The various forms of pyrite, and how to give curvature to the cube. Thesquare Alhamra net and the nodal gyroid surface.

Sten AnderssonSandforsk, Institute of SandvikS-38074 Löttorp, Sweden

www.sandforsk.se

Abstract The exponential mathematics of the rhombic dodecahedron is used todescribe the various crystal forms of pyrite. Ordinary mathematics is used todescribe the nodal gyroid surface, which is shown to be built of interpentrating andnon-intersecting planar square Alhambra net. The use of bending a planarAlhambra net to obtain the shape of curved pyrite crystals has its background ingiving curvature to a cube. The boundaries consist of eight planar curves and arederived from the rhombic dodecahedron.

1 Introduction

The bending of the square Alhambra net very precisely builds the curved pyritecrystal(Ref 1). Doing that the planar net gives curved boundaries for the crystal.We have found that the mathematical background is the symmetry of the rhombicdodecahedron and we shall here explore some remarkable topologies.

2 The rhombic dodecahedron and related polyhedra

The rhombic dodecahedron is the archtype for what in daily talk is called the bccsymmetry. The mathematical description is from its group below.

12i in P432, which has the following permutations:

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;

Which gives the exponential equation 1, which gives the rhombic dodecahedron infig 1a.

†

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

1

We generalize to equation 2

†

e(y + bz) + e(bx + z) + e(x + by) +

e-(y + bz) + e-(bx + z) + e-(x + by) +

e(y - bz) + e(-bx + z) + e(x - by) +

e-(y - bz) + e-(-bx + z) + e-(x - by)

= 1013

2

By varying b several crystal shapes show up in a continous connected spacebetween the rhombic dodecahedron and the cube as shown in Fig 1a-f. A similarstudy was done in Ref 2. For b=t in fig 1c we obtain the pentagonal dodecahedronwhich occurs as quasicrystals for AlFeCu (M.Audier in ref 3). Also theintermediate shape in Fig1b for b=1.25 occurs as AlCuRu quasicrystals(H.U.Nissen in Ref 3). The rhombic dodecahedron itself does not seem to exist as a ironsulfide mineral but it is wellknown in alloy systems, and is a common form ofgarnet, a picture of which is shown in Fig2 a. The pyritohedron in fig 1d for b=2 isa very common crystal shape for pyrite and shown in Fig 2b. The curved pyrite inFig 1e and 2c shows up here as a topological product between the rhombicdodecahedron and the cube.

Fig1a Rhombic dodecahedron, b=1 Fig1b, b=1.25

Fig 1c Pentagonal dodecahedron b=t Fig1d Pyritohedron b=2

Fig1e Curved pyrite b=4 Figf Cube b=20

Below we see mineral species of the shapes calculated above. We acknowledgeProfessor Hans Georg von Schnering for the gift of the garnet crystals in Fig2a,and Professor Carlos Otero Diaz for the pyrite minerals in Fig 2.

Fig 2a Rhombic dodecahedron b Pyritohedron

c Curved pyrite d Pyrite cube

Recently (Ref 1)we used the bending of a planar Alhambra net to obtain curvedpyrite and to give curvature to a cube. We now wish to show that the boundariesgiving this kind of curvature to the cube are easily obtained from the rhombicdodecahedron. This polyhedron contains a cube which we give curvature using itssymmetry as shown in fig 3a and b. Note that each curve is planar as derived inthis way.

Fig 3a. Model in thin Al. Fig 3b. Model in paper.

Curved cubes with different wavelength and amplitudes of waves may be obtainedfor any polyhedron, as was shown in Ref 1. In the cubic case this is particular easyusing the rhombic dodecahedron. And each curve always stays planar.

3 Translation structures of the rhombic dodecahedral symmetry

We shall now study translation structures from 12i of the group above with thebackground as the rhombic dodecahedron:

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,0;

We write an ordinary cyclic function as shown in equation 3 and if we use the repart we get the nodal IWP surface(in order to avoid singularities and get thesimilarity with the minimal surface we need a constant of -1).

ix+y + ix+z + iy+z + iy-z + i- x+z + ix- y +

i- x-y + i-x-z + i- y- z + i-x+ y + ix-z + i-y+ z = 0 3

Adding a phase to equation 3 gives equation 4 which is the same thing as theordinary trigonometric equation in 5. This we use we to get the beautiful nodalgyroid shown in two different projections after the cubic axes in figure 4. Equation5 is also identical with the nodal gyroid mathematics as derived by von Schneringand Nesper.

The figures reveal that the nodal gyroid surface contains planar square Alhambranets, that interpenetrate, without intersections. There are four planar and parallelsuch nets in each cubic unit cell and direction, of which we show two in fig 5.

ix+y +1 + ix+z+1 + iy+ z+1 + iy-z+1 + i-x+ z+1 + ix-y+1 +

i- x-y -1 + i- x-z-1 + i-y -z-1 + i-x+ y-1 + ix-z-1 + i-y+z-1 = 0 4

†

sinp(x + y) + sinp(x + y) + sinp(x + y) +

sinp(x + y) + sinp(x + y) + sinp(x + y) = 0 5

Fig 4 Projections of the nodal gyroid surface along cubic z and x.

Fig 5. Alhambra nets in nodal gyroid at sections z=.25 and z=.75 (1/8 and 3/8 of unit cell)

In the real gyroid, the minimal surface, these nets are not planar, they are spacecurves.

References.1. S. Andersson, Z. f. Anorganische und Allgemeine Chemie. In print.

2. M. Jacob and S. Andersson, THE NATURE OF MATHEMATICS AND THEMATHEMATICS OF NATURE, Elsevier, 1998.

3. C. Janot, Quasicrystals, pages 198-199, Oxford Science Publications, 1994.