This is a good question. I have seen this over looked a lot of times when seeing examples of planes made online. Why isn’t obvious really.
I am going to assume that you know little about quarter sawn and plain sawn lumber other than you know how it looks to the eye. I think that is where we all started in woodworking. We understood that a lot of older furniture had quarter sawn lumber in it but the only thing we saw was that it looked different. We didn’t realize until later that it not only looked different but it acted differently as well.
Probably oak is the wood that comes to mind when we think of quarter sawn. It is no surprise because basically all use of oak originally was “quarter sawn”. I put that in quotation marks because it wasn’t sawn at all. It would have been riven not sawn. Anyways I digress. If you want to know more about how oak built the world civilizations I recommend you read the book “Oak The Frame Of Civilization” by William Bryant Logan.
But traditional planes are made of beech almost exclusively. I get a few eye brows raised at me when I tell people that beech is related to oak. In fact oaks are in the family of the beeches (Fagaceae). They share many qualities. One important one is that they both have distinct rays.
I think it is important to understand what ray fibers are in order to really grasp how they affect wood movement, especially in quarter sawn wood. Rays are the fibers in the tree that run perpendicular to the long grain fiber that run the length of the tree. In other words the ray fibers run from the inside to the outside of the tree while the long grain fibers run from top to bottom. Rays actually work to move moisture and nutrients throughout the tree in times of drought, etc. Ray fibers are not common in tropical woods, by the way.
So what do these rays do in wood. Essentially they reduce wood movement during moisture cycling. That is why most hardwoods essentially have a 2:1 ratio of tangential shrinkage to radial shrinkage. So that means that the radial plane of wood moves less compared to the tangential, about half. Just think of it this way. The annual rings are the tangential plane while the rays are the radial plane.
So we can think of rays creating a lattice work when they cross the long grain fibers. In fact the rays are essentially long grain fibers as well. And we know that a piece of wood along its length (long grain) basically doesn’t change in length. That is why rays reduce wood movement in that plane. Therefore a piece of wood that is quarter sawn is more stable across its width. That is why you choose quarter sawn boards for the frame of a door and it is the same reason you choose quarter sawn wood for picture frame mitered corners. I have drawn an illustration to show this latter example. This also illustrates why a wedge in a plane benefits from being in the quarter sawn plane.