spherical geometry and the structure of a dodecahedral crystal Recently we brought the symmetry codes into the Schrödinger solution of the quantized harmonic oscillator to learn if we
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could use the Hermite mathematics as a finite periodic function [9]. The result was rewarding and we continued in order to find out if there was any bearing on the quasi crystal structure problem. Again the result was rewarding [10] and we had also proposed earlier to use spherical geometry to describe the extraordinary stellations of the dodecahedron and the icosahedron [4,11]. We have now taken the complete step for a thorough study of this problem, in combining the symmetry codes with dodecahedral symmetry, and doing the mathematics with the new periodicity of spherical geometry. We showed above simple examples of spherical geometry with radial periodicity. The spherical geometry allows all the symmetry elements with the 5- or 10-fold axes of rotation as essential as the 2-, 3- and 4-fold axes of rotation in the description of structures. So it is only natural to add the symmetry code for the dodecahedron to the others given above: x, τ x, 0; 0, x, τ x; τ x, 0, x; x, −τ x, 0; 0, x, −τ x; −τ x, 0, x; −x, −τ x, 0; 0, −x, −τ x; −τ x, 0, −x; −x, τ x, 0; 0, −x, τ x; τ x, 0, −x. This gives the exponential equation (7.1), which yields the dodecahedron in Fig. 13. e(x+τy) + e(x−τy) + e(y+τ z) + e(y−τ z) + e(z+τ x) + e(z−τ x) + e−(x+τy) + e−(x−τy) + e−(y+τ z) + e−(y−τ z) + e−(z+τ x) + e−(z−τ x) = 1013 .
(7.1)
An excellent example of the power of the concept of spherical geometry with radial periodicity is given in a comparison of the dodecahedral structure with that of the rhombic dodecahedral. 2 2 2 2 Sum e−(x+y−n) + e−(x−y−n) + e−(y+z−n) + e−(y−z−n) 2 2 + e−(z+x−n) + e−(z−x−n) , {n, m, −m, −2} − 5.25 = 0.
(7.2)
The equation in (7.2) gives all the geometry for bcc in spherical manner. A crystal is growing radially with increasing m. We show the cases m = 2, 4 and 6 below. It is easy to see how the faces of the rhombic dodecahedron are growing. At m = 2 one face contains 4 bodies, while at m = 6 one face contains 16 bodies. We have changed the constant slightly, to 5.12 for m = 6, to make the polyhedron more visible. The case for m = 4 was also shown above. We have shown the close relation between the rhombic dodecahedron and the dodecahedron before [12]. The development of the dodecahedral symmetry is very similar as in Fig. 14, and the ‘crystals’ shown in a sequence of pictures below all have dodecahedral shapes. The equation in (7.3) gives the geometry for all the shapes and structures below. 2 2 2 Sum e−(x+τy−n) + e−(x−τy−n) + e−(y+τ z−n) + e−(y−τ z−n) + e−(z+τ x−n) + e−(z−τ x−n) , {n, mτ, −mτ, −τ } − const = 0. 2
2
2
(7.3)
The polyhedra that grow in the sequence of crystal sizes with m = 1–3, with a constant of 6.55 in Eq. (7.3), are shown in order of appearance in Figs. 15. For m = 3 and higher it is getting increasingly difficult to dissect the pictures, which is due to the Cartesian cut in our calculations. sin x sin[x cos π/5 + y sin π/5](sin[x cos 2π/5 + y sin 2π/5] × sin[x cos 3π/5 + y sin 3π/5] × sin[x cos 4π/5 + y sin 4π/5] = 0.32. Fig. 13. The planes from Eq. (7.1) bend over and form the dodecahedron.
(a)
(b)
(7.4)
We saw above how small crystals of the rhombic dodecahedron were formed layer by layer (mathematically) in Figs. 14. The
(c)
Fig. 14. The rhombic dodecahedra. (a) m = 2. (b) m = 4. (c) m = 6.
S. Andersson / Solid State Sciences 8 (2006) 730–739
(a)
737
(b)
(c)
(d)
(e)
(f)
(g)
Fig. 15. (a) m = 1. (b) m = 2. (c) m = 3. (d) m = 4. (e) m = 4. After 5 fold axis. (f) After Eq. (7.4). (g) After 5 fold axis.
