Dresden Codex

Section in Pages 69 - 74

Map of Section

Page 69 Page 70 Page 71 Page 72 Page 73
Number of Columns
2 2 2 2 5 2 7 5 1
Introductory Remarks Dates Dates Table of Values Table of Values Ref. Page 74
Number of Units in Column
1 2 2 5 3 2

The Section starts on Page 69 denoted by a red line that runs vertically from the top to the bottom of the page. The Section ends on Page 74. Pages are divided into columns with sub-sections in each column. Page 69 columns 2 and 3 are preface material. On Page 70 the most left two columns are Long Count dates that are stacked two-on-two. Then appear two other columns with other information, before the tabular data begins in columns 5 and 6.  The first set of tabular date then continues across Pages 70 to 73, with a change in format on Page 71. Pages 70 columns 5-6 and Page 71 columns 1-5 have five sub-sections per column. Page 71 columns 6 - 7 through page 73 columns 1 - 5 have three sub-sections per column. Page 73 column 6 is independent of the previous tabular data. The Section ends on Page 74 with a postscript.

In the following Tables I have identified individual pages, sections, and columns. Refer to individual photographs. Click on each thumbnail to obtain a full display of the Page.

69 70 71 72 73 74

The following Table shows the introductory columns on Pages 69 and 70 with technical data. I offer it only for information purposes, not for analysis. I have not attempted a translation. The web site

http://www.mayacodices.org

offers limited translation of text, which must be taken with a grain of salt since it reflects common false notions of the meaning of the Codex.

I append a description of the subject of Page 74 at the bottom of the following Table One produced by the authors of the Maya Codices site.

Table One

Col 69-2

Col 69-3

Col 70-1

Col 70-2

Col 70-3

Col 70-4

Col 73-6

caption 1: (__ __ 03 joined? gods.)
caption 2: (The piktun was formed.)
caption 3: (18 b'aktuns white Kimil Pawah white rabbit? Pawah [0] k'atuns 8 tuns 16 k'ins.)
caption 4: (The winal was formed from 19 [plus 01] days.)
caption 5: (04 Ahaw 08 Kumk'u.)
caption 6: (The fifth arrival? from the water; 01 piktun entered, 03 winals entered, 01 day entered.)
caption 7: (15 k'atuns 09 tuns 04 winals and 04 days were formed.)
caption 8: (09 K'an 12 K'ayab')
caption 9: (13 Mak)
caption 10: (__'s holy footsteps stopped.)

Introductory Glyphs

Introductory Glyphs

Introductory Glyphs

Introductory Glyphs

Introductory Glyphs

Introductory Glyphs

Rain god Chaak is perched on the open mouth of a serpent whose body alternates with red and black numbers. Chaak is painted black and wears a peccary head as a headdress. He holds a spear and also has a shield. (Interwoven in this serpentine body are numerical figures shown below.)

 

9

9

10

10

1

13

19

17

11

19

12

11

13

3

0

10

13

12

19

0

0

0

12

14

This is a numeric value, 14040, that fits the 65 modulo shown below. The calculated modulo is 216 with prime factors of   2 · 2 · 2 · 3 · 3 · 3 · 5 · 13

9 Ix

9 Ix

9 Ix

9 Ix

1

4

   

12

10

   

Ring 6

Ring 6

   

4 Ahau

4 Ahau

8 Kumku

8 Kumku

Glyphs

Glyphs

Black

Red

9 Ix

9 Ix

Red

1

9

4

8

8

14

17

19

6

6

16

2

2?

13

0

16

19

16

14

12

13

12

10

12

This is a numeric value that does not calculate properly to any modulo. The script is confused with red numbers inserted.

8

10

0

0

4

10

Ring 6

Ring 8

4 Eb

9 Ix

4 Ahau

3 Ahau

9 Ix

Text

Eroded

8 Kumku

8 Kumku

12?

