Data

Tools

Indexes

Applications

Experiments

Awesome

An open API service indexing awesome lists of open source software.

https://github.com/mindful-ai-assistants/lacan-psychoanalysis-math-graphs

𝚿 This repository offers a detailed exploration of Lacan's psychoanalytic theory with a focus on the **Graph of Desire** and associated *mathemes*. Included
https://github.com/mindful-ai-assistants/lacan-psychoanalysis-math-graphs

Last synced: 30 days ago
JSON representation

𝚿 This repository offers a detailed exploration of Lacan's psychoanalytic theory with a focus on the **Graph of Desire** and associated *mathemes*. Included

Awesome Lists containing this project

README 

          




## 
 𝚿 [Lacan’s Graph of Desire]() 𝛷 [Mathematical Formalization \& Computational Psychoanalysis]()  ⚤







[Jacques Lacan’s](), [ **Graph of Desire**]() (French: *graphe du désir*) is a central conceptual tool in his psychoanalytic theory. It is a **topological model** that graphically represents the intricate structure of human desire and subjectivity as mediated by language and the Symbolic Other . Developed across his seminars, notably in [*Les formations de l'inconscient* (1957-58)]() and [*The Subversion of the Subject and the Dialectic of Desire* (1960)](), the graph illustrates the "logical moments" of the speaking subject's formation, rather than a chronological development .








#### 
 [![Sponsor Mindful AI Assistants](https://img.shields.io/badge/Sponsor-%C2%B7%C2%B7%C2%B7%20Mindful%20AI%20Assistants%20%C2%B7%C2%B7%C2%B7-brightgreen?logo=GitHub)](https://github.com/sponsors/Mindful-AI-Assistants)








https://github.com/user-attachments/assets/7380e439-6547-400b-8ef1-d03d2f9bf2e0








## Table of Contents



- [Overview](#overview)

- [What’s Inside](#what’sinside)

- [Background on Lacan and Computational Psychoanalysis]()

- [Key Concepts](#key-concepts)

- [Recent Computational Formalization](#recent-computational-formalization)

- [Repository Structure](#repository-structure)

- [Installation](#installation)

- [Usage and Examples](#usage-and-examples)

- [Visualization of Lacanian Concepts Using Python]()

- [Limits of Formalization: Gödel \& Lacan](#limits-of-formalization-g%C3%B6dels-incompleteness-theorems-and-lacanian-theory)

- [Lacan’s Graph of Desire: A Comprehensive Visual Guide]()

- [Lacan and the Möbius Strip - Topology, Psychoanalysis, and Representation]()

- [Lacan’s Topological Figures Beyond the Möbius - TORUS]()

- [Resources and References]()

- [Contributing](#contributing)

- [Let the data flow... Ping Me !]()

- [License](#license)








## [Overview]():



This repository offers a deep exploration of [**Jacques Lacan’s psychoanalytic theory**](), focusing on the [**Graph of Desire**]() and associated [*mathemes*]() formal symbolic and topological representations of unconscious subjectivity and desire.



By linking psychoanalysis with symbolic mathematics and computational frameworks such as the [**Free Energy Principle (FEP)**](), this project bridges psychoanalysis, linguistics, cognitive science, and [**Humanistic AI**]().








## [What’s Inside]()



   - [**LaTeX/TikZ visualizations**]() of Lacan’s key graphs, including the Graph of Desire, with elements such as the [barred subject]()  ($\bar{\textdollar}$) , [Other]() ($A$), and [objet petit]() ($a$).



   - [**Python Jupyter Notebooks**]() - simulating Lacan-inspired dynamics using:



   - [**NetworkX**]() - graph nodes and edges visualization



   - [**Matplotlib**]() - plotting the vector of desire (horseshoe curve)



   - [**SymPy**]() - symbolic logic to express Lacan’s sexuation mathemes







### - [Theoretical discussions on]():







[-]() Alienation, truth, and transference



[-]() Embodiment and unconscious material formalization



[-]() Relationships between Lacanian theory and computational neuroscience



[-]() Latest research insights formalizing Lacan’s psychoanalysis with the **Free Energy Principle** approach:



[-]() Modeling the **Real, Symbolic, Imaginary (RSI)** triad as autonomous, interacting units minimizing free energy



[-]() Formalizing **desire** as dyadic synchronization of symbolic orders



[-]() Modeling **the Other** as an emergent triadic collective dynamic



[-]() Educational and research resources for linguists, psychoanalysts, AI researchers, and Humanistic AI data scientists.