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(a)
(b)
(c)
(d)
Fig. 16. (a) m = 129.44 or 80τ , content about 1000 bodies, from Eq. (7.3). (b) Real dodecahedron Bulk Al65Cu20Fe15 annealed at 845 ◦ C for 48 h. From Prof. A.P. Tsai [13] Tohoku University. (c) Inside of figure (a). (d) Projection of bigger part of inside (a), the dodecahedron from (c) is clearly visible.
‘Gedankenspiel’ is that all the 12 faces of dodecahedral symmetry develop simultaneously with the spherical periodicity, all the way up to the structures in Figs. 16, where the calculated shape of a dodecahedron is compared to a real dodecahedral crystal. From Fig. 15(a) we obtain Fig. 15(b) by adding pentagons, with bodies in anti-positions on the pentagons in Fig. 15(a). In Fig. 15(c) the third layer contains a central pentagon in antiposition with the pentagon in the layer below. This continues with every layer when a dodecahedron is growing. From the observations above we conjecture that a crystal growth must have a strict frame of rules: (a) These crystal growth principles of dodecahedral symmetry are in analogy with those of the orthorhombic rhombohedral symmetry (in alloy terminology body centered cubic). (b) The central pentagon is rotated ±π/5 passing each layer radially from center of the crystal. In each layer there is a structure built of different sized pentagons that seems to be identical, or nearly so, with an ordinary cyclic, two dimensional, calculation (without translation of course) from Eq. (7.4). (If you take cos instead of sin in Eq. (7.4) there is a perfect agreement with the common dif-
fraction patterns of the icosahedral Al phase.) Eq. (7.4) is made finite in Fig. 15(f) via a Photoshop cut, and which is compared with a complete dodecahedron of m = 4 in Fig. 15(e). The white dots in this figure mark the 3-fold axes, as well as corners of the dodecahedron. These white dots are easily recognized in Fig. 15(f), and 12 such filled pentagons build the outside of the structure for m = 4. The inside we already know. Such layers grow in size with the size of the dodecahedron, and their structures follow a gradual expansion from Eq. (7.4). We show one more member in Fig. 15(g) for m = 5. We conclude that the mathematics and the symmetry codes allow for the strict planar character of the faces of the dodecahedral crystals. The complete a 3-dimensional structure, of any crystal size, is extraordinarily simple. We like to show excellent agreement between the shapes of this calculated dodecahedron and the natural one from Professor A.P. Tsai’s laboratory [13] as shown in Fig. 16. The calculated dodecahedron from 80τ (two hours computing with my Mac) contains about 1000 bodies and the size factor is 104 as compared to Tsai’s crystals. It is obvious that with a computer
S. Andersson / Solid State Sciences 8 (2006) 730–739
big enough we could calculate a dodecahedron with edges as sharp as those of the crystals prepared in the Tsai [13] laboratory. Indeed the observations in Fig. 16 give great support for the model derived. If we study the inside of a crystal with m = 80, we see exactly the same pattern as shown in Figs. 15. In order to see a part of this we have used a constant of 6.69 to get the dodecahedral center as in Figs. 16(c) and (d), the expansion of which is easier to dissect than that of an icosahedral center which is there at a constant of 6.5. We have illustrated how crystal growth occurs via the faces, which of course is normal in the alloy systems where the quasi crystal phase is formed fast, as with those first made [14], the mechanism might be better described as martensitic [11]. Still the method of crystal structure description would be the same. We note that simple mathematics from Eq. (7.3) describe some seemingly complicated structures. The crystals of dodecahedral symmetry, from a few taus up to 80τ in size, are built in spherical space with radial periodicity. Simple mathematics and simple symmetry describe some beautiful science. The model derived here has nothing in common with various models as derived from Kowalewski’s early work (as discussed in Ref. [15]). In a forthcoming paper we shall extend this study to the icosahedral polyhedron and also elliptic geometry.
