A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

Picture of a square

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 bavg × R gives b*avg where the bavg 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 8th 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 5th row having half integer values as in Part IF. The method for this conversion employs the following rules:

Average of two Tables
Tables N Tables N*
aavgi × Requals a*avgi
bavg × Requals b*avg
cavg × Requals c*avg

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.

Table T1
Tables (N1 + N2)
Tables (V + VI)
Tables (VII + VIII)
Tables (IX + X)
Tables (XI + XII)
Tables (XIII + XIV)
Tables (XV + XVI)
Tables (XVII + XVIII)
Tables (XIX + XX)
Tables (XXI + XXII)
Tables (XXIII + XXIV)
Tables (... + ...)
Table T1a (8th row average)
aavgi aiavg × R a*avgi
72i419i408i
408i2378i 2376i
2376i13848.3i 13848i
13848i80712i 80712i
80712i470424i 470424i
470424i2741832i 2741832i
2741832i15980568i 15980568i
15980568i93141576i 93141576i
93141576i54268888i 54268888i
54268888i3164071752i xxx
...i...i ...i

Table T1b (8th row average)
bavg bavg × R b*avg
1481.684
84489.6490
49028562856
28561664616646
166469702097020
97020565474565474
56547432958243295824
32958241920947019209470
19209470111960996111960996
111960996652556506xxx
.........
Table T1c (8th row average)
cavg cavg × R c*avg
75437425
42524772475
247514425.414425
144258407584075
84075490025490025
49002528560752856075
28560751664642516646425
166464259702247597022475
97022475565488425565488425
5654884253295908075xxx
.........

The table shows that as the averages aavg, bavg and cavg get bigger the products aiavg × R, bavg × R and cavg × 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 8th 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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Copyright © 2016 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com