1. Background

The narrative surrounding the first DOC, HeTu, takes us back to ancient China, where King Fu Xi was chosen to lead. One fateful day, he witnessed a dragon-horse emerge from the Yellow River, adorned with a distinctive black-and-white pattern on its back. Inspired, Fu Xi replicated this design, and shortly thereafter, the dragon-horse vanished. This diagram, known as HeTu (river map), became a source of inspiration for Fu Xi, who later rose to legendary status.

Centuries later, during the Xia dynasty (circa 2000 BC), catastrophic floods plagued China. A man named Yu was tasked by the king to manage the crisis. While working, Yu observed a turtle ascend from the Luo River, bearing a similar black-and-white pattern. He adapted this design into a strategy for flood control, ultimately succeeding and earning the title of king. This diagram became known as LuoShu (the Book of Luo).

2. Developing 3-D and Spacetime Models

The 3-D and spacetime models of HeTu and LuoShu are predicated on the following hypotheses:

1. Each diagram signifies a balanced energy system.
2. Each numeral in HeTu and LuoShu symbolizes:
   1. an element at that position,
   2. the dynamic energy associated with that element, and
   3. the sequence in which that element contributes to the system.

For instance, the number 3 indicates:

1. There exists an element designated as 3 at that position.
2. This element possesses the third lowest energy within the system.
3. Element 3 is the third entity incorporated into the system.

A. 3-D Models of HeTu

The 3-D representations of HeTu manifest in three distinct structures:

1. a body-centered cubic octahedral structure (BCC octahedrons),
2. two concentric tetrahedrons, and
3. two tetrahedrons aligned at their centers.

(Refer to Figures 2, 3, and 4).

1. BCC Octahedrons

Figure 2c illustrates HeTu as a BCC octahedral structure, where the number 5 is concealed behind 10. The 2-D HeTu provides a frontal view of this arrangement. Upon removal of 10, the structure transforms into a body-centered cubic formation.

Figure 2. (a) Original HeTu (b) Black-and-white dots substituted with numbers (c) 3-D HeTu as BCC octahedron (d) 3-D HeTu as body-centered cubic structure.

2. Two Concentric Tetrahedrons

The second model of 3-D HeTu, depicted in Figure 3, comprises two concentric tetrahedrons. The numbers 1, 2, 3, and 4 represent the vertices of the smaller tetrahedron, while 6, 7, 8, and 9 denote the vertices of the larger one. Here, 5 is at the center of the smaller tetrahedron, and 10 occupies the center of the larger tetrahedron, encircling 5.

Figure 3. 3-D HeTu as a concentric tetrahedron.

3. Two Tetrahedrons Aligned at the Center

The subsequent 3-D model proposes that HeTu symbolizes two tetrahedrons aligned at their centers, with tetrahedron 6789 positioned in front of tetrahedron 1234. The 2-D HeTu serves as the frontal perspective of these structures.

Figure 4. 3-D model of HeTu as two tetrahedrons aligned at their centers.

B. The Space-Time Models of HeTu

The space-time models of HeTu, illustrated in Figure 5, showcase the evolution of the 3-D structures over time. Following the numerical sequence, we observe that (a) the BCC octahedrons and (c) the aligned tetrahedrons cyclically repeat, suggesting a numerical growth pattern, while (b) the concentric tetrahedrons expand dramatically, indicating growth in size. Together, they depict a growing fractal system with individual subunits representing smaller versions of the entire structure.

Figure 5. The space-time model of HeTu. (a) Space-time model of BCC octahedron (b) Space-time model of concentric tetrahedron (c) Space-time model of aligned tetrahedrons.

C. The Complex Building Sequence of HeTu

It is evident that HeTu is constructed through intricate steps. It begins with 1 at the base, progresses to 2 at the top, then to 3 on the left, 4 on the right, culminating with 5 at the center, adhering to the energy sequence. This indicates that the space with the lowest energy is filled first, beginning with 1 and moving upward. Notably, the system transitions from 1 to 2 rather than to 3 in a circular fashion, suggesting that the energy gap between 1 and 2 is significantly smaller than that between 1 and 3. Similar principles apply to 6789 and 10.

