Christoffel Jacob Bouwkamp (CJB) (1915 - 2003)

C J Bowkamp

Figure 1; C.J. Bouwkamp

Christoffel Jacob Bouwkamp (CJB) was born on June 26, 1915, at Hoogkerk, then near to (absorbed by) the city of Groningen in the north of the Netherlands. He studied at the University of Groningen, where he graduated in mathematics, theoretical physics and mechanics in 1938. He became assistant to F. Zernicke (Nobel-prize winner, physics, 1953), and after his military service (September 1939 - June 1940) he completed his doctoral thesis. On January 21, 1941, the degree of Doctor of Science was conferred upon him. His dissertation is titled, "Theoretical and Numerical Treatment through a Circular Aperture" [translation from Dutch]. After two months as a lecturer in mechanics and quantum mechanics at the University of Groningen, he joined the staff of Phillips Research Laboratories. N.V. Philips Gloeillampenfabrieken, Eindhoven, The Netherlands, March 1, 1941.

In 1969, he was appointed Scientific Advisor for Applied Mathematics at Phillips Research Laboratories, retiring July 1, 1975. From 1955, he was Professor Extraordinary of Applied Mathematics, first at the University of Utrecht, The Netherlands, until 1958 and since then at the Technical University of Eindhoven , retiring again in 1980. CJB was elected member of the Koninklijke Nederlandsche Akadamie van Wetenschappen (1960). A member of the American Mathematical Society since 1947, he was a reviewer for Mathematical reviews from 1947 to 1980. At various times he was visiting lecturer, research associate and research mathematician, spending half a year (1952/53) at New York University at the Division of Electromagnetic Research under Morris Kline. Also he was consultant to the Bureau of Standards, Washington, D.C., with Abramowitz and has lectured at Harvard and Washington Universities. In 1955 he spent three months at the Institute of Electrotechnology, Berkeley and several weeks at the Institute of Geophysics, Los Angeles. In Europe he has lectured on many occasions in Sweden and Italy for several weeks.

On October 14, 1943 he married Alida Hendrika van der Veer. The couple has one daughter, Christine Margriet, born 1949.

In his dissertation, CJB compared numerical values of rigorous diffraction functions to the approximation obtained by Huygens-Kirchhoff Formulation. Naturally, he would have been exposed to the Kirchhoff laws and as his sphere of influence increased, it would intersect the work of the Trinity Four. CJB and WTT would first interact at scientific meetings shortly after the formal conclusion of the Second World War. By then CJB was generating a picture catalog of most of the nets discovered by the Trinity Four, although CJB had generated his set independently. He was quick to see the value of using computing machines on the problem. There was too much opportunity for error. Every early publication of the subject contains numerical errata and some rectangles were overlooked. At the time, WTT and CJB had a degree of difficulty in communicating as each had a different native tongue. This led to a misunderstanding on CJB's part regarding a claim by the Trinity Four of an uncrossed simple perfect square. A perfect square is said to be crossed if the corners of four elements meet at a point. This communication difficulty found its way into print with CJB on one side and WTT on the other . It was unclear to CJB that the method of construction of the square required only one square to be deleted to eliminate the overlap. When the problem was clearly explained to him, CJB immediately published a retraction of his views and quickly constructed a solution differing from WTT's.[5]

But CJB's course was set, and while the two would publish separately for years, eventually CJB's vision of computer generated simple perfect rectangles from c-nets would be realised. In characteristic cooperation, WTT supplied all the 15 wire nets. In the interim CJB had twice published tables of simple perfect rectangles for order 9-13, second for orders 9-14. The table for orders was a very useful, not to mention well thought out and constructed document. It is a major publication on the subject. In the early sixties CJB with Duijvestijn would press on through order 19 in search of squared squares. The computer generated c-nets and squared rectangles of CJB and AJWD provided independent experimental verification of the graph theoretical results of Tutte and pioneered the use of computers in mathematical research.

Bouwkamp enthusiastically embraced the Transform Techniques approach of Willcocks and Federico. Using transform techniques, new squared squares were able to be discovered from existing ones without the use of computers.

Bouwkamp remained interested throughout his career in squared squares and squared rectangles, as well in as other mathematical tiling pursuits such as pentominoes, where he contributed new results also. He contributed many new squares and several publications resulting from the explosion of squared square discoveries in the 1990's.

