This ring by designer John Strobel was made using the lost wax casting process. John began by carving a full-sized model of his design using hard jeweler's carving wax. This model was then encased in high-temperature plaster, called investment, in a metal canister. After the plaster hardened, the canister was placed in a small kiln, called a burnout oven, and the wax was melted and burned out--hence the name lost wax process--leaving a perfect "negative" of the ring.
After John melted the gold for the ring, he "cast" it into the canister using a centrifugal casting machine. (Because the molten gold, like all metals in their liquid state, likes to stay in a ball--think of mercury--either centrifugal force or vaccuum pressure must be used to force the metal into the mold and hold it there until it cools to a solid state.)
When the newly-cast ring was no longer red hot, John quenched the flask by putting it in a bucket of water. The rapidly cooling plaster sputtered and dissolved in the bucket, revealing the gold ring. But it didn't look like gold--its surface was dark from oxidation, so it sat in a warm acid bath until the surface brightened.
Hours spent filing, sanding and polishing the surfaces brought the ring to the crisp perfection you see in the image, and only then could the stones be set. First, holes for the diamonds were drilled, using specially shaped setting burs. Then John set the diamonds in the holes, working gold all around the edges of each stone. This type of setting is called "gypsy" setting, or "hammer" setting.
Then John used a small compressed-air driven sandblaster to put the soft texture on the ring. (Because diamonds are the hardest natural substance, they were undamaged by this process.)
Finally came the setting of the tanzanite. John had fashioned the ring of gold, called a bezel, to the exact measurements of the stone. Using a series of tools, he worked the gold edge evenly over the edge of the stone and then sanded and polished the rim.