Disclination in atomic structures described in curved spaces J.F. Sadoc, R. Mosseri

To cite this version:

J.F. Sadoc, R. Mosseri. Disclination density in atomic structures described in curved spaces. Journal de Physique, 1984, 45 (6), pp.1025-1032. ￿10.1051/jphys:019840045060102500￿. ￿jpa-00209832￿

HAL Id: jpa-00209832 https://hal.archives-ouvertes.fr/jpa-00209832 Submitted on 1 Jan 1984

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Physique 45 (1984) 1025-1032 JUIN 1984, 1025

Classification Physics Abstracts

61.00 - 61.70

Disclination density in atomic structures described in curved spaces

J. F. Sadoc (*) and R. Mosseri (**) (*) Lab. de Physique des Solides, 91405 Orsay, France (**) Physique des Solides, CNRS, 1 Pl. A. Briand, 92190 Meudon Bellevue, France

(Reçu le 28 octobre 1983, accepte le 8 février 1984)

Résumé. 2014 La courbure de l’espace et la densité de disinclinaisons sont deux grandeurs connectées. Il y a une relation exacte à 2 dimensions. Nous montrons comment utiliser une relation approchée à 3 dimensions. Les appli- cations à la structure W-03B2 et aux phases de Laves sont présentées. La coordinance, dans les structures denses aléatoires, est expliquée à partir de la notion de densité de disinclinaison.

Abstract. 2014 The curvature of a space and the density of disclinations are two related quantities. There is an exact relation in 2-D spaces. We show how an approximate solution can be used in 3-D space. Applications to the 03B2-W structure and the Laves phase are presented The coordination number in dense random structures is explained in terms of disclination density.

1. Introduction. deficit of a surface topologically equivalent to a sphere, and the sum of the Gaussian curvature on the In a 2-D space the curvature of space and the density surface. In fact this is a simple example of the Gauss- of disclinations are related to each other exactly by Bonnet theorem, and this relation is applicable to the Gauss-Bonnet theorem [1]. An attractive example all the surfaces approximated by a polyhedron occurs when disclinations are not infinitesimal, with flat faces. Consequently in 2d space, there is but correspond to a finite angular deficit (or excess) : an entire equivalence between the disclination density the surface is everywhere flat except on point defects (with a weighting factor equal to their angular deficit) to a local concentration of curvature. corresponding and the curvature. In this case the 2-D space is the surface of a polyhedron Unfortunately in 3-D space no such simple result [2] a since it is not a closed (in general sense, necessarily exists. surface). In the case of regular polyhedra simple The aim of this paper is to relate disclination density defined two Schlafli indices { p, q } the angular by and curvature in 3-D spaces. (For example polytopes deficit 6 of each vertex is 6 = 2 7C 2013 where qap, ap embedded in 4-D Euclidean space.) Application to is the vertex angle of a p-gonal face. some complex periodic structure [3] which can be So described in corrugated 3-D space (with a zero mean curvature) is presented. We show how positive and negative disclinations approximatively annul each The number of vertices for regular polyhedra is other, if positive and negative curvatures balance mutually in to give a zero mean curvature. Coordination numbers in dense structures are explai- ned in terms of disclination density. (from the Euler relation). Consequently the total angular deficit for a poly- hedron is Lb = 4 7L Now consider a polyhedron 2. Polytopes and disclinations. to be an approximation of a sphere with a constant A polytope is an approximation by flat spaces (cells) gaussian curvature K. We can write Kda = 4n of a 3-D spherical space. The curvature is concen- ff trated on edges acting as disclinations. and conclude to the equality between the total angular Polytopes are very well described in the books

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045060102500 1026

by Coxeter [4, 5]. They are characterized by three Schlafli indices I p, q, r } : I p, q } is characteristic of the polyhedral cell and r is the number of cells sharing a common edge. In the following paper the word polytope is used with two different meanings : some times it is an ensemble of cells with straight edges, and some times it is a spherical space containing vertices connected by geodesic edges; in this case we call it a spherical polytope.