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References [1] H. Weyl, Symmetry, Princeton Univ. Press, Princeton, NJ, 1952, p. 129. [2] A.V. Shubnikov, V.A. Koptsik, Symmetry in Science and Art, Plenum, New York, 1974, p. 73. [3] S. Andersson, Group and structure, Chapter 8, Living book www. sandforsk.se. [4] S. Andersson, Stellations, compounds, periodicity and the exponential scale, www.sandforsk.se, November 2004. [5] K. Larsson, J. Phys. Chem. 93 (1989) 7304. [6] S. Andersson, K. Larsson, M. Larsson, M. Jacob, Biomathematics, Elsevier, Amsterdam, 1999. [7] International Tables for X-Ray Crystallography, The Kynoch Press, 1969. [8] A.F. Wells, Three-Dimensional Nets and Polyhedra, Wiley, New York, 1977, p. 35. [9] S. Andersson, Crystal structures, the quantized harmonic oscillator and the exponential scale, www.sandforsk.se, October 2005. [10] S. Andersson, Quasi structure and quantized harmonic oscillator, www.sandforsk.se, October 2005. [11] S. Andersson, Quasi structure, spherical geometry and interpenetrating icosahedra, www.sandforsk.se, January 2005. [12] S. Andersson, The various forms of pyrite, and how to give curvature to the cube. The square Alhamra net and the nodal gyroid surface, www.sandforsk.se, November 2004. [13] A.P. Tsai, A. Inoue, T. Masumoto, Jpn. J. Appl. Phys. 27 (1987) L1505. [14] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [15] S. Andersson, S. Lidin, M. Jacob, O. Terasaki, On the quasicrystalline state, Angew. Chem. Int. Ed. 30 (1991) 754.
Crystal structure and elliptic periodicity. Cubes and dodecahedra. Without π Sten Andersson Sandforsk, Institute of Sandvik, S-38074 Löttorp, Sweden Received 5 January 2006; received in revised form 24 February 2006; accepted 18 April 2006
Dedicated to Professor Hans Georg von Schnering on the occasion of his 75th birthday
Abstract A new periodic and elliptic function, based on the exponential scale, is described. This space contains classic as well as icosahedral crystallography. Crystal structures and crystal shapes of cubic and dodecahedral symmetries are discussed in the spherical space of the new periodic function. © 2006 Elsevier SAS. All rights reserved.
0. Preface I have enjoyed an excellent friendship with Hans Georg von Schnering for almost half a century. We have had great many extraordinary discussions, normally very tough but never unfriendly. Whatever science idea I propose, diffuse and/or theoretical, he understands immediately. He tells me, also immediately, if it is stupid, or not. Normally he is right in his judgement. It has indeed been a remarkable association. Hans Georg is a brilliant man, in science and in friendship. I believe our collaboration has been so fruitful because we have done careful thinking on “What harm a wind too strong at sea might do”, because we are sailors. But it might be more adequate to quote Snorre’s Edda: ett vet jag som aldrig dör, domen över död man.1 Snorre wrote his Edda on Iceland 300 years before The merchant of Venice was written. 1. Introduction The Arabs derived planar groups with the notion of symmetry and applied translation to get structures to describe patterns present on walls or in gardens or parks. Some of these patterns had been around long before the Alhambra. But the Arabs E-mail address: [email protected] (S. Andersson). URL: http://www.sandforsk.se. 1 B.G. Hyde has translated this for me: One thing I know which never dies, the judging of a dead man. Which is much better than what is in the trade. 1293-2558/$ – see front matter © 2006 Elsevier SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2006.04.012
found all the 17 planar groups to describe any planar pattern [1]. Thereby, and with other findings they invented modern mathematics. This was the beginning of many things, among them crystallography. But it took 1000 years for the Western world to rediscover it all. The need was there not only to describe mineral crystals but also to handle the findings of von Laue, and the Braggs. Crystallography had its peak in science during the first half part of last century. Geometry and structure formed the raison d’etre for crystallography during the time up to now with biology and chemistry, the DNA and the protein structures that gave the clue to the understanding of life. And of course, in solid state science where it has been the main tool from the beginning up till now. The geometry we describe in this article is in elliptic space. We may as well say spherical geometry, or just spherical trigonometry for the simpler spaces. The elliptic space allows five-fold symmetry, as well as the ordinary symmetry operations from Euclidean 3D space. The tool to describe it is to stay in Euclidean space and use special functions. We can, for example, describe 20 points regularly spaced on a sphere with a simple function of six terms. This distribution corresponds to a dodecahedron. Another possibility is 20 faces regularly spaced on a sphere, which is the dual, or the icosahedron. Yet another distribution, eight points equally spaced on a sphere, is described with three terms. Earlier we described polyhedra like the great dodecahedron, or the great stellation of the icosahedron in this way using exponential scale methods. Shells were
