The text of the Book as the spatial coordinate grid

Russian researcher Spirin V.S. (1929-2002) in his studies of ancient Chinese classical texts proposed a hypothesis of the existence of “easy” and “difficult” texts which had two-dimensional, detailed in the plane, and three-dimensional, detailed in the space, structures. The Ancient Chinese “Book of Changes” I Ching relate to the three-dimensional classical text. In the text of I Ching among hieroglyphs denoting names of hexagrams we can also distinguish a group of so called “structure hieroglyphs”. Unlike other hieroglyphs they in addition to conveyance of the meaning also carry indication on certain geometrical structure. These hieroglyphs from the text of the “Book of Changes” include: 師, 升, 井, 坎, 節, 屯, 比, 離, 旅, 震.

The way of reconstruction of three-dimensional geometric model of spatial representation in hieroglyphics is the application of basic elements of hieroglyphs on the structure of the Diagram of the “Supreme Ultimate” (see Figure 1):

Figure 1. The spatial representation of Chinese hieroglyphs on the structure of the Diagram of the “Supreme Ultimate”.

five	a field	a mountain

Correspondence between these elements of ancient Chinese hieroglyphics and the structure of the Diagram of the “Supreme Ultimate” in the I Ching is not a single or accidental instance, it is confirmed in Ming Tzu’ (孟子, 372-289 BC) philosophical text in which spatial characteristics of ancient Chinese hieroglyphs are associated with the description of man’s qualities: “deep-well – deep-well” his (a man’s) order, “very strict – very strict” his ability of thorough knowledge of himself, “separate – separate” his sequence, “joyful-joyful” his following the Tao.” (“ 井井 兮其有理也, 嚴嚴 兮其能敬己也, 分分 兮其有終始也, 猒猒 兮其能長久也, 樂樂 兮其執道不殆也, 炤炤 兮其用知之明也, 脩脩 兮其用統類之行也). Such structures are widespread in the Islamic and Chinese art of calligraphy where words, their meaning and visual three-dimensional geometric model compose the art form.

Figure 2. Islamic and Chinese art of calligraphy (based on

Geometric patterns created by hieroglyphs correlate with volumetric sound patterns which manifest themselves in homophonous or similar sounding words. Confucian philosopher Dong Zhongshu (董仲舒, 179-104 BC) in his book “A Heavy Dew of Spring and Autumn Annals” (春秋繁露) wrote: “Perfect men of wisdom in ancient times listening to loud sounds of nature (xiao) comprehended through them the Sky and the Earth and called these sounds as titles (hao). Thus the title (hao) expresses the sound (xiao) and comprehension (xiao)”. Examples of these rhythmic patterns in Chinese language are the concept of the Tao in the meaning of the "way" and the concept of the dao in the meaning of the "mirror symmetry", in Arabic language it is “a pupil” and “a man” in Ibn Arabi’s philosophy.

The geometric pattern and the sound pattern can be accompanied by the mathematical pattern. In Chinese culture numerical schemes He Tu and Lo Shu should be considered as such mathematical patterns. Let us follow the logical reorganization of these schemes in order to clarify their role in the formation of three-dimensional space (see Table 1). Bilateral inversion Lo Shu reflects two principles of mirror symmetry under the laws of which hexagrams are formed from trigrams. The first rule (1) is the mirror symmetry with the opposite disposition of elements reflected in the mirror. The second rule (2) is a simple duplication when elements are doubled without inversion. In ancient China this kind of symmetry was shown by “magic mirrors”. These two forms of symmetry reflects two principles of hexagram formation from trigrams

Table 1. Reorganization of Lo Shu pattern into He Tu pattern

  Lo Shu pattern

  ༤ 4 ༩ 9 ༨ 8 ༣ 3 ༥ 5 ༧ 7 ༢ 2 ༡ 1 ༦ 6

5 (in place of 9) and 1 as a symbol of the centre which is oriented to the Sky

    83 1 72   From square to cross

He Tu pattern

  4 5 8 3 1 7 2 1 6

  Mirror symmetry (1)  

  Duplication (2) of central cross

This pattern has continued in the form of a matrix with symmetrical distribution of prime numbers (see Figure 3).

Figure 3. The matrix with symmetrical distribution of prime numbers.

This distribution of numbers has a cyclic structure. One cycle includes six square matrixes with numbers from one to forty-four. Then numbers get their primary positions (for example, 1 and 41; 3 and 43; 5 and 45; 6 and 46). One cycle has the following geometry in three-dimensional space (in scale 1:2):

Figure 4. Geometry of one cycle in three-dimensional space

This mathematical structure is confirmed by ornaments at the bottom of the ceramic dish of Dasi culture (Ancient China, beginning of the 4^th century BC) and on Persian carpet for prayer (Iran, 16^th century).

Figure 5. Ornament of Ancient Chinese dish and Persian carpet (a fragment) (based on

There is the reconstruction of the ornament and its comparison with the matrix of prime numbers:

Figure 6. The matrix of prime numbers on the Ancient Chinese ornament (based on Karapetyants A.M., 2015, p. 480).

It is important to note that this symmetry is described in the I Ching and it is opposed to another type of symmetry which archaeological artifacts correspond to the Mesopotamian culture. Changing of symmetry of this geometric pattern indicates altering cosmos into chaos.

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