3. Orthoscheme and symmetry sphere.

The orthoscheme tetrahedron is a very efficient way to describe a The faces of this tetrahedron polytope. Fig. 2. - The orthoscheme tetrahedron for the {4, 3, 4 } are mirrors. A given vertex is reproduced in these tessellation (the euclidean cubic tessellation). Point Po mirrors like in a kaleidoscope. All these images are correspond to a vertex, P1 is the centre of edge, P2 is the the vertices of the polytope. The faces of the ortho- centre of face, P3 the centre of cell. scheme tetrahedron are rectangular triangles defined in the spherical space. An Euclidean tetrahedron can also be defined by analogy. A M6bius spherical triangle (which is equivalent to the orthoscheme tetrahedron in 2-D spherical space) is presented in figure 1. All identical triangles obtained by reflection in their sides form a pattern covering the sphere just once. If a vertex is placed on one of the vertices of the M6bius triangle, its images form a polyhedron.

Fig. 3. - The unfolded orthoscheme. The value of all angles as functions of p, q, r are given in the appendix.

2-sphere. These geodesic lines are the reflection lines of the polyhedron. In 3-D spherical space a face of the spherical orthoscheme is part of a 2-sphere which is Fig. 1. - A Mobius spherical triangle VCE, and other called « symmetry sphere of the polytope ». identical triangles obtained by reflection in their sides to form a pattern covering the sphere just once. The angles of this triangle are V = nlp, C = nlq, E = n/2 if the poly- 4. Disclinations and symmetry spheres. hedron is { p, q }. Consider a spherical 3-D space with an inscribed spherical polytope. This polytope is characterized by The orthoscheme tetrahedron of {4, 3, 4 } (the its symmetry spheres. If the curvature of the space is Euclidean cubic tessellation) is represented on figure 2. then concentrated on the edges of the polytope, a The relation between a polytope cell and the ortho- symmetry sphere is distorted and becomes a poly- scheme tetrahedron is clearly defined on this figure. hedron (but we always call it « symmetry sphere »). In order to define angles of the orthoscheme tetra- The basic idea of this paper is to study the effect of hedron faces, it is drawn unfolded on figure 3. The disclinations going through a symmetry sphere in vertex of the polytope { p, q, r } is called Po, the centre order to understand the combination law of these of cell is P3, the centre of face is P2 and the centre of disclinations. We take advantage of the existence of an edge is Pl. In the appendix we give the different ele- exact solution for a symmetry sphere (which is a 2-D ments of the Mobius triangle and of the orthoscheme space) and we extend results to the spherical 3-D tetrahedron following the Coxeter notation. space characterized by the symmetry spheres. Looking at figure 1 we observe that sides of the Disclination lines are edges of the polytope, conse- Mobius triangle are parts of great circles of the quently they go through a symmetry sphere with 1027 respect to the mirror symmetry. A disclination is perpendicular to this surface, if there is no vertex of the polytope corresponding to its intersection with the symmetry sphere. If there is a vertex on the symmetry sphere it is a node for disclinations. Some of these disclination segments are in the surface, others are disposed symmetrically with respect to the mirror. In this case they can be considered as a continuous discli- nation line with a cusp where it goes through the symmetry sphere.

- 4.1 EXAMPLE OF THE { 5, 3, 3} POLYTOPE. - The Fig. 5. Positive disclinations relative to a pentagonal face of the of the 5, 3, { 5, 3, 3 } is a packing of dodecahedra in which an symmetry sphere { 3 } polytope. edge joins three cells. The angular deficit, leading to disclinations, is due to the deviation from 2 7r/3 of the there are five disclinations a dihedral angle. More formally this going through pentagonal face with a bend at the crossing point point). deficit is related to the deviation of the 7r/r (r = 3) (Po We can estimate the contribution to the curvature angle in the spherical orthoscheme, from the same of the different disclinations crossing the symmetry angle in the Euclidean orthoscheme arcos sphere. (Keeping in mind that the total curvature is (2013 2013 4 n). Disclinations to the surface = x perpendicular symmetry cos nlq The° disclination angle is 6 2 n - 6 sIn n p )) contribute with their exact angular deficit 6 to the 1.01722 or 6 - 0.1798 rd curvature. Disclination lines (edges of the polytope) form a Contribution bb of disclinations bent at the crossing tetracoordinated network since there are four dode- point Po can be obtained from the sum of the vertex cahedra sharing a vertex. angles of triangular faces of the orthoscheme which The sphere of symmetry of the { 5, 3, 3 } can be cover the symmetry sphere. obtained from an icosidodecahedron. This polyhedron Disclinations through P 1 point have a contribution contains all equatorial vertices of the dual ’polytope b = 2 ropq - n and through Po point (vertex of the { 3, 3, 5 }. Consequently the icosidodecahedron is the polytope) set of all the P3 points of the { 5, 3, 3 } symmetry sphere (Fig. 4a). On figure 4b the positions of other points (Po, I P I, P2) are shown in a pentagonal and a triangular face of the icosidodecahedron. using the appendix notation (see also Fig. 3). Applied with p = 5, q = 3, r = 3

6 = 0.1798 rd (10.30°) and bb = 0.1494 rd (8.56°) .

