Stuart Errol Anderson

A Short Biography

Born New Zealand, 1959. Lived in both North and South Islands, (mostly North Island) during the 1960's.

In 1970 I emigrated with my father and brothers to Newcastle Australia and attended high school there, studying English (3 unit), Mathematics (4 unit), Physics and Chemistry in senior years.

After high school I attended at Newcastle University. I studied Classical Civilisations, English, Philosophy, Mathematics, then moved to Sydney and switched studies to the Visual Arts; studying Sculpture, Sound, Film, Multimedia and Computer Programming at Sydney College of the Arts (now part of Sydney University).

I worked in the Australian Public Service (Taxation Office) , I spent much of my time there in industrial, organisational and technological change roles.

I worked briefly as web designer and pension officer at Veterans Affairs, then left the public service for a variety of positions, including a stint as a Google Quality Rater.

I currently work at the Reserve Bank Sydney in a security role.

Since 2000 I have been doing research and study in a range of art related and mathematics topics.

My most developed research has been in the area of squared square dissections. I also maintain this website (www.squaring.net) with it's many tilings and related information.

My Research into Squared Squares

At high school I read Martin Gardner's Mathematical games column in the Scientific American. Later, I came across his articles on squared squares. In the early 1980s I incorporated a squared square into a performance art piece. In the years 2000 to 2003 I did some initial research into squared squares, learning how to produce my own, using graph theory and electrical network analysis. I wrote to William Tutte, received some replies from him, and began collaborating and exchanging discoveries and information with Jasper Dale Skinner II, who had already done extensive work in this area. At that time the majority of my discoveries were in order 28 SPSSs. I discovered several dozen new dissections.

In 2010, after a hiatus of several years, I recommenced work on the enumeration of squared squares in a collaboration with Ed Pegg. We completed order 28 SPSSs, finding the total of 3001 in that order, and also completely enumerated all CPSSs up to and including order 28, and then started work on order 29 SPSSs and CPSSs. In the 1990's Duijvestijn had enumerated all SPSSs from order 21 to order 26. Jasper Skinner had enumerated order 27 SPSSs and over 95% of order 28 SPSSs. Until this time the enumeration of complete CPSS orders had been an outstanding unsolved problem. Computer assisted enumeration of CPSSs was done in 1982 by Duijvestijn, Federico and Leeuw, and order 24 was the only order that achieved a conclusive enumeration result. Gambini's thesis included the computer assisted production of squared rectangles (and squared squares as a subset) to order 26. Gambini's CPSS isomer counts for CPSS orders 24, 25 and 26 (4, 12 and 100 respectively) provide evidence that he completely enumerated these orders, however he did not publish any dissection code for the order 26, 512 side, 8 isomer CPSS, which was unknown and unpublished at the time. Duijvestijn, Federico and Leeuw probably discovered the same CPSS in 1979/1982, as it was within the scope of programs used, but did not publish the discovery either.

The CPSS enumeration began as a collaboration between Richard K. Guy, Ed Pegg Jr and myself to extend the known solutions to the Mrs Perkin’s quilt problem. Mrs Perkin’s quilts include all combinations of simple, compound, perfect and imperfect squared squares. Using Brendan McKay and Gunnar Brinkmann’s planar graph generation software plantri and electrical network tiling software written with C++ standard libraries and Boost Ublas library (by myself), Pegg and I enumerated all perfect squared squares and simple imperfect squared squares (SISSs) to order 28. As a subset of the quilt enumeration Pegg and I produced all CPSSs up to and including order 28. In isomer classes of CPSS, orders 24 to 28, the final counts are;

1. 1 CPSS of order 24, with 4 isomers, (24:175 THW)
2. 2 CPSSs of order 25, with 12 isomers, (25:235 PJF (4) & 25:344 PJF (8))
3. 16 CPSSs of order 26, with 100 isomers, including 1 CPSS, with 8 isomers, with side 512 not previously identified, (discovered in 1979, but not published by Leeuw and rediscovered but not identified by Ian Gambini in 1999) which completed this order.
4. 46 CPSSs of order 27, with 220 isomers, including 4 CPSSs not previously known (with sides 345a, 624a, 648a, 857a), and 3 CPSSs which had been discovered by Leeuw in 1979, but never published, (27:804a, 27:820a and 27:824a) which completed this order.
5. 143 CPSSs of order 28, with 948 isomers, including 50 CPSSs not previously known, which completes this order. Paul Leeuw found 23 new CPSSs in this order in 1979 but only 1 was published. Out of the remaining 22, Anderson and Pegg rediscovered 14 CPSSs in the scope of Leeuw's program (28:753a, 28:811a, 28:1032a, 28:1049a, 28:1069a, 28:1075a, 28:1078a, 28:1093a, 28:1131a, 28:1164b, 28:1170a, 28:1170b, 28:1208a and 28:1229a). These discoveries have been attributed to DFL (Duijvestijn, Federico and Leeuw) 1979.

