A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

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 6^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:

*Average of two Tables*
Tables N		Tables N^*
a_avgi × R	equals	a^_avg*i
b_avg × R	equals	b^_avg*
c_avg × R	equals	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 T1_a (6^th row average)
a_avgi	ai_avg × R	a^_avg*i
36i	209.8i	210

204i	1189i	1189i

1188i	6924.2i	6924.2i

6924i	40356i	40356i

40356i	235212i	235212i

235212i	1370916i	1370916i

1370916i	7990284i	7990284i

7990284i	46570788i	46570788i

46570788i	271434444i	271434444i

271434444i	1582035876i	xxx

...i	...i	...i

Table T1_b (6^th row average)
b_avg	b_avg × R	b^_avg*
10	58.3	58

60	349.7	350

350	2040	2040

2040	11890	11890

11890	693008	69300

69300	403910	403910

403910	2354160	2354160

2354160	13721050	13721050

13721050	79972140	79972140

79972140	466111790	xxx

...	...	...

Table T1_c (6^th row average)
c_avg	c_avg × R	c^_avg*
39	227.3	227

221	1288	1288

1287	7501.2	7501.2

7501	43719	43719

43719	254813	254813

254813	1485159	1485159

1485159	8656141	8656141

8656141	50451687	50451687

50451687	294053981	294053981

294053981	1713872199	xxx

...	...	...

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 IIF using the sixth rows of two tables for averages. To see the results using the eighth row of two table averages Part IIIF. See what started it all go to Part IG which lists the new set of magic square of seven squares.

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