Pythagorean Triples Level II--The Side Angles
by Richard Brown
First came the 3,4,5 right triangle of antiquity, and of course, the ancient cuneiform tablet Plimpton 322. Then the concept of filling angle space. This page looks at "angle filling" using Pythagorean triples.
First, lets look at the Plimton 322 Triples. Again, refer back to Dr. David Joyce's web page referenced in the previous section(View web page). Also of interest is a picture of the original tablet (Just_click_here). Exhbit I. below is a list of the Pythagorean triples used in the original tablet but formatted like the the other tables in this series and includes angles in degrees. Angle measure was not fully developed for almost another 2000 years!
Exhibit I. The Primitive Pythagorean Triples of the Plimpton 322 Cuneiform Tablet (Circa 1800BC)
| Number | Generators u,v | Side A | Side B | Diagonal | Angle,Change | |
|---|
| 26 | 5 | 12 | 119 | 120 |
169 | 44.7603 | -0.2397 |
| 768 | 27 | 64 | 3367 | 3456 | 4825 | 44.2527 | -0.5076 |
| 1058 | 32 | 75 | 4601 | 4800 | 6649 | 43.7873 | -0.4654 |
| 2951 | 54 | 125 | 12709 | 13500 | 18541 | 43.2713 | -0.5160 |
| 16 | 4 | 9 | 65 | 72 | 97 | 42.0750 | -1.1963 |
| 75 | 9 | 20 | 319 | 360 | 481 | 41.5445 | -0.5305 |
| 563 | 25 | 54 | 2291 | 2700 | 3541 | 40.3152 | -1.2293 |
| 199 | 15 | 32 | 799 | 960 | 1249 | 39.7703 | -0.5449 |
| 122 | 12 | 25 | 481 | 600 | 769 | 38.7180 | -1.0523 |
| 1299 | 40 | 81 | 4961 | 6480 | 8161 | 37.4372 | -1.2808 |
| 1 | 1 | 2 | 3 | 4 | 5 | 36.8699 | -0.5673 |
| 466 | 25 | 48 | 1679 | 2400 | 2929 | 34.9760 | -1.8939 |
| 46 | 8 | 15 | 161 | 240 | 289 | 33.8550 | -1.1210 |
| 514 | 27 | 50 | 1771 | 2700 | 3229 | 33.2619 | -0.5931 |
| 8 | 2 | 7 | 45 | 28 | 53 | 31.8908 | -1.3711 |
There are many interesting points to make about this table. First, the triples are ordered from angles of about 45 degrees
to about 30 degrees. Since a correspondence was not developed between degree measure and triples, the tablet maker used the formula (diagonal/side)^2 to order the set.
Second, the
use of that angle measure meant the angle numbers would all be between 1 and 2 for angles between 0 and 45 degrees. Since, it is believed we don't have the full tablet, other entries may have completed the table to zero, we will give a proposed completed table below.
The third fact is that the larger side had to be composed solely of the prime factors of 60(2,3, and 5) so that the fractions would terminate in base 60 arithmetic and not be repeating fractions. That is believed to be the reason why such a wide range of triples were used in the tablet with diagonals ranging up to 18,541 or about the first 3,000 primitive Pythagorean triples sorted by diagonal side.
Completing Plimpton 322--Ideas of David Joyce and John Conway and Richard Guy
As mentioned above, it is believed that the Plimpton 322 Tablet is not complete, in fact the tablet appears boken off below the last entry. John Conway and Richard Guy in their "Book of Numbers" proposed a completion of the tablet, that is shown below in the same format as Exhibit I. Also included in Exhibit II as the first entry is David Joyce's addition to the tablet to fill the angle space below the (3,4,5)triangle.
Exhibit II. Completion of Plimpton 322?
