Diamond crystal. Solid carbon comes in different forms known as allotropes depending on the type of chemical bond. The two most common allotropes of pure carbon are diamond and graphite. In graphite the bonds are sp orbital hybrids and the atoms form in planes with each bound to three nearest neighbors 120 degrees apart. In diamond they are sp and the atoms form tetrahedra with each bound to four nearest neighbors. Tetrahedra are rigid, the bonds are strong, and of all known substances diamond has the greatest number of atoms per unit volume, which is why it is both the hardest
Pure subgroup Pure subgroup In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas. A subgroup formula_1 of a (typically abelian) group formula_2 is said to be pure if whenever an element of formula_1 has an formula_4 root in formula_2, it necessarily has an formula_4 root in formula_1. Formally, formula_8 is solvable in formula_2 formula_10 is solvable in formula_1. Pure subgroups are also called isolated subgroups or serving subgroups and were first investigated in Prüfer's 1923 paper