S. Andersson / Solid State Sciences 8 (2006) 730–739
formed as a case of spherical geometry, and a great number of points were described from very simple functions [4]. To this we add the radial spherical periodicity in Section 2 below with which we build crystals containing atoms, or bodies. As we use Mathematica we need to establish a few habits: In the Cartesian system of Mathematica, an implicit function of the three spacial coordinates x, y, z will have the cubic symmetry of the coordinate system if the expression is homogeneous in x, y, z. To generate a general position (x, y, z in crystallographic literature) we have to distinguish between the three, and use an expression of a form such as x, 2y, 3z. Similarly, in permutations below for a position x, x, x in crystallographic literature we simply use x, y, z. Example: For the permutation of the axes for a general point (x, y, z → z, x, y in the crystallographic literature) we use x, 2y, 3z → 3x, y, 2z. All the 47 Shubnikov polyhedra [2] from the point groups have been described in our book, Group and Structure, published in Sandforsk, Chapter 8 [3]. The one below, pentagonal tristetrahedron, belongs to the space group P23, and the last one from group Pm3m is called hexoctahedron and has 48 identical triangular faces (non-regular), and shown in Fig. 3. This is some spherical periodicity indeed, without π and τ as well! Those polyhera where first produced in the von Federov kaleidoscope. In order to obtain these polyhedra von Federov recommends pouring mercury into his kaleidoscope [2]! The pentagonal tristetrahedron we use as example and calculate its surface. We use the general positions as planes which added together, on the exponential scale, bend over to form the surfaces of the group. This yields the shape of the Shubnikov polyhedron or the mineral crystal. f (x,y,z) = The basic exponential form used is simply e const, where the function f (x, y, z) is the symmetry-distinguishing sum of the components x, y, z, and the sum is taken over all symmetry equivalent permutations in an orbit. The size of the constant is the value of the iso-surface generated, and this value has been determined experimentally for each case shown below. Other values will give surfaces of the same symmetry, but with different curvature, and even different topology. The spacegroup for the pentagonal tristetrahedron is P23 (No. 195) with the general position given by x, y, z; z, x, y; y, z, x; x, −y, −z; z, −x, −y; y, −z, −x; −x, y, −z; −z, x, −y; −y, z, −x; −x, −y, z; −z, −x, y; −y, −z, x; This translates to a set of functions f (x, y, z) x + 2y + 3z; 3x + y + 2z; 2x + 3y + z; x − 2y − 3z; 3x − y − 2z; 2x − 3y − z; −x + 2y − 3z; −3x + y − 2z; −2x + 3y − z; −x − 2y + 3z; −3x − y + 2z; −2x − 3y + z; The general position is 12 j and the first equation is in (1.1). ex+2y+3z + e3x+y+2z + e2x+3y+z + ex−2y−3z + e3x−y−2z + e2x−3y−z
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Fig. 1. The pentagonal tristetrahedron of group P23.
Fig. 2. The final stellation of the icosahedron, equation in Ref. [4].
+ e−x+2y−3z + e−3x+y−2z + e−2x+3y−z + e−x−2y+3z + e−3x−y+2z + e−2x−3y+z = 1016 .
(1.1)
This polyhedron as shown in Fig. 1 is called the pentagonal tristetrahedron by Shubnikov (No. 38) and has 12 faces of identical and non-regular pentagons. So this is our starting example of finite periodicity, which of course is spherical. Of course all polyhedra are examples of periodic spherical geometry. The stellated cases are also excellent cases as reported in Sandforsk [4]. We give one example here, the great stellation of the icosahedron in Fig. 2 with altogether 92 points in a formidable periodic and spherical geometry described in Ref. [4] with only 10 terms. A spherical and periodic geometry without π , but with τ . 2. Background to the radial spherical finite periodicities We continue now with the radial spherical periodicity. A number of GD (Gauss distribution) terms added together with a proper constant give the new periodicity. It was first developed to describe the Larsson cubosomes [5,6]. The key is that each term is peaked. One term consists of two infinite
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a detailed description of the dodecahedral structures here, we leave the icosahedral structures for a forthcoming paper. The cubic symmetry codes x, 0, 0; 0, x, 0; 0, 0, x; −x, 0, 0; 0, −x, 0; 0, 0, −x; x, x, 0; 0, x, x; x, 0, x; x, −x, 0; 0, −x, −x; x, 0, −x; −x, x, 0; 0, x, −x; −x, 0, −x; −x, −x, 0; 0, −x, x; −x, 0, x; x, x, x; x, −x, −x; −x, x, −x; −x, −x, x; x, x, x; x, −x, −x; −x, x, −x; −x, −x, x; −x, −x, −x; −x, x, x; x, −x, x; x, x, −x; The dodecahedral symmetry code x, τ x, 0; 0, x, τ x; τ x, 0, x; x, −τ x, 0; 0, x, −τ x; −τ x, 0, x; Fig. 3. The hexoctahedron of group Pm3m after Shubnikov (No. 47). The corresponding equation is given in Sandforsk [3].
