There isn’t a quadruple ring. There’s 0, 1, 2, and then 3 ring, although the rings on the 3-ringed ones look more like a spinning clover. These normally guarantee a legendary, but if you’ve caught all the legends in that wormhole color, it’ll greatly increase your chance of getting a shiny, up to a 36% chance depending on your distance.
Keep in mind that this shiny rate doesn’t apply for legendaries or UBs. Normal Pokémon have their shininess determined once you’ve entered the wormhole, while legends and UBs are determined once you enter the battle, which makes them full/charm odds.