Simple Perfect Squared Rectangles with Isomers

Isomers are squared tilings with the same elements arranged differently. They have an important historical role to play in the research conducted by Brooks, Smith, Tutte and Stone.

Figure 74; 112 x 75A Smith Diagram

Figure 74; 112 x 75A Smith Diagram

Tutte wrote, "It was a pleasing recreation to work out perfect rectangles corresponding to networks with a high degree of symmetry. We considered, for example, the network defined by a cube, with corners for terminals and edges for wires. This failed to give any perfect rectangles. However, when complicated by a diagonal wire across one face, and flattened into a plane, it gave the Smith diagram of Figure 74 and the corresponding squared rectangle of Figure 75. This rectangle is especially interesting because its reduced elements are unusually small for the thirteenth order. The common factor of the elements is six. Brooks was so pleased with this rectangle that he made a jigsaw puzzle of it, each of the pieces being one of the component squares."

Figure 75; 112 x 75A Squared Rectangle

Figure 75; 112 x 75A Squared Rectangle

"It was at this stage that Brooks's mother made the key discovery of the whole research. She tackled Brooks's puzzle and eventually succeeded in putting the pieces together to form a rectangle. But it was not the squared rectangle which Brooks had cut up!

Brooks returned to Cambridge to report the existence of two different perfect rectangles with the same reduced sides and the same reduced elements."

Figure 75; 112 x 75B Squared Rectangle

Figure 75; 112 x 75B Squared Rectangle

This led the researchers to examine the electrical networks of the two rectangles, they found they were related by symmetry. Further exploration of symmetry led to networks which could produce rectangles of the same size with all elements different (or with a single corner element the same), these can be assembled, with 2 other squares, to form a compound perfect squared square.

Duijvestijn cataloged isomers.  In order 25 SPSS he found 3 pairs of isomers [1].

Isomers can be classified by the number in each class;

The following counts are a final tally of sorted adjacent isomers by order
CPSS always exist as isomers, the breakdown into equivalence classes is not shown (yet). It is also possible for simples and compounds of the same dimensions and order to form isomers. CPSRs have not been produced except at low orders.

THE NUMBERS OF SQUARED RECTANGLE AND SQUARED SQUARE ISOMERS

Order SPSR
SISR
SPSS
CPSS
SISS
CPSR
9
-
-



 
10
-
-




11
-
-




12
-
-


-

13
1 pair
-


-

14
-
-


-

15
-
2 pair


-

16
1 pair
5 pair


1 triple

17
11 pairs
21 pairs
1 triple



1 pair

18
45 pairs
38 pairs


2 pair
1 triple


19
68 pairs
1 triple
91 pairs
3 triples
1 quintuple


4 pairs

20
176 pairs
2 triples
237 pairs
10 triples
1 quadruple


5 pairs
4 triples

21
428 pairs
5 triples

-

25 pairs
1 triple
2 quadruples

22


-

28 pairs
11 triples
2 sextuples

23


-

112 pairs
10 triples
14 quadruples
1 quintuple
1 sextuple

24


-
4
180 pairs
35 triples
9 quadruples
10 sextuples
1 octuple
2 duodecaples

25


3 pairs
12
562 pairs
39 triples
70 quadruples
5 sextuples
3 octuples

26


1 pair
100
945 pairs
123 triples
89 quadruples
37 sextuples
1 septuple
4 octuples
10 duodecaples

27


4 pairs
220
2438 pairs
122 triples
346 quadruples
3 quintuples
28 sextuples
1 septuple
21 octuples
2 duodecaples

28


7 pairs
1 triple
948
4286 pairs
364 triples
499 quadruples
10 quintuples
133 sextuples
3 septuples
30 octuples
43 duodecaples

29


25 pairs
10580 pairs
387 triples
1702 quadruples
8 quintuples
131 sextuples
10 septuples
153 octuples
1 nonuples
2 decaples
19 duodecaples
4 sexdecaples
1 octdecaples