planes in space, two terms give four planes that collaborate to form a cylinder, and with three, there is a sphere. These properties of the GD function in space is the backbone of the 2D periodic properties around the sphere. The radial periodicity in 3D space completes a finite elliptic periodicity, which we find very useful for crystal structure descriptions. The equations in (2.1) and (2.2) define the periodicities. e
−x 2
e
−(x−δ)2
+e
−y 2
−z2
+e
+e
−(y−δ)2
= const, +e
−(z−δ)2
= const.
−x, −τ x, 0; 0, −x, −τ x; −τ x, 0, −x; −x, τ x, 0; 0, −x, τ x; τ x, 0, −x. We give a number of examples of the new periodicity. 3. Primitive cubic We start with the simplest permutation possible, using the position 6e in Pm3m, with the first symmetry code x, 0, 0; 0, x, 0; 0, 0, x; −x, 0, 0; 0, −x, 0; 0, 0, −x
(2.1)
exponentially it becomes
(2.2)
ex + ey + ez + e−x + e−y + e−z = const
Eq. (2.1) is of the type used to describe the stellations that gave excellent examples of spherical periodicity, as the great stellation of the icosahedron above. With δ this function is now growing radially in space in a true periodic, and also finite, manner. The δs must take the same value in x, y, z to stepwise increase the radius for the spherical periodicity. The motion has 3-fold symmetry. In a cube you have four diagonals of such symmetry and in the dodecahedron, there are ten. The motion now generates the 3D periodicity as said above. The general 3D radial periodicity is elliptic and we will handle this special spherical case of 3D radial periodicity that. The elliptic periodicity is easily developed within group theory, and we hope to do so in a forthcoming article. There are a few simple permutations in space that occur in one form or another in the groups [7], which we call symmetry codes. The symmetry codes on the exponential scale used here make shapes (or polyhedra) which after the crystallographic jargon are: the cube, the rhombic dodecahedron, the diamond, the octahedron, the dodecahedron and the icosahedron. Using GD terms and the new periodicity we have in order as above: pc or primitive cubic structure, bcc or body centred cubic structure, diamond structure, fcc cubic close packing or octahedral structure, the dodecahedron for the dodecahedral structure, and the icosahedron for the icosahedral structure. While we shall give
(3.1)
which is a cube (Fig. 4). We now switch to the GD function, and write periodic functions, which are finite. This makes sense since crystals are of finite periodicity. So we write in Eq. (3.2) a function with only three terms diagonally that give 27 bodies in Fig. 5(a). e−x + e−y + e−z + e−(x−2) + e−(y−2) + e−(z−2) 2
2
2
2
2
+ e−(x+2) + e−(y+2) + e−(z+2) = 2.65. 2
2
2
2
(3.2)
Eq. (3.3) contains seven terms which, due to the properties of true periodicity, gives 343 bodies (73 ) in Fig. 5(b). But the equation itself is a summation and can, equivalently, be written in
Fig. 4.
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Mathematica manners as in Eq. (3.4) below
4. Body centered cubic(bcc) and 3-connectivity(gyroid)
e−x + e−y + e−z + e−(x−2) + e−(y−2) + e−(z−2) 2
2
2
+e
−(x−4)2
+e
−(y−6)2
+e
−(z+2)2
2
+e
−(y−4)2
+e
−(z−6)2
+e
−(x+4)2
2
−(z−4)2
+e
+e
−(x+2)2
+ e−(y+2)
+e
−(y+4)2
−(z+4)2
+e
+ e−(x+6) + e−(y+6) + e−(z+6) 2
2
2
In our next example the symmetry code is 12i, P432, which has the following permutations:
−(x−6)2
x, x, 0; 0, x, x; x, 0, x; x, −x, 0; 0, −x, −x; x, 0, −x;
2
+e
−x, x, 0; 0, x, −x; −x, 0, −x; −x, −x, 0; 0, −x, x; −x, 0, x.
