The natural shape of a mineral would be its crystalline structure.
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There is no general shape of a mineral. Each mineral has its own shape, even though two different minerals might have the same shape. A few examples of shapes of minerals are: Rhomboid, rock, and cubic.
diamond
cube
Diamond is the hardest mineral at Mohs hardness 10, ten being the hardest. There is no natural mineral substitute for processes requiring diamond. Diamond is four times as hard as corundum, the mineral constituting rubies and sapphires. Although diamond is the hardest naturally occurring mineral, it is easily fractured, a characteristic which allowed early jewellery makers to facet this crystal.
No shape is mathematical really unless it has been created by a mathematical formula, but is certainly a geometric shape. But anything which is a 2D or 3D shape is geometric. My improvement: A catenary curve from a mathematical equation such as cosh x, is a mathematical and natural shape. Maby each other arch can be approximated by a mathematical formula.