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Dresden Codex

Section in Pages 43b - 45b

Pages 43 to 45 of the Dresden Codex are each divided vertically by horizontal red lines into three sections. In the mid-section, understood as 43b, 44b, and 45b, is a Table that shows multiples of the number 78. This Table is read from left to right across the first two pages. The third page is dedicated to an augury or divinatory presentation. Various attempt have been made to relate the Table to the orbital periodicity of Mars, or to natural phenomena, such as rainfall, but with serious scholarly debate on all suggestions.

Starting in 1924 R. W. Willson proposed that the Table offers data to permit the tracking of the planet Mars. This idea was picked up by Victoria and Harvey Bricker, but objected to early on by J. E. Thompson, with support from Bruce Love. Thompson, Love, et. al. thought the data might be related to rain fall, or seasonal weather phenomena. The idea of a connection to the planet Mars is based on the value of the numbers found in the Table, with many showing multiples of 780, while the synodic period of Mars is 779.80875 days. However, this division is not possible with the last nine positions of the Table, and the authors of this theory do not offer an explanation. Other difficulties exist with this theory. Division by 780 leaves five positions with an uneven number, a 1/3 fraction in two cases, a 2/3 fraction in two cases, and 1/13 fraction in the other case. Except for a unique one and the last nine the numbers are also evenly divisible by 260, which is 1/3 of 780. Another difficulty with the Martian theory is that 780 is the 10X multiplier of 78, which divides into almost all of the numbers. Further, there is a descending order of the quotients which follows that of the descending order of the numbers as one follows the Table. Translating this information into consistent theories about Mars or about the weather becomes an inordinate intellectual burden.

The web site, http://www.mayacodices.org, dedicated to a translation of the Maya Codices, following the Brickers, attempts to show the meaning of the text accompanying the four beasts on pages 44 and 45 but again without success. The translation is too fragmentary and uncertain to make sense. Speculation continues in the face of the lack of precise understanding of the text.

Explicit tabulation of the columns of numbers does not support either of these theories. The various authorities all seem to ignore the mathematical nature of the data. The data are simply a mathematical table, in descending order, and with properties that reinforce a mathematical theory.

An example of negligent treatment of the numbers is in the fact that 78 has a factor of 13 as a prime number. 78 = 2 X 3 X 13. This means that 260 contains 13 as a prime number, and hence 780 does also. None of the authorities discuss this fact.

Numbers that are tabulated into a mathematical format, and that contain 13 as a prime number, are shown as well in Sections in Pages 69-74 of the Dresden Codex. This Table cannot be treated with intellectual honesty without consideration of those other Tables, and placing it into a consistent context.

Here is a reproduction of Part B of the three pages. Parts "A" and "C" of these pages are devoted to non-mathematical information.

Page 43 B

Page 44 B
Page 45 B

Each of the parts of the Table are divided into two sections, top and bottom, with 5 columns each on page 43b and 6 columns each on page 44b. The first column of six on page 43b is taken up with a Long Count date. Every column, including the LC,  is ended by a religious dedicatory symbol, always 3 Lamat, except the following: the last set of numbers on page 44b, running from #9 to #1 at the end, (#10 falls between #5 and #6), has the following dedication symbols:

Refer to the glyph table on the left.

Clearly there is a regular pattern to the order of the number columns in the Codex and the order of the numbers in the glyph table. #9 (3 Oc) is two places in the table before #8 (3 Eb), #8 is two places in the table before #7 (3 Ix), and so on round the glyph table for every second location of 10 positions out of a possible 20. Only #10 is out of order in the Codex Table, but still fits within the glyph order scheme. #10 uses 3 Lamat, the same as the other number columns on 43b and 44b.

This reason for this regular order of glyph assignments, jumping every second one, assigned to the single digits of multiples of 78, but not to higher multiple digits, is not evident to us. We see this unusual assignment to low-order digits on the Modulo 54 Table also. Refer to that paper. Since all the other columns end in 3 Lamat, and since the Long Count date has the same 3 Lamat dedication, this method may be a routine adjusted to single-digit numbers obeyed by the scribe(s) to meet their rules of scribal notation. Perhaps the reason is religious. Hence, all other supposed date dedicatory assignments in the Dresden Codex come under the same rule but with a meaning which escapes us. It seems that these are not real date assignments, per se, as has been proposed by the several authorities, but merely religious dedicatory date assignments.

See especially, Thompson, John Eric Sidney A Commentary on the Dresden Codex; a Maya Hieroglyphic Book , Philadelphia, American Philosophical Society, 1972.

Here is a tabulation for quick comparison.