Page 74: Water spews from the open mouth of a crocodilian figure at the top of the page, whose body forms a skyband (including the glyphs for “eek’” = ‘star, Venus’, “ka’an” = ‘sky’, a variant of the “k’in” = ‘sun” glyph, and “ak’b’al” = ‘night, darkness’). A solar and lunar eclipse glyph are attached to the skyband, with streams of water falling from each of them. The central part of the scene is dominated by the old goddess Chak Chel, here in her destructive aspect (as indicated by her clawed feet and hands and the crossed bones on her skirt). She wears a serpent headdress and overturns an olla of water, which contains a distance number of 5.1.0 (5 tuns, 1 winal, and 0 days) and the day glyph Eb’. This refers to the date 4 Eb' recorded in the table immediately preceding this page of the codex. At the bottom of the page is the black God L, here carrying weapons identifying him as a Venus god (a spear and atlatl darts). Perched on his head is an owl, a bird identified with the underworld. This scene has been associated with the flood referenced in colonial Maya sources said to have destroyed the previous creation of the world (Taube 1988).

If we were to assume that these columns all give Long Count dates we would obtain:

Table Two

Col 69-3b Black 

9-19-13-12-8  

December 23, 823

Col 69-3b Red

4-6-0-13-10  

August 13, 1417 BC

Col 70-1a

9-13-12-10-0  

August 1, 704  

Col 70-1 Date Extension

1-12

32 days

Col 70-1b

8-6-16-12-0-4  

May 20, 176 + 4 days

Col 70-2a

9-19-11-13-0  

January 14, 822

Col 70-2 Date Extension

4-10  

90 days

Col 70-2b

8-16-19-10-0-10  

May 12, 376 + 10 days

Col 70-3a

10-17-13-12-12  

October 27, 1178

Col 70-4 

10-11-3-19-14  

January 28, 1051

Col 70-4 Date Extension

14-2-16-12

101,852 days

 

Table of Values, Pages 70 - 73

Generally, the top section of the columns are read across the pages 70 to 73, with return to Page 70 for the second section, and so on. However, the scribe did not strictly adhere to this arrangement. Some sections are interlaced with others. I have arranged the Table to reflect the order of the data, and not the exact scribal order. The top portion of the columns is devoted to a multiplier of 54 (modulo 54); the bottom portion to a multiplier of 65 (modulo 65). The number in parenthesis in the Table after the location identifier shows the multiplier for that set of data, either modulo 54 or modulo 65. I do not show the religious date dedication symbol at the bottom of each column of data, since this does not affect conclusions drawn from the data. These dedication symbols are mostly 4 Eb interlaced with other Tzolkin date symbols. I have been unable to understand the rationale behind these assignments, and feel they may be logically random.

The authors of the Maya Codices web site borrow their interpretation from previous authorities, without critical examination. Victoria and Harvey Bricker received considerable attention; their interpretations are repeated without rigorous assessment. The Brickers note that some of the columnar data show blue vertical lines indicating rainfall. Those with such lines they say mean rain for that segment. Those without mean drought. Furthermore, and much more critical, is their assumption that each of the tabular data point to a specific Long Count date. They do not appear to understand the mathematical nature of the data. The result of my examination would see their attempts at interpretation mostly as nonsense.

The Fascinating Number 13

Every number in the data set of both modulo 54 and modulo 65 is evenly divisible by 13, except for the special numbers in the last group of modulo 54.

This is a most remarkable fact. The numbers had to be consciously chosen to meet that criteria. For example, consider the number 45360. This is a relatively large number but is not evenly divisible by 13.

I have given the prime factors for all numbers so that this remarkable numeric feat can be readily visible.

We should also examine the prime factors of 54 and 65. 54 has 2 X 3 X 3 X 3, while 65 has 5 X 13. These prime factors will always appear in the modulo 54 and modulo 65 numbers respectively, except for the last group in modulo 54.

Modulo 54 Discussion

All numbers in the modulo 54 group are divisible by both the numbers 54 and 13, except as noted.

54

108

162

216

270

324

418

452

486

540

594

648

11

13

2

4

6

8

10

12

1

3

5

7

The last group in modulo 54 are numerically too small to be divisible by both 54 and 13. But they are marked special with a ring and a tied ribbon. Enclosed in this ring is another number, each uniquely different.  I have shown these assignments here in order to recognize that there appears to be no rationale behind the ring number, although they are ranked.

The unique numbers assigned within the rings do not include 9, which is missing from the group. The number 9 is assigned to the bottom of an extraneous column, 73-6, with a confusing numeric value.

Furthermore, the order in these special small ring numbers (not the numbers within the ring) increases in numeric size as one scans across the Pages, rather than decreasing as do the other numbers in modulo 54. The range of these numbers in modulo 54 is one short of 13. Above 12 the numbers then all become evenly divisible by both 54 and 13.

