Common stones

This page shall answer:
Given a gem-gum factory what could you make from the material in a stone
(or inorganic dirt) that you randomly pick up from the ground.

Granite => quartz and stishovite

Most stones are to a high degree silicates. One especially common one is granite.
From this you can make all sorts of polymorphs of SiO₂
Including not only quartz and similar hardness modifications but also
the ultra hard ambient pressure stable high pressure modification gems stishovite and seifertite.

Using the metals that are present concentratedly in the feldspar and mica components of the granite
many other high performance gems can be made by mechanosynthesis. Including …

• harder topaz (orthorhombic, Mohs 8 per definition, containing Al) - sapphire using no silicon
• cubic crystal symmetry gems like spinels and garnets.
• orthorhombic olivines/peridots (Mg/Fe)
• perhaps tourmaline
• many more

Chalk-stone => calcite and aragonite

Many stones (entire mountain ranges) are metamorphized ancient sediments from ea life.
One especially common one is microcrystalline chalk stone.
From this you can make all sorts of polymorphs of CaCO₃
Including: Calcite (aka bifringent calc-spar), Aragonite, …

Clay => sapphire

Clays typically contain quite a bit of aluminum dioxide
From this you can make (among other things) Al₂O₃

• Sapphire
• Deltalumite - a barely known polymorph

Iron ore => wüstite, hematite, and magnetide

Pick up a some random brownish looking stone an chances are good there's iron inside.
With this you can make wüstite, hematite, and magnetite.
Iron (unlike titanium) unfortunately does not like to make
gemstones with super awesome mechanical and thermal properties.

Salt => periclase

There are plenty of places to find magnesium (including seawater)
With magnesium one can make periclase (MgO).
Simple cubic rock salt structure. But unlike rock salt / table salt (NaCl) it is barely water soluble.
Just a tiny bit making it environmentally friendly degradable.
Passivatability of surfaces for moving sliding/rolling interfaces is questionable. May need some lining.
Thin quartz based lining can preserve degradability.

Rocks in space

Moon rocks => rutile, anatase, brookite, tistarite, hongquiite

If you so happen to be on the moon there is a good chance that your rock contains a lot of Titanium
This element is especially useful for making structural gemstone like compounds.
You can make the oxidic gemstones:

• rutile (TiO₂)
• anatase (TiO₂)
• brookite (TiO₂)
• hongquiite (TiO)
• tistarite (Ti₂O₃)
  

More "synthetic moon titanium gemstomes" are:

• titanium phosphide (TiP) – here ad hoc invented name: "phosphotanite"
• titanium silicide (many forms) – here ad hoc invented name: "silitanite"
• titanium diboride (TiB₂ – here ad hoc invented name: "diboritanite"

since these partner elements are quite non-volative
and likely still on the moon in quantities.

No shortage back here on Earth:
Not to worry here on Earth there is plenty of titanium too. Just not that much.
Beside the oxidic titanium gemstones there are plenty excellent others
Titanium combing in a pure way with other elements rarely happens in nature.
So these come without pretty mineral names.

• The golden titanium nitride (TiN) – here ad hoc invented name: "nitrotanite"
• Titanium carbide (TiC) – here ad hoc invented name: "carbonitanite"

Rocks dunes and lakes on Saturns moon Titan => diamond and lonsdaleite

On Saturns moon Titan your rock will most likely be water ice.
That is not too useful as structural building material.
Go to the tholine dunes and methane lakes instead.
All the stuff there is probably mostly organic volatile elements.
Hopefully not too sludge like (we will see when amazing the dragonfly mission will arrive)
So the products of titaninan gem-gum tech will likely be combustible if
heated enough in the presence of for that purpose synthesized oxygen.
See: Diamondoid waste incineration

Related

• Rock digestion chamber, Gem-gum waste dissolution
• Atomically precise disassembly
• Diamondoid waste incineration – does not work for pretty much all stones except diamond and carbon notride
• Passivation layer minerals
• Gemstone like compounds with high potential
• Gemstone