Comparison of Modulo 54 and Modulo 78 Single Digit Numerical and Calendar Assignments
Modulo 54 1 (54) 2 (108) 3 (162) 4 (216) 5 (270) 6 (324) 7 (378) 8 (432) 9 (486) 10 (540) 11 (594) 12 (648)
11 13 2 4 6 8 10 12 1 3 5 7
Modulo 78 1 (78) 2 (156) 3 (234) 4 (312) 5 (390) 6 (468) 7 (546) 8 (624) 9 (702) 10 (780)    
3 Cimi 3 Kan 3 Ik 3 Ahau 3 Etznab 3 Cib 3 Ix 3 Eb 3 Oc 3 Lamat    

At the beginning of page 43, after a Initial Series dedication, is the Long Count date, 9.19.8.15.0, with an addendum period appended below the date, 17.12. Again, the date has an introductory glyph with a value of 3 Lamat. Another introductory glyph relates the Long Count date and the following number Table to the four "beasts" that appear on pages 44b and 45b.  All modern authors seem to believe that the 3 Lamat provides a date on which hangs the other calculations, but obviously this explanation is simplistic and not real. The ubiquitous presence of 3 Lamat in most of the columns of the Table suggests it is a religious dedication symbol, and not a calendar date. Furthermore, all Tzolkin symbols are assigned the number 3, throughout the Table. Again this appears as a scribal dedication device, without rationale apparent to us.

My calculated date for 9.19.8.15.0 is March 11, 819. (Other authors assigns a date if March 26, 818.) The Tzolkin date is 4 Ahau. The Haab date is 13 Zip. By some twist I do not understand Thompson adds the 17.12 to 13.0.0.0.0 4 Ahau 8 Kumku to obtain a date of 9.19.7.15.8. Actually, this is merely the subtraction of the numbers.

Following is a Table of the numbers as shown in the Codex with Page assignments indicated.

Table Pages 43b-44b
As shown in Dresden Codex
Page/Column Calendar Value Mars 780 Theory Mars 260 Theory

14400

7200

360

20

1

Total

43B-1

 

15

3

6

0

109200

140 420
 

18

4

0

0

131040

168 504
 

10

2

4

0

72800

93 1/3 280

 

 

9

13

6

0

69600

1160/13 3480/13

 

 

4

5

17

0

30940

39 2/3 119

43B-2

1

1

0

6

0

151320

194 582
 

10

15

0

3900

5 15

 

 

9

7

0

3380

4 1/3 13
 

6

9

0

2340

3 9
 

4

6

0

1560

2 6

44B-1

 

2

3

6

0

15600

20 60
 

1

16

2

0

13000

16 2/3 50
 

1

17

2

702

  27/10
 

1

13

4

624

  12/5
 

1

9

6

546

  21/10
 

1

5

8

468

  9/5

44B-2

 

2

3

0

780

1 3
 

1

1

10

390

  3/2
 

15

12

312

  6/5
 

11

14

234

   
 

7

16

156

   
 

3

18

78

   

Obviously, the scribe did not maintain a consistent pattern in his arrangement of the numbers; they are not in strict numeric order. Thompson rearranged them but in increasing order instead of decreasing, as the scribe mainly presented them. See Thompson's Figure 64. As I studied the numbers I saw that part of the Table was fractured in places, had numbers missing, had errors, and sometimes could not easily be followed as one could in the Modulo 54 and Modulo 65 Tables. Here I rearranged the Table to bring it into the same order as the scribe, and ranking the numbers as Thompson had done. I then created substitute lines, or modified some lines, to more readily see the pattern of the numbers. The pink lines below show them as they appear originally. The grey lines show a recreation or additional material to bring out the intention of the original Designer (not the scribe) of the Table.

Thompson noted two errors: 9.13.6.0 should have been written 8.13.6.0. 4.6.12.0 was written as 4.5.17.0, 260 days earlier. I shall discuss.

Division by 260 fits all numbers within the calculation range, correct or not, except for 69,600. This implies that this particular number is mistaken. Compare to division with 780, the number used by so many mistaken theories.

I now analytically review all entries of the Table.

Lines 1 & 2

Lines 3 & 4

Line 5

Lines 6 & 7

Lines 8 & 9

Lines 10 & 11

Line 12

Lines 13 & 14

Line 15

Lines 16 & 17

Lines 18 to 29

Table Pages 43b-44b
Rearranged

Line #

Page/
Column
Calendar Value

Prime Factors

Total Ratio

Difference from preceding cell.

