Edge wrap is the foundation on which wrap-around behavior at the edges of a world can be built.
Many games, turn based strategy and role playing games in particular, have made use of worlds and levels that wrap at their outer boundaries, such that entities moving across the boundary on one side will appear instantly on the far side. Make It Tile is fully capable of generating topologies that also behave in this fashion. However, because of certain aspects of modern game engines (in particular, the rasterized rendering of triangles rather than the blitting of sprites), this can become tricky in the details. To account for and hide this complexity, wrapping behavior is stored for every single edge in the topology, and in most cases, this data allows the library to do the expected right thing automatically.
One of the difficulties of topologies that wrap is that certain elements can seem to be in multiple places at once. For example, in the diagram below which wraps both horizontally and vertically, you can see that vertex 10 on the left side, a corner of face 5, is also on the right side as a corner of face 9. This is because faces 5 and 9 are directly next to each other, as far as the topology is concerned, and thus it is only inevitable that they share a couple of vertices. But there is only one vertex which can only have one position. The question becomes how to get the "ghosted" position on the right when appropriate, without overly complicating the topology itself, such as by duplicating vertex 10 and somehow treating both vertices as distinct and yet representative of the same vertex.
Make It Tile accomplishes this by tracking wrapping behavior between various neighboring elements per edge, and then using the context of how an element is being accessed to adjust data appropriately. For example, when getting the position of a vertex, it matter which face is contenxtually relevant. If you are trying to to draw a square for face 5, then you want the position reported for vertex 10 to be the position on the left. However, if you are drawing face 9, you want the position on the right. This context can be communicated to the system by accessing the vertex position not directly, but through the appropriate face edge. Accessing vertex 10 through face edge 21 says that you want the attribute for vertex 10 relative to the near face of that edge, face 5. If you use face edge 38 instead, you can get the vertex attribute relative to face 9.
To get a bit more detailed about the process, when accessing a vertex attribute using a face edge, the question to ask is if a wrapping boundary is crossed when transitioning from the near face of the edge to its far vertex, and if so, along which axes and in which directions did the boundary crossing go. When accessing vertex 10 using face edge 21, no such boundary crossing occurs, and so the vertex attributes can be accessed directly. But accessing it from face 9 through edge 38, a boundary is crossed in the positive direction of the first (horizontal/x) axis. Because of this, the original position needs to have the horizontal axis vector added to it to get the final position.
The edge wrap data stores such relations for both axes, tracking both positive and negative directions, for the relations between any two kinds of topology elements except for edge-to-edge. So in addition to the above example, other combinations of accessing vertex, face, or edge attributes using either vertex edges or face edges will do the intelligent thing concerning edge wrapping. If you have a face attribute collection of the centers of all the faces, and you wish to compare the center position of face 9 with all of its neighbors, faces 4, 8, and 14 would do the ordinary thing, and then if looking up face 5 through face edge 39, the center of face 5 could be returned as if it were immediately to the right of face 9, rather than far to the left. If you use vertex edges instead of face edges, then the same kind of logic will apply, but will look for a boundary crossing when transitioning from the near vertex of the edge to the far vertex or face of the edge or the edge itself, depending on what type of attribute collection you access.
To get the behavior described above, you must use an attribute type that knows how to apply wrapping behavior to the values stored within the attribute collection. Make It Tile comes with three attribute collections to do this for standard three-dimenstional positions: PositionalVertexAttribute, PositionalFaceAttribute, and PositionalEdgeAttribute. If you have data other than basic positional data that needs to react to wrapping boundaries, you can use the above types as examples for constructing your own implementation of IVertexAttributeT, IFaceAttributeT, or IEdgeAttributeT.