## [Background on Lacan and Computational Psychoanalysis]()







> **Jacques Lacan’s Graph of Desire** graphically illustrates the interaction between language (signifying chain), desire, and subjectivity, entwining:







- The [**signifying chain**]()  ($S \to S'$), representing linguistic structure



- The [**vector of desire**](), a horseshoe-shaped curve intersecting the chain



- The [**barred subject**]()  ($\bar{\textdollar}$) , or the divided self mediated by language



- The [**Other**]()  ($A$), symbolic authority shaping subject formation



- The [**objet petit a**]()  ($a$), the elusive cause of desire perpetually sought







> Recent computational work (e.g., Heins et al. 2025) embeds these concepts within the **Free Energy Principle (FEP)** framework, modeling living and cognitive systems as minimizing prediction error or "free energy."







### [This formalization reveals]():



- Desire as [**generalized synchronization**]() between subjects’ symbolic orders



- The [**incompleteness and instability of the Other**](), aligning with Lacan’s maxim “the Other does not exist”



- How intersubjectivity and unconscious drives can be simulated as dynamical systems








## [Key Concepts]()



- [**Graph of Desire:**]() A structure showing how unconscious desire traverses and is shaped by language (the signifying chain), symbolic law (the Other), and internal division (the barred subject).



- [**Mathemes:**]() Lacan’s formal logic of key psychoanalytic notions, like sexuation, subjectivity, and the structure of desire.



- [**Symbolic Formalization:**]() Use of logic, topology, and graph theory to model the dynamics of the unconscious.







## [Recent Computational Formalization]()



A recent study ([Heins et al., 2025](https://doi.org/10.3389/fpsyg.2025.1574650)) **formalizes Lacanian psychoanalysis using the Free Energy Principle (FEP)**, providing computational simulations embodying concepts such as desire, the Other, and their dynamic relations.







### ⚡️ [Highlights]()







- Models the [Real](), [Symbolic](), and [Imaginary (RSI)]() orders as interacting autonomous systems minimizing free energy.



- Simulates structures like the [**Borromean knot**](), [**dyadic synchronization**]() (desire as metaphorical convergence), and [**triadic collective behaviors**]() (the Other as emergent).



- Demonstrates that the subject’s desire is ultimately the [*desire of the Other*](), and that [*“the Other does not exist”*](); mirroring [**Gödelian incompleteness**]().







> ***This work connects psychoanalysis, computational neuroscience, and [Humanistic AI](), supporting the rigorous modeling of subjectivity and unconscious processes.***







## [Repository Structure]()







| [Path]() | [Contents]() |

| :-- | :-- |

| `/latex/` | LaTeX/TikZ code for Lacanian graphs |

| `/notebooks/` | Jupyter Notebooks for visualization and math |

| `/python/` | Python scripts for simulation |

| `/references/` | Papers and psychoanalytic bibliography |







## [Installation]()







[1](). Install Python 3.8+ (Anaconda recommended).



[2](). Install dependencies:



```bash

# Installation of required libraries

%pip install networkx sympy matplotlib numpy

```



[3](). Launch Jupyter Lab or Notebook:



```bash

jupyter lab

```







[4](). Launch Jupyter and open the notebooks.







## [Usage and Examples]()







> This repository enables visualization and symbolic analysis of **Lacan’s Graph of Desire**:







### [Visualizing Lacan’s Graph of Desire]()



- Directed graph of key Lacanian nodes and symbolic edges

- Horseshoe-shaped vector of desire intersecting the signifying chain

- Annotation of barred subject, Other, desire point







### [Symbolic Formalization]()



- Logical symbolic formulas representing Lacan’s sexuation mathemes, e.g.:

    

    - [Masculine sexuation](): $\forall x \neg \Phi(x)$

    - [Feminine sexuation](): $\neg \exists x \neg \Phi(x)$







### [Computational Psychoanalytic Modeling]()



   [-]() Simulation scripts demonstrating basic dynamics inspired by Lacanian theory and FEP

   [-]() Potential to extend with interactive widgets or agent-based models for deeper research







## [Visualization of Lacanian Concepts Using Python]()



This section includes visualizations and symbolic computations of key Lacanian concepts: the [graph of desire}(), the [horseshoe curve}() (vector of desire), and [sexuation mathemes}().