It is obvious that 4 7r = 20 6 + 12 x 5 bb*

4.2 EXAMPLE OF THE SIMPLEX POLYTOPE { 3, 3, 3 }.- The frame of the symmetry sphere is a spherical tetra- hedron with non-equilateral triangular faces; angles of these triangles appear on figure 6. A disclination line goes through this surface on P 1 point with a contribu- tion equal to ð = 2 n - 3 C(T = 2.5903 rd (otr is the Fig. 4a. - The symmetry sphere (polyhedral approximation) for the { 5, 3, 3 } polytope can be deduced from an icosido- decahedron.

Fig. 4b. - Positions of (Po, P1, P2, P3) points in icosido- decahedron triangular and pentagonal faces.

There are 20 disclinations perpendicular to the symmetry sphere in the centre of triangular faces (P 1 points). The arrangement of other disclinations close to the symmetry sphere is described on figure 5. There are Fig. 6. - The unfolded symmetry sphere (polyhedral five disclination segments in each pentagonal face and approximation) for the simplex polytope f 3, 3, 3 }. 1028

dihedral angle of a regular tetrahedron). Three other disclination lines go through Po points making an angle n/3 at these points. Contribution to the symmetry sphere curvature is 6b = 3.325 rd. In this example we observe an increase of the disclination contribution on bending. So any approxi- mation of 6b by bb = 6 sin a, in which a is the angle between the disclination with the symmetry surface, is completely false. (In the case of the { 5, 3, 3} with a = 108°/2 bb/b = 0.809, compared to the exact value 6bl6 = 0.83119.)

4. 3 APPLICATION TO #-TUNGSTEN STRUCTURE. - The Fig. 8. - Projection on a [100] plane of the fl-W structure. circles are for a of atoms in the fl-W (A 15) structure can be described [3] by disclina- Heavy family just projection Some Voronoi Cells are shown. Points tions in a tetrahedral packing. The tetrahedral packing plane. Po, P1, P2, P3 defined on a symmetry surface are noted 0, 1, 2, 3. is built up in a first step by atoms put on the vertices of a { 3, 3, 5 } polytope or in a dual description by atoms in the center of cells of a { 5, 3, 3 } polytope This results from the (these cells are Voronoi cells of the structure). This geodesic triangles. supposition that all tetrahedral interstices are tetrahedra tetrahedral packing is in a spherical curved space. In a regular second step it is transformed into the fl-W structure by with equal edges. In this description A 15 structural disclinations along the A 15 linear chains (Fig. 7). This space is Euclidean only in average. In paragraph 4 the { 3, 3, 5 } symmetry sphere is described using an icosi- dodecahedron which is a tiling by pentagons and triangles. The symmetry surface of the A 15 structure can be deduced in a similar way from a tiling of triangles and (see in the figure P3 points). This tiling correspond to an icosidodecahedron trans- formed by negative disclinations, which change penta- gons into hexagons. Account of the positive and negative disclinations in fl-W structure We describe fl-W structure using a mixing of positive and negative disclinations. The positive disclinations (edge of the Voronoi froth) are identical to { 5, 3, 3 } edges. Their contribution is determined in paragraph Fig. 7. - Cubic cell of a f3- W structure. 4.1. Disclinations going through symmetry sphere orthogonally on P 1 points contribute with 6 = 0.1798 rd; disclinations structure can be considered as a mixing of positive going through point Po a bend on Voronoi contribute to the disclinations on the edges on the Voronoi cells, and (with vertices) curvature with = 0.1494 rd. The of the negative disclinations on the straight lines going 6 b part sym- in the cubic cell is a through 14 coordinated atoms (A 15 chains). The metry plane crystallographic which contains two and six Voronoi froth is a packing of regular dodecahedra P 1 points Po points. with polyhedra characterized by 12 pentagonal faces The positive contribution to the curvature of this sur- and 2 hexagonal faces (a disclinated dodecahedron). face in this square is C+ = 1.257 rd The curvature is due to disclina- On figure 8 is presented an orthogonal projection of negative negative tions to the surface in the fl-W structure. On the right part of the figure some orthogonal symmetry P2 points. curvature at these is 6 = 27r - Voronoi cells appear. The projection plane is a mirror Negative points of the structure. On the left some part of the figure = 1.257 rd The negative curvature exactly points are quoted with the notation used to characte- 12 5 n rize orthoscheme vertices. annuls the positive curvature. This mirror of the structure can be considered as a symmetry sphere distorted by negative disclinations, as 4.5 LAVES PHASE EXAMPLE. - A similar description the A 15 structure is obtained from a { 3, 3, 5 } polytope can be done with the Laves phase structure (Cu2Mg) distorted to an Euclidean structure by a disclination [3]. This structure can also be described using a procedure. { 5, 3, 3 } polytope with atoms inside dodecahedral Notice that in this description the symmetry surface cells, and inserting negative disclinations in order to is not exactly flat, but is a corrugated surface tiled by reduce this curvature to zero. But in this case the 1029 negative disclination network is a diamond network. 5. Disclination density and the Regge calculus [6]. A symmetry plane of the structure appears in figure 9. In paragraph 4 we have described exactly how discli- It is a [ 110] crystallographic plane (Fig.10). This plane nations contribute to the curvature of particular can be obtained from an icosidodecahedron by geodesic surfaces which are mirrors of the structure. changing some triangles into pentagons in order to Intuitively, it seems that the disclination density have the same number of pentagons and triangles on the of geodesic lines going from one (this is needed to follow the Euler relation for the depends length of these surfaces to another close surface. This can be plane). Disclinations are represented on the left part explained more precisely. of the figure. In a unit cell of this plane there are A disclination segment goes from a point to 10 positive disclinations of type through points, Po bb Po another point (vertices of the polytope) following 2 positive disclinations of ty-pe 6 through P 1 points and Po an of the This disclination is ortho- 2 negative disclinations through P3 points occupied by edge polytope. to a face of a cell at a point. The disclination large atoms. gonal P, concentrates all the curvature of the bipyramidal volume built with this face and the two Po points (Fig. 11). (Called an E-Cell).

