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Quasi structure, spherical geometry and interpenetrating icosahedra
Sten Andersson
Sandforsk, Institute of Sandvik
S-38074 Löttorp, Sweden
www.sandforsk.se
Abstract
Spherical geometry, fivefold symmetry and finite periodicity are described.
Transformations between quasi structures and alloy or steel phases as bcc or fcc
are described as martensitic.
The icosahedral interpentration structure (iis) for a quasi crystal structure is
described in terms of the exponential scale method. Atomic positions from (iis)
are used to show a martensitic transformation from bcc.
1 Introduction
When you take a number of pentagons and put them on a plane, they do not fit
together. When you spread them out on a sphere of proper curvature they fit
perfectly well together. If there are 12 of them you build a dodecahedron and we
say this is finite periodicity in a spherical space. Or if you with twelve atoms form
the corners of an icosahedron, you need 20 for a dodecahedron.
Fivefold symmetry is natural for the spherical geometry.
A polyhedron is easily described as a periodic structure on or in a spherical
surface. The periodicity is of course finite, and good examples are the great
stellated dodecahedron and the final stellation of the icosahedron. So it is
tempting to use the mathematics of spherical geometry to describe these
polyhedra, and in the elongation to try and also describe the structure of a quasi
crystal of fivefold symmetry.
Finite periodicity and spherical geometry
First we repeat the foundation of finite periodicity using the Euclidean plane(ref
1).
We show the equation for a cube.
2
2
y2
e x + e + ez = const
1
We make it to a GD function which for the used constant is hyperbolic as shown
in fig 1a.
†
2
2
-y2
e-x + e
+ e-z = 2
2
And as e-x
structure:
†
†
2
is a wellknown peaked function we construct a finite periodic
2
2
2
2
2
-y2
-(y-4)2
e-x + e
+ e-z + e-(x-4) + e
+ e-(z-4) +
2
2
-(y-8)2
e-(x-8) + e
+ e-(z-8) = 2
3
Equation 3 means adding three octahedra along the 3-fold axes, which gives the
finite periodic structure containig 27 octahedra in fig1b.
†
Fig1a
Fig1b Finite periodicity
In Sandforsk(ref 2) we recently reported the structures of the mentioned
stellations using this mathematics of finite periodicity. For the great stellated
dodecahedron we used equation 4, and for the final stellation of the icosahedron
we used equation 5. In figs 2a and b we see projections along the fivefold axes of
the two polyhedra.
-(tx+ y) 20
20
-(-ty+z) 20
20
†
-(tz+ x) 20
+ 20
-(t (x+ y+z))10
10
-(x+ t 2 y)10
10
-(y+ t 2 z)10
10
†
-(-tx+ y) 20
+ 20
-(ty+z) 20
+ 20
-(-tz+ x) 20
+ 20
-(t (-x+ y+z))10
+ 10
-(-x+ t 2 y)10
+ 10
-(y-t 2 z)10
+ 10
+
4
= 5/ 2
-(t (x+ y-z))10
-(t (x-y+z))10
+ 10
+ 10
+
2 10
2 10
+ 10-(z+ t x) + 10-(z-t x) +
5
=2
Starting with the great stellated dodecahedron there are six terms in its equation.
And there are first 12 icosahedral corners, and then 20 dodecahedral corners in the
outer shell. In all 32 Fibonacci points repeated in a spherical geometry.
The final stellation of the icosahedron has three shells in a 3D Fibonacci
expansion. Outside 20 dodecahedral corners in the inner shell, there are 12 points
at corners of an icosahedron in the next shell. In the final third shell there are the
60 points at the corners of a truncated icosahedron. Alltogether 92 points in a
3
formidable periodic and spherical geometry described by equation 5 with its 10
terms. A spherical geometry without π, but with t.
In the spherical non Euclidean geometry the fivefold axis of symmetry is added to
the two- and threefold symmetries. This is an answer as good as any to the
question ‘Why five fold symmetry?’ And five fold symmetry is very common in
biology.
We have found finite periodic mathematical functions to describe Fibonacci
expansion in 3D. Is it possible to create the kind of expansions as seen in
diffraction projections in 2D with this kind of mathematics? As Fibonaccis
starting up again, and again to fill space. There are at least 59 icosahedra around
to use(ref 3), and by adding the many periodic functions on the exponential scale
we are a bit on the way to get a multi spherical shell structure. But the
administration seems to be overwhelming and there must be simpler ways to get
structure proposals for pentagonal quasi crystals.