### [Features]()



- [**Graph visualization**](): Using NetworkX and Matplotlib to plot key Lacanian nodes (barred subject, the Other, etc.) and edges (signifying chain, vector of desire).



- [**Horseshoe curve**](): Plotting the iconic vector of desire intersecting the signifying chain.



- [**Mathemes with SymPy**](): Generating and printing symbolic formulas, such as Lacan's sexuation mathemes (masculine and feminine formulas).








### 1. [Graph of Desire Visualization]()







```python

# Import libraries for graph visualization

import matplotlib.pyplot as plt

import networkx as nx



# Constructing the directed graph with Lacanian signifiers

G = nx.DiGraph()



# Define node positions for clarity in visualization

positions = {

    "$S$": (0, 0),        # Subject

    "$S'$": (4, 0),       # Barred subject

    "$A$": (1.8, 0),      # The Other

    "$d$": (2.7, 0),      # Desire

 _networkx_edges(G, positions, edgelist=[("$S$", "$S'$")], arrowstyle='-|>', arrowsize=20, edge_color='blue')

nx.draw_networkx_edges(G, positions, edgelist=[("$\\$$", "$d$"), ("$\\$$", "$A$")], style='dashed', edge_color='red', arrowstyle='-|>', arrowsize=15)



plt.axis('off')

plt.title("Simplified Lacan's Graph of Desire")

plt.show()   "$\\$$": (2.25, 1.5), # Barred Other (split subject)

    "$s(A)$": (2.1, 0.9)  # Signifier of the Other

}



# Add nodes to the graph

for node in positions:

    G.add_node(node)



# Add edges representing signifying chain and vectors of desire

G.add_edge("$S$", "$S'$")

G.add_edge("$\\$$", "$d$")

G.add_edge("$\\$$", "$A$")



# Plot the graph

plt.figure(figsize=(8,5))

nx.draw_networkx_nodes(G, positions, node_size=700, node_color='lightblue')

nx.draw_networkx_labels(G, positions, font_size=12)



# Directed edges with styles to distinguish types

nx.draw

```


















> ### [Explanation for Image Placement (Graph of Desire)]()








 

 This graph  depicts a simplified model of Lacan’s [*Graph of Desire*](), illustrating key structural relations in psychoanalysis:



 



- The [**horizontal line** represents the **signifying chain**](), the unfolding of linguistic meaning through signifiers from the subject ($S$) to the barred subject ($S'$), who is divided by language and unconscious processes.



 



- The [**curved vectors**]() from the barred Other, $\bar{\textdollar}$, toward desire ($d$) and the Other ($A$)) represent the [**vector of desire**](), the dynamic movement of the subject’s desire mediated by the symbolic Other.



 



- The [**barred subject**]() (the split Other) $\bar{\textdollar}$ symbolizes the subject split by the unconscious and language, emphasizing that subjectivity is never unified but always marked by a lack or division.



 


 

- This visualization captures how [desire and language are intertwined](): the subject’s desire is not autonomous but [arises]() in relation to the [Other’s symbolic]() position.








### 2. [Horseshoe Curve — Vector of Desire and Signifying Chain]()







```python

# Additional imports

import numpy as np



# Define the horseshoe curve (vector of desire)

t = np.linspace(0, np.pi, 200)

x = 2 + np.cos(t)

y = 1 + np.sin(t)



# Plotting signifying chain and vector of desire

plt.plot([0, 4], [0, 0], color='blue', label='Signifying chain')

plt.plot(x, y, color='red', label='Vector of Desire')



# Key points on the curve

plt.scatter([1.8, 2.7], [0, 0], color='black')



# Labeling the nodes with LaTeX

plt.text(1.8, -0.1, '$A$', fontsize=12, ha='center')

plt.text(2.7, -0.1, '$d$', fontsize=12, ha='center')

plt.text(2, 1.4, '$\\$$', fontsize=14, ha='center')



plt.title("Simplified Lacan's Graph of Desire: Horseshoe Curve")

plt.legend()

plt.axis('equal')

plt.axis('off')

plt.show()

```


















> ### Explanation for Image Placement (Vector of Desire and Signifying Chain)







This plot visualizes [*Lacan’s Graph of Desire*](), showing the interaction between two fundamental elements:







- The [**blue horizontal line**]() represents the **signifying chain**, the sequential unfolding of linguistic signifiers that constructs meaning in language over time.