Fig. 9. - Projection on a [110] plane of the Laves phase structure. Positive disclinations go through this plane on points Po and P,, negative disclinations go through this plane on points P3 occuped by large atoms. Fig. 11. - An edge Po Po between two connected vertices of a polytope, and a face P3, P3, P3 define a bipyramid If the curvature of the space is concentrated on a Po Po edge, this given edge contained all the curvature of the bipyramid

The disclination contribution to the curvature depends on the length of the Po Po disclination seg- ment and on the volume of the pyramid The difficulty comes from the evaluation of this volume in a non- Euclidean space. Nevertheless in a space of constant curvature the area of the face can be evaluated : it is 2 r times the area of the P, P3 P2 triangle (Fig. 3). In non-Euclidean space the area of this triangle is Fig. 10. - Projection plane in the cubic cell and negative disclinations. (Î + ; + ’" pq - 1t). (The radius of curvature is used as unit). We deduce the contribution in the cell positive The area of the face is S = r 2 r 7r - 2 qlpq - 7r’ it is very important to note that this area is also the The negative contribution of negative disclinations angular deficit corresponding to the disclination. is If we suppose the volume calculated by V = ’3 S. h (as it is in Euclidean space) the disclination density 3 M . p = is equal to 3. usirig the appendix notation. We observe the exact S. h compensation of positive and negative curvature This result can be compared with a relation intro- (there are 2 negative disclinations in the cell). duced by Regge [7] in order to study an approximation 1030 of space-time by tetrahedral Euclidean cells connected 6. Coordination number in dense packing with only by their faces and edges as in a honeycomb : tetrahedral sites. The { 3, 3, 5 } polytope is an example of a dense packing in a spherical space [8]. In this case the coordination number is 12. This polytope can be used to build where (3)R is the scalar curvature and where £ extends different models of Euclidean dense structures. Nega- i tive disclinations are introduced in order to change the to all disclinations segments bi of length li in the volume curvature. Numerous structures are obtained if various of summation. Applied to a 3-sphere with (3)R = 6/R2 networks of negative disclinations are used. With this (R is the radius of curvature) gives : description positive disclinations run on {3, 5, 5 } edges (1), and negative disclinations are superposed to some of the edges. They appear on a tetrahedron edge if 6 tetrahedra share this edge (in the following dis- cussion we suppose that all tetrahedra are regular, we deduce consequently the space is corrugated). The coordina- tion number of these structures can be determined if the relative length of positive disclinations and nega- tive disclinations is known. We have = with I to the or with a sphere of unit radius : defined p 161V equal geo- desic length of the E-Cell height and V equal to the volume of the E-Cell in curved space. This volume can all disclinations volume of the 3-sphere be calculated if the space is spherical. But in the pre- sent case the space is corrugated with positive and which is equivalent to the 2-D relation negative curvature which cannot be taken as constant in a whole E-Cell. An exact calculation will be, if it is possible, extremely difficult We only give an estima- all disclinations area of the 2-sphere tion of the density of disclinations. But the 3-D result is an approximation. This approxi- The density of positive disclinations p+ = L+ 6+ mation is (in 3-D) similar to the approximation (in is p+ = 31R 2if we suppose infinitesimal disclinations. In R 2-D) of the area of a 2-sphere by the area of a poly- the following discussion the radius of curvature hedron. But in the present description it is the geodesic of the spherical space is the unit length, then p + = 3. length and the spherical surface of the bipyramid which In a spherical space approximated by a