At his offices in one of Chelsea's historic factory buildings, the ebullient Danish architect Bjarke Ingels is hovering over an oddly shaped Taskboard model. One of the latest designs to emerge from his upstart firm, the brashly named BIG (Bjarke Ingels Group), this vision of a New York apartment complex has a startling profile totally at odds with the grids of crew-cut, square-shouldered towers dotting the Manhattan skyline.

Angling upward to a point, like a child's drawing of an alp, it is not quite a high-rise. (Ingels...