In August 2011 Stephen Johnson and I finalised enumeration of SPSSs and SISSs of order 29. We found a total of 7901 SPSSs and 326037 SISSs in order 29. We also commenced order 29 CPSSs, and processed all 2-connected min degree 3 graphs with up to 15 vertices. This left the largest graph class, the 16 vertex class, still to be processed.

In May 2012 Thom Sulanke and I collaborated to re-investigate squared cylinders. The investigation by myself and Sulanke established that there are 18 Simple Perfect Squared-Cylinders (SPSCs) in order 20, 16 were new discoveries and were all 80x80 in size and can be obtained by combining two cyl-nets and so were not found by Augusteijn and Duijvestijn. This confirms that order 20 is the lowest order in which SPSCs can appear, and that the 79x79 SPSC found by Augusteijn and Duijvestijn in 1983 is the smallest size in the lowest possible order .

2012 April, I produced millions of CPSSs in orders 41 to 60 by substituting PSSs into PSSs. Out of the millions of CPSSs produced, only those with small sides are shown on this site, (depending on the quantity produced in each order). Order 41 CPSSs is the first order where this can be done (Duijvestijn's order 21 :112 inserted into its elements and scaled). This method can produce unlimited CPSSs as it can be applied iteratively. By substituting the catalog of PSSs as it existed in 2012 from this website into itself just once, well over 10 billion CPSSs could be produced. If the operation was repeated twice, approximately 10^22 CPSSs could be produced, this is about the number of grains of sand on the Earth.

2012 July, I produced over 100,000 CPSSs in orders 34 to 44 by combining same size pairs of SPSRs of orders 16 - 21 with no element in common (disjoint n-tuple SPSRs). If there are more than several thousand in an order, only the smaller CPSS of that order are shown on this website.

In October/November 2012 I used the Amazon EC2 supercomputer to enumerate the remaining CPSSs of order 29. I found a total of 412 CPSSs and 2308 CPSS isomers in order 29

2013 March, June; I published (March) to the Arxiv and updated (June) a paper on 'Compound Perfect Squared Squares of the Order Twenties' which detailed the history of the enumeration of low-order CPSSs.

2013 March and April; Lorenz Milla and I enumerated simple squared squares of order 30. Lorenz used plantri (McKay/Brinkmann) to generate graphs, and my sqfind program to find squared squares and my sqt to encode the dissections. Lorenz ran the programs on 17 dual core computers over the Easter school holidays. Some 6756 new SPSSs were found, combined with the known 13810 SPSSs of order 30 (including 9189 SPSSs found in Jan/Feb 2013 by James Williams) there are 20566 order 30 SPSSs in total. Simple Imperfect Squared Squares (SISSs) of order 30 were also enumerated. The total for order 30 SISSs is 667403.

2013 May, Lorenz Milla and I enumerated CPSSs of order 30. Lorenz and processed over 44 billion graphs to find 941 CPSSs, with 5668 CPSS isomers in order 30. Lorenz and I both wrote new software; we implemented a technique of Tutte's, used by Duijvestijn in his thesis, that factors the Kirchhoff matrix determinant of a graph into a product of 3 numbers as follows; 2, a square free number and a square number (this is a necessary condition for a graph to produce a squared square). This method resulted in a 3x speed up of the search. This part of the search was implemented as a program called sqfree, available on the downloads page. Lorenz also produced new software to produce the all isomers of the most common types of CPSSs and identify the canonical tablecode representative from the isomers of a CPSS.

2013 June, July, August; Lorenz Milla rewrote my software to improve the efficiency of the determinant factorisation technique recommended by William Tutte. Lorenz also wrote a plantri plugin to filter graphs using the determinant factorisation technique as they were produced, he was also able to speed up the routines in my sqfind and sqt programs and combine the plantri plugin and sqfind into a single program mandrill (a combination of the names Milla and Anderson). The end result was squared square software 35 times faster than what was used several months ago. This made it possible to complete the enumeration of order 31 and 32 compound perfect (CPSSs), simple perfect (SPSSs) and simple imperfect squared squares (SISSs) in under 2 months. The computations were done by Lorenz on his computers. There are 54541 SPSSs in order 31 and 144161 SPSSs in order 32. There are 17351 order 31 CPSS isomers and 2788 order 31 CPSSs (isomer classes), 52196 order 32 CPSS isomers, and 7941 order 32 CPSSs (isomer classes).