Analysis of the nazca mandala

Analysis of the Nazca Mandala... – Jayro Luna

Analysis of the Nazca Mandala: Some Mathematical Considerations and Others not so much ...

Jayro Luna

ABSTRACT: In this article, we analyze the enigmatic figure of the "Nazca Mandala" that differs from the standard of the other figures of the Nazca plateau, because it is neither figurative nor anthropomorphic or zoomorphic. It presents mathematical and geometrical interpretations regarding the structure of the figure as well as aspects related to the esoteric meaning of the figure. The article is still a partial result of research done on the subject, but already has sufficient elements for this text to present it with academic and scientific substance. In this article, we also analyze and critique interpretations that have been tried for an understanding of the figure. As a requisite for understanding what is set forth here, a certain familiarity with the mathematical and esoteric meaning of the magic squares is necessary.

Keywords: Nazca Plateau, Nazca Mandala, Semiotic Neo-structuralism, Magic Squares, Esotericism.

Note of Warning: First, I apologize for the quality of my English language mastery, still somewhat primal. The text originally written in Portuguese. However, I insist on presenting this troublesome version in terms of the English language in order to acquaint a larger number of people with the results of my research.

1. Introduction The Nazca Plateau is a world-known place, located in southern Peru, cut by the PanAmerican Highway. The place, of desert characteristic has become famous for the figures drawn in the soil of such size that can only understood from a position in the sky, that is to say, flying. Figures that resemble various animals (dog, hummingbird, monkey, whale, etc ...) and other representations (the astronaut, for example) provoke the imagination of scholars, scientists and curious in search of its real meaning, the purpose of their work and of who actually would have done them.

Here we will focus on a particular figure whose nature escapes the pattern of Nazca inscriptions, it is the figure commonly known as the "Nazca Mandala", also known as "Star" or "Nazca Cross."

The region is divided into two sets of figures, one, larger and located in the flattest part, called "Nazca Plateau" because it is situated in the direction of the city of Nazca. A second set, smaller, on more rugged and elevated terrain, called "Palpa" because it is located near this village. The Mandala is located in this second set.

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Analysis of the Nazca Mandala... – Jayro Luna

Figure 1

The Map above is relatively simplified, not showing all the figures, the following map, more detailed, refers only to the Palpa, so that we can perceive the set of figures in which the mandala is located:

Figure 2

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Analysis of the Nazca Mandala... – Jayro Luna

Note the position of the Mandala ("La Estrela"). The set of Palpa in relation to that of Nazca, has more anthropomorphic figures (Paracas Family, The Traveler, The Woman, the Fertility). We will not dwell here on the anthropological questions about the theories that determine the mode of making, or what people did them and why, nor do we turn to the fantastic assumptions about extraterrestrial travelers. It is easy to find videos, documentaries and other material on these assumptions. Our goal is simply to show some observations we have made about the structure of the Nazca Mandala, which in our view presents some very specific peculiarities correlated with researches that we have developed about the nature and characteristics of so-called "magic squares". Nor will we give basic explanations of magic squares, usually - as mathematical curiosities or as a very peculiar kind of matrix. Anyway, let us get right to the point.

2. The Structure of the Nazca Mandala The Nazca Mandala differs from the other local figures by its non-anthropomorphic or zoomorphic constitution. A grid of squares and lines that cross this grid at different angles. Another aspect is also that its lines, although visible, are fainter than the lines that make up the other figures. Located in the denominated part of Palpa is located a little above the terrain of the Nazca lines.

Joseph E. Mason1 in an article published on a non-academic page presents an interesting observation on the structure of the mandala in question, noting its resemblance to an 8x8 magic square, known in the cabalistic and alchemical texts as the "magic square of Mercury." In fact, it is easy to recognize an 8x8 grid in the Nazca Mandala:

Figure 3

Above (fig.3), we have an aerial view of the Nazca Mandala, below (fig.4) are marked the lines of the grid, including the lines that cut the great in different angles:

1

MASON, Joseph E. The Nazca Sun-Star as the Magic Square of Mercury. Available in: http://www.greatdreams.com/numbers/nazca/nazca.htm

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Analysis of the Nazca Mandala... – Jayro Luna

Figure 4

The figure may bewilder our perception a bit, but continuous observation allows us to see various aspects of the figure. The 8x8 grid has squares marked in a darker tone, because on the floor of Palpa they appear in a lighter tone than the others do. There are three for each corner of the grid and twelve central squares that form a cross. The association of the Nazca Mandala with the magic square of Mercury occurs only in principle by the number of squares of the grid (64), since the numerals corresponding to the magic square of Mercury not inscribed in each square. Thus, we insert the numerals and from the very indications of the lines and the colors of the squares, we prove that in fact it is a magic square 8x8:

Figure 5

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Analysis of the Nazca Mandala... – Jayro Luna

The magic square of Mercury - hereinafter referred to as this - gives as a sum of each column, line or diagonal the value 260. But this is a basic condition of the magic square.