(Two numbers in parenthesis in the Modulo 54 Tables show where I corrected scribal errors. These become discernible when tabulated.)

The Modulo 54 numbers begin with 70,200/54 = 1300. Note that 70,200 = 54 X 13 X 100; this is a decimal relationship. They then decrease in size by modulo steps of 260 = Tzolkin Round Number = 20 X 13 until they reach 7020. Between 28080 and 7020 a jump occurs with a modulo difference of 390 instead of 260. 390 = 260 + 130 = 260 + 1/2(260).Thereafter, all numbers decrease in modulo steps by 13 to 702 (54 X 13). Again note that 7020 is 10 X 702. Even more, the numbers 70200, 7020, and 702 are all 10X multiples of the lower number. These occur at the three major break points of the number sequence.

This decimal relationship is not discernible in the vigesimal system. One must convert to decimal to see the 10X relationship.

Even more revealing of the scheme of the Designer is the Ratio of the numbers as one scans down the list. The ratios are calculated by the initial number (70,200) divided by the individual number. Here the total array of twenty-six numbers divided into three groups is more easily discernible. From 70200 to his first break at 7020 the ratios are 5/5, 5/4, 5/3, and 5/2. This defines the first group. Then the ratios are 100 starting with 100/10 and going in continuous steps to 100/1. This defines the second group. The ratios in the third and last group start at 1300/1 and progress to 1300/12. Thus we see how the special ring numbers are listed as a unique group. (If we wished we could include the first group in the second group calculated as 100/100, 100/80, 100/60, and 100/40.)

In this mode of calculation the break of the groups at 7020 and 702, is far more dramatic.

The decimal grouping of the numbers is now clearly evident. Not only do we have 70200/7020 and 7020/702 but the entire array has a decimal definition: the first group of multiples of 5 (X 20), the second group of ratios of 100, and the third group of ratios of 1300 = 13 X 100.

Quite clearly there is a decimal purpose behind the numbers. They are not just random numbers in some indiscernible ranking, nor are they random numbers pointing to Long Count dates.

Why would the Designer create a table of numbers that carry meaning when translated to the decimal number system, but not be visible in the vigesimal number system?

These cannot be accidental numbers. The 10X relationship, the 100 and 1300 ratios, and the break points in the sequence, all speak to an intentional design. The Designer knew how the numbers would relate to one another in the decimal system. He did not merely throw numbers around. Which means he had an understanding of number systems in general, and placed the design now shown in the Codex in a manner that would become discernible by some later translator and someone familiar with the decimal number system. Then, scholars who later would see only the mythological aspects of the Codex, would produce mythological nonsense, such as Victoria and Harvey Bricker. Which means further that he anticipated some translator performing a manipulation that would open the meaning of the numbers - in the decimal system.

In other words, he anticipated such event.

How did the Designer of this number sequence reach his values?

He started with the modulo 54 most basic number, 702. (54 X 13 = 702.) He then incremented by modulo 54 numbers of 13 until he reached 7020. Actually, for 702 to reach 7020 required exactly 10 steps. That is, 54 X 13 had to repeat in additive value 10 times until it reached 540 X 13 = 7020. He listed those steps individually, which means he was showing us how he reached 7020 from 702. That was his major break point in the sequence. But now look at what was necessary to reach 70200 from 7020. The difference is 70200 - 7020 = 63180. This is 1170 by modulo 54. (1300 - 130 = 1170.) To reach this value did not require 100 X 13 but rather 90 X 13. 260 + 260 + 260 + 390 = (20 X 13) + (20 X 13) + (20 X 13) + (20 X 13) + (10 X 13) = 90 X 13 = 1170.

Every indication in the number sequence shows he understood the decimal number system, not merely the vigesimal number system. Further, for someone who was acute enough to follow his steps, he was showing us that he and that other person both understood the decimal number system, and could convert from vigesimal to the decimal. By his choices he converted that knowledge into a vigesimal presentation.

We can see that the "jump" in the data from 7020 to 28080, followed by 42120, occurs in the midst of a group of numbers at 71-2 Col 1-5, and hence had no intervening or missing numbers.

The data set is complete. There are no missing gaps. Regardless of the quality of the Dresden Codex in toto, or of Section 69-74, this set is self contained.