| Number | Generators u,v | Side A | Side B | Diagonal | Angle,Change |
|---|
| 3138 | 64 | 125 | 11,529 | 16,000 | 19,721 | 35.7751 | -1.0948 |
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| 54 | 9 | 16 | 175 | 288 | 337 | 31.2844 | -0.6064 |
| 3 | 1 | 4 | 15 | 8 | 17 | 28.0725 | -3.2119 |
| 1442 | 50 | 81 | 4061 | 8100 | 9061 | 26.6273 | -1.4452 |
| 15 | 5 | 8 | 39 | 80 | 89 | 25.9892 | -0.6381 |
| 140 | 16 | 25 | 369 | 800 | 881 | 24.7615 | -1.2277 |
| 2 | 2 | 3 | 5 | 12 | 13 | 22.6198 | -2.1417 |
| 974 | 45 | 64 | 2071 | 5760 | 6121 | 19.7760 | -2.8438 |
| 485 | 32 | 45 | 1001 | 2880 | 3049 | 19.1659 | -0.6101 |
| 151 | 18 | 25 | 301 | 900 | 949 | 18.4922 | -0.6737 |
| 180 | 20 | 27 | 329 | 1080 | 1129 | 16.9423 | -1.5499 |
| 4 | 3 | 4 | 7 | 24 | 25 | 16.2602 | -0.6821 |
| 7 | 4 | 5 | 9 | 40 | 41 | 12.6803 | -3.5799 |
| 9 | 5 | 6 | 11 | 60 | 61 | 10.3888 | -2.2915 |
| 279 | 27 | 32 | 295 | 1728 | 1753 | 9.6880 | -0.7008 |
| 23 | 8 | 9 | 17 | 144 | 145 | 6.7329 | -2.9551 |
| 28 | 9 | 10 | 19 | 180 | 181 | 6.0256 | -0.7073 |
| 106 | 1 | 26 | 675 | 52 | 677 | 4.4052 | -1.6204 |
| 76 | 15 | 16 | 31 | 480 | 481 | 3.6952 | -0.7100 |
| 19 | 24 | 25 | 49 | 1200 | 1201 | 2.3383 | -1.3569 |
The Full Plimpton 322 Tablet
Listed below are the original values of the Plimpton 322 tablet from Exhibit I, the proposed values of Conway and Guy from Exhibit II, and the missing values added by the author. The author's values are in bold, the original values are those above the space in the table, Conway and Guy's values are those below the the space in the table not bolded.
The entries are sequenced by row number which matches the numbering of the Plimpton tablet for the first 15 entries. Dr. Joyce's entry was not included because it is assumed that the tablet is composed of entries from the first 3,000 triples only.
The single asterisk delineates that the other side of the triangle is a 2,3,5 composite. The double asterisk denotes that
the side A or "odd side" is the larger of the two sides and hence the divisor in the (diagonal/side)^2 equation. Of the 43
entries 39 are based on "even side" divisors, and only 4 on "odd
side" divisors.
Exhibit III. The Author's Completed Plimpton 322 Tablet
| Number | Generators u,v | Side A
| Side B | Diagonal | Angle, Change | row |
|---|
| 26 | 5 | 12 | 119 | 120 | 169 | 44.7603 | -0.2397 | 1 |
| 768 | 27 | 64 | 3367 | 3456 | 4825 | 44.2527 | -0.5076 | 2 |
| 1058 | 32 | 75 | 4601 | 4800 | 6649 | 43.7873 | -0.4654 | 3 |
| 2951 | 54 | 125 | 12709 | 13500 | 18541 | 43.2713 | -0.5160 | 4 |
| 16 | 4 | 9 | 65 | 72 | 97 | 42.0750 | -1.1963 | 5 |
| 75 | 9 | 20 | 319 | 360 | 481 | 41.5545 | -0.5305 | 6 |
| 563 | 25 | 54 | 2291 | 2700 | 3541 | 40.3152 | -1.2293 | 7 |
| 199 | 15 | 32 | 799 | 960 | 1249 | 39.7703 | -0.5449 | 8 |
| 122 | 12 | 25 | 481 | 600 | 769 | 38.7180 | -1.0523 | 9 |
| 1299 | 40 | 81 | 4961 | 6480 | 8161 | 37.4372 | -1.2808 | 10 |
| 1 | 1 | 2 | 3* | 4 | 5 | 36.8699 | -0.5673 | 11 |
| 466 | 25 | 48 | 1679 | 2400 | 2929 | 34.9760 | -1.8939 | 12 |
| 46 | 8 | 15 | 161 | 240 | 289 | 33.8550 | -1.1210 | 13 |
| 514 | 27 | 50 | 1771 | 2700 | 3229 | 33.2619 | -0.5931 | 14 |
| 8 | 2 | 7 | 45** | 28 | 53 | 31.8908 | -1.3711 | 15 |
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| 54 | 9 | 16 | 175 | 288 | 337 | 31.2845 | -0.6063 | 16 |
| 157 | 16 | 27 | 473 | 864 | 985 | 28.6987 | -2.5858 | 17 |
| 3 | 1 | 4 | 15** | 8* | 17 | 28.0725 | -0.6262 | 18 |