This, like those of another order, holds a series of internal relations involving 4, 8, or 16 numbers that add up in each specific relation 260. The ancients knew of these intrinsic properties of the magic squares. In the case of Mercury the following figure is very illustrative of these internal relations:

Figure 6

The detailed explanation of the meaning of this illustration would give another article, but the reader can get some interesting information in the book of Henry Cornelius Agrippa of Nettesheim2, pages 945, 946 and 950.In the case of the numerals marked in the Nazca Mandala, we note that if we add two opposite sets of trios of number marked, plus the two central numbers perpendicular to the diagonal formed by the two sets, we obtain 260. Example: a) b) c) d)

9 + 2 + 64 = 75 1+ 63 + 56 = 120 29 + 36 = 65 75 + 120 + 65 = 260

If the reader wants to exercise his ability to make head count, try the other numbers in the other set. In the central cross, the vertical rod adds 260, as well as the horizontal bar of the cross. From these aspects, it seems clear to us that the 8x8 grid of the Nazca Mandala is not only a grid of empty squares, but a square of magic square.

Then, we will demonstrate how the knowledge of who did this figure in Nazca went beyond the knowledge until now divulged about the use of the magic squares. We hope, perhaps, to put in the reader a question mark about the usefulness of this figure as well as the amount of knowledge required to produce it and especially what it can actually mean.

3. The intersections of magic squares and the dynamics of their transformations.

What we are going to emphasize in this section of this text involves the markings that exist in the Mandala of Nazca in the mentioned four corners and in the central units.

There are several non-academic articles on the Internet about the interpretation of these markings, but in our view, they are in conjecture without much support, making constant

2

NETTESHEIM, Cornelio Agrippa de. Three Books of Hidden Philosophy. SĂŁo Paulo, Madras, 2012.

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Analysis of the Nazca Mandala... – Jayro Luna

references to the symbology of the cross and the astronomical relations that can drawn from the interpretation of the angles formed by the lines that cross the grid.

Note that if we look closely at the figure, there are two overlapping grids, the second, with respect to the first one a 45 ° angle of slope in the southwest-northeast direction. From this orientation it is perfectly identifiable a smaller central grid of 4x4 extension. In the esoteric scope, known as "magical square of Jupiter". A basic square of this type has 16 units in total and the sum of each line, column or diagonal gives 34. If we look at the magic square of Mercury again, that cross turns into a magic square of 4x4 if we only add the units of the chants (values: 19, 22, 43 and 46). In analyzing this new magic square contained in the grid of Mercury we observe that it keeps the same properties of the magical square of Jupiter, in the sense that each line, column or diagonal adds the same amount, but now the value is 130:

22

44

45

19

35

29

28

38

27

37

36

30

46

20

21

43

Figure 7

O quadrado mágico assim obtido, guarda também todas as relações intrínsecas a um típico quadrado mágico de Júpiter, no sentido de que todas as relações existentes internamente entre conjuntos de 4 e 8 unidades do quadrado estão mantidas. Elas podem ser exemplificadas pelo seguinte conjunto de diagramas (fig.8):

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Analysis of the Nazca Mandala... – Jayro Luna

Figure 8

Most internal transmissions of the product of 4 units added to 34 are placed there, but there are still, we do not intend to extend at that point, it is only to give an idea of what we call internal relations. Note that the 4x4 magic square obtained from the magic square of Mercury in the Nazca Mandala maintains all these properties, only now with the value of 130, that is, half the value obtained by a set of 8 units without a square of Mercury.

But the most interesting, to our understanding, comes now, for we observe the three units placed in the corners of the square of Mercury. It does not seem to us that their goal is merely to compose values within the Mercury square. For, just as we can obtain a 4x4 magical square from the central part of Mercury, we can also obtain a square of order 6x6 if we follow the directions of the Nazca Mandala itself.