polytope are used to approximate volume, and not their Eucli- p + = N16,12 n 2. The quantities N, I and 6 + are dean counterparts. This increases the accuracy of the defined in table I. For the particular case of the approximation. Table I shows application of this { 3, 3, 5 } polytope p + = 2.942. approximation for the regular polytopes. The density of negative disclinations (with the same unit length) in the approximation of infinitesimal disclinations is p_ = L - 6_ = - 3 in order to ba- Table I. - Approximative volume V for {p, q, r} lance positive disclinations. If we suppose the E-Cell polytope in relation to the angular deficit 6 on edge and space to have a negative constant curvature, then to the edge length I. N is the number of edges. s is the I p - I :> 3. The exact calculation of the volume of a ratio between V and the volume of the 3-sphere 2 n2. tetrahedron in hyperbolic space is not an easy pro- blem [9]. In the present case the reference hyperbolic structure is { 3, 3, 6 } which is a very special honey- comb with all vertices at infinity and consequently the hyperbolic approach does not offer any advantage. But the reader can easily convince himself that the approximation of the tetrahedron volume by V = -116 is an under estimate in spherical space ( p + 3) and an over-estimate in hyperbolic space. (For example : in hyperbolic space the tetrahedron with point at infinity has a finite volume.)

(1) In paragraph 4 we have described A15 structure with positive disclinations on the { S, 3, 3} edges. The present treatment offers a dual description to that given previously and gives completely equivalent results for the coordination number. 1031

If the space is corrugated there are necessarily of disclinations as is the case for Laves phase. Oppo- departures from a constant curvature in each E-Cell. sitely if negative disclinations are not connected The effect is an increase of geodesic length which (fl-W) z increases up to 13.50. The fl-uranium decreases p (defined by p = 611V where I and V are structure is an intermediate case. But I p-I fluctua- height and volume of the E-Cell in the space with tions around 3 cannot explain large a z value, because spread curvature, and not in the space approximated it is unrealistic to supposes p- ! value very different by Euclidean cells). from 3. Consequently p+ is smaller than 3, but p can be The body centred cubic structure is helpful to greater or smaller than 3 depending on the exact understand this problem. In this case all positive configuration of the structure. disclinations correspond to 6+ = 2 n - 4 at In order to determine the coordination number we (5+ = 1.359 rd). Using the polytope { 3, 3, 4} then needs the length of positive and negative L+ and L_ L = 1.909 and p + = 2.59. With p_ = - 3 we disclinations per unit volume. They are related to p + deduced nt = 5.17 and z = 14.51. This is a too large and p- by p+ = 6+ L+ and p- = 6- L- (using z value compared to that of 14 in the b.c.c. structure. 6+ =0.1283 rd and 6- = 2 n - 6 C(t (6_ = 1.1026 rd) p- have to be reduced to a value lower than 3 in for negative disclinations corresponding to six tetra- relation to the great connectivity of disclination hedra sharing an edge). networks. From this example we conclude that high The mean number of tetrahedra sharing an edge is coordination numbers are related with strong positive disclinations. In random gas of points, disclinations corresponding to 3 tetrahedra sharing an edge can The coordination number is related (9) to nt by appear and increase z. These results have to be com- 12 pared to N. Rivier’s [12] results. In the N.R. paper, = =- =- zZ 12 _. With IP p I P+ 3 we getet ( b-n large z values are related to Voronoi cells with a low symmetry. In the present paper this is related to the strong positive disclinations. Both analyses are close to each if we which are the values obtained by Coxeter [10] using other consider that strong positive trigonometric arguments. disclinations completely distort the icosahedral order