| 1442 | 50 | 81 | 4061 | 8100 | 9061 | 26.6273 | -1.4452 | 19 |
| 15 | 5 | 8 | 39 | 80 | 89 | 25.9892 | -0.6381 | 20 |
| 140 | 16 | 25 | 369 | 800 | 881 | 24.7615 | -1.2277 | 21 |
| 1766 | 22 | 103 | 10125** | 4532 | 11093 | 24.1135 | -0.6480 | 22 |
| 2 | 2 | 3 | 5* | 12 | 13 | 22.6199 | -1.4936 | 23 |
| 370 | 27 | 40 | 871 | 2160 | 2329 | 21.9613 | -0.6586 | 24 |
| 307 | 25 | 36 | 671 | 1800 | 1921 | 20.4443 | -1.5170 | 25 |
| 974 | 45 | 64 | 2071 | 5760 | 6121 | 19.7760 | -0.6683 | 26 |
| 485 | 32 | 45 | 1001 | 2880 | 3049 | 19.1659 | -0.6101 | 27 |
| 151 | 18 | 25 | 301 | 900 | 949 | 18.4922 | -0.6737 | 28 |
| 180 | 20 | 27 | 329 | 1080 | 1129 | 16.9423 | -1.5499 | 29 |
| 4 | 3 | 4 | 7 | 24 | 25 | 16.2602 | -.06821 | 30 |
| 262 | 25 | 32 | 399 | 1600 | 1649 | 14.0025 | -2.2577 | 31 |
| 1696 | 64 | 81 | 2465 | 10368 | 10657 | 13.3738 | -0.6287 | 32 |
| 7 | 4 | 5 | 9* | 40 | 41 | 12.6804 | -06934 | 33 |
| 2636 | 81 | 100 | 3439 | 16200 | 16561 | 11.9851 | -0.6952 | 34 |
| 9 | 5 | 6 | 11 | 60 | 61 | 10.3889 | -1.5962 | 35 |
| 279 | 27 | 32 | 295 | 1728 | 1753 | 9.6880 | -0.7009 | 36 |
| 1547 | 64 | 75 | 1529 | 9600 | 9721 | 9.0495 | -0.6385 | 37 |
| 23 | 8 | 9 | 17 | 144 | 145 | 6.7329 | -2.3166 | 38 |
| 28 | 9 | 10 | 19 | 180 | 181 | 6.0256 | -0.7073 | 39 |
| 106 | 1 | 26 | 675** | 52 | 677 | 4.4052 | -1.6204 | 40 |
| 76 | 15 | 16 | 31 | 480 | 481 | 3.6952 | -0.7100 | 41 |
| 19 | 24 | 25 | 49 | 1200 | 1201 | 2.3383 | -1.3569 | 42 |
| 2062 | 80 | 81 | 161 | 12960 | 12961 | 0.7117 | -1.6266 | 43 |
Implications of the "Completed" Plimpton 322 Table
Of course we don't know if this is what the completed table would have included. We are basing this completion on four facts: first, the table included entries from the first 3,000
primitive Pythagorean triples only. That all the permissible entries from the first 3,000 triples were included in the Plimpton tablet and none others is bolsterd by the fact that Dr. Joyce's triple which is slightly over number 3,000 (or 50:00 in base 60 arithmetic)was excluded from the table.
Second, the original table is so consistent, the ordering of the
triples is correct, and the table appears broken off right below the 15th row.
Third, rows 11 and 15 are of particular interest. Row 11 is
the only triple in the original table where both sides are 2,3,5
composites. Row 15 is the only triple in the original table that uses the "odd side" divisor in the equation. Both of these rows use a diffrent formatting than the other thirteen. The other thirteen triples all use even side divisors and the odd side is not a 2,3,5 composite.
Fourth and finally, substantially increasing the size of the source table of triples does not significantly increase the
number of triples in the Plimpton table. Increasing the table
size to include the first 3,600 triples(or 1:00:00 in base 60 arithmetic) increases the number of entries in the Plimpton table by only three! Forty three entries would fit on one tablet and be an ordered summary of fifty or more source tablets of the full list of the first 3,000 primitive triples.
The full tablet is a well-ordered list of well-defined angles, which could be used for other purposes. The table could be used
to organize the full set of 3,000 triples by angle measure by
using approximate angle measures to find where each triple would fit in the Plimpton tablet. The 45 degree angle would be at the
top of the table, the 30 degree angle would be between rows 16 and 17.