- The [**red curved "horseshoe" line**]() depicts the [**vector of desire**](), a non-linear, metaphorical trajectory that crosses and intersects the signifying chain, embodying the dynamic nature of desire.



- Key points on the plot, such as **$A$ (the Other)** and **$d$ (desire)**, mark places where desire confronts and engages with the symbolic structure of language.



- The [**barred Other**]() $\left(\text{the split Other }\bar{\textdollar}\right)$  , positioned above symbolizes the split or divided nature of the Other as related to the subject.



  

This visual representation clarifies how, within Lacanian theory, desire is not a straightforward, linear process but one that loops and cuts across language’s structural order, highlighting the subject’s divided nature and the complex interplay between desire and signification.








### 3. [Symbolic Mathemes with SymPy]()







```python

# Import SymPy for symbolic logic

from sympy import symbols, Function

from sympy.logic.boolalg import Not



# Define symbolic variables

x = symbols('x')

Phi = Function('Phi')



# Print Lacan's sexuation mathemes formulas

print("Masculine sexuation formula: ∀x ¬Φ(x)")

print("Feminine sexuation formula: ¬∃x ¬Φ(x)")

print("\nSymbolic expressions:")



# Demonstrate function and negation using SymPy notation

print(f"Phi(x): {Phi(x)}")

print(f"Not Phi(x): {Not(Phi(x))}")

```







###  [The Output]() from the Code







```

Masculine sexuation formula: ∀x ¬Φ(x)

Feminine sexuation formula: ¬∃x ¬Φ(x)



Symbolic expressions using SymPy:

Phi(x): Phi(x)

Not Phi(x): Not(Phi(x))

```







>  ### Explanation for Output Placement







This output provides symbolic representations of [Lacan’s]() well-known [*sexuation formulas*]() in formal logical notation:









### 
  ♂︎  [**The Masculine Formula**]()







$$\Huge \forall x \neg \Phi(x)$$







```latex

\Huge \forall x \neg \Phi(x)

```



### 
 [Where]() the universal quantifier ($\forall\$) applies to all $x$, negating the property $\Phi$.








    ───────── ⚤ ─────────








###  
 ♀︎  [**The Feminine Formula**](): 







$$\Huge \neg \exists x \neg \Phi(x)$$







```latex

 \neg \exists x \neg \Phi(x)

```



### 
 [Negating]() the existence of any $x$ not having property $\Phi$.









[Both formulas]() express fundamental [logical structures]()  underlying Lacan's conceptualization of [sexuation]()  and [sexual difference]() , where $\Phi(x)$ can be read as the [phallic function]()  or [castration law]()  applied to the [speaking subject]()  $x$.







The SymPy expressions printed (`Phi(x)` and `Not(Phi(x))`) demonstrate how these [logical functions]() are represented [symbolically in Python]() , linking [Lacan's psychoanalytic]() theory with [formal symbolic logic]().









## [Limits of Formalization: Gödel’s Incompleteness Theorems and Lacanian Theory]()







This repository not only formalizes Lacan’s psychoanalytic theory, but also acknowledges the [**limits of all formal systems**]() as demonstrated by [**Gödel’s incompleteness theorems**](). No [axiomatic model—whether]()  in mathematics, logic, or psychoanalysis can be both complete and consistent. There will always be aspects of [reality]() (the “real” in Lacanian terms) that escape symbolic capture.







This parallel reinforces Lacan's insight that the unconscious and human subjectivity remain fundamentally open and incomplete, echoing the [**lack in the Other**]()  and the inherent boundaries of any modeling approach—critical for anyone working in [Humanistic AI]().







Below, I develop this parallel including the most relevant mathematical formalizations in LaTeX and its relation to Lacanian theory.







### 1. [Gödel and the limitations of formalization]():



Gödel proved that [**no sufficiently powerful formal system**]() (such as basic arithmetic) can be both *complete* and *consistent*.







### - [**First Incompleteness Theorem**](): There exist true propositions in arithmetic that can never be proven within the formal system itself.