The first part of the data set is contained on Pages 70 and 71. In sequence it is at 70-2 (2 columns) and 71-2 (5 columns), and then exactly parallel 70-3 (2 columns) and 71-3 (5 columns). These reside one beneath the other to complete that part of the set. He then turned to Pages 71, 72, and 73, First Level, to provide that part of the data set marked by a special ring, and shown at the bottom of the set in Table 3, but in reverse order. By this arrangement he forces our attention to the meaning of the numbers, and the choice of 54 X 13 = 702. The modulo 54 comes about because of his choice of a number that would be meaningful in his display. The base rationale is 13. He then used a number, 54, that would give him the modulo results he desired.

(The base 13 shows in other tabular data from the Codex, to maintain a relationship throughout the Codex. I shall discuss.)

54 is not a magical number. Other numbers could have been used to bring about the decimal display. It was merely a useful number.

He could have carried the theme to higher numbers in the display, to reinforce recognition of the decimal relationship, but then the numbers would have become unwieldly and would have consumed precious space. If one leads to discovery of this relationship, further reinforcement seems unnecessary.

To say that he was a mathematical genius might be an understatement.

Table Three - Modulo 54

Page/Column Calendar Value

Prime Factors

Total

Ratio

Difference from preceding cell.

7200

360

20

1

Total

Total/54

Total/Data
70-1 and 71-1 Seven Columns Eroded Text. The remnant of this material indicates that it is different from and does not belong to the main tabular data. The columns all end in 0,0 with columns 5 through 7 containing digits 12, 13 and 14 in sequence respectively. They seem to have no effect on the tabular data below. They all end with 4 Ix.

70-2 C5-6 (54) 

9

15

0

0

70200

2 · 2 · 2 · 3 · 3 · 3 · 5 · 5 · 13

1300

5/5          

7

16

0

0

56160

2 · 2 · 2 · 2 · 2 · 3 · 3 · 3 · 5 · 13

1040

5/4 260

71-2 C1-5 (54)

5

17

0

0

42120

2 · 2 · 2 · 3 · 3 · 3 · 3 · 5 · 13

780

5/3 260

3

18

0

0

28080

2 · 2 · 2 · 2 · 3 · 3 · 3 · 5 · 13

520 5/2 260
 

19

9

0

7020

2 · 2 · 3 · 3 · 3 · 5 · 13

130 100/10 390
 

17

9

18

6318

2 · 3 · 3 · 3 · 3 · 3 · 13

117 100/9 13
 

15

10

16

5616

2 · 2 · 2 · 2 · 3 · 3 · 3 · 13

104 100/8 13

70-3 C5-6 (54)

 

13

11

14

4914

2 · 3 · 3 · 3 · 7 · 13

91

100/7 13
 

11

12

12

4212

2 · 2 · 3 · 3 · 3 · 3 · 13

78

100/6 13

71-3 C1-5 (54)

 

9

13

10

3510

2 · 3 · 3 · 3 · 5 · 13

65

100/5 13
 

7

14

8

2808

2 · 2 · 2 · 3 · 3 · 3 · 13

52

100/4 13
 

5

15

6

2106

2 · 3 · 3 · 3 · 3 · 13

39

100/3 13
 

3

16

4

1404

2 · 2 · 3 · 3 · 3 · 13

26

100/2 13
 

1

(17)

2

702

2 · 3 · 3 · 3 · 13

13

100/1 13
71-1 C6-7 (54)

 

  2 14 54 2 · 3 · 3 · 3 1 1300/1

 This data set has a special Ring marker at the bottom of the columns rather than the normal Tzolkin date dedication symbol, giving in sequence the numbers 11, 13, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7. These data do not have 13 as a prime factor. 

  5 8 108 2 · 2 · 3 · 3 · 3 2 1300/2

72-1 C1-7 (54)

 

8

2

162

2 · 3 · 3 · 3 · 3

3 1300/3
 

10

16

216

2 · 2 · 2 · 3 · 3 · 3

4 1300/4
 

13

10

270

2 · 3 · 3 · 3 · 5

5 1300/5
 

16

4

324

2 · 2 · 3 · 3 · 3 · 3

6 1300/6

1

0

18

378

2 · 3 · 3 · 3 · 7

7 1300/7

1

3

12

432

2 · 2 · 2 · 2 · 3 · 3 · 3

8 1300/8

1

6

6

486

2 · 3 · 3 · 3 · 3 · 3

9 1300/9

73-1 C1-3 (54)

1

9

0

540

2 · 2 · 3 · 3 · 3 · 5 10 1300/10

1

11

14

594

2 · 3 · 3 · 3 · 11 11 1300/11

1

14

(8)

648

2 · 2 · 2 · 3 · 3 · 3 · 3 12 1300/12

Modulo 65 Discussion

Why associate a Modulo 65 grid with a Modulo 54? In the Codex script they are intimately positioned beneath one another from Page 71 to 73. Hence, our attention seems to be forced to recognize them in that relationship.