Another View of Plimpton 322
To compare the Plimpton tablet with an alternative list of primitive Pythagorean triples the author has constructed an ordered list of triples
by the small side.
Exhbit IV. An Alternative Plimpton Tablet
Using The Small Side
| Number | Generators u,v | Side A | Side B | Diagonal | Angle,Change | row |
|---|
| 5 | 2 | 5 | 21 | 20 | 29 | 43.6029 | -1.3971 | 1 |
| 12 | 3 | 8 | 55 | 48 | 73 | 41.1121 | -2.4908 | 2 |
| 131 | 10 | 27 | 629 | 540 | 829 | 40.6463 | -0.4658 | 3 |
| 56 | 8 | 17 | 225 | 272 | 353 | 39.5977 | -1.0486 | 4 |
| 1 | 1 | 2 | 3 | 4 | 5 | 36.8698 | -2.7279 | 5 |
| 109 | 8 | 25 | 561 | 400 | 689 | 35.4893 | -1.3805 | 6 |
| 18 | 3 | 10 | 91 | 60 | 109 | 33.3985 | -2.0908 | 7 |
| 127 | 8 | 27 | 665 | 432 | 793 | 33.0087 | -0.3898 | 8 |
| 175 | 9 | 32 | 933 | 576 | 1105 | 31.4173 | -1.5914 | 9 |
| 55 | 5 | 18 | 299 | 180 | 349 | 31.0483 | -0.3690 | 10 |
| 39 | 4 | 15 | 209 | 120 | 241 | 29.8629 | -1.1854 | 11 |
| 3 | 1 | 4 | 15 | 8 | 17 | 28.0725 | -1.7904 | 12 |
| 104 | 6 | 25 | 589 | 300 | 661 | 26.9915 | -1.0810 | 13 |
| 13 | 2 | 9 | 77 | 36 | 85 | 25.0577 | -1.9338 | 14 |
| 96 | 5 | 24 | 551 | 240 | 601 | 23.5366 | -1.5211 | 15 |
| 2 | 2 | 3 | 5 | 12 | 13 | 22.6198 | -0.9168 | 16 |
| 41 | 3 | 16 | 247 | 96 | 265 | 21.2394 | -1.3804 | 17 |
| 61 | 11 | 16 | 135 | 352 | 377 | 20.9829 | -0.2565 | 18 |
| 6 | 1 | 6 | 35 | 12 | 37 | 18.9247 | -2.0582 | 19 |
| 102 | 4 | 25 | 609 | 200 | 641 | 18.1806 | -0.7441 | 20 |
| 167 | 5 | 32 | 999 | 320 | 1049 | 17.7614 | -0.4192 | 21 |
| 65 | 3 | 20 | 391 | 120 | 409 | 17.0616 | -06988 | 22 |
| 118 | 4 | 27 | 713 | 216 | 745 | 16.8540 | -0.2076 | 23 |
| 37 | 2 | 15 | 221 | 60 | 229 | 15.1893 | -1.6647 | 24 |
| 10 | 1 | 8 | 63 | 16 | 65 | 14.2501 | -0.9392 | 25 |
| 51 | 11 | 14 | 75 | 308 | 317 | 13.6855 | -0.5646 | 26 |
| 7 | 4 | 5 | 9 | 40 | 41 | 12.6803 | -1.0052 | 27 |
| 17 | 1 | 10 | 99 | 20 | 101 | 11.4212 | -1.2591 | 28 |
| 164 | 3 | 32 | 1015 | 192 | 1033 | 10.7117 | -0.7093 | 29 |
| 22 | 1 | 12 | 143 | 24 | 145 | 9.5273 | -1.1844 | 30 |
| 100 | 2 | 25 | 621 | 100 | 629 | 9.1479 | -0.3794 | 31 |
| 117 | 2 | 27 | 725 | 108 | 733 | 8.4728 | -0.6751 | 32 |
| 19 | 7 | 8 | 15 | 112 | 113 | 7.6281 | -0.8447 | 33 |
| 40 | 1 | 16 | 255 | 32 | 257 | 7.1527 | -0.4754 | 34 |
| 52 | 1 | 18 | 323 | 36 | 325 | 6.3597 | -0.793 | 35 |
| 64 | 1 | 20 | 399 | 40 | 401 | 5.7249 | -0.6348 | 36 |
| 94 | 1 | 24 | 575 | 48 | 577 | 4.7715 | -0.9534 | 37 |
| 50 | 12 | 13 | 25 | 312 | 313 | 4.5812 | -0.1903 | 38 |
| 58 | 13 | 14 | 27 | 364 | 365 | 4.2421 | -0.3391 | 39 |
| 141 | 1 | 30 | 899 | 60 | 901 | 3.8184 | -0.4237 | 40 |
| 162 | 1 | 32 | 1023 | 64 | 1025 | 3.5799 | -0.2385 | 41 |
| 206 | 1 | 36 | 1295 | 72 | 1297 | 3.1823 | -0.3976 | 42 |
| 255 | 1 | 80 | 1599 | 80 | 1601 | 2.8642 | -0.3181 | 43 |
| 160 | 22 | 23 | 45 | 1012 | 1013 | 2.5460 | -0.3182 | 44 |
The curious thing about this table is that it was constructed with only the first 255 primitive Pythagorean triples. The reason for this is that using the
small side, more triangles would be composites of the 2,3, and 5. The diagonal could not be used as a divisor because only powers of 5 would be 2,3,5 composites and hence terminating fractions and a tablet including the first
44 powers of five would be very large indeed!
