I have two Beta fish - a.k.a. Siamese Fighting Fish - I’ve had them for more than a year and they have grown about a centimeter in every direction. Recently one of them - Archimedes - has developed a line 3 millimeters from the tip of his dorsal fin, and another in the same place on his abdominal fin. The lines look like dotted lines running perfectly across the tip of each fin, like this was meant to happen. 
Whats happening? Do their fins fleck off if they get too big? Anyone had a beta for a while and seen this phenomenon?
The only two times that my Beta lost fins- rot, and when the other one ate them.
Both resulted in the slow death of the fish.
Bettas do not moult. Sounds like fin rot. Or it could be a fungus that is attacking the fin. Have you done a full water change lately? Are you adding aquarium salt to the water on a regular basis?