The point of the algebra is that one can describe what cannot be visualized directy.
Think about it: Einstein revolutionized the world with his Gedanke Versuch with trolley cars and motorcycles traveling at relativistic speeds; the vector of these objects travel essentially being the normal vector of infinite two dimensional planes, traveling everywhere in no time at all and having infinite mass, their trajectory essentially being infinitely curved (flat space essentially being an infinite circle of infinite radius).
Because of his work, we can understand why we see muons at the Earth's surface when their creation is so high in the upper atmosphere, and their half life is so short, they'd decay before we could detect them.