A partially pandiagonal order-9 magic square has recently been brought to my attention by Peter D. Loly (Senior Scholar of the Department of Physics and Astronomy of the Retired Faculty of Science at the University of Manitoba):
|Order-9 partially pandiagonal square dko9a, first published by Dame Kathleen Ollerenshaw|
This order-9 square, initially published in 2006 by the late Dame Kathleen Ollerenshaw in "Constructing pandiagonal magic squares of arbitrarily large size," was later analysed in 2013 by Ian Cameron, Adam Rogers and Peter D. Loly, in their paper "Signatura of Magic and Latin Integer Squares: Isentropic Clans and Indexing." Cameron, Rogers and Loly have attributed the reference number dko9a to the square, and have described its characteristics in their table 4. In a recent correspondance, Peter D. Loly has pointed out that the square has a rank 5 matrix, which is the lowest he has seen for the order-9.
The dko9a is a single square viewpoint of a partially pandiagonal torus, which in this case displays a total of 26 other essentially different partially pandiagonal squares, plus 54 essentially different semi-magic (although still partially pandiagonal) squares. It is practical to consider the dko9a square as a torus, in order to define its construction by a formula, and thus generate its torus ascendants and descendants throughout its respective lower and higher-orders. To find out more about the approach and the method, please refer to "Magic torus coordinate and vector symmetries."
After observing the symmetries of the ninth-order partially pandiagonal square dko9a, we can deduce that if N is the order of the magic torus that displays this square, and if B represents any of the numbers of the torus from 1 to N², then the modular coordinates of position vectors from the number 1 to any of the numbers B can be expressed as follows:
|Formula for the position vectors of the numbers of the magic torus that displays the dko9a square|
Generated by this equation, the following figures show partially pandiagonal tori of direct lineage in the first, fourth, ninth, sixteenth and twenty-fifth perfect-square-orders:
|First-order (1+1)/2 pandiagonal torus T1, ascendant of T4.173|
|Fourth-order (2+2)/8 pandiagonal torus T4.173|
|Displaying the dko9a square, the ninth-order (3+9)/18 pandiagonal torus, descendant of T4.173|
|Sixteenth-order (4+8)/32 pandiagonal torus, descendant of T4.173|
|Twenty-fifth-order (5+25)/50 pandiagonal torus, descendant of T4.173|
Apart from the first-order torus T1, the immediate torus ascendant of the dko9a is the fourth-order semi-pandiagonal torus with index number T4.173 and type number T4.02.2.11. This torus displays eight semi-pandiagonal squares with Frénicle index numbers 89, 183, 323, 368, 464, 539, 661, and 698. The T4.173 torus also displays eight semi-magic squares and is entirely covered by 8 sub-magic 2x2 squares. For details of the classification of the 255 fourth-order magic tori by type numbers, please refer to "255 fourth-order magic tori, and 1 third-order magic torus." For details of the classification of the 255 fourth-order magic tori by index numbers, (which, by using normalised squares is in a way a homage to Bernard Frénicle de Bessy), please refer to "Table of fourth-order magic tori."
Please note that other formulae may well generate different torus ascendants or descendants. However, the formula shown above is ideal, as it theoretically generates an infinite number of similarly constructed partially pandiagonal torus descendants throughout the higher-perfect-square-orders.
Observing the partially pandiagonal characteristics of these perfect-square-order magic tori suggests that there are recurring patterns: In the odd-orders, the patterns of the magic diagonals are (1+1)/2 for the first-order, (3+9)/18 for the ninth-order, and (5+25)/50 for the twenty-fifth-order. This suggests a possible recurring pattern of (√N+N)/2N for the magic diagonals of the odd-orders that are generated by the perfect-square-order formula described above. In the even-orders, the patterns of the magic diagonals are (2+2)/2 for the fourth-order, and (4+8)/32 for the sixteenth-order. This suggests a possible recurring pattern of (√N+N/2)/2N for the magic diagonals of the even-orders that are generated by the perfect-square-order formula described above. Testing in higher-orders, or a mathematical proof, will validate or invalidate these hypotheses.