144000

7200

360

20

1

Total

Total/78 Total/260 Total/Data
1

43B-2

1

1

0

6

0

151320

2 · 2 · 2 · 3 · 5 · 13 · 97

1940 582  
2   1 1 13 6 0

156000

2 · 2 · 2 · 2 · 2 · 3 · 5 · 5 · 5 · 13 2000 600 1/1  
3     18 8 6 0 132600

2 · 2 · 2 · 3 · 5 · 5 · 13 · 17

1700 510 2000/1700 300
4 43B-1  

18

4

0

0

131040

2 · 2 · 2 · 2 · 2 · 3 · 3 · 5 · 7 · 13

1680 504 2000/1680  
5

 

 

15

3

6

0

109200

2 · 2 · 2 · 2 · 3 · 5 · 5 · 7 · 13

1400 420 2000/1400 300
6     11 18 6 0 85800

2 · 2 · 2 · 3 · 5 · 5 · 11 · 13

1100 330   300
7  

10

2

4

0

72800

2 · 2 · 2 · 2 · 2 · 5 · 5 · 7 · 13

933 1/3 280 15/7  
8

 

 

9

13

6

0

69600

2 · 2 · 2 · 2 · 2 · 3 · 5 · 5 · 29

11600/13 3480/13 65/29  
9  

8

13

6

0

62400

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

800 240 2000/800 300
10  

4

6

12

0

31200

2 · 2 · 2 · 2 · 2 · 3 · 5 · 5 · 13

400 120 2000/400 400
11

 

 

4

5

17

0

30940

2 · 2 · 5 · 7 · 13 · 17

396 2/3 119 600/119  
12

44B-1

 

2

3

6

0

15600

2 · 2 · 2 · 2 · 3 · 5 · 5 · 13

200 60 2000/200 200
13  

1

16

2

0

13000

2 · 2 · 2 · 5 · 5 · 5 · 13

166 2/3 50 12/1  
14  

1

1

12

0

7800

2 · 2 · 2 · 3 · 5 · 5 · 13

100 30 2000/100 100
15 43B-2  

10

15

0

3900

2 · 2 · 3 · 5 · 5 · 13

50 15 2000/50 50
16

 

 

9

7

0

3380

2 · 2 · 5 · 13 · 13

43 1/3 13 600/13  
17  

8

12

0

3120

2 · 2 · 2 · 2 · 3 · 5 · 13

40 12 2000/40 10
18  

6

9

0

2340

2 · 2 · 3 · 3 · 5 · 13

30 9 2000/30 10
19  

4

6

0

1560

2 · 2 · 2 · 3 · 5 · 13

20 6 2000/20 10
20

44B-2

 

2

3

0

780

2 · 2 · 3 · 5 · 13

10 3 2000/10 10
21 44B-1  

1

17

2

702

2 · 3 · 3 · 3 · 13

9 2 + 7/10 2000/9 1
22  

1

13

4

624

2 · 2 · 2 · 2 · 3 · 13

8 2 + 2/5 2000/8 1
23  

1

9

6

546

2 · 3 · 7 · 13

7 2 + 1/10 2000/7 1
24  

1

5

8

468

2 · 2 · 3 · 3 · 13

6 2 + 1/5 2000/6 1
25 44B-2  

1

1

10

390

2 · 3 · 5 · 13

5 3 2000/5 1
26  

15

12

312

2 · 2 · 2 · 3 · 13

4 1 + 1/2 2000/4 1
27  

11

14

234

2 · 3 · 3 · 13

3 1 + 1/5 2000/3 1
28  

7

16

156

2 · 2 · 3 · 13

2   2000/2 1
29  

3

18

78

2 · 3 · 13

1   2000/1 1

The Design Rules

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

The scheme for the design is from the bottom up, not the top down. Yet the numeric presentation (after corrections) is from the top down, from left to right on the pages of the Codex. This is the same scheme as Modulo 54 and Modulo 65.

Choose a Modulo number 78 = 2 X 3 X 13.

This includes the prime number 13.

All subsequent mathematical operations include this prime number.

The base number is the Modulo number.

Increment in linear steps of 1 from 1 to 10. This is a series additive operation. (10 steps.)

Increment in linear steps of 10 from 10 to 50. This is a series multiplicative operation. (4 steps.) (Note that the position of 10 can fit within both series.)

Increment in steps of progressive mathematical series of double previous value from 50 to 800. (4 steps.) This shows a mathematical progressive series of 1, 2, 4, and 8, similar to that found in Modulo 65 series. Note also that if the Designer had carried this one step more the double of 800 would have been 1600, which would have interfered in his higher numbers.

Increment in linear steps of 300 from 800 to 2000. (4 steps.) Note that the Designer ended the Table on the round number of 2,000.

There seems to have been a definite intent to limit each series to 4 steps above the additive group.

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