= = which is one the best With p_JI 3 and p + 2.942 (spherical space) it of polyhedral approximations of the In N.R. z comes nt = 5.106, z = 13.42. spherical symmetry. the paper, small nt and z values for some crystalline structure which value are related to large volume fluctuations of can be described with this approach are presented in Voronoi cells (or to large coordination number table II. We observe variation of z from 13.33 to 14. fluctuations). In the present paper it is related to high Larger value for z are observed with completely connectivity of the negative disclination network. random structures [11] (z = 15.54 for random gas of Both conclusions are equivalent if we suppose the points). second effect is not being screened by the first one These values can be explained in terms of disclina- (strong positive disclinations). tion density. z values smaller than 13. 39 are related to I p -[ 3. This is the consequence of large variations of the 7. Conclusion. curvature in the corresponding E-Cells. It is the case if Glassy structures, but also some crystalline structures positive and negative E-Cells are strongly mixed This [1-13] can be described from curved structures dis- corresponds to a highly connected negative network torted by disclination lines in order to change the

Table II. - Coordination in metal structures. Notice that the two numbers n and fit are related by n(6 - nt) =12 (from Euler relation). If regular tetrahedra are packed with a common edge there is place for 5.1043 tetrahedra. 1032

curvature. We have determined the defect density. Following Coxeter [5] we give the value of angles There is not an exact mathematical solution in 3-D defined on the figure 3 in the spherical case. space, but there are approximate solutions. Using opql qlpql Xpq (resp. Xqr, t’lqr, qqr) are the sides of the symmetry surface the relation of curvature and Mobius triangle of the cell f p, q } (resp. { qr })(Fig. 1). disclinations is reduced to a 2-D problem easy to They are obtained from the relations : solve. A disclination perpendicular to a surface acts on this surface as a point disclination with the same angular deficit If disclinations cross together on a surface the intuitive law leading to a contribution depending on the sine of the angle with the surface (as will be the case if the effect of disclinations depended on the flux of a disclination vector through the surface) is false. Using the exact calculation in 2-D, we extend it to and and r indices. 3-D. The density of disclinations can be estimated if respectively with q the whole volume is divided into elementary volumes 0, t/J, X are edge-lengths of the orthoscheme tetra- associated with disclination segments. This density hedra I p = Lb where L is the length of disclination in a unit volume is close to p = 3 1 R is the radius of curva- ture). But in 3-D space this result is an approximation; the exact value depends on the precise of the disclination network. It is used to explain the fluctua- tion of the coordination number in dense packing.

where is given by Appendix. - The orthoscheme tetrahedron (Figs. 2-3), hpq This figure defined in a curved space works like a kaleidoscope : by reflection into its mirror faces, for given values of p, q, r, some very regular figures appear. These figures are polytopes if the space curvature is with these relations all elements of the orthoscheme positive, or honeycomb if the curvature is negative. tetrahedron can be obtained in terms of p, q, r.

References

[1] LELONG-FERRAND, Géométrie Différentielle (Masson [6] REGGE, T., Il Nuovo Cimento, XIX (1961) 558. Paris), 1960. [7] MISNER, C. W., THORNE, K. S., WHEELER, J. A. Gra- [2] GASPARD, J. P., MOSSERI, R. and SADOC, J. F., in vitation (W. H. Freeman and Company, San Proc. of « Structure of Non-Crystalline Mate- Francisco). rials » (Edited by Gaskell, P. H. et al.) 1982, [8] SADOC, J. F., J. Non-Cryst. Solids 44 (1981) 1. p. 550. [9] THURSTON, W. P., Geometry and topology of 3-Manifold GASPARD, J. P., MOSSERI, R. and SADOC, J. F., à Lecture Notes (Princeton University). paraitre dans Philos. Mag. (1984). [10] COXETER, H. S. M., Illinois J. Math. 2 (1958) 746. [3] SADOC, J. F., J. Physique Lett. 44 (1983) L-707. [11] HANSON, H. G., J. Stat. Phys. 30 (1983) 591. [4] COXETER, H. S. M., Regular Polytopes (Dover Publ. [12] RIVIER, N., J. Physique Colloq. 43 (1982) C9. New York) 1973. [13] NELSON, D., Phys. Rev. Lett. 13 (1983) 983. [5] COXETER, H. S. M., Regular Complex Polytope (Cam- bridge University Press) 1974.