Agrippa presents a method for constructing a square of the Sun (6x6) based on a series of inversions from a vector taken from a signal found in the square of Saturn (3x3).

Figure 9 – The Saturn’s Seal

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Analysis of the Nazca Mandala... – Jayro Luna

This gravure called the saturn seal is the result of marking on the cells of a 3x3 square. The diagonal sum 15, the two compass-shaped signals, add different values (6 and 24, respectively the smallest and largest possible sum of 3 cells), but whose average is 15. Note the following figure, as Agrippa uses this "Seal" to promote inversions in a 6x6 grid, in which the cells are filled in the order of 1 to 36 starting from the cell of the lower left corner, and after it promoted the inversion of the diagonals3.

Figure 10

In the case of the Nazca Mandala, we have an 8x8 grid and we want to produce a grid of a magic square 6x6. Thus we apply the solution of Agippa in the grid of Mercury, observing the limit of 6x6 and then we inverted only the tips of the diagonals of the square of the Sun:

Figure 11

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For a more detailed explanation we indicate: AGRIPPA, H. C. Three Books of Occult Philosophy. SĂŁo Paulo, Madras, 2012, p. 938.

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Analysis of the Nazca Mandala... – Jayro Luna

Or by presenting otherwise the changes made, for a more objective presentation:

Figure 12

Note that the square of the Sun now formed gives as sum of each line, column or diagonal the value 195 (3/4 of the sum 260 of Mercury). The square of the standard Sun gives as sum 111.

The grid now formed starts at number 10 and jumps from six to six numbers two numbers (10, 11, 12, 13, 14, 15, -, -, 18, 19, 20 ...).

Now, about the three cells marked at each corner of the Mercury grid, as well as the twelve-cell cross at the center, we can ask why this is so, given that the Mandala is known to have been produced in a time prior to colonization and Christianization. In this sense, the community that produced it, in principle did not have the cross as a religious symbol. Assumptions raised by scholars, academics, and non-scholars. But careful observation of the properties of the 8x8 Mandala solves the problem in a way that requires no symbolic interpretation other than the mathematical characteristics of the Mercury grid.

In a first step let us approach the three cells from each corner (sum 130 each set) such that they close a 4x4 square:

8

58

63

1

49

56

9

16

64

2

7

57

Figure 13

We have a 4x4 grid (Jupiter) that lacks the values of the four central cells. Each diagonal gives as value 65 (1/4 of Mercury), the two lateral columns and the first and last complete lines that are, add 130 (1/2 of Mercury). We are immediately tempted to place in the four central cells the values of the Mercury central and in doing so we obtain:

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Analysis of the Nazca Mandala... – Jayro Luna

130

8

58

63

1

130

49

29

28

56

162

9

37

36

16

98

64

2

7

57

130

130

126

134

130

130

260

260

Figure 14

The sum of the second and third lines does not give 130, but the sum of the two gives 260, as well as the second and third columns added also gives 260. Moreover, if we replace the four central cells by the values adjacent to them in the diagonals of the great one of Mercury we maintain the same pattern, until reaching the corners: 15

130 8

58

63

1

130

49

22

19

56

146

9

46

43

16

64

2

7

130

128

132

130 8

58

63

1

130

49

15

10

56

130

114

9

55

50

16

130

57

130

64

2

7

57

130

130

130

130

130

130

130

130

260

260

260

260

Figure 15

As for the figure of the central cross, if we now replace the four central cells by the numbers above and below the central cells, and then by the numbers to the right and left of these cells we have the same pattern: 130

120

8

58

63

1

130

8

58

63

1

49

44

45

56 194 260

49

35

38

56 168 260

9

20

21

16

9

27

30

16

64

2

7

57 130

64

2

7

57 130

66

130 124 136 130 130

130

82

130 122 128 130 130

260

260 Figure 16

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Analysis of the Nazca Mandala... – Jayro Luna

In this way, the figure of the cross appears, since they refer to the numbers that complete this 4x4 grid. As well as marking the numbers of the corners of the 8x8 grid.

