Solution of the Rigth Triangles according to Gudea Theorem

CARLOS CALVIMONTES R.

From the remote antiquity in the Sumerian culture of Mesopotamia was known the basic configuration of right triangles with integers to solve practical problems of construction and surveying, this application was transmitted to later cultures1 and has reached the present day in the daily practice of the construction to square the elements of a work. Having been recognized the important significance of this configuration2, 41 centuries ago the testimony of how the relationship between the sides of these right triangles in a simple and elegant way, described in the evidence I found in 1991 in the famous sculpture, was left. of the oldest known architect, and which I called the Gudea Theorem3 in fair recognition of the creator of the algorithm by which you can deduce the dimension of one of the sides of a right triangle if you know the measurements of the other sides that are integers.

26 years after my discovery scientists from the University of New South Wales in mid-2017 gave the news that they had found triplets of right triangles on the Plimpton 3225 tablet but they did not go so far as to point out that these triplets are the result of an admirable formula whose knowledge I have spread in various media.

ESCULTURA DEL ARQUITECTO DEL PLANO
AUTHOR

Gudea, 'the chosen one', is the paradigmatic character of the Sumer Renaissance. Ensi o Patesi, was at the same time prince, ruler, and priest of the city-state of Lagash (he governed between 2144 - 2124 or 2122 a.) At the time of the greatest splendor of the ancient Mesopotamian civilization. Lagash is the current Tello or Al Hiba, on the banks of the Tigris River, 300 km southeast of Baghdad in Iraq.

Of him and his work has been commented extensively, it has even been said that he was a god. He was the most portrayed man of antiquity and of the many sculptures that were made the best known is the seated person who has been called 'the Architect of the Plane' and which is preserved in the Louvre Museum.

In one of the commemorative documents that he wrote he states that he invented "a new way of building, not used before by any sovereign". It has been assumed so far that it referred to a new type of brick, but it is more likely that it alluded to a useful geometric resource. This, attending to a problem of all times, makes it easy to put 'square' the rethinking of design in the field, a construction site or task of surveying, in cases of the different ratio between the lengths of the legs of triangles rectangles with sides of whole numbers.

TABLERO EN EL 'ARQUITECTO DEL PLANO'
SUPPORT

On the knees of the Architect of the Plane, there is a drawing board, in which the plan of a supposed fortified temple is engraved, accompanied by a drawing instrument and another measuring instrument. This has the same shape and size as the one currently used for technical drawing and is known as a scaler. In the inner band of the instrument, there is a measuring rule with sexagesimal basis. In the outer strip, there are marks that indicate successively magnitudes proportional to 1, 5 and 12, which make a total of 18 between the extremes. The necessary and meticulous measurements were made on the sculpture.

PURPOSE

The dimensions found in the outer band of the instrument show the quantitative characteristics of the right triangle with sides 5, 12 and 13: the difference between the major leg and the hypotenuse, 1, the minor leg that is odd, 5, the long leg, 12, and the sum of the minor leg and the hypotenuse, 18. The disposition of the measures in the rule allows us to infer that we deliberately wanted to show the linear development of that triangle and suggest the procedure for the design of this and similar ones. The set of these characteristics defines, generalizing, what can be called Gudea Triangles: being rectangles, their sides are measured with integers, and their solution does not require the use of square roots.

REGLA GRADUADA CON SOLUCIÓN DEL TRIÁNGULO RECTÁNGULO

GUDEA THEOREM

 For the verifiable merit of the proposed geometric solution

PROPOSITION

In right triangles whose sides are measured with integers if: the hypotenuse is odd, one leg is odd with the other pair and the difference between the largest and the hypotenuse is 1 or 2; the value of the hypotenuse is equal to the major leg plus 2 if the minor leg is even and 1 if it is odd.

DEMONSTRATION Y REPERTOIRE

SMALL

CATHETUS

MAJOR

CATHETUS

E-B

A + B + C

HYPOTENUSE 

D – A  or B + C

B + D

B + E

A

B

C

D

E

F

G

3

4

1

8

5

12

9

5

12

1

18

13

30

25

7

24

1

32

25

56

49

8

15

2

25

17

40

32

9

40

1

50

41

90

81

12

35

2

49

37

84

72

16

63

2

81

65

144

128

AXIOMATIC CONDITIONS

COROLLARIES

________________________________________________________

1. In a drawing found in the tomb of Ramses IX, the pharaoh who lived ten centuries after Gudea, a similar solution was described in a right triangle 3-4-5, with an additional 1 in a prolongation of the major leg. Six centuries later Pythagoras (who nurtured knowledge in Mesopotamia and Egypt) became famous for demonstrating the quadratic relations between the sides of right triangles. See: Geometry in ancient Egypt.

2. It should take 16 long centuries for the Samos School to be expressed in several mathematical discoveries the Pythagorean Theorem: in a right triangle "the sum of the squares of the legs equals the square of the hypotenuse" in a singular demonstration of what the mathematicians of Babylon and India were doing in the configuration of right triangles.

3. The discovery of Gudea's Theorem was made in 1991 after the measurements made by the author in the sculpture of the Architect of the Plane in the Oriental Arts Room of the Louvre Museum, and has the record of intellectual property in favor of its author and owner Carlos Calvimontes Rojas, with the title "Theorem for the solution of right triangles without the use of square roots", in the Copyright Directorate of the Ministry of Economic Development of Bolivia, with Administrative Resolution No. 1-1001-386 / 2001 on May 29, 2001 in the city of La Paz.

4. The web page “Mathematical mystery of ancient clay tablet solved” states that “UNSW scientists have discovered the purpose of a famous 3700-year-old Babylonian clay tablet, revealing it is the world’s oldest and most accurate trigonometric table.” Noting that it was “…possibly used by ancient mathematical scribes to calculate how to construct palaces and temples and build canals.” And that knowledge “…reveals an ancient mathematical sophistication that had been hidden until now.” See also: “Mathematical secrets of ancient tablet unlocked after nearly a century of study” and “The Babylonian tablet Plimpton 322

5. The Plimpton 322 tablet - by the number of the George Arthur Plimpton collection - was found in the early 1900s in Larsa (Tell Senkereh), Iraq. It was dated between 1820 and 1762 a. C., that is to say after three centuries of which Gudea instructed to engrave in the sculpture with his effigy the key of the solution of the triangles rectangles with whole numbers.

Although it is the oldest written testimony that is known, it is unquestionable that there may be other tablets with similar or referent information among those of half a million found or in others not yet discovered. In addition, 26 years before the study of the scientists of the University of New South Wales -in another medium and in graphic form recorded three centuries before the Plimpton 322-, the unquestionable testimony of the procedure was found to know the relationship of the sides of triangles rectangles with whole numbers in the sculpture of the Architect of the Plane. That discovery and its interpretation are verifiable.

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