To construct the Plimpton tablet the old Sumerians had to have a full listing of all the first 3,000 primitive Pythagorean triples. Which begs the question:
"How did the Sumerians empirically discover the triples?" Certainly, discovering the first primitive triple would not be difficult, even the first few dozen triples, after that the Sumerians needed some general rules to go searching for larger and larger triples.
In constructing the primitive Pythagorean tables, the Sumerians must have acquired a deep understanding of prime numbers and the use of squared numbers. On another page(still in development) the author will propose a way to empircally find the full table of primitive triples and "discover" the Pythagorean theorem.
One final point, it has been proposed that the Plimpton tablet might be the first example of an attempt to construct trigonometic tables. Certainly having a rigorous way of order the triples by angle size was a first step. When the trigonometic tables were first developed by Hipparchus about 100 AD, the Pythagorean triples were abandoned for their first cousins, right triangles with square root sides. An interesting fact about Pythagorean triples is that they are all non-cycling triangles--by that I mean that no multiple of any triple ever cycles to any multiple of 360 degrees. The building blocks of the trigonometric tables are 30, 36, and 45 degree right triangles and their combinations and succesive half angles and these right triangles are all cyclic. A separate page will show the first trigonometic table and it's construction.
Side Angle Measures Of Pythagorean Triples
To compare the Plimpton tablet with the tables of triples exhibited in the previous section, we have ordered the first the
first 17 triples by small side angle for comparison.
Exhibit V. The First 17 Pythagorean Triples Ordered by Small Side Angle
| Number | Generators u,v | Side A | Side B | Diagonal | Angle,Change |
|---|
| 5 | 2 | 5 | 21 | 20 | 29 | 43.6029 | -1.3971 |
| 16 | 4 | 9 | 65 | 72 | 97 | 42.0750 | -1.5579 |
| 12 | 3 | 8 | 55 | 48 | 73 | 41.1121 | -0.9626 |
| 1 | 1 | 2 | 3 | 4 | 5 | 36.8698 | -4.2423 |
| 8 | 2 | 7 | 45 | 28 | 53 | 31.8908 | -4.9718 |
| 11 | 4 | 7 | 33 | 56 | 65 | 30.5102 | -1.3806 |
| 3 | 1 | 4 | 15 | 8 | 17 | 28.0725 | -2.4377 |
| 15 | 5 | 8 | 39 | 80 | 89 | 25.9892 | -2.0833 |
| 13 | 2 | 9 | 77 | 36 | 85 | 25.0977 | -0.8915 |
| 2 | 2 | 3 | 5 | 12 | 13 | 22.6198 | -2.4779 |
| 6 | 1 | 6 | 35 | 12 | 37 | 18.9247 | -3.6951 |
| 4 | 3 | 4 | 7 | 24 | 25 | 16.2602 | -2.6645 |
| 10 | 1 | 8 | 63 | 16 | 65 | 14.2501 | -2.0101 |
| 7 | 4 | 5 | 9 | 40 | 41 | 12.6803 | -1.5698 |
| 17 | 1 | 10 | 99 | 20 | 101 | 11.4212 | -1.2591 |
| 9 | 5 | 6 | 11 | 60 | 61 | 10.3888 | -1.0324 |
| 14 | 6 | 7 | 13 | 84 | 85 | 8.7974 | -1.6214 |
Go to Pythagorean Triples Level III