Note that there is a major scribal error in the third number position at 71-4, column 3. The numbers are incorrect to fit the sequence scheme. This must have been recognized by a scribe who copied the Codex; he inserts other red-colored numbers to indicate his confusion, and perhaps attempt to correct the error. However he does not seem to understand the scheme of presentation because his substitute numbers do not fit. We should give special attention where this extraneous red color appears in other tabular matter in the Codex. Refer to my discussion on the scribal history of the Codex.

The sequence of steps here follows that of Modulo 54 pattern above. The sequence of the difference in the modulo steps has 224 (4), 112 (5), 140 (1), 28 (2), and 1 (27). Except for the last set, these are all exact multiples of a base 7 multiplier. 8 X 4 X 7 = 224, 5 X 4 x 7 = 140, 4 X 4 X 7 = 112, and 4 X 7 = 28.

On the other hand the ratios again are more revealing. 15/15 in steps of two down to 15/7, then in steps of one down to 15/2, then in steps of (15 x 112) divided by 84 to 56 to 28, and then by steps of one (15 X 112)/27 to the end at (15 x 112)/1.

We can see from the ratios that the array is divided into four parts: 224 (2 X 112), (Page 71-4 Columns 1-5 complete within the Page); 112 down to 71-5 C1-3; 28 down to 71-5 C5-8 (112/4); and finally in steps of one to the end. (Here the presentation is not clearly delineated in sequence, but moves back to earlier Pages to continue.)

What do all these numbers mean?

I have been unable to find cyclic patterns in nature to explain these numbers, either in the sun, moon, or planets visible to the Mayan eye, or in other natural phenomena. This leads me to believe the numbers are strictly mathematical.

In the Modulo 54 we found a clear decimal relationship. Here the numbers seem to be 15 and 112.

Again, as in the Modulo 54 set, the data appear to be self-contained, with no extraneous or missing numbers.

 

Table Four - Modulo 65

Page/Column Calendar Value

Prime Factors

Total

Ratio

Difference from preceding cell.

7200

360

20

1

Total

Total/65

Total/Data

71-4 C1-5 (65)

15

3

6

0

109200

2 · 2 · 2 · 2 · 3 · 5 · 5 · 7 · 13

1680

15/15  

13

2

16

0

94640

2 · 2 · 2 · 2 · 5 · 7 · 13 · 13

1456

15/13 224

10

2

4

0

72800

2 · 2 · 2 · 2 · 2 · 5 · 5 · 7 · 13

*1120

   
*This column has two red #12 inserted in the 2nd and 4th positions, with no #0 at the bottom. Reconstructing by the pattern of the numbers yields the line below.
11 2 8 0 80080

2 · 2 · 2 · 2 · 5 · 7 · 11 · 13

1232 15/11 224
9 2 0 0 65520

2 · 2 · 2 · 2 · 3 · 3 · 5 · 7 · 13

1008 15/9 224
7 1 10 0

50960

2 · 2 · 2 · 2 · 5 · 7 · 7 · 13

784

15/7 224

70-5 C5-6 (65)

6

1

6

0

43680

2 · 2 · 2 · 2 · 2 · 3 · 5 · 7 · 13

672

15/6 112

5

1

2

0

36400

2 · 2 · 2 · 2 · 5 · 5 · 7 · 13

560

15/5 112

71-5 C1-7 (65)

4

0

16

0

29120

2 · 2 · 2 · 2 · 2 · 2 · 5 · 7 · 13

448

15/4 112

3

0

12

0

21840

2 · 2 · 2 · 2 · 3 · 5 · 7 · 13

336

15/3 112

2

0

8

0

14560

2 · 2 · 2 · 2 · 2 · 5 · 7 · 13

224

15/2 112
 

15

3

0

5460

2 · 2 · 3 · 5 · 7 · 13

84

(15X112)/84 140
 

10

2

0

3640

2 · 2 · 2 · 5 · 7 · 13

56

(15X112)/56 28
  5 1 0 1820

2 · 2 · 5 · 7 · 13

28 (15X112)/28 28
  4 15 15 1755

3 · 3 · 3 · 5 · 13

27 (15X112)/27 1

72-3 C1-7 (65)

