To understand both of the structures you asked about, you have to know something about the valence electrons of the atoms. Valence electrons are at outermost electrons in an atom, and are the ones that can participate in chemical bonds with other atoms. Diamond is made out of just carbon, which has 4 valence electrons. Most atoms want to have 8 valence electrons, which means that each carbon can make 8-4 = 4 other bonds. Since every carbon can make a bond with 4 other carbons, then each of those carbons can bond with 4 other carbons and so on, you get a repeating pattern which makes a crystal. Here is a picture of what I mean:diamond
In the picture, each ball is a carbon atom. If you look at one ball in the middle of the picture, you can see that each ball is connected to four other balls by a stick (which represents the chemical bond). The other important thing to point out is that all of the atoms want to be equally far away from each other. Thats why you get a 3-dimensional (3D) structure with all of the atoms evenly spaced.
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