Tight and taut immersions are a living and growing part of contemporary mathematics largely due to the legacy of Nicolaas Kuiper. He made central contributions to many different areas of mathematics during his long and productive career, but it is in tight and taut immersions that his geometric style showed forth in a special way. In that subject, his personal enthusiasm and extraordinary geometric insight combined to bring forth examples and theorems of great conceptual and visual appeal. He delighted in discovering new phenomena, and in presenting his examples using sketches and in cardboard or wire-frame models. He found surprising connections among apparently unrelated areas of mathematics, creating entirely new methods for handling a range of geometric structures: analytic, differentiable, once-differentiable, combinatorial, and topological. He was the first to appreciate the essentially geometric character of tightness, exploiting the relationship between the minimal total curvature condition for smooth submanifolds and critical point theory so that the notion could be extended to non-smooth objects. He guided generations of mathematicians who have followed his lead.

In several other subjects his contributions were necessarily abstract, for example in the embedding theorems he produced with John Nash, or the surprising result that the unit sphere in Hubert space is contractible. Often he would listen to lectures on a new subject, read about it and study it, and come up with a crucial insight that no one else was close to realizing. He would then leave the field to other mathematicians, encouraging their efforts. He was particularly supportive of young colleagues from many different countries, especially while he was director of the Institut des Hautes Etudes Scientifiques.