Say I'm given two simplices: one is a 4-simplex and the other is a 3-simplex, and I've proven their intersection is non-empty. How do I now explain what the dimension of their union is?
I understand that it can never be higher than 4, and I'm trying to visualize the situation, but thats difficult in 4 dimensions.
I thought of a simpler example: the union of a 1-simplex and a 2-simplex (so let's say a line and a triangle), must have dimension 2, right?
Am I then correct in saying that the dimension of the union of a 3-simplex and a 4-simplex will be 4?
You're right, by the definition, the dimension of a simplicial complex is defined as the maximal dimension of its faces.