Mathematically, given a consistent and recursively enumerable formal system $F\$ that includes arithmetic, there is a statement $G_F\$ (called the Gödel sentence) such that:








$$\Huge

F \not\vdash G_F \quad \text{and} \quad F \not\vdash \neg G_F

$$







```latex

\Huge

F \not\vdash G_F \quad \text{and} \quad F \not\vdash \neg G_F

```







### 
 [That is](): neither $G_F\$ nor its negation can be proven in $F\$, rendering $F\$ incomplete.








### 
   ───────── ⋆⋅[✮]()⋅⋆ ───────── 








### - [**Second Incompleteness Theorem**](): The system $F\$ cannot prove its own consistency, formally expressed as:








$$\Huge

F \not\vdash \mathrm{Consis}(F)

$$







```latex

\Huge

F \not\vdash \mathrm{Consis}(F)

```







### 
 [Where](): $\mathrm{Consis}(F)$ is a formula within the system expressing the “non-contradiction” of $F\$.








### 2. [Conceptual translation to Lacan and psychoanalysis]():







The psychoanalytic subject’s experience, according to Lacan, is marked by a structure of *lack*, which he calls the [**“Real”**](); a domain resisting full symbolization by the [**“Symbolic”**]()  (language and its rules). This Real is precisely what escapes any complete and closed formalization.







 - The [**“Other”**]()  (in Lacanian theory, the symbolic instance) contains a fundamental [*lack*]() , a kind of **structural incompleteness** that prevents the subject from being perfectly represented or captured in totality.



 - Just as [Gödel]()  shows that every formal system has inevitable limits (undemonstrable truths), the Lacanian subject experiences their unconscious as marked by an [“excess” beyond the symbolic]() , not fully representable or accessible.



 






### 3.[ Formalism and the subject's experience — mathematical/psychoanalytic parallel]():







We can express analogously the paradox of subject and formal system as follows:







- Let $F\$ be a formal system representing the Symbolic (language, culture, law).



- The sentence $G_F\$, true but unprovable in $F\$, corresponds to the Lacanian Real;  that which is always beyond the reach of the symbolic, not captured by language.








### 4. [Note on the syntactic construction of the Gödel sentence]():



The [Gödel sentence]()  $G_F\$ is constructed to indirectly refer to itself via arithmetic on natural numbers, employing a method of arithmetization of syntax (Gödel numbering), which enables encoding meta-mathematical statements inside the formal system itself.







## [Lacan’s Graph of Desire: A Comprehensive Visual Guide]()







Jacques Lacan's [**Graph of Desire**]() (or Graph of the Analytic Act) is a formal, schematic diagram he developed to represent the complex structures of subjectivity, desire, and the unconscious in psychoanalysis. It elaborates how desire functions in relation to the subject and language, revealing the dynamics of the split or barred subject (*$*) as situated within the Symbolic order.







### ➠ [Key points about Lacan's Graph of Desire include]():







- [It is a **"flattened" representation** of the crossing between two pathways](): the [**signifying chain**> (sequence of linguistic signs with meaning) and the **vector of desire** (the will or volition of the subject, expressed metaphorically and timelessly). The graph thus captures how desire operates amidst language and symbolic limitation.







- [The barred subject (*$*) is central](): it designates the split or conflicted human subject formed in the process of individuation beginning in infancy through the loss of symbiotic relations (with the mother), mediated by language. Desire is thus marked by a fundamental **lack** or absence never fully satisfied, only substituted by other objects.







- [The points where the vector of desire and the signifying chain cross](), represent moments of **Freudian double inscription**, where conscious and unconscious meanings coincide and constrain each other, reflecting the structure of psychoanalytic experience.







- [Lacan used several versions or steps of these graphs](), beginning from a basic "elementary cell," progressively elaborating the formation of the ego ideal and the subject's positioning relative to the Other and the Law; notably, the [“Name-of-the-Father”]() as [symbolic prohibition]().







- The graph illustrates how [**desire is always the desire of the Other**]() , meaning the subject seeks desire validated or recognized by the [symbolic Other]() — the social or linguistic Otherness that shapes subjectivity.







- [Lacan describes the subject]() not as a fixed, autonomous individual but as constituted through language and desire, with the ego as an object rather than true subjectivity.

