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While not intended to be used mathematically, the I Ching hexagrams developed around 1000AD represented the binary numbers from 0-63 (2^6), as discovered by Lei

While not intended to be used mathematically, the I Ching hexagrams developed around 1000AD represented the binary numbers from 0-63 (2^6), as discovered by Leibniz who was quite fond of binary himself.

19eightyfour 9y ago
That is weird, right? Where did the precise use of broken and unbroken lines, in all the combinations necessary to represent binary 0 to 63 originate? It's a compact representation to label 64 different things using 6 "sticks"...but why did they do it like this? Were they aware of binary numbers?
- MBCook 9y ago
  It's not that hard. If you limit yourself to two symbols and try to think of ways to represent as many things as you can, you just have to keep adding additional columns/rows. As long as you realize that you can use different combinations of the "first" spots with each additional spot you quickly come up with what is effectively binary. I distinctly remember in early middle school trying to think of ways to enumerate things where you could never confuse one for the other based on the symbols. I was using base 10 because I wanted to store the combination of things in a variable in BASIC on a program I was trying to write on my calculator. As I was playing around on paper I came up with a great system. I would only use one and zero because you could always tell them apart, where as if you use the other numbers then you can't tell if you meant five or if that five was a combination of two and three. When you needed to make another thing and you were out of places you would just add a digit on the left. Then about five seconds later it hit me that what I had was just binary. All of a sudden the idea of using individual bits in masks to represent things (which I'd seen before) made a lot more sense.
- jes5199 9y ago
  the sticks are literal sticks, used in fortune telling. It's roughly equivalent to flipping a coin. If you had a game where you flipped six coins, and each combination had a different meaning, you would be close to inventing binary
- chandler 9y ago
  > ...but why did they do it like this? ...because people are smart. People have _always_ been smart, it's one of the defining attributes of humanity. It strikes me oddly (not specifically your comment, but the general color accompanying these kinds of articles)--there's a kind of ground assumption that by looking into the past, one sees nothing but a gradually descending IQ.
  - pilsetnieks 9y ago
    > looking into the past, one sees nothing but a gradually descending IQ Actually, it quite literally might be so: https://en.wikipedia.org/wiki/Flynn_effect https://en.wikipedia.org/wiki/Flynn_effect Though it's questionable as to how IQ is related to actual intelligence over time.
    - 19eightyfour 9y ago
      Agree with that too. Maybe IQ is out of date? Also, the huge effect that all these new technologies must be having is surely not going to be factored into that.
  - 19eightyfour 9y ago
    I agree. People are also arrogant. We like to look back and think "how quaint" and we are better than them. It's humbling to think that we actually weren't. But it's probably not very adaptive to think that way. Since, believing we are better now, by falsely diminishing the whatever metric of the past, probably helps us keep going forward to create the bright future. Hopefully without repeating the mistakes of the past tho! Even that phrase "mistakes of the past" is telling, right? I mean it's not like, IMHO, you hear a similar amount of talk about, "the brilliance of the past", except it a sort of quaint, dismissive way: "oh, look, plumbing in ancient Rome, weren't they sort-of clever!"
    - jamesrcole 9y ago
      I think that it's fair to say that, in the past, people were (overall) a lot more ignorant about the character of the world and the universe, and that this led them to (overall) believe a lot more incorrect things. This is not to beat our chests in a "we're better" kind of fashion, but just to acknowledge that we have the benefits of the knowledge that people who came before us built.
natch 9y ago
Came here to say this. https://en.m.wikipedia.org/wiki/I_Ching https://en.m.wikipedia.org/wiki/I_Ching And as you know each hexagram was composed of two groups (trigrams) of three pairs. Similar to bytes and nibbles. Even though the superstition is unfounded, the trigrams could combine in any way to refer to concepts and expositions that could give a lot of food for thought. Quite a nice system and very elegant because of its basis in binary numbers. Edit: and I think you meant to say 1000 BC, not AD. In short a LONG time ago!
- 1001101 9y ago
  I was thinking of the Shao Yang (binary) arrangement, rather than the original King Wen arrangement, which would have been created during SY's lifetime (1012-1077). This is the version Leibniz examined [1]. But, yes, as you point out, if you go back to the King Wen version, even earlier. If only Leibniz was around today! [1] https://www.jstor.org/stable/1399337?seq=1#page_scan_tab_contents https://www.jstor.org/stable/1399337?seq=1#page_scan_tab_con...
handojin 9y ago
The I Ching hexagrams don't represent numbers. They're not ordered numerically and any of the traditional sequences and they're not used to perform arithmetic. The individual lines represent (AFAIK) 2's, 6's, 3's, and 9's depending on whether the line is male or female, moving or static. The correspondence between the hexagrams and the binary numbers nothing more than a necessary consequence of the representation chosen: the number of states of a string of 6 characters in a system of 2 symbols.

