The rule that minerals must be inorganic may be the strictest one.
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The substances that make up coal, for instance, are different kinds of hydrocarbon compounds derived from cell walls, wood, pollen and so on. These are called macerals instead of minerals (for more see Coal in a Nutshell). But if coal is squeezed hard enough for long enough, the carbon sheds all its other elements and becomes graphite. Even though it is of organic origin, graphite is a true mineral, carbon atoms arranged in sheets. Diamond, similarly, is carbon atoms arranged in a rigid framework. After some 4 billion years of life on Earth, it's safe to say that all the world's diamonds and graphite are of organic origin even if they aren't strictly speaking organic.
A few things fall short in crystallinity, hard as we try. Many minerals form crystals that are too small to see under the microscope. But even these can be shown to be crystalline at the nano-scale using the technique of X-ray powder diffraction, though, because X-rays are a super-short-wave type of light that can image extremely small things.
Having a crystal form means that the substance has a definite recipe, or chemical formula. It might be as simple as halite's (NaCl) or complex like, say, epidote (Ca2Al2(Fe3+,Al)(SiO4)(Si2O7)O(OH)), but if you were shrunk to an atom's size, you could tell what mineral you were seeing by its molecular makeup and arrangement.
But a few substances fail the X-ray test. They are truly glasses or colloids, with a fully random structure at the atomic scale. They are amorphous, scientific Latin for "formless." These get the honorary name mineraloid.
Mineraloids are a small club: strictly speaking it includes only opal and lechatelierite. Opal is a nearly random combination of silica (SiO2, the same as quartz) and water formed under near-surface conditions, while lechatelierite is a quartz glass formed by the shock of a meteorite impact or lightning striking the ground.
Other substances considered mineraloids include the gemstones jet and amber, which are respectively high-quality fossils of coal and tree resin. Pearl goes here too, although I disagree because by that logic, seashells should be included. The last mineraloid is rather like the rusty car I mentioned earlier: limonite is a mixture of iron oxides that, while it may assume the shape of a proper iron-oxide mineral, has no structure or order whatever.