2
= 2.78, 2 2 2 Sum e−(x−n) + e−(y−n) + e−(z−n) , {n, 6, −6, −2} − 2.75 = 0.
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(3.3)
This gives the exponential equation (4.1) and the rhombic dodecahedron in Fig. 6(a).
(3.4)
The Mathematica script used follows as below Sum f, {n, nmin, nmax, di} the sum with n increasing in steps of di. This is of course identical to using .
(a)
(a) (b)
(b) (c) Fig. 5. (a) Three terms in Eq. (3.2) form a small crystal. (b) Bigger crystal from seven terms in Eq. (3.3). Or m = 6 in Eq. (3.4).
Fig. 6. (a) Rhombic dodecahedron. (b) m = 4. (c) m = 4.5.
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e(y+z) + e(x+z) + e(x+y) + e−(y+z) + e−(x+z) + e−(x+y) + e(y−z) + e(x−z) + e(x−y) + e−(y−z) + e−(x−z) + e−(x−y) = 1013 .
(4.1)
We give structure and periodicity directly in the summation below: 2 2 2 2 Sum e−(x+y−n) + e−(x−y−n) + e−(y+z−n) + e−(y−z−n) 2 2 + e−(z+x−n) + e−(z−x−n) , {n, m, −m, −2} − 5.25 = 0.
(4.2)
(a)
(b) Fig. 8.
Eq. (4.2) gives a small crystal of bcc, with the outer shape of a rhombic dodecahedron in Fig. 6(b). A phase shift (m = 4.5) gives a new structure as shown in Fig. 6(c). This structure has a 3-connected net originally derived by Wells [8] and this is a very beautiful structure indeed. This has been discussed several times in our books in connection with the γ -Si and SrSi2 structures. Each body is surrounded in a planar way by three other bodies. If the plot is made to include finite borders, the crystal shape is again that of a rhombic dodecahedron. The group for bcc is Im3m, and for Wells’ net the group is I 41 32 with bodies in 8(a). 5. Diamond structure This structure has the Wykoff position 4e as it occurs in the cubic groups with the positions
(a)
x, x, x; x, −x, −x − x, x, −x; −x, −x, x. The points are used to describe the planes forming a tetrahedron (Fig. 7) with the equation ex+y+z + ex−y−z + e−x+y−z + e−x−y+z = 108 .
(5.1)
Due to the non-centric structure of the tetrahedron we have to make the summations as in Eq. (5.2) producing a small diamond, or the adamantane structure in Fig. 8(a) and a bigger piece of diamond in Eq. (5.3), and shown in Fig. 8(b) 2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, 1.5, −0.5, −2} = 3.7, (5.2)
(b) Fig. 9. (a) Octahedral diamond. (b) Tetrahedral diamond.
2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, 1.5, −2.5, −2} = 3.7.
(5.3)
We make a still bigger piece of diamond after Eq. (5.4) and show it in Fig. 9(a) where we clearly see that the shape is octahedral, and will remain so whatever size we build.
Fig. 7.
2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, 3.5, −4.5, −2} = 3.7.
(5.4)
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735
(a)
Fig. 10. Inner part of a diamond crystal sized 3 microns.
Fig. 11. Octahedron.
Only using positive phases as in Eq. (5.5) we obtain a shift in crystal shape as in Fig. 9(b). 2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, −0.5, −8.5, −2} = 3.75. (5.5) The summation concept makes it easy to extend to any crystal size, and in Eq. (5.6) below the crystal size is about three microns. We have with these boundaries for the crystal calculated a small part in the interior of the crystal as is obvious in Fig. 10. The calculation is carried out under rather low resolution—still it took almost an hour on my Mac. A normal high resolution picture like the one in Fig. 9(b) takes 35 seconds. 2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, −0.5, −104 , −2} = 3.83. (5.6) 6. Octahedral code or ccp And finally for the octahedron we take the 8g in Pm3m, which has the following permutations: x, x, x; x, −x, −x; −x, x, −x; −x, −x, x; −x, −x, −x; −x, x, x; x, −x, x; x, x, −x. And from the octahedral planes we obtain the octahedron via Eq. (6.1) in Fig. 11.
(b) Fig. 12.
ex+y+z + ex−y−z + e−x+y−z + e−x−y+z + e−(x+y+z) + e−(x−y−z) + e−(−x+y−z) + e−(−x−y+z) = 108 .