D. 3-D Models of LuoShu

The 3-D models of LuoShu, illustrated in Figures 6, 7, and 8, consist of:

1. the foundational unit of a face-centered cubic (FCC) system,
2. two interlocking tetrahedrons,
3. a double-helix spiral-coil structure.

1. Face-Centered Cubic System

The face-centered cubic (FCC) structure is characterized by a cube with an additional element at the center of each face. In Figure 6, the numbers 1, 3, 7, and 9 denote the vertices of the back surface, while 5 occupies the center of that surface. The numbers 2, 4, 8, and 6 correspond to the centers of the remaining four surfaces. The front surface of the unit cell is omitted here.

Figure 6. 3-D LuoShu (a) The fundamental unit of a face-centered cubic (FCC) structure.

2. Two Interlocking Tetrahedrons

The second 3-D model suggests that LuoShu represents the 2-D frontal view of two interlocking tetrahedrons, as shown in Figure 7.

Figure 7. 3-D LuoShu as two interlocked tetrahedrons. (a) Tetrahedron formed by elements 1, 9, 3, 7, and 5 (b) Tetrahedron formed by 8, 2, 6, and 4 (c) The two interlocked tetrahedrons, with element 5 at the center of the entire structure or the front tetrahedron.

3. Double-Helix Spiral Coil

The construction of the LuoShu double-helix spiral coil involves three steps:

1. Adjusting the center: To simplify demonstration, the center is reset from 5 to 0, termed NewLuoShu. This system now incorporates both positive and negative integers. Consequently, Figure 8c becomes intriguing as each number mirrors its counterpart on the opposite side of the center. The left half (from -4 to -1) mirrors the right half (from 4 to 1), with a twist along the 3/0/-3 axis.
2. Unwinding NewLuoShu: The numerical arrangement within NewLuoShu reveals a clockwise circulation. Unwinding the structure clockwise yields two chains: the original chain commencing with -4, and its counterpart beginning with 4, as depicted in Figures 8d and 8e.
3. Creating the twist: The two matching chains intertwine between 3 and -3. NewLuoShu evolves into a double helix. As the structure expands, it resembles a spiral coil, as shown in Figure 8f.

The double helix is a prevalent configuration in liquid-crystal DNA. Generally, liquid-crystal molecules exhibit diffusion similar to liquid molecules but retain a degree of positional and orientational order like solid crystals. Thus, 3-D LuoShu implies a structure that can exist as either a solid or a state beyond solid. When in a solid crystal form, it takes on FCC or interlocked tetrahedral configurations. However, at elevated energy states, such as that of liquid crystals, the structure may manifest as a double-helix spiral coil or a combination of FCC, interlocked tetrahedrons, and double helices. This mirrors the structure of the nucleosome, where double-helix DNA wraps around histones, which are octamers formed by two interlocking tetrahedrons.

Figure 8. 3-D LuoShu as a double-helix structure. (a) LuoShu (b) Replace black-and-white dots with numbers (c) Reset center to 0, naming it NewLuoShu (d) Unwind NewLuoShu (e) NewLuoShu as a double-chain structure (f) 3-D model of NewLuoShu as a double helix.

E. The Secret Behind the Twist and Perfect Pairing

How do we explain the twist? The analysis in Table 1 reveals that only with a twist along the 3/0/-3 axis can the system achieve the lowest energy (0 energy) across all dimensions. This arrangement signifies the system's most stable state, indicating that the twist is crucial for structural equilibrium. We can conclude that perfect pairing (or specificity) at each step, along with the twist, are essential for maintaining systemic balance.

Note that both the center and the overall system energy remain at 0.

Table 1. Energy comparison with possible twists along different axes.

F. The Yin-Yang Diagram and LuoShu

The Yin-Yang diagram exemplifies the energy cycle of LuoShu. The left side represents the increase of positive energy (Yang energy), while the right side symbolizes the rise of negative energy (Yin energy). They mirror each other, with the black and white dots indicating the locations of the twist (i.e., t = 4.5/8T).

Figure 9. The Yin-Yang diagram.

