Why is Pentagonal Symmetry present in a molecule but not in crystal?

Copper crystals of pentagonal symmetry have been reported by A. A. Vikarchuk and A. P. Volenko. These crystals are grown by electrolytic deposition method. The sizes of the pentagonal crystals reported are small, of the order of 200-300 ?m. Symmetry of the crystals are determined by various factors viz. non covalent interactions, solvent used for crystallization, inclusion of foreign particle etc. Hence a molecular symmetry and crystal symmetry would not be a match. By adjusting the conditions, crystals of different space group could be formed. read less

CRYSTALS AND FIVE FOLD SYMMETRY Mamoona posted a question on | 11 Nov, 2014 pentagonal symmetry is present in a molecule but not in crystal why? - See more at: http://www.urbanpro.com/tuition-questions#sthash.AO8lPEMN.dpuf Dear Mamoona (1) Ignore my previous answer to you (the lengthy one) (2) (a) It is the crystallographic restriction theorem that answers your question. A crystal has inherent in it certain rotational symmetry features. Yet again, the atoms forming a crystal structure are accounted by a set of periodically spaced points forming lattices (merged into each other, as in the case of two face centered cubic cells merging in, to account for the diamond crystal structure). These lattices have their points with spatial periodicity and for that matter discretely spaced. The crystallographic restriction theorem therefore observes the simple fact that the group representing the rotational symmetries of a point in the lattice of discrete points must be finite. In a given space of a given dimension (we human beings can visualize one dimensional, two dimensional and three dimensional spaces and have only the mathematical schemes for higher dimensional spaces) all finite rotational groups are not compatible with a discrete lattice – there can be a finite number of rotational groups which are compatible. This is because a unit cell formed by the base atoms of a crystal forms the entire lattice by translations along vectors formed by superposition of integer multiples of vectors representing the edges of the mentioned unit cell. A pentagon as a unit cell doesn’t form a periodic, 2 dimensional lattice with a minimum limit of separation of lattice points, because, the very idea of discrete separation of lattice points collapses while attempting to build up such a lattice. The proof comes by adding the five base vectors of the unit cell (pentagon, the vertices of which are lattice points)l starting from one of its lattice point and adding the base vectors (the vectors representing the edges of the unit cell), each one having its tail at the head of another base vector and five of the distinct base vectors of the cell taken in a such a way that it leads to a five point star, returning to the original lattice point after the vector addition of all the 5 base vectors are complete. The vertices that result in the vector addition give way to another pentagon. As per the properties of the Bravais lattice, each among the vertices of the new pentagon represent a lattice point and hence the vertices joined together form yet another cell that is a unit cell. It is also true that the newly obtained unit cell is smaller than the original. When a similar unit cell of lesser size is constructed with the newly formed cell, it is again smaller. This being repeated, it leads to the conclusion that unit cells of arbitrarily small size is possible. This leads to the paradox that two points with arbitrarily small separation is possible against the discrete separation required of any two lattice points in a Bravais lattice. [Wkepedia is what I referred for arriving at this answer]. A molecule need not pave way to a Bravais lattice and hence can have a 5 fold symmetry. (b) For a molecule not forming a crystal lattice there is no restriction about a 5 fold rotational symmetry read less