(6.1)
As we use a symmetric summation we may use the tetrahedral summations shown in Eq. (6.2). Bodies meet to form the ccp packing in Fig. 12(a) with a constant of 2.45 and it is clear that each body has 12 next neighbours. 2 2 2 Sum e−(x+y+z−n) + e−(x−y+z−n) + e−(x+y−z−n) 2 + e−(−x+y+z−n) , {n, 6, −6, −3} = const. (6.2) And for a constant of 2.97 we see the complete finite periodicity for this symmetry in the form of a cubic close packed arrangement of 75 bodies in an outer octahedral shape in Fig. 12(b). 7. The dodecahedral code, spherical geometry and the structure of a dodecahedral crystal Recently we brought the symmetry codes into the Schrödinger solution of the quantized harmonic oscillator to learn if we
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could use the Hermite mathematics as a finite periodic function [9]. The result was rewarding and we continued in order to find out if there was any bearing on the quasi crystal structure problem. Again the result was rewarding [10] and we had also proposed earlier to use spherical geometry to describe the extraordinary stellations of the dodecahedron and the icosahedron [4,11]. We have now taken the complete step for a thorough study of this problem, in combining the symmetry codes with dodecahedral symmetry, and doing the mathematics with the new periodicity of spherical geometry. We showed above simple examples of spherical geometry with radial periodicity. The spherical geometry allows all the symmetry elements with the 5- or 10-fold axes of rotation as essential as the 2-, 3- and 4-fold axes of rotation in the description of structures. So it is only natural to add the symmetry code for the dodecahedron to the others given above: x, τ x, 0; 0, x, τ x; τ x, 0, x; x, −τ x, 0; 0, x, −τ x; −τ x, 0, x; −x, −τ x, 0; 0, −x, −τ x; −τ x, 0, −x; −x, τ x, 0; 0, −x, τ x; τ x, 0, −x. This gives the exponential equation (7.1), which yields the dodecahedron in Fig. 13. e(x+τy) + e(x−τy) + e(y+τ z) + e(y−τ z) + e(z+τ x) + e(z−τ x) + e−(x+τy) + e−(x−τy) + e−(y+τ z) + e−(y−τ z) + e−(z+τ x) + e−(z−τ x) = 1013 .
(7.1)
An excellent example of the power of the concept of spherical geometry with radial periodicity is given in a comparison of the dodecahedral structure with that of the rhombic dodecahedral. 2 2 2 2 Sum e−(x+y−n) + e−(x−y−n) + e−(y+z−n) + e−(y−z−n) 2 2 + e−(z+x−n) + e−(z−x−n) , {n, m, −m, −2} − 5.25 = 0.
(7.2)
The equation in (7.2) gives all the geometry for bcc in spherical manner. A crystal is growing radially with increasing m. We show the cases m = 2, 4 and 6 below. It is easy to see how the faces of the rhombic dodecahedron are growing. At m = 2 one face contains 4 bodies, while at m = 6 one face contains 16 bodies. We have changed the constant slightly, to 5.12 for m = 6, to make the polyhedron more visible. The case for m = 4 was also shown above. We have shown the close relation between the rhombic dodecahedron and the dodecahedron before [12]. The development of the dodecahedral symmetry is very similar as in Fig. 14, and the ‘crystals’ shown in a sequence of pictures below all have dodecahedral shapes. The equation in (7.3) gives the geometry for all the shapes and structures below. 2 2 2 Sum e−(x+τy−n) + e−(x−τy−n) + e−(y+τ z−n) + e−(y−τ z−n) + e−(z+τ x−n) + e−(z−τ x−n) , {n, mτ, −mτ, −τ } − const = 0. 2
2
2
(7.3)
The polyhedra that grow in the sequence of crystal sizes with m = 1–3, with a constant of 6.55 in Eq. (7.3), are shown in order of appearance in Figs. 15. For m = 3 and higher it is getting increasingly difficult to dissect the pictures, which is due to the Cartesian cut in our calculations. sin x sin[x cos π/5 + y sin π/5](sin[x cos 2π/5 + y sin 2π/5] × sin[x cos 3π/5 + y sin 3π/5] × sin[x cos 4π/5 + y sin 4π/5] = 0.32. Fig. 13. The planes from Eq. (7.1) bend over and form the dodecahedron.