G. Transformation Between HeTu and LuoShu (BCC–FCC Transformation)

The transformation between HeTu and LuoShu illustrates the structural alteration between BCC and FCC. Recent advancements in technology have unveiled that BCC is a prevalent form of metal crystals at room temperature. However, as temperature rises to a certain threshold, these materials undergo structural transformations, such as BCC-to-FCC, to attain a new equilibrium in a higher energy environment. The reverse transformation may occur when temperature decreases back to lower levels.

To scientists, BCC, FCC, and double helix represent three common forms of DNA. Understanding the interchange process between these structures is paramount for deciphering nature's hidden codes.

Figure 10 outlines the significant steps in the transition from HeTu to LuoShu:

1. A 180-degree octahedral twist along the Z-axis,
2. A 45-degree octahedral twist along the X-axis,
3. A transposition of one-half l in the -X direction, where l denotes the length of the cube.

Figure 10. HeTu-LuoShu (BCC–FCC) transformation. (a) 3-D HeTu as BCC (b) Following 180-degree octahedral twist along Z-axis (c) After 45-degree octahedral twist along X-axis (d) Transposition of one-half l in -X direction. HeTu morphs into a fundamental FCC unit (e) LuoShu in a diagonal position (f) 2-D LuoShu.

H. The Spacetime Model of LuoShu (NewLuoShu)

In Figure 11, the element of TIME is incorporated into NewLuoShu, a system with energy circulating clockwise as time progresses. Each cycle can be segmented into eight phases, rendering the spacetime model of NewLuoShu a double-helix spiral coil, akin to its 3-D structure.

In essence, we are enveloped by this energy cycle in our daily lives: the cycle within a day (midnight -> morning -> noon -> afternoon -> midnight), a month (new moon -> half-moon -> full moon -> half-moon -> new moon), and a year (winter -> spring -> summer -> autumn -> winter).

Figure 11. TIME element assigned to NewLuoShu.

The notion of fractal LuoShu is depicted in Figure 12. The yellow ring within the red spiral coil signifies one cycle of LuoShu, with each segment of that cycle representing a smaller version of the complete LuoShu system.

Figure 12. Spacetime LuoShu as a fractal.

One might ponder, "What occurs when 'time' forms a circle?" The answer is that the spacetime model of LuoShu transforms into a spiral ring, signifying a four-dimensional closed-loop system with its center and total energy remaining at zero.

Figure 13. Spacetime model of LuoShu as a spiral ring when time becomes a circle.

I. Summary

In summary, HeTu and LuoShu embody a singular fractal composed of five fundamental elements. When the system exists in a crystalline state, its unit cells are arranged in BCC octahedral and/or tetrahedral forms. However, structural changes can occur as energy levels rise, prompting the system to evolve into liquid crystals. These liquid crystals may manifest as FCC cubes, interlocked tetrahedrons, double helices, or combinations thereof.

3. Are River Maps the Blueprints of DNA?

Remarkably, deoxyribonucleic acid (DNA), the genetic blueprint of all living organisms, exhibits characteristics akin to the River Maps. Notably:

1. DNA can be found in BCC, FCC, and double-helix spiral forms.
2. The fundamental packing unit of chromosomes is known as a nucleosome, which comprises DNA double helices wrapped around an octamer formed by five distinct proteins called histones (i.e., two interlocked tetrahedrons and double helices coexist simultaneously, mirroring the concepts suggested by 3-D LuoShu).
3. DNA incorporates twists in every cycle of the coil.
4. DNA comprises five fundamental elements:
   • Four base-pair components: adenine (A), thymine (T), cytosine (C), guanine (G)
   • The sugar-phosphate backbone.
5. Pairing specificity is evident in both DNA and the River Map. For instance, in DNA, A must pair with T, and C must pair with G. Similarly, in LuoShu, 1 must pair with -1, 2 with -2, and so forth.

All these observations indicate that the River Map may indeed represent the schematics of DNA. Consequently, DNA should also be considered a fractal, and the organism it symbolizes must also be a fractal.

Further discussions regarding the River Map DNA Model, the implications for infectious diseases, symptom prediction, and pathogen genome analyses are elaborated in another publication titled The River Map DNA Model and Its Applications by Mai Wang, 2013.

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