Fig 2a The great stellated dodecahedron
Fig 2 b. The final stellation of the icosahedron
We search other routes.
Our model and quasi
Five planes multiplied in the saddle way and going cyclic beautifully produce the
diffraction pattern of the decagonal quasi phase(ref 4). We do not describe the
strucure in this way, but it might be an entrance to find the route for a detailed
description of the icosahedral interpentration structure (iis) for a quasi crystal
structure we derived some time ago (ref 5,6,7).
In this structure we use the interpenetration geometry as described in figs 3a and
b.
4
Fig3a
b Kepler rhombus
Local symmetry operations like these repeated in space and incorporating random
and steric selection of present or absent vertices, recovers the proposed model of
the icosahedral quasi crystal(ref 6). With a structure of 400 interpentrated
icosahedra or 3900 points, numerical calculations of Fourier transforms of the
projections along the two-, three-, and five-fold axes were carried out. Such
calculated Fourier transforms show excellent agreement with observed ones in
form of electron diffraction patterns from real quasi crystals of the P-type.
An excellent description of the interpenetration principle, and its extension to
derive a complete quasi structure is given by Jacob in his thesis(ref 8), and in his
article in ref(6). Jacob has also proposed that the interpenetration principle
suggests a simple mechanism of crystal growth when crystals are formed from
the melt(ref 7,8).
The structure of a quasi crystal from Jacob’s thesis is shown in fig 4.
Fig 4
This structure model is described in a simple manner below using an exponential
equation that describes the conversion of a rhombic dodecahedron to a
dodecahedron. The corners of the rhombic dodecahedron plus its center represent
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a small piece of a bcc arrangement of atoms. By varying b in equation 6 as shown
in figure 5 a,b,c,d and e we arrive to fivefold symmetry. The rhombic
dodecahedron in fig 5c has 14 vertices and the dodecahedron 20 in figs 5a and 5e.
The end points in the transformation below in fig 5, 1/t and t correspond then to
the Fibonacci positions as derived in the icosahedral structure in ref (5,6,7,8).
Note that the cube of corners in both the polyhedra stay intact during the
transformation(the metric change is considerable).
e
e
(y + bz)
(y - bz)
4
4
+e
+e
(bx + z)
4
(-bx + z)
4
+e
(x + by)
+e
4
(x - by)
+
4
= 10
6
6
†
Fig 5a
Dodecahedron b= t
b Mode 1 b= 1/2(t+1/t)
c Rhombic dodecahedron b=1
d Mode 2 b= t/2
e Dodecahedron b= 1/t
6
We also do the turning of the dodecahedron into an icosahedron in fig 6 via
equation 7.
10
10
(y + bz)
10
(y - bz)
10
a(x - y - z)
10
a(x + y + z)
+ 10
10
+ 10
10
10
(bx + z)
10
(-bx + z)
+ 10
+ 10
+ 10
= 10
+
10
(x - by)
10
a(-x - y + z)
10
10
(x + by)
+
10
+ 10
a(-x + y - z)
7
+
6
†
Fig 6
a=1
b=2
a=1.5
b=2
a=1.5
b=2.25
For a=1.618 and b=2.168 there is the icosahedron.
The martensite transformation and its relation to quasi
The alloy phases having quasi structures are formed in temperature regions where
the classic structures like bcc, fcc, hcp or even w phases exist. There are structural
relationships between the traditional phases and we assume such relationships also
exist with the the quasi crystal phases. Such relationships are also often useful for
describing transformation mechanisms on the atomic level. We shall start to give
one example which is useful as a foundation in our further discussion.
The martensite transition was once explained by a Bonnet transformation
operating on the fcc structure to form a bcc packing of atoms(ref 9). In terms of
differential geometry the transformation is isometric meaning it occurs under
constant Gaussian curvature. There is no energy exchange. The transformation
occurs with extremely high velocity(close to speed of sound). Which means that it
starts at various sites within the crystal, and that different regions may transform
simultaneously. It can be said to happen everywhere at the same time. The
relationships between the quasi structure and the bcc and fcc metal alloy
structures make us propose simple martensite-similar mechanisms for the
transformations between these structures. The martensite transformation offers a
route for the understanding of the structure of a quasi crystal.