He passed away in 2003


  1. 'Squared Squares, Who's Who & What's What', p52-54 Jasper Dale Skinner, II, Ph.D.
  2. On the dissection of rectangles into squares (Nederl. Akad. Wetensch. Proc., Ser. A, 49, 1946, pp. 1176-1188)
  3. On the dissection of rectangles into squares (Nederl. Akad. Wetensch. Proc., Ser. A, 50, 1947, pp. 58-71).
  4. On the dissection of rectangles into squares (Nederl. Akad. Wetensch. Proc., Ser. A, 50, 1947, pp. 72-78).
  5. On the construction of simple perfect squared squares (Nederl. Akad. Wetensch. Proc., Ser. A, 50, 1947, pp. 1296-1299).
  6. C.J. Bouwkamp, A.J.W. Duijvestijn and P. Medema, "Catalogue of simple squared rectangles of orders nine through fourteen and their elements", Technische Hogeschool, Eindhoven, The Netherlands, May 1960, 50 pp
  7. C.J. Bouwkamp, A.J.W. Duijvestijn and P. Medema, "Tables relating to simple squared rectangles of orders nine through fifteen", Technische Hogeschool, Eindhoven, The Netherlands, Aug 1960, 360 pp (Reprinted January, 1986 EUT Report 86-WSK-03.)
  8. C.J. Bouwkamp, A.J.W. Duijvestijn and P. Medema, "Table of c-nets of orders 8-19 inclusive", 2 vols. Philips Research Laboratories, Eindhoven, the Netherlands 1960; unpublished available in UMT file of Mathematics of Computation.
  9. A.J.W. Duijvestijn, Electronic computation of squared rectangles. Dissertation, Technische Hogeschool, Eindhoven, The Netherlands, 1962; also in Phillips Res. Reports 17 (1962) 523-612
  10. C.J. Bouwkamp, A.J.W. Duijvestijn and J. Haubrich, Catalogue of simple perfect squared rectangles of orders 9 through 18, Philips Research Laboratories, Eindhoven, The Netherlands, 1964 (unpublished) vols 1-12, 3090 pp.
  11. C.J. Bouwkamp, Table of Duplets of order up to 18 inclusive according to increasing vertical to horizontal side. (1965) 177pp. (unpublished).
  12. C.J. Bouwkamp, Review of Table of c-nets of orders 8-19 inclusive, Math. Comp. 24 (1970) 995-997.
  13. C.J. Bouwkamp, On some special squared rectangles, J. Combinatorial Theory Ser. B, Vol 10, Issue 3 (1971) 206-211.
  14. C.J. Bouwkamp, Note on some new simple perfect squared squares, Discrete Mathematics 100 (1992) 67-75.
  15. C.J. Bouwkamp and A.J.W. Duijvestijn, Catalogue of simple perfect squared squares of orders 21 through 25, EUT Report 92-WSK-03, Eindhoven, The Netherlands, 207 pp.
  16. C.J. Bouwkamp and A.J.W. Duijvestijn Report 94-WSK-02 Eindhoven, July 1994 Album of Simple Perfect Squared Squares of order 26
  17. C.J. Bouwkamp EDT Report 98-WSK-01 Eindhoven, July 1998, Simple Perfect Squared Squares of order 27
  18. C.J. Bouwkamp, Simple Perfect Squared Squares of order 27, EUT Report 98-WSK-01, Eindhoven, July 1998 227 pp
  19. C. J. Bouwkamp, An old pentomino problem revisited, Simplex Sigillum Veri, Technische Universiteit Eindhoven (1995), pp. 87-96 ISBN 90-386-0197-2
  20. C. J. Bouwkamp, An Unusual Pentomino Problem, CFF-41, Oct. 1996, pp. 39-40
  21. C. J. Bouwkamp, On Benjamin's Pentomino Cube, EUT Report 97-WSK-01, Eindhoven, Dec. 1997
  22. C. J. Bouwkamp, Tiling the surface of the cube by 12 identical pentominoes, EUT Report 98-WSK-02, Eindhoven, Nov. 1998
  23. J Haubrich, "C.J. Bouwkamp en de Squared Squares" (in Dutch) Nederl. Akad. Wetensch. 5/4 nr. 4 December 2003
  24. A.T. de Hoop, Levensbericht C.J. Bouwkamp, in: Levensberichten en herdenkingen, ('life message CJ Bouwkamp', In Dutch) 2004, Amsterdam, pp. 18-25