19eightyfour 9y ago

That is weird, right? Where did the precise use of broken and unbroken lines, in all the combinations necessary to represent binary 0 to 63 originate? It's a compact representation to label 64 different things using 6 "sticks"...but why did they do it like this? Were they aware of binary numbers?

MBCook 9y ago
It's not that hard. If you limit yourself to two symbols and try to think of ways to represent as many things as you can, you just have to keep adding additional columns/rows. As long as you realize that you can use different combinations of the "first" spots with each additional spot you quickly come up with what is effectively binary. I distinctly remember in early middle school trying to think of ways to enumerate things where you could never confuse one for the other based on the symbols. I was using base 10 because I wanted to store the combination of things in a variable in BASIC on a program I was trying to write on my calculator. As I was playing around on paper I came up with a great system. I would only use one and zero because you could always tell them apart, where as if you use the other numbers then you can't tell if you meant five or if that five was a combination of two and three. When you needed to make another thing and you were out of places you would just add a digit on the left. Then about five seconds later it hit me that what I had was just binary. All of a sudden the idea of using individual bits in masks to represent things (which I'd seen before) made a lot more sense.
jes5199 9y ago
the sticks are literal sticks, used in fortune telling. It's roughly equivalent to flipping a coin. If you had a game where you flipped six coins, and each combination had a different meaning, you would be close to inventing binary
chandler 9y ago
> ...but why did they do it like this? ...because people are smart. People have _always_ been smart, it's one of the defining attributes of humanity. It strikes me oddly (not specifically your comment, but the general color accompanying these kinds of articles)--there's a kind of ground assumption that by looking into the past, one sees nothing but a gradually descending IQ.
- pilsetnieks 9y ago
  > looking into the past, one sees nothing but a gradually descending IQ Actually, it quite literally might be so: https://en.wikipedia.org/wiki/Flynn_effect https://en.wikipedia.org/wiki/Flynn_effect Though it's questionable as to how IQ is related to actual intelligence over time.
  - 19eightyfour 9y ago
    Agree with that too. Maybe IQ is out of date? Also, the huge effect that all these new technologies must be having is surely not going to be factored into that.
- 19eightyfour 9y ago
  I agree. People are also arrogant. We like to look back and think "how quaint" and we are better than them. It's humbling to think that we actually weren't. But it's probably not very adaptive to think that way. Since, believing we are better now, by falsely diminishing the whatever metric of the past, probably helps us keep going forward to create the bright future. Hopefully without repeating the mistakes of the past tho! Even that phrase "mistakes of the past" is telling, right? I mean it's not like, IMHO, you hear a similar amount of talk about, "the brilliance of the past", except it a sort of quaint, dismissive way: "oh, look, plumbing in ancient Rome, weren't they sort-of clever!"
  - jamesrcole 9y ago
    I think that it's fair to say that, in the past, people were (overall) a lot more ignorant about the character of the world and the universe, and that this led them to (overall) believe a lot more incorrect things. This is not to beat our chests in a "we're better" kind of fashion, but just to acknowledge that we have the benefits of the knowledge that people who came before us built.

natch 9y ago

Came here to say this. https://en.m.wikipedia.org/wiki/I_Ching https://en.m.wikipedia.org/wiki/I_Ching And as you know each hexagram was composed of two groups (trigrams) of three pairs. Similar to bytes and nibbles. Even though the superstition is unfounded, the trigrams could combine in any way to refer to concepts and expositions that could give a lot of food for thought. Quite a nice system and very elegant because of its basis in binary numbers. Edit: and I think you meant to say 1000 BC, not AD. In short a LONG time ago!

1001101 9y ago
I was thinking of the Shao Yang (binary) arrangement, rather than the original King Wen arrangement, which would have been created during SY's lifetime (1012-1077). This is the version Leibniz examined [1]. But, yes, as you point out, if you go back to the King Wen version, even earlier. If only Leibniz was around today! [1] https://www.jstor.org/stable/1399337?seq=1#page_scan_tab_contents https://www.jstor.org/stable/1399337?seq=1#page_scan_tab_con...

handojin 9y ago

The I Ching hexagrams don't represent numbers. They're not ordered numerically and any of the traditional sequences and they're not used to perform arithmetic. The individual lines represent (AFAIK) 2's, 6's, 3's, and 9's depending on whether the line is male or female, moving or static. The correspondence between the hexagrams and the binary numbers nothing more than a necessary consequence of the representation chosen: the number of states of a string of 6 characters in a system of 2 symbols.