  





In sum, Lacan’s Graph of Desire is a conceptual and formal tool to map the [relations]() between [language]() , [desire](), the [split subject](), and the [unconscious]() in psychoanalytic theory. It is often studied to understand how [desire]()  eludes fulfillment while [structuring]()  human subjectivity within [symbolic and imaginary dimensions]().








| [Aspect]()                    | [Description]()                                                                                  |

|----------------------------- |--------------------------------------------------------------------------------------------------|

| [Signifying chain (S → S')]()   | Linguistic progression from sign to meaning with duration                                         |

| [Vector of desire]()              | Metaphorical, atemporal representation of the subject’s volition and desire (*$*)                 |

| [Split subject (*$*)]()         | Barred subject formed by loss and language mediation, fundamentally divided and lacking          |

| [Crossings]()                   | Points of Freudian double inscription—interaction of conscious and unconscious meaning             |

| [Ego ideal and Law]()           | Formation of ego ideal and imposition of symbolic Law (e.g., Name-of-the-Father) outline subject’s position |

| [Desire]()                      | Always the desire of the Other; subject’s desire mediated by symbolic Other                        |







### [Lacan constructs the complete graph in **four stages**](), each building in complexity to illustrate the full structure. These stages are pedagogical devices, as the graph always exists as a whole.








###  1.[ The Elementary Cell]()



The [**Elementary Cell**]()  is the foundational stage of the Graph of Desire. It depicts the fundamental intersection of the [**signifying chain**]()  and the [**vector of the subject's intentionality**]()  (or desire).







- The [**horizontal line**]()  represents the *[*diachronic signifying chain**]()  ($S \to S'$), which moves from a linguistic sign to a signification or meaning. It has duration.



- The [**horseshoe-shaped line**]()  represents the [**vector of the subject's desire**]()  ($\$$), which is expressed metaphorically and has no duration. This vector starts from the split or barred subject ($\$$).



The [**double intersection**]()  of these two lines illustrates the **nature of retroaction or [**point de capiton***]()  , where [meaning]()  is determined [retroactively]()  .







 

 [**Visual Representation of the Elementary Cell:**]()







```

      _________ Vector of Subject's Intentionality ($)

     /         \

    /           \

   /             \

S ------------------ S' Signifying Chain

```









###  2. [The Elementary Cell with the Locus of the Other]() (A)







In the second stage, Lacan introduces the **locus of the Other (A)**, representing the big Other, the symbolic order, or the "treasure trove" of signifiers. This stage illustrates how the message, located at $s(A)$, is retroactively punctuated and given meaning by the Other (A).







- The [**prelinguistic mythical subject of pure need**]()  (represented by a triangle, or a rudimentary subject prior to language) must pass through the ["defiles of the signifier"]() .



- This process produces the [**divided subject**]()  $\bar{\textdollar}$, which is the result of entering the Symbolic order.







[**Visual Representation adding**]() A and $s(A)$







```

      _________ Vector of Subject's Intentionality ($)

     /         \

    /           \

   /----(s(A))---\

S ---------A------- S' Signifying Chain

```









### 3. [The Graph of Demand and Desire]()







This stage introduces the concepts of **demand** and **desire**, distinguishing them within the graph. It often resembles a question mark, as desire, for Lacan, is fundamentally a question.







- [It features two new symbols]():

    

    - The **lozenge ($\diamond$)**, or *poinçon*, which indicates a relation of envelopment, conjunction, or disjunction between two elements. Lacan describes it as "soldering together" the mathematical symbols for "less than" and "more than".

    

    - A lowercase **$d$** which indicates **desire**.

 

  



- This stage highlights how **demand** (often represented as uppercase D or as the subject's relation to demand, `(D)` ) and **desire** are interconnected and shaped by the Other. Desire, for Lacan, is always "the desire of the Other" ($A \diamond d$).







[**Visual Representation (adding Demand and Desire)**]():







```

      _________ Vector of Subject's Intentionality ($)

     /         \

    /           \

   /----(s(A))---\

S ---------A------- S' Signifying Chain

        /   \

       /     \

      d ----- D (Demand)

```









### 4. [The Complete Graph]()







The **Complete Graph** is a complex [topological representation]() with [**two signifying chains**]90 and multiple intersecting vectors. It integrates all the elements from the previous stages and adds new dimensions related to the [**unconscious**](), [**enunciation**](), [**jouissance**](), and the [**drive**]().







- The [**lower chain**]() (from signifier to voice) represents the [**conscious signifying chain**](), the **level of the statement**.



- The [**upper chain**]() (from jouissance to castration) represents the [**signifying chain**]() in the unconscious**, the [**level of the enunciation**]().