Again, we have a number of rhombic dodecahedra that fill space in a bcc manner
of packing atoms. There is one rhombic dodecahedron cluster in which the atoms
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move to Fibonacci coordinate numbers after fig 5 above, and the interpenetration
principle. This kind of flipping happens at same time and everywhere in the
crystal.
Conclusion
Jacob has found that the iis for a quasi crystal structure can be described as two
identical interpentrating networks. We show this with figs 7a and b from Jacob’s
thesis. Jacob says the surface between these two icosahedral nets has icosahedral
symmetry. How do we arrive at a mathematical description for this surface, that
also would become the mathematics for a quasi crystal? We can sketch how to get
that. We know the clear reationships between the the rhombic dodecahedron, the
dodecahedron and the icosahedron. The bcc structure is described by the O,C-TO
minimal surface for which we give the beginning in fig 8. The GD mathematics in
eq 8 give us directly the relationship to quasi symmetry via fig 9. where the
Cartesian plotting gives the possibility via boundaries to spell out the relation
between a bcc surface and a quasi surface on spee.
Fig 7a
Fig 8
b
Fig 9
The two surfaces for figs 8 and 9 are in equation 8 with a=1, const=1.75 for fig 8
and a=1/t, const=2.2 for fig 9.
8
4
4
-(ax+ y) 4
+4
-(-ay+z) 4
-(-ax+ y) 4
+4
-(az+ x) 4
+4
+4
-(ay+z) 4
-(-az+ x) 4
+
8
= const
The bcc surface is easily obtained by using the approach of hand made
periodicity(ref 4), which means that we add the positions of spheres in bcc
coordinates in a 3D summation, as given in equation 9, and shown in fig10. This
surface has the same topology as the O,C-TO minimal surface as said above. The
complete derivation of this formula in eq 9 is given in the next forthcoming article
in Sandforsk.
†
Ê 1
2
2
2 ˆ
Á - 4 ((x - n) + (y - m) + (z - p) ) ˜
SumÁ e
˜,{n,-3,10,6},{m,-3,10,6},{p,-3,10,6} +
Á
˜
Ë
¯
Ê 2
ˆ
2
2
2
Á - ((x - n) + (y - m) + (z - p) ) ˜
SumÁ e 5
˜,{n,0,10,6},{m,0,10,6},{p,0,10,6} = .21
Á
˜
Ë
¯
9
†
Fig 10
Jacob has concluded that his pictures in figs 7 a and b have a surface between the
red and blue. Such a surface would be a description of the quasi structure. The
way to proceed to obtain the icosahedral surface of Jacob is to construct a
summation as in eq 9, but with the Fibonacci numbers as derived by Jacob to
construct the (iis). This surface would have flat points like a periodic minimal
surface, but be distributed in Fibonacci numbers in space. It would indeed mimic
an icosahedral minimal surface.
References
1 M. Jacob and S. Andersson, THE NATURE OF MATHEMATICS AND THE
MATHEMATICS OF NATURE, Elsevier, 1998, pages 156-190.
2 S.Andersson Stellations, compounds, periodicity and the exponential scale,
Sandforsk, 2004.
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3 H.S.M. Coxeter, P. du Val, H.T.Flather, J.F.Petrie. The fifty-nine icosahedra.
University of Toronto 1951.
4 M. Jacob and S. Andersson, THE NATURE OF MATHEMATICS AND THE
MATHEMATICS OF NATURE, Elsevier, 1998, pages 137, and 205-228.
5 S. Andersson, S. Lidin, M.Jacob, O.Terasaki, On the quasicrystalline state,
Angewandte Chemie Int.Ed. 30 (1991) 754.
6 M. Jacob, A routine for generating the structure of an icosahedral quasicrystal,
Z. Kristallogr. 209 925 (1994).
7 M. Jacob, S. Lidin, S. Andersson, On the Icosahedral Quasicrystal Structure,
Z. anorg. allg. Chem.
619 (1993) 1721.
8 M. Jacob, Order, disorder and new order in the solid state. Thesis, Lund, 1994.
9 S.T. Hyde and S. Andersson, The martensite transition and differential
geometry. Z. Kristallogr. 174, 225 (1986).