 

4

12

10

1690

2 · 5 · 13 · 13

26

(15X112)/26 1
 

4

9

5

1625

5 · 5 · 5 · 13

25

(15X112)/25 1
 

4

6

0

1560

2 · 2 · 2 · 3 · 5 · 13

24

(15X112)/24 1
 

4

2

15

1495

5 · 13 · 23

23

(15X112)/23 1
 

3

17

10

1430

2 · 5 · 11 · 13

22

(15X112)/22 1
 

3

14

5

1365

3 · 5 · 7 · 13

21

(15X112)/21 1
 

3

11

0

1300

2 · 2 · 5 · 5 · 13

20

(15X112)/20 1

73-3 C1-5 (65)

 

3

7

15

1235

5 · 13 · 19

19

(15X112)/19 1
 

3

4

10

1170

2 · 3 · 3 · 5 · 13

18

(15X112)/18 1
 

3

1

5

1105

5 · 13 · 17

17

(15X112)/17 1
 

2

16

0

1040

2 · 2 · 2 · 2 · 5 · 13

16

(15X112)/16 1
 

2

12

15

975

3 · 5 · 5 · 13

15

(15X112)/15 1
71-2 C6-7 (65)   2 9 10 910 2 · 5 · 7 · 13 14 (15X112)/14 1
  2 6 5 845 5 · 13 · 13 13 (15X112)/13 1

72-2 C1-7 (65)

 

2

3

0

780

2 · 2 · 3 · 5 · 13

12

(15X112)/12 1
 

1

17

15

715

5 · 11 · 13

11

(15X112)/11 1
 

1

14

10

650

2 · 5 · 5 · 13

10

(15X112)/10 1
 

1

11

5

585

3 · 3 · 5 · 13

9

(15X112)/9 1
 

1

8

0

520

2 · 2 · 2 · 5 · 13

8

(15X112)/8 1
 

1

4

15

455

5 · 7 · 13

7

(15X112)/7 1
   

19

10

390

2 · 3 · 5 · 13

6

(15X112)/6 1

73-2 C1-6 (65)

   

16

5

325

5 · 5 · 13

5

(15X112)/5 1
   

13

0

260

2 · 2 · 5 · 13

4

(15X112)/4 1
   

9

15

195

3 · 5 · 13

3

(15X112)/3 1
   

6

10

130

2 · 5 · 13

2

(15X112)/2 1
   

3

5

65

5 · 13

1

(15X112)/1 1
 

Summary Discussion

The two data sets offered in the Modulo 54 and Modulo 65 are both self-contained without missing or extraneous numbers. The arrangement of the tabular matter on the Pages and in the columns as continuous numbers clearly shows this. The data are strongly mathematical in nature and do not reflect natural cycles.

Two errors in the Modulo 54 material are easily corrected. The one set of data in the Modulo 65 material is obviously badly mistaken, and can be replaced with a high degree of certainty. (We should mention that the scribe's erroneous material follows through on the scheme of the prime number 13. However, this may have been totally accidental.)

I have been unable to find reasonable constructs for 70-4, C 5-6 and 73-1, C 4-5 that fit within the numerical scheme indicated here. I do not include them in the above tabulations. Columns 4-5 of Page 73-1 offer oddball values that do not fit within the scheme of the other data. They have prime factors of 2 X 41737 and 2 X 2 X 19 X 457 respectively. Similarly with other odd numbers that appear on these Pages.

Hence we must conclude that the other (religious or weather) information given in the Pages 60 to 74 do not reflect upon, nor modify these two sets. That other information may have been "space fillers" placed there by the scribe who copied these data from another source. The references to the Rain God Chaak or other religious mentions simply do not seem compatible with the mathematical presentation and strongly suggest that the scribe was blending two different kinds of information. Or he was offering a context for the mathematical information that was his own unique devising, without fully understanding the material. We saw his lack of understanding in the third line of the Modulo 65 material.

How does one extract from vigesimal data decimal information? Not only do we have decimal information clearly present in the Modulo 54 data set, we also have it in the Modulo 65 data set. The ratio of the numbers, the presence of the prime number 13, and the multipliers all show a decimal orientation.