(a)
(b)
(7.4)
We saw above how small crystals of the rhombic dodecahedron were formed layer by layer (mathematically) in Figs. 14. The
(c)
Fig. 14. The rhombic dodecahedra. (a) m = 2. (b) m = 4. (c) m = 6.
S. Andersson / Solid State Sciences 8 (2006) 730–739
(a)
737
(b)
(c)
(d)
(e)
(f)
(g)
Fig. 15. (a) m = 1. (b) m = 2. (c) m = 3. (d) m = 4. (e) m = 4. After 5 fold axis. (f) After Eq. (7.4). (g) After 5 fold axis.
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(a)
(b)
(c)
(d)
Fig. 16. (a) m = 129.44 or 80τ , content about 1000 bodies, from Eq. (7.3). (b) Real dodecahedron Bulk Al65Cu20Fe15 annealed at 845 ◦ C for 48 h. From Prof. A.P. Tsai [13] Tohoku University. (c) Inside of figure (a). (d) Projection of bigger part of inside (a), the dodecahedron from (c) is clearly visible.
‘Gedankenspiel’ is that all the 12 faces of dodecahedral symmetry develop simultaneously with the spherical periodicity, all the way up to the structures in Figs. 16, where the calculated shape of a dodecahedron is compared to a real dodecahedral crystal. From Fig. 15(a) we obtain Fig. 15(b) by adding pentagons, with bodies in anti-positions on the pentagons in Fig. 15(a). In Fig. 15(c) the third layer contains a central pentagon in antiposition with the pentagon in the layer below. This continues with every layer when a dodecahedron is growing. From the observations above we conjecture that a crystal growth must have a strict frame of rules: (a) These crystal growth principles of dodecahedral symmetry are in analogy with those of the orthorhombic rhombohedral symmetry (in alloy terminology body centered cubic). (b) The central pentagon is rotated ±π/5 passing each layer radially from center of the crystal. In each layer there is a structure built of different sized pentagons that seems to be identical, or nearly so, with an ordinary cyclic, two dimensional, calculation (without translation of course) from Eq. (7.4). (If you take cos instead of sin in Eq. (7.4) there is a perfect agreement with the common dif-
fraction patterns of the icosahedral Al phase.) Eq. (7.4) is made finite in Fig. 15(f) via a Photoshop cut, and which is compared with a complete dodecahedron of m = 4 in Fig. 15(e). The white dots in this figure mark the 3-fold axes, as well as corners of the dodecahedron. These white dots are easily recognized in Fig. 15(f), and 12 such filled pentagons build the outside of the structure for m = 4. The inside we already know. Such layers grow in size with the size of the dodecahedron, and their structures follow a gradual expansion from Eq. (7.4). We show one more member in Fig. 15(g) for m = 5. We conclude that the mathematics and the symmetry codes allow for the strict planar character of the faces of the dodecahedral crystals. The complete a 3-dimensional structure, of any crystal size, is extraordinarily simple. We like to show excellent agreement between the shapes of this calculated dodecahedron and the natural one from Professor A.P. Tsai’s laboratory [13] as shown in Fig. 16. The calculated dodecahedron from 80τ (two hours computing with my Mac) contains about 1000 bodies and the size factor is 104 as compared to Tsai’s crystals. It is obvious that with a computer
S. Andersson / Solid State Sciences 8 (2006) 730–739
big enough we could calculate a dodecahedron with edges as sharp as those of the crystals prepared in the Tsai [13] laboratory. Indeed the observations in Fig. 16 give great support for the model derived. If we study the inside of a crystal with m = 80, we see exactly the same pattern as shown in Figs. 15. In order to see a part of this we have used a constant of 6.69 to get the dodecahedral center as in Figs. 16(c) and (d), the expansion of which is easier to dissect than that of an icosahedral center which is there at a constant of 6.5. We have illustrated how crystal growth occurs via the faces, which of course is normal in the alloy systems where the quasi crystal phase is formed fast, as with those first made [14], the mechanism might be better described as martensitic [11]. Still the method of crystal structure description would be the same. We note that simple mathematics from Eq. (7.3) describe some seemingly complicated structures. The crystals of dodecahedral symmetry, from a few taus up to 80τ in size, are built in spherical space with radial periodicity. Simple mathematics and simple symmetry describe some beautiful science. The model derived here has nothing in common with various models as derived from Kowalewski’s early work (as discussed in Ref. [15]). In a forthcoming paper we shall extend this study to the icosahedral polyhedron and also elliptic geometry.
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