The 16 Coated Amethysts are ingredients for the Amethyst Multishard - which itself crafts the Megacake.
There's 4 extra secret bonus Coated Amethysts that aren't related to the Amethyst Multishard.
All amethysts are obtained by combining 1 Amethyst Shard with 1 specific material at a Lapidary Table.
The Lapidary Table is crafted with 1 Amethyst Block, 1 Opal Core, 1 Polished Rhyolite, and 6 Quartz Blocks.

Lapidary table interface:
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The list of possible combinations are listed on the Lapidary Table's List of fusables menu.
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The 4 secret ones are SECRET. You can read about them at the bottom of this wiki page if you don't care about spoilers.
Combine all 16 main Coated Amethysts at an
Amefuser to obtain the
Amethyst Multishard. Wielding the Multishard gives the
Multishine advancement.
Amefuser recipe:

Amefuser interface:
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(each slot corresponds to a Coated Amethyst Shard)
Here's all the SECRET Lapidary Table recipes:
End Energy Coated Amethyst Shard. It plays a very special song when used.
Purple Magma Coated Amethyst Shard. This only exists because there's a magma and soul magma one so... gotta complete the collection!
Perfect Emerald Coated Amethyst Shard. Plays a partying sound when used.
Crystal Shard. It's simultaneously ammo for the Ribbon + Crystal and the crafting ingredient for the Ribbon + Crystal - a very powerful ranged weapon.