Ian Gambini

In 1999 Ian Gambini published his doctoral thesis Quant aux carrés carrelés on squared squares. He used several different methods to enumerate perfect squared rectangles and squares.

He implemented his version of what he called the classical method. That is, he generated non-isomorphic 2-connected planar graphs (with minimum degree three to ensure perfect dissections) and solved the Kirchhoff equations for electrical networks of the graphs to find the sizes of the squares in the dissection corresponding to edges with unit resistances. His graph generation method, unlike Duijvestijn's, did not use Tutte's wheel theorem. Gambini was able to generate graphs with up to 25 edges and produce simple and compound perfect squared rectangles (SPSRs and CPSRs) to order 24. Within these solutions he found the known CPSSs and simple perfect squared squares (SPSSs) up to and including order 24. He published a table of SPSR and CPSR counts up to and including order 24.

Gambini observed that a perfect squared square can only have one side with a minimum of two squares along an edge. Hence only one of the polar vertices in the graph, or its dual, can have a degree of three. He thereby constrained the graph generation algorithm and eliminated some graphs from production which could not produce squared squares. Gambini continued the 'classical method' beyond order 24 for perfect squared squares and produced all to order 26. Table 2.6 of Gambini's thesis listed SPSSs and CPSS isomers counts up to and including order 26. In the SPSS counts Gambini obtained the same results as Duijvestijn. In the CPSS counts Gambini identified;

  1. four isomers of order 24 CPSS
  2. 12 isomers in order 25 CPSSs
  3. 100 isomers in order 26 CPSSs

Gambini did not associate the isomers with particular CPSSs, however we can match them up with known discoveries of that time.

  1. The four isomer counts in order 24 corresponded to T.H. Willcock's 24:175a CPSS (four isomers).
  2. The 12 isomer counts in order 25 corresponded to P.J. Federico's 25:235a (four isomers) and 25:344a (eight isomers).
  3. The 100 isomer counts of order 26 corresponded to a total of 92 isomers derived from 15 known order 26 CPSSs (isomer counts in parentheses); 288a(four), 360a(four), 360b(four), 384(four), 429a(four), 440a(four), 480a(four), 483a(four), 492a(four), 493a(four), 500a(16), 608a(16), 612a(four), 638a(eight), 648a(four) and an additional eight isomers not associated with any CPSS(s) known at the time.

The eight isomer discrepancy was not resolved until 2010. The additional CPSS which completed the order has a side of 512 and has eight isomers. This CPSS was deduced to have been discovered by Duijvestijn, Federico \& Leeuw in 1979 but not published and not identified until 2010 by Anderson and Pegg. This CPSS completes the catalogue of order 26. Please see the Order 26 CPSS menus or pdfs for an illustration of CPSS 26:512a.

Gambini also developed new methods of producing perfect squared squares using several tiling algorithms. He improved the efficiency of his algorithms by proof of theoretical bounds he established on the minimum sizes possible for elements on both the boundary sides (size of five) and corners (size of nine) of a perfect squared square.

He was able to produce a large number of SPSSs across an unbroken range of orders from order 21 to order 128.

He proved that the three SPSSs with sides of 110, originally found by Duijvestijn and Willcocks, are the minimum possible size for a perfect squared square.

He produced only one new CPSS (of order 52, side 976).

Using a variation on his tiling algorithm Gambini was also able to find perfect squared cylinders and a perfect squared torus (of order 24 with side 181).