SPHERICAL IMAGERY WITH MAGIC SQUARES

SPHERICAL IMAGERY WITH MAGIC SQUARES
COPYRIGHT 2016 Tasha Lindsay

Tuesday, December 6, 2011

Magic Squares and Prime Numbers, Part Three

Taking a closer look at the 5x5 magic square, Part One

The fact that there are nine ( a square number) prime numbers in the 5x5 magic square that sum to another square number (100) and that the sum of the prime numbers in just the cruciform portion of the magic square is 61 (part of the 11-60-61 Pythagorean triplet) is cause to take a closer look at the 5x5 magic square.

Examining this square will reveal insights into the School of Pythagoras and what is referred to as the “secret math”.  Pythagoreans were understandably amused by Pythagorean triplets that satisfied the Pythagorean Theorem, dubbed the most important of all mathematical formulae as this was a formula that described space and was represented symbolically by the carpenter’s square.    The carpenter’s square was also known as the gnomon.

The gnomon established a long tradition for thousands of years as an astronomical instrument that could identify summer and winter solstice and is forever linked as a model for time.  Time, space, and math:  these were the concepts that allowed humankind to evolve and prosper (via agriculture) and thus were also of great importance to Plato, Pythagoras, the early Chinese, and the early Christian hierarchy.   Important symbols for these concepts have survived for thousands of years and are relevant, sometimes, to the understanding of early Christian art and architecture (see below).  [for more on the gnomon, see Websters Dictionary and Needham’s Science and Civilisation in China, Vol. III p. 19-22].

The 5x5 magic square

The first point of interest is that this square features two Pythagorean Triplets that are in a gnomonic or right angle relationship, 5-12-13 and 7-24-25.  This is the smallest order magic square in the Luo Shu format that features two Pythagorean triplets.  Note that all the odd components of the two Pythagorean triplets fall within the cruciform of odd numbers.

The second point of interest is the number 39.  The above 5x5 magic square that is colored shows four groups of numbers in the corners of the square shaded in beige that are also in the shape of a gnomon or right angle.  The total sum of the numbers shaded beige is 156; however, each gnomonic component sums to 39.
  
 A closer look at the internal 3x3 grid within the 5x5 magic square
12
25
8
5
13
21
18
1
14
demonstrates that each column, row, and diagonal that pass thru the central axis (13) sums to 39.

Concerning the blue colored numbers: 7, 14 and 18 sum to 39 as do 8, 12 and 19; 9, 12, and 18; and lastly 8, 14, and 17.  When two of these groups of three numbers are connected by lines, this is the image:


It was pointed out earlier that the total sum of all the prime numbers in the square is 100 and the total sum of the prime numbers in just the cruciform portion of the square is 61, the difference of these two numbers being….39.

The third point of interest concerns the sum total of each of the three groups of numbers identified by color.  The sum total of the numbers shaded beige is 156, when divided by 13 gives us 12, and there are indeed twelve numbers that comprise this sum.  The sum total of the numbers shaded light blue is 104, when divided by 13 gives us 8, and there are indeed eight numbers that comprise this sum.  Lastly, the sum total of the numbers shaded bronze is 65, when divided by 13 gives us 5, and indeed there are five numbers that comprise this sum.

Math can be fun, or as my nephew says “cool”.  (He did say weird at first but I corrected him).  More relevant, this is how math was taught thousands of years ago. 

Pythagoreans as well as the Chinese thousands of years prior to Pythagoras were interested in functional math that explained Time and Space.  Things such as Pythagorean triplets, the gnomon, and calendrical numerology excited the Pythagoreans and the mathematically minded early Chinese; Luo Shu magic squares possessed all these characteristics and were part of the esoteric tradition of what some refer to as magic but was probably none other than math and wisdom.

More detailed  explanations than can be given here are available in my e-book, The Language of Numbers Dymystified.  Scroll to top of page, click on "a book for the library", and order for only $15.

The carpenter's square as a Christian Symbol
Jesus and the four apostles, Ravenna mosaic - 7th century
The use of the "gammadia" symbol in early Christian art has raised some controversial issues, mainly that art historians throughout the ages have refused to acknowledge any meaning to this symbol and that its usage is strictly ornamental.  The gammadia or carpenter's square is symbolic of the gnomon, time, space, and math.  The gammadia's use on several hundred mosaics throughout Italy but especially in Ravenna over the course of five hundred years (between 350AD and 850AD) has been very consistent: the symbol only appears on clothing, altar cloths, or altar curtains to identify people or things of religious significance.  The gammadia appears on the clothing of Jesus, the apostles, evangelists, and others who had status within the church.  Furthermore, it is well documented that Popes during the Italian Renaissance had an obsession for Egyptian obelisks, another representation of the venerated gnomon.  Therefore, the symbolic role of the gnomon and carpenter's square is consistent through out Christian art and architecture.  Subjects such as the Christian use of symbols that relates to the early Chinese culture and their veneration of the Luo Shu are discussed in detail in my book.

