Why are abrasives hard?
Diamond is made from carbon atoms linked by unusually strong chemical bonds in a rigid three-dimensional lattice. This structure is very difficult to deform so is ‘hard’. Diamond is, of course, the hardest substance known. Silicon carbide has a similar structure to diamond except that every second carbon atom in the lattice is replaced with a silicon atom, silicon and carbon being very similar chemically. Again this forms a rigid three-dimensional lattice, but the atoms are not as strongly bonded together as in diamond, so the structure is less hard.
Silica, again, has a similar structure to diamond. It has silicon atoms linked by oxygen atoms to also form a rigid three-dimensional structure. This is, however, less rigid than either diamond or silicon carbide, so silica, while hard, isn’t as hard as the other two. As there is an oxygen atom between all the silicons there are twice as many oxygens as there are silicons.
Aluminium oxide and rouge are similar to each other, but different from the first three because they are ‘ionic’. ‘Ions’ are atoms that have an electrical charge. Both aluminium oxide and rouge contain metal ions with an unusually large charge. Rouge has iron III ions and aluminium oxide has aluminium III ions and both have oxide (oxygen) ions. In both, the small metal ions surround the larger oxide ions, which in turn surround the metal ions. Due to the high charge on the metal ions, the forces holding the ions together are very strong and so the ionic structure is rigid. Again this means the substances are hard.
Other metals which form ions with a high charge, such as chromium, would have oxides that are hard and would, perhaps, be suitable as abrasives. Iron III and aluminium III ions do have the advantage over many other metal ions in that they are not toxic (as far as we know).
All of the discussed form crystals with straight edges and sharp points suitable for abrading metal or stone particles from a surface to form a cutting edge.