The producers of this figure observed an intrinsic property to the grid. For a better visualization of like done, let us see the grid schematized below:

Figure 17

Now observing the figure of the Nazca Mandala from another angle, note that under the Mercury grid there is another one, which is located at 45 ° in the NE-SO direction (fig.4). The grid that now appears formed by the magic square of order 12x12, thus containing 144 cells. These are easily observable, since they are more prominent a central part, forming a new square of Jupiter (4x4) and on the outside, contiguous to this grid 12x12, located inside the two central rows and columns, two cells, thus forming a species of cross. This magical square of Jupiter is totally inserted in the area of the previous magical square of Jupiter, since the size of its cells is smaller. The larger magical square of Jupiter, whose sum gives 130, in terms of ascending numerical order, begins with cell number 19, and every four numbers, jumps another four, then resumes the count (19,20,21,22, , 27,28,29,30 ...) ending at number 46. A magic square with regular order that goes from 130 would start at number 30 and end at 45. Note that 45 is the number of degrees of slope of this second square of Jupiter. Let us see what is the regular numbering of a square of order 12x12:

Figure 18

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Analysis of the Nazca Mandala... – Jayro Luna

The sum of each line, column or main diagonal gives 870. In the figure above, we have already divided the grid 12x12 into 9 squares of Jupiter (4x4), with the center giving 290 for each row, column or main diagonal. The others, even though they always have the same value (290) for row or column, vary the value of the main diagonals to less than or greater than the value 290. It happens that if we add up and average the two opposing Jupiter squares (p. the light blue in the upper right corner of the figure and the white in the lower left corner) we have the average value of 290. Thus, it seems to us, the emphasis given to the central square 4x4.

91

90

181

80

66

67

77 290

68

78

79

65 290 290

55

54

109

148 290 290 142 290

290 290

Figure 19

From the central cells, by adding the cells that placed adjacent to the columns and central lines, we obtain by summing up rows and columns the value of 290.

Moreover, by adding laterally these adjacent cells (the four columns or the four of the lines), we also have 290 (1/3 of the sum of the 12x12 grid).

Thus, we have here exposed in this text how the grids of the Nazca Mandala seem produced from a deep knowledge of the properties of the magic squares.

The last point that we will address in this text, refers to the double circle circumscribed in the grid of 8x8, as can be observed in the following figure:

Figure 20

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Analysis of the Nazca Mandala... – Jayro Luna

The circle, or wheel, as we call it, can represented by a sequence of twelve cells of the 8x8 grid, namely:

8

58

59

5

4

62

63

1

49

15

14

52

53

11

10

56

41

23

22

44

45

19

18

48

32

34

35

29

28

38

39

25

40

26

27

37

36

30

31

33

17

47

46

20

21

43

42

24

9

55

54

12

13

51

50

16

64

2

3

61

60

6

7

57

Figure 21

As the cells are selected, they enter the wheel, the sum of them gives 390 (or 260 +130) and if we want to sum them in groups of 4, alternating one sequence, get 130 for each group. Other options for forming the cells composing the wheel can attempted, show results with the values (130, 260, 390).

We still have a lot to talk about the Nazca Mandala, for example, about the several lines that leave the center, including and especially the curved lines that form rosettes, velica pisces, stars and that from the analysis of cells of the two great superposed grids (Mercury - 8x8 and 12x12) we can define its origin in relation to the same grids.

And yet, all the anthropomorphic and zoomorphic figures of the Nazca plateau (including Palpa) can be understood from the formulation of griffins circumscribed to Mandala. In this sense, we assume that the Mandala was the first figure to be built on the site, or at least the first to be planned. But this is a subject for other texts.

4. References: FARRAR, Mark S. Magic Square. USA, ed. Author, 2007 MASON, Joseph E. The Nazca Sun-Star as the Magic Square of Mercury. Disponível em: http://www.greatdreams.com/numbers/nazca/nazca.htm LUNA, Jayro. “Alguns Apontamentos Semióticos Breves Sobre a Obra Esotérica de Raji Guenol: Selos de Planetas e Quadrados Mágicos” em: Revista Diálogos, Garanhuns (PE), n.° 10, nov. 2013, p. 119-144. Disponível em: http://www.revistadialogos.com.br/Dialogos_10/Luna_Guenol_Selos.pdf NETTESHEIM, Cornelio Agrippa de. Três Livros de Filosofia Oculta. São Paulo, Madras, 2012 PASLES, Paul C. Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey.

New Jersey, Princeton University Press, 2003.

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