Saturday, October 1, 2011

Magic Squares and Prime Numbers, Part Two

Table 1: examples of prime numbers in magic squares


This discussion of the occurrence of prime numbers in magic squares will mostly be limited to magic squares that can be constructed with the following formula and the resultant cruciform of odd numbers:

Table 2: The Luo Shu, formula, and cruciform of odd numbers

4
9
2

Y-1
X2
Y-X

4
9
2
3
5
7

Y
2Y-X

3
5
7
8
1
6

X+Y
1
Y+1

8
1
6
3X3 magic square

formula

cruciform of odd #'s



Magic squares based on the above formula are called magic squares in the Luo Shu format.  X will always be located to left of center and will equal the order of the square; Y will always represent the center number. When magic squares are constructed in this manner, a unique Pythagorean triplet of numbers appears at the heart of the square involving the odd numbers, X and Y. 

Table 3: The Luo Shu and Pythagorean Triplets
 
cycle of
Pythagorean

Pythagorean
Luo Shu
Triplets

Theorem




X
Y-1
Y

X2 + (Y - 1)2 = Y2

1

3
4
5

9 + 16 = 25

2

5
12
13

25 + 144 = 169

3

7
24
25

49 + 576 = 625

4

9
40
41

81 + 1600 = 1,681

5

11
60
61

121 + 3600 = 3,721

6

13
84
85

169 + 7,056 = 7,225

7

15
112
113

225 + 12,544 = 12,769

Another feature of magic squares based on this formula is that a cruciform of odd numbers runs through the horizontal and vertical axis of the square.  

Table 4: Higher Order Luo Shu magic squares, Pythagorean triplets, and the cruciform of odd numbers






9x9 magic square






37
78
29
70
21
62
13
54
5






6
38
79
30
71
22
63
14
46
3x3 magic square

47
7
39
80
31
72
23
55
15
The Luo Shu

16
48
8
40
81
32
64
24
56






57
17
49
9
41
73
33
65
25

4
9
2


26
58
18
50
1
42
74
34
66

3
5
7


67
27
59
10
51
2
43
75
35

8
1
6


36
68
19
60
11
52
3
44
76






77
28
69
20
61
12
53
4
45















5x5 magic square


7x7 magic square








22
47
16
41
10
35
4

11
24
7
20
3


5
23
48
17
42
11
29

4
12
25
8
16


30
6
24
49
18
36
12

17
5
13
21
9


13
31
7
25
43
19
37

10
18
1
14
22


38
14
32
1
26
44
20

23
6
19
2
15


21
39
8
33
2
27
45








46
15
40
9
34
3
28


This post will examine two considerations of prime numbers that occur in magic squares in the Luo Shu format:
  • The sum total of all the prime numbers (∑ primes) in the first four magic squares in the Luo Shu format, and,
  • The sum total of all the prime numbers in the cruciform portion    
    (cruciform primes) of the first four magic squares in the Luo Shu format.
The sum total will then be broken down to the basic common denominators that compose the number.

Table 5:  Sum of Prime Numbers in Luo Shu Magic Squares

magic square
∑ primes
broken down
cruciform  primes
broken down
3x3
17
17
15
5*3
5x5
100
(5*2)2
61
61
7x7
328
41*23
208
13*24
9x9
791
113*7
305
61*5

A center number from each magic square is a component of the number that represents a total sum (with the exception of a sum that is prime and not a centered number).   Whether it is the total sum of all the prime numbers in a magic square or the total sum of the prime numbers that occur just in the cruciform portion of the magic square that is to be considered, one will discover a “centered” number that is a component of the total entity.  

The only center number (25) not represented is from the 7x7 magic square (in the Luo Shu format), however; it really is represented as 52.  Therefore, the 3x3, 5x5, 7x7, and 9x9 magic squares all have prime numbers whose total sum and/or cruciform sum have numerical components that correspond to the center numbers of all of the magic squares from order 3 to order 11 and order 15.  

Conclusion

There exists a relationship between magic squares in the Luo Shu format and prime numbers.  This relationship has mostly to do with the sum of prime numbers and the center number of magic squares in the Luo Shu format.

Magic squares in the Luo Shu format:
  1. Are based on a formula
  2. Reveal a Pythagorean Triplet of numbers
  3. Reveal a cruciform of odd numbers
  4. Reveal a relationship to prime numbers
  5. Form a three dimensional torus
  6. Are symmetrical, that is, any two numbers equidistant from the center add up to the same number