Traits Library
Once a Keeper names a newly discovered Chaos Trait, that Trait will become an entry in the Traits Library. Well this is where you will find it. The Library contains every PKS named Trait (all the ones you already know such as Horse Tail, Furry Material, Hermes Plumage etc. etc.) and every Keeper named Chaos Trait—yes, every single one, worldwide.
Traits can be filtered by type (e.g. Pattern, Plumage etc.), sorted by name or popularity and searched by text input.
If you tap on a particular Chaos Dot, you will also see some stats including the Keeper that discovered the Trait, when it was discovered, where in the world it was discovered, and how many times the Trait has been Hatched on other Dots.
If you happen to see a Dot with a Chaos Trait that you want, you have the option to spend 300 Peridough for a Dot with that Trait to visit you. The Dot does not appear in the Visitor Dots screen but in a tab called ‘purchased’ in the Traits Library screen (we know—that doesn’t make 100% sense but it is what it is). The visiting Chaos Dot will remain there until you use it to Hatch another Dot to get the Chaos Trait. The Dot that appears is only guaranteed to have the Chaos Trait you wanted - all other Traits are likely to be random - here's an example.
So let's imagine that we want the Ears from the middle Dot - tap on the Dot.
Here you will get a bigger image of the Dot and its Traits - tap on the pink bar.
This is simply a confirmation screen but notice the wording: Are you sure you want to have a Dot with this Trait visit you? Tap on the Yes bar.
So, the next time you go into the 'Purchased' section of your Trait Library, you will see a Dot there with the Trait you selected previously.
This Dot will have the Trait you selected but all other Traits will be random, as will the Dot colour. Carefully read the highlighted words under the previous image here.
And here's your Puchased Dot, with the Trait you wanted. Now you can try to obtain that Trait on one of your Dots by Hatching.
One other thing to remember with the Traits Library, under the 'Discovered' tab, you will see all the Chaos Traits that you have discovered. No big deal right? - Wrong, Whilst you cannot Hatch-a-Dot with one of your own Dots (that would just be wrong), you can summon a Dot with one of your Discovered Traits. That Dot is not classed as one of your Dots therefore you can now spread your Discovered Traits around your Dot collection without relying on finding another Keeper's Dot that had inherited the Trait.
Only Chaos Traits can be summoned in this manner - PKS Traits for Archetypes cannot, which maintains the 52 Archetypes in the game as somewhat elusive.