The greatest possible division: 1/e — Işık Barış Fidaner

When I explained the meaning of Lacan’s torus [1] I made a mathematical claim: The fraction 1/e ≈ 37% achieves the greatest possible division of a combination. Now let’s ask two questions: what does this mean, and why is this the case?

What does this mean?

A practical example: Say you are a political party who aims to break the present political consensus and lead the people to a new consensus. Until you gain 37% support you remain to be a divisive force, an opposition party. But when you surpass 37% your party becomes a force that actively unites the people. According to common sense this limit should be 50% but mathematically this is not the case.

A smaller scale example: Say there is a political cell of three comrades and the secret police aims to break this cell. If the police manages to turn just one of the three partisans against the other two, the police have almost reached 37% infiltration and they have already succeeded in breaking the cell. When one out of three is lost, the link of the remaining two gets broken too. This is also the principle of Lacan’s Borromean knot where any two of the three rings hold together only thanks to the third ring.

This also works the other way around: Say there is a team of three co-workers and you are a political organizer. If you manage to convince just one of the workers, this act already divides the remaining two. If you manage to convince a second worker, you have successfully established a new unity because the third person is practically isolated. It’s like Hegel’s idea of repetition: While the first occurrence serves as contrarian evidence that is still contingent and accidental, the second occurrence serves as the proof that retroactively confirms and validates the necessity of the first occurrence.

Why is this the case?

Why is 1/e the greatest possible division? This claim is not based upon any external tests or statistics, it is just a direct mathematical consequence of Shannon’s entropy equation. Let me explain.

According to information theory and probability theory, a coin toss contains exactly 1 bit of information. This sounds like an innocent neutral statement that is objective and scientific, but it carries an implicit ideological model of the symbolic universe.

Formally, the geometric shape of the coin and the physical act of tossing ensures that Heads and Tails naturally exclude each other as two possible events, staging a competition between two Master-Signifiers. As to their content, the coin’s two-faced shape resembles an imaginary identification in the mirror stage as if Heads and Tails were meant to render each other meaningful in a dual relation.

In practice, this means that the gain of one implies the loss of the other: Formally, when the coin yields Heads, it didn’t yield Tails. As to their content, the probabilities of Heads and Tails must sum up to the “full probability = 1” which represents an inevitability. In both levels, Heads and Tails are supposed to balance each other out and achieve an innocent neutrality. This model serves as the scientific basis for the design of computers, where a bit yields either 1 or 0, traditionally illustrated by a light bulb that’s either on or off.

The seemingly innocent image of the coin toss that decides between Heads and Tails is in fact ideologically loaded because it embodies the disavowal of another choice between the useful and the useless. The real 1 bit of information resides in the decision to keep or dismiss an entity based on its usefulness. Such information gets processed both in consciousness and in the unconscious, and it marks their boundary. Thus the two-sided coin model secretly presupposes a clear-cut distinction between the useful and the useless. This leads to a mode of thinking that always tries to grab key elements and extract resources, which I call “decryption” [2].

But as the idiom goes, Heads and Tails, useful and useless “are two sides of the same coin”. Here one can evoke Lacan’s Möbius strip [3] and say that what appears like two separate sides are in fact parts of one and the same surface. To transform the active mode of thinking into a “decipherment” of truth, one needs the model of a one-sided coin [4]. We get the one-sided coin when we say “Tails does not exist.” In contrast to the two-sided coin whose faces stage the competition between exclusive signifiers, the one-sided coin stages the enactment of a unary trait [5]. The two-sided regular coin enacts two signifiers that divide a subject between themselves. The one-sided coin instead enacts the bar of the subject’s division as such.

Finally let’s cover the maths. Here is Shannon’s entropy equation that computes the information or entropy that is contained in a two-sided coin:

H(p) = -(p)log(p)-(1-p)log(1-p)

The first term is for Heads and the second term is for Tails. When p = 1/2 this entropy takes its maximum value, which is 1 bit.

To obtain the equation for the one-sided coin, we simply remove the second term that was intended for Tails:

H(p) = -(p)log(p)

When p = 1/e this entropy takes its maximum value, which is also 1/e.

Işık Barış Fidaner is a computer scientist with a PhD from Boğaziçi University, İstanbul. Admin of Yersiz Şeyler, Editor of Žižekian Analysis, Curator of Görce Writings. Twitter: @BarisFidaner


[1] See “Lacan’s Torus is the 3D Spatial Approximation of the Combinatorial Unworld”

[2] See “Decryption and Decipherment”

[3] See “The Möbius Strip is an Island”

[4] See “One-sided coin”

[5] See “One-sided Coin is the Unary Trait”


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