real case. The Bi$_{2}$Fe$_{4}$O$_{9}$ real lattice is therefore topolog- ically equivalent to the Cairo one after renormalization of the J$_{3}$ and J$_{5}$ interactions with respect to the J$_{4}$ one. To summarize, we have found that the compound Bi$_{2}$Fe$_{4}$O$_{9}$ displays a very original non-collinear magnetic structure, made of four Fe$_{1}$ moments and four Fe$_{2}$ ones forming interpenetrating patterns of four-fold spin rota- tions. This magnetic arrangement is very peculiar for a pentagonal lattice with nearest neighbour interactions only and is not a mere propagation of the magnetic ar- rangement minimizing the energy in each individual pen- tagon, contrary to the case of triangles based lattices. It actually results from two ingredients, a high geometrical frustration and an additional complex connectivity (two kinds of sites with different coordinations). This first materialisation of a pentagonal Cairo lattice opens new perspectives in the field of magnetic frustration. Work at Sofia was supported by the Bulgarian Science Foundation under Grants No. TK-X-1712/2007. The authors are grateful to Michael Mikhov for the high tem- perature susceptibility results and to Luc Valentin for the help with the tilings. FIG. 4: (color online) Magnetic structure. First and second magnetic domains. (Yellow): Fe 1 sites. (Orange): Fe 2 sites. The second domain is deduced from the first one by apply- ing the symmetry operation ( x + 1 / 2 , ¯ y + 1 / 2 , ¯ z ) lost by the magnetic structure, taking into account the translational part that shifts atoms from one cell to a neighbouring one. the magnetic configuration is always made of antiparal- lel Fe$_{1}$ pairs of moments and parallel Fe$_{2}$ ones. On a same site, the pairs of moments are perpendicular to each other. This is exactly what is observed experimen- tally. The only parameter that could vary as a function of the particular values of the J$_{i}$ ’s is the phase angle be- tween the two sites Fe$_{1}$ and Fe$_{2}$. The ∼ 155 ◦ observed value requires J$_{3}$ , J$_{4}$ , J$_{5}$ to be all negative, with a ratio J$_{3}$ / J$_{5}$ =2.15 for | J$_{4}$ |≥ | J$_{3}$ | . The observed magnetic struc- ture of Bi$_{2}$Fe$_{4}$O$_{9}$ and in particular the 90 ◦ phase angle between pairs of atoms at the same site, is thus recovered by this model with some realistic set of the five super- exchange interactions J$_{i}$ . In the Bi$_{2}$Fe$_{4}$O$_{9}$ real geometry, the Fe 3+ lattice differs from the Cairo pentagonal one by the presence of two stacked Fe$_{2}$ atoms, instead of a sin- gle atom, at two vertices in each pentagon. Performing the calculations in the real geometry yield very similar results than those of the ideal case. The perpendicular ferromagnetic and antiferromagnetic Fe$_{1}$ and Fe$_{2}$ pairs is a robust recovered feature. The only difference between both lattices is the domain of existence of this phase that extends to lower | J$_{3}$ | and | J$_{5}$ | with respect to | J$_{4}$ | in the ∗ ressouche@ill.fr [1] X. J. Kong, Y. P. Ren, L. S. Long, Z. Zheng, R. B. Huang, and L. S. Zheng, J. Am. Chem. Soc. 129 , 7016 (2007). [2] J. Kepler, Harmonices Mundi (J. Planck for G. Tampach, Linz, Austria, 1619). [3] R. Moessner, A. P. Ramirez, Physics Today, 59 (2) , 24 (2006). [4] A. K. Singh, S. D. Kaushik, B. Kumar, P. K. Mishra, A. Venimadhav, V. Siruguri, S. Patnaik, Appl. Phys. Lett., 92 , 132910 (2008). [6] M. H. Waldor, W. F. Wolff, J. Zittartz, Phys. Lett. 106A , 261 (1984); Z. Phys. B: Condens. Matter 59 , 43 (1985). [5] N. Shamir, E. Gurewitz, Acta Crystallogr. Sect. A 34 , 662 (1978). [7] U. Bhaumik, I. Bose, Phys. Rev. B 58 , 73 (1998). [8] R. Moessner, S. L. Sondhi, Phys. Rev. B 63 , 224401 (2001). [9] E. Ressouche et al. , in preparation. [10] D. M. Gianchinta, G. C. Papaefthymiou, W. M. Davis, H. C. Zur Loye, J. of Solid State Chem. 99 , 120 (1992). [11] A. G. Tutov, I. E. Mylnikova, N. N. Parfenova, V. A. Bokov, S. A. Kizhaev, Sov. Phys. Solid State 6 , 963 (1964). [12] J. Brown, J. Matthewman, Cambridge Crystallography Subroutine Library , Report No. RAL93-009 (1993). [13] P. Wolfers, J. Appl. Crystallogr. 23 , 554 (1990). [14] J. Rodriguez-Carvajal, Physica B 192 , 55 (1993). [15] P.J. Becker, P. Coppens, Acta Crystallogr. Sect. A 30 , 129 (1974). [16] V. C. Rackhecha, N. S. Sayta Murthy, J. Phys. C.:Solid State Phys. 11 , 4389 (1978). [17] E. F. Bertaut, J. Phys. Chem. Solids 21 , 256 (1961). 4