Types of Investiture Hi All,

Had a quick question that might not be clear yet, but do we know why investiture manifests in different ways (allomancy, surgebinding, biochromatic breaths, etc.) 

Does it have to do with the Shard or the World or a mixture? For instance if Odium came to Scadrial would he be able to use allomancy or feurochemy? If a shard leaves a world does it "invest" on other worlds in the same way? Was voidlight something Odium used before he came to Roshar?