Zhou bi suan jing

 
The Zhou bi suan jing - Oldest Math Book Known
Pythagorean Theorem Proof

The Zhou bi suan jing is the oldest of the ancient Chinese Mathematical classics (as early as 1100 BC to late sixth century BC) and has a sacred place in Chinese cosmology. The translation of the Zhou bi suan jing is “The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven” . This classic text concerns itself with how to use the right angle triangle to measure heights and distances, earthly and celestial.

The gnomon, or a stick in the ground, was the most sophisticated mathematical instrument to humankind for thousands of years

A discussion of the right angle triangle takes place in the beginning of the text when a student asks his teacher, Fu-Hsi, how the degrees of the celestial spheres were learned. The answer concerns the use of the gnomon or carpenter's square. The teacher describes a proof of the Pythagorean Theorem using the 3 – 4 – 5 triad of numbers. In this dialogue the Chinese word chu is used interchangeably with gnomon, carpenter's square, and T-square. Then the teacher is asked about the Tao and the use of the right angle triangle (gnomon). The mathematics of the right angle triangle, the teacher offers, allows one to calculate distances on land as well as in the heavens.

The art of numbering or the arrangement of numbers in a square that the Chinese referred to as the Luo Shu also generates a Pythagorean triad of numbers that describe a right angle triangle (see later section in this chapter), thus the art of numbering may esoterically mean the Luo Shu and its higher orders.

Knowledge is derived from the use of the gnomon and its straight line or shadow, the straight line is derived from the right angle. Practical understanding and use of the right angle triangle is what guides mankind in understanding the earth, and those who could were considered wise. Higher understanding of the right angle triangle and the Luo Shu is what guides mankind in understanding the heavens, and one who could would be considered a sage.

 THE PYTHAGOREAN THEOREM AND THE QUINCUNX

Courtesy of Charles Johnson at www.earthmatrix.com

One can see from these images how the quincunx design is a sacred geometric design that incorporates the:

square
circle
the right angle triangle

The quincunx design was adopted by the Chinese, Armenians, Persians (Sassanian Fire Temples), and Christians and incorporated into temple and church architecture.  Leonardo da Vinci also used this very design he learned from Lucas Pacioli in his chrch drawings. 

                                    LEONARDO DA VINCI - QUINCUNX CHURCH DRAWING

Having been a student of the great mathematician Lucas Pacioli, Leonardo da Vinci learned how to incorporate the square, the circle, the ya or cruciform mandala like design for the ground plan of church design which is based on the Lo Shu.   The sacred center is domed by a cupola and geometrically balanced.  It is well documented that Lucas Pacioli studied Magic Squares and was a tremondous influence on Leonardo Da Vinci and Albrecht Durer, both of whom enjoyed mathematics and incorporated proportion in their famous works.  Lucas Pacioli wrote the famous The Divine Proportion and Leonardo Da Vinci was the illustrator for this classic book.

CRUCIFORM OF ODD NUMBERS, THE CIRCLE, AND THE SQUARE

The cruciform of odd numbers combined with the square and the circle incorporated the symbols of Heaven, Earth, male, female, odd, even, the four cardinal directions, the math of the gnomon, the four seasons, and the calendar.  The Lo Shu or cruciform of odd numbers became the basis of the mandala, Chinese temple design, and early church architecture.   

The Right Angle Triangle in Art

This is a plate discovered by Sir Leornard Woolley.  This plate was discovered in Ur and is over 4,000 years old.  The design on the plate fits nicely in the model for the Pythagorean Theorem as described in the Zhou bi suan jing. 

The swastika symbol is a curious choice for a significant symbol for the center of this design.  However, Lowenstein and Granet have connected the swastika with the Luo Shu.  Most likely, the symbol is connected with the Sun and its cycle, Time and Space, the four cardinal directions, the concept of spin, and fractal growth.

Overlaying the Pythagorean Theorem on a 4,000 year old plate, Ur.

This is a great example of what Lucas Pacioli would refer to as Divine Proportion, that is, using sacred geometry in the layout of a creation of art.  Both Leonardo da Vinci and Albrecht Durer spent consideralbe time with Lucas Pacioli in order to learn how to incorporate divine proportion into art.  In other words, using the square, circle. polygons, triangles, or combinations thereof, as a template for a design and removed when the art piece is completed.  This will give mathematical balance to the piece and be more pleasing to the visual senses. 

The swastika is also known as the Greek gammadion; a gammadia is a single right angle.  The element that make up the swasitka, four right angles, can also form the Ya Shape, also known as the voided Greek cross. The gammadia was a common symbol used in early Christian art, as was the swastika.  Both of these symbols may be related to the math of the gnomon, the calendar, and a model for Time and Space. 

The swastika pictogram was also a Chinese character "wan" which translates to "the ten thousand things" - a philosophical reference to the finite yet uncountalbe sum of things. The Ya shape, similar to the voided Greek cross, translates to the center, where the enchanted one lives and is a model for the sacred Ming Tang temple.