The evidence is positive proof that an originator, not the scribe who prepared this document, knew the decimal number system, and devised a scheme in the vigesimal number system to convey his knowledge. The scribe who prepared the document did not know the decimal number system; the evidence shows that he was merely copying from some other source.

In both the Modulo 54 and Modulo 65 the Designer presented a list of numbers necessary to show his knowledge.

  # of groups Value #1 Decrement Value #2 Decrement Value #3 Decrement Value #4 Decrement Total # of Steps
Modulo 54 3 4 - 260   10 - 13 12 - 1 26
Modulo 65 4 5 - 224 5 - 112 3 - 28 27 - 1 40

 

The Design Rules

We are in a position now to define the rules that guided the Designer.

The scheme for both designs is from the bottom up, not the top down. (The base for Modulo 54 is at 702, going both ways, up and down.) Yet the numeric presentation is from the top down, from left to right on the pages of the Codex.

Modulo 54

Choose a Modulo number 54 = 2 X 3 X 3 X 3.

Chose a prime number 13.

Chose a base number 54 X 13 = 702.

Increment in 10 steps of 702 to 7020 (2X 3X 4X etc.) (This is a multiplicative operation.)

[The multiplicative increment of 10 steps above 702 is divided into 2X, then 3X, then 4X(2 X 2), then 5X, then 6X(2 X 3), etc.]

Next increment in 4 larger (4X 6X 8X 10X) steps to 70200. (Increase the multiplier in each step to reach the higher number without showing all the possible intervening incremental steps.)

Note that the Modulo 54 highest number is 1300, or 13 X 100.

The decrement from the highest number is by modulo 13 for 260 = 13 X 20, to the dividing number of 7020, then the decrement is by steps of 13 to the base number 702. In this manner the Designer reinforced our attention to the number 13.

These increments always have 13 as a multiplier because they are multiples of the base number.  That is the method the Designer used to preserve the prime number 13.

Since 702 was his base number he could not decrement to numerical values below 54 X 13 and still preserve both the Modulo 54 and the 13. Therefore, he sacrificed 13 to the Modulo 54. Since 702 was his starting point for the design he went in the reverse direction -- starting at 54 to show us the minimum Modulo 54, and then increasing in 12 steps, the number needed to reach 702. In this manner he bifolded the presentation, to emphasize his starting point of 702. This also showed that there were only 12 possible whole-number steps to reach 702 from 54 with Modulo 54. At the next step, 13, he once again reinforces our attention to that number.

Modulo 65

Chose a Modulo number 65 = 5 X 13

This Modulo Number includes the prime number 13 by definition.

The base number is the Modulo number.

Increment in single steps from 65 to 1820. Dividing by Modulo 65 the resulting numbers run from 1 to 28 (this is an additive operation). Each step carries with it the prime number 13.

28 has 7 as a prime number: 4 X 7 = 28.

1820 now has two prime numbers, 13 and 7.

Increment in multiplier steps of 2X -- three stages --- 1820, 3640, and 5460 for Modulo 65 numbers: 28, 56, 84. When reduced to their arithmetic components, these numbers contain the Modulo 65 and (1) X 4 X 7,  (2) X 4 X 7,  (3) X 4 X 7.

Then jump to 8 X 4 X 7, and increment by 4's: 12 X 4 X 7, 16, 20, 24, and 28 X 4 X 7 for 14560 to 50960 respectively.

Then jump to 36 X 4 X 7, and increment by 8's: 44, 52 to 60 X 4 x 7 for 65520 to 109200 respectively. Note that 36 is 8 above 28, the preceding level.

The scheme of increase in the numbers is 1+, 28 places, 2X, 3 places, 4X, 6 places, and 8X, 4 places. This shows a mathematical progressive series of 1, 2, 4, and 8.

Curiosities

On the Modulo 54 the vigesimal numbers are in order, descending for the 7200's in steps of two: 15, 13, 11, 9 and 7. The 360's are ascending in steps of two for the same four numbers: 15, 16, 17, 19.This scheme runs from 70200 of the first division to 7020 of the second division.

The second division numbers run in steps of two in descending order of the 360's: 19, 17 . . . 3, 1 to 702, the base number.

For the Modulo 65 the 7200's run in steps of two in descending order from 15 to 7. Thereafter they run in steps of one from 7 to 2. These mark the jumps in difference from the preceding cell of 224 and 112 respectively.