Chapter 39 - 17: Eight Trigrams and Binary

Chapter 39: Chapter 17: Eight Trigrams and Binary

Yu Xian wondered if he had misheard, or if a whole morning of researching papers had simply scrambled his brain. In his understanding, the relationship between binary and the Eight Diagram was like the relationship between Shi Zuozhou and eight-treasure congee—which was to say, there was no relationship at all.

When Wen Xiao just told him that binary was actually inspired and refined by the Eight Diagram, the shock in his heart was no less than when he first heard that Jin Yong, Qiongyao, and Xu Zhimo were relatives.

"You didn’t mishear!" Wen Xiao seemed quite satisfied with Yu Xian’s shell-shocked expression. She fiddled with the white sugar packets on the table and continued, "This is a true story in the history of computer science. When Leibniz saw Fuxi’s Sixty-Four Trigrams Diagram brought back to Europe by missionaries, he was beyond thrilled. He discovered that it was completely logically identical to the binary arithmetic he was researching!"

Seeing Yu Xian’s still-bewildered face, Wen Xiao explained further, "Let’s put it this way. In binary, the most basic units are ’0’ and ’1’, right? In the Eight Diagram, the most basic units are the ’Yin lines’ and ’Yang lines’."

Yu Xian had seen the trigrams in the Eight Diagram before. He only knew they were composed of many horizontal lines—some broken, some continuous—but he didn’t know anything beyond that.

Wen Xiao picked up a slender white sugar packet and showed it to Yu Xian. "A continuous horizontal line like this is called a Yang line. We treat the Yang line as ’1’ in binary."

Next, she tore an empty white sugar packet in half from the middle. "A broken horizontal line like this is called a Yin line. We treat the Yin line as ’0’ in binary."

She then arranged three intact white sugar packets from bottom to top. "Like this, three Yang lines combined together is called the ’Qian Trigram’, which corresponds to ’111’ in binary. Converted into a decimal number, it becomes 7."

As a science student, Yu Xian naturally understood the principles of converting between binary and decimal. It was actually quite easy to comprehend. The calculation method for binary could be imagined as a special abacus. The beads on this abacus were made of different materials in each column, and their corresponding "value" varied as well.

The rightmost bead was a Stone Bead, and that position represented the value "1". Moving one column to the left, the bead became a Scrap Iron Bead, its value doubling to represent "2". Moving left again, it became a Bronze Bead, doubling again to represent "4".

And so on. A silver bead represented the value "8"; a gold bead, "16"; a platinum bead, "32"; a diamond bead, "64"; a starlight bead... wait, no.

In short, starting from the far right, for every column moved to the left, a bead’s "value" doubled. The "total value" represented by this abacus depended on the position of the beads in each column. A "1" meant the bead was pushed up, and a "0" meant it slid down.

Starting from the rightmost bead, if a column’s bead was pushed up, its value was factored into the total; if it slid down, the bead in that column was ignored. Therefore, when Wen Xiao mentioned the binary "111", it was equivalent to a Bronze Bead on the left with a value of 4, plus a Scrap Iron Bead in the middle with a value of 2, plus a Stone Bead on the far right with a value of 1.

Their total value, 4+2+1, corresponded to the decimal number 7. In more refined mathematical language, this was 1×2² + 1×2¹ + 1×2⁰, which equaled 7.

Wen Xiao then arranged three torn sugar packets from bottom to top. "Like this, three Yin lines combined together is called the ’Kun Trigram’, which corresponds to ’000’ in binary. Naturally, the corresponding decimal number is 0."

All three beads were "0", meaning they had slid down, so none of their values were calculated. 0×2² + 0×2¹ + 0×2⁰ equaled 0.

Yu Xian was somewhat shaken. "Aren’t you just forcing a connection between the shape of the Eight Diagram and binary? Could this just be a coincidence?"

"Not at all! If you calculate each trigram of the Eight Diagram using binary, and then arrange the results in descending order—that is, from extreme Yang to extreme Yin—into a sequence of natural numbers..." Wen Xiao extended a slender finger and traced on the table. "...you’ll find that their order perfectly matches the numbering rule of the ’Innate Trigrams’. That is: Qian 1, Dui 2, Li 3, Zhen 4, Xun 5, Kan 6, Gen 7, and Kun 8. It fits flawlessly, without a single discrepancy."

Looking at the sugar packets on the table, Yu Xian felt as if something was sprouting in his mind. As a science student, he had always believed in the objective reality of this world. He had never imagined that the Eight Diagram, which he had long dismissed as "superstition," was fundamentally a 3-bit binary coding system.

How was this possible? Wait, there seemed to be a logical flaw here.

"Is this... a chicken-or-egg question?" Yu Xian asked. "Rather than saying the Yi Jing Trigrams match binary perfectly, wouldn’t it be more accurate to say that Leibniz designed binary based on the Eight Diagram?"

"Then have you ever considered why the ancients from thousands of years ago would use binary to record and calculate numbers? Or why they would use binary coding to discuss mysticism and philosophy?" Wen Xiao seemed to see through Yu Xian’s shock, a hint of smugness in her tone. "Moreover, the divination rules in the *Book of Changes*—like changing lines and moving lines—are actually an incredibly complex set of logic gate circuits. A changing line is equivalent to an XOR gate, and a moving line is equivalent to a NOT gate."