A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

CONVERSION TABLES OF RIGHT DIAGONAL TO HIGHER TABLES (Part IIIF)

Method of Table Conversions using the Magic Ratio (R)

The tables of partial imaginary tuples listed in   Part IB,   Part IC,     Part IIIC,   Part ID   and Part IE are useful, when squared, as right diagonals for magic square of squares. I will show that the averages of each of two tuple numbers of two lower level tables when multiplied by the magic ratio (R) of (1 + √2)² = 5.8284268.... affords the average of each tuple numbers of the next higher level tables. Thus, for example b_avg × R gives b^*_avg where the b_avg stands for the average of two values of from the starting two tables and b^*_avg stands for the calculated new average b values of the next higher two tables.

Let me note here that the second tuple (±i,b₂,c₂) in each table only the b₂ and c₂ are multiplied by R. The i, however, is the only numeral which remains as either +i or -i throughout its appropriate table, in a sense initializing the table.

Furthermore, because these tables involve an infinite set of numbers we will use only the 8^th row of each table in the conversion of one set of averages into a second set. Note that the numbers using this even number row are more integer-like as opposed to the 5^th row having half integer values as in Part IF. The method for this conversion employs the following rules:

where N = N₁ + N₂ equals the average of two Roman Numeral tables (Table T₁) for the a, b or c values. I have separated out the as, bs and cs into three different tables for readability.

The table shows that as the averages a_avg, b_avg and c_avg get bigger the products ai_avg × R, b_avg × R and c_avg × R approach and equal the next higher values. a^*_avgi,   b^*_avg   and c^*_avg. In addition, those values in xxx are undetermined since they belong to the next higher table. However, these values should follow the same trend. In addition, the ellipsis (...) at the end implies "going on forever".

As I said previously the magic ratio (R) behaves similarly to the Fibonacci golden ratio, but however, on a much larger scale since it involves multiplying an infinite number of three part tuples by the the magic ratio (R) within an infinite number of tables.

This concludes Part IIIF using the eighth rows of two tables for averages. To go to generation of geometric progressions based on the 8^th row of Tables from this page see Part IVF. Finally to see what started it all go to Part IG which lists the new set of magic square of seven squares.

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