## 5-fold Crystal Symmetry

Corannulene is a hydrocarbon molecule (C20 H10) that is formed from a cyclopentane ring (C5 H10) surrounded by five benzen (C6 H6) rings. Its curved geodesic surface puts it in the company of fullerenes including buckballs and nanotubes and for this reason the bowl shaped molecule is often called a “buckybowl”. Because of the five sided core to the molecule, it is considered to have a five fold symmetry axis, meaning it can be rotated 1/5 of a revolution and keep the same appearance. Scientists are interested in studying this molecule because they think it will be useful in designing self-assembling molecular sized components that can be used for a variety of purposes, including targeted delivery of very small bio-medical agents and construction of a variety of nano-scale mechanisms.

**The impossible crystal** – [empa.ch]

Five-fold symmetry is considered to be impossible in crystallography for the same reason that pentagonal tiles do not exist – it is not possible to cover a floor or wall simply using tiles with five sides of all the same length. The only way around the problem is to use other geometrical shapes to fill in the gaps, a principal used by the builders of mosques as long ago as the 15th century. The complex ornamental structure was “rediscovered“ by mathematicians last century. Roger Penrose demonstrated a pattern named the Penrose Parquet, which achieves complete coverage following simple rules using two periodically repeating geometrical forms.

Chemists are faced with a similar problem, since, by analogy, molecules with five-fold symmetry cannot completely cover a surface without leaving gaps. Despite this, they strive to pack themselves together in crystals or on surfaces as densely as possible, as do other molecules. The question is, how do they do it?

**Introduction and Symmetry Operations** – [tulane.edu]

Symmetry

Crystals, and therefore minerals, have an ordered internal arrangement of atoms. This ordered arrangement shows symmetry, i.e. the atoms are arranged in a symmetrical fashion on a three dimensional network referred to as a lattice. When a crystal forms in an environment where there are no impediments to its growth, crystal faces form as smooth planar boundaries that make up the surface of the crystal. These crystal faces reflect the ordered internal arrangement of atoms and thus reflect the symmetry of the crystal lattice. To see this, let’s first imagine a small 2 dimensional crystal composed of atoms in an ordered internal arrangement as shown below. Although all of the atoms in this lattice are the same, I have colored one of them gray so we can keep track of its position.

If we rotate the simple crystals by 90o notice that the lattice and crystal look exactly the same as what we started with. Rotate it another 90o and again its the same. Another 90o rotation again results in an identical crystal, and another 90o rotation returns the crystal to its original orientation. Thus, in 1 360o rotation, the crystal has repeated itself, or looks identical 4 times. We thus say that this object has 4-fold rotational symmetry.

**Symmetry of Quasicrystals** – [tau.ac.il]

We saw in the introduction that the facets of a (quasi)crystal as well as its diffraction diagram clearly reveal a certain kind of symmetry. This symmetry is expressed by the set of rotations that leave the directions of the facets unchanged (Figure 1), or the set of rotations that leave the positions of the Bragg peaks in the diffraction diagram unchanged (Figure 2). But what exactly is the nature of the symmetry exhibited by the microscopic arrangement of atoms in the crystal itself? What do we really mean when we say that a crystal has the symmetry of a certain rotation?

**Artificial Viral Shells Could Be Useful Nano-Containers** – [nanitenews.com]

The researchers decided to pick apart the construction of viral capsids to determine exactly how their identical parts come together. They built a handful of pentagonal tiles with magnetic edges that mimic the chemically-bonding edges of natural capsid proteins. In some of their first experiments, they simply shook the magnetic tiles together in a plastic jar and watched the pieces snap together to form a sphere.

“Although intellectually we knew that this type of self-organization occurs spontaneously, watching it happen from random shaking on the macroscopic scale was inspirational,” Keinan and colleagues write in their paper.

The researchers then turned to computer simulations of capsid construction, working with the dish-shaped chemical compound called corannulene. Also called the buckybowl, corannulene has a five-sided symmetry and rigid curve that makes it a potentially good building block for an artificial capsid.

In the simulations, Keinan and colleagues experimented with different chemical “sticky edges” to the corannulene building blocks to determine the conditions under which the corannulene units would self-assemble into a ball. They created a half-sphere in the simulation, and expect to have a full sphere soon.

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