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whatever type of rock your research indicates is best to use, with a ceramic tile saw--these are fitted with diamond embedded blades with water circulating to the cutting surface to keep them cool. Diamond blades are a available for different materials, sometimes the diamonds are embedded in a softer metal, and sometimes in a harder one, these differing constructions offer specialized benefits in certain materials over others.
These tile saws can be rented or bought. You are limited to cutting smaller rocks. A brick saw is similar, but with a larger diameter blade.
There were certain times when water-soluble oil mixed with water is used instead of pure water, it may have been with the diamond cutting blade--my memory fails me. Water-soluble oil can be purchased at an auto supply house, although its not used as frequently as it used to be. The old timers used to put it in the cooling systems to stop rust in iron engine blocks.
Carbide based grinding wheels can be used for rough shaping of most rocks except diamond, but it's done "wet" with water rather than dry. A true knife-sharpening machine, one that uses water, can probably be used.
Polishing and fine shaping is a different matter. There are a number of different polishing compounds, with different levels of coarseness, some more suitable for certain rocks than others. This is sometimes done with a felt pad and compound or with an animal skin and compound. The compound is usually a white powder mixed with water to be a paste. Some of these polishing compounds may be available at dental supply houses.
I would think that granite would be a poor choice of rock to use, it has many micro fractures in its composition, it's a composite of several different ores. Obsidian might work, but in thin slabs I'd think it might be somewhat brittle as a pick. Quartz might also be brittle. Agate might be used, as long as there are no fractures in it.
You might end up making lots of them out of different types of rocks until you find one that works well thin. Most rocks are not a pure mineral.
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