The structure is duplicated, with the upper part of the graph mirroring the lower part's structure. This complete graph forms a series of interacting "systems" (both inter-subjective and intra-subjective), where elements can participate in multiple systems simultaneously.







[**Key elements in the Complete Graph:**]():



- [**$s(A)$**](): The signified of the Other, the subject's position in relation to the meaning produced by the Other.



- [**$I(A)$**](): The Ego Ideal, the point from which the subject is seen as likable by the Other.



- [**(D)**](): The subject in relation to demand, leading to the Drive.



- [**$a$**](): *Objet petit a*, the object-cause of desire, linked to jouissance.







[**Visual Representation (Conceptual overview of the Complete Graph):**]()







```

      _______ (Unconscious Chain: Jouissance --> Castration) _______

     /                                                              \

    /     I(A)                Desire (⋄a) = the unfulfilled shimmer that drives all.

   /     /                                                         \

  /     /                                                           \

 d     /       S(A)                                                \

 |    /       /                                                       \

 |   /       /                                                         \

 \  /       /                                                           \

  \_______/________ Vector of Desire ____________________________________\

           |

           |

           |

           | ($) -------- S(A) ------ S' (Conscious Chain: Signifier --> Signification)

           |

           |

           |

           m (ego)

```







***[Note](): The actual complete graph is much more intricate with specific paths and loops for fantasy, drive, etc., forming a dense network of vectors and points.***








## The [Torsion]():







https://github.com/user-attachments/assets/9915b330-9845-421d-97ca-6f733d455db0



##### 🎶  ***Creation  by Fabi***  🖤








##   Lacan and the [Möbius Strip - Topology, Psychoanalysis]() and Representation







#### [Möbius Strip in Lacan's Psychoanalysis]():







The Möbius strip is a mathematical surface with a half-twist, forming **one continuous side and one boundary**.







- [**Key property:**](): Traveling on it covers what seem to be “two sides” without lifting the pen since it really has only one side.



- [**Lacan’s metaphor:**](): It illustrates how the subject’s inside and outside, or self and Other, are continuous and intertwined rather than opposed.



- [**Psychoanalytic significance**](): Desire, language, and unconscious processes similarly cross and fold these boundaries, making the psyche a dynamic and paradoxical structure.



- [**Clinical insight**]()(: Symptoms and identifications are points of twist or cut in this continuous flow, and psychoanalysis is a process of traversing and understanding this topology.








## [Lacan and the Möbius Strip: Topology, Psychoanalysis, and Representation]()







### What Is the [Möbius Strip]() ?



The **Möbius strip** is a topological surface formed by taking a strip of paper, giving it a half-twist, and joining the ends to make a loop. It has the remarkable property of having only _one side_ and _one continuous edge_. If you trace along the surface, you pass “both sides” without ever crossing an edge, revealing a space where the usual binary oppositions (inside/outside) collapse.

















### [Lacan’s Use of the Möbius Strip in Psychoanalysis]()







Jacques Lacan employed the Möbius strip as a privileged topological metaphor to represent the paradoxical, continuous, and intertwined nature of subjectivity, language, and desire:







- [**Ambiguity of Inside and Outside**](): The strip illustrates that the subject is not clearly divided between internal (self, unconscious) and external (Other, language, society); these aspects are dynamically continuous, as shown by the single surface of the Möbius strip.







- [**Structure of Desire**](): Desire is conceptualized as an ongoing loop traversing between the subject and the Other, never fixed to one side but constantly moving and folding back on itself, like traveling along the Möbius strip’s twisted surface.







- [**Unconscious as Language Structure**](): Just as the strip defies simple spatial logic, the unconscious defies the neat inside/outside division. The strip symbolizes how language and subjectivity are folded and inseparable, with “signifier” and “signified” linked in a continuous chain.







- [**Clinical Implications**](): Symptoms and psychic phenomena can be seen as “twist points” or “cuts” on the strip that break but also constitute the continuous flow of signification and desire. The analytic process can be conceptualized as “traversing the fantasy,” moving along the strip beyond fixed identifications.







- [**Relation to Other Topological Figures**](): Lacan’s broader topological toolbox includes the torus, Klein bottle, cross-cap, and Borromean knot, each representing other complex structures in psychoanalysis (imaginary, symbolic, real, and their interrelations).







### - [Representations of the Möbius Strip]()







Below are several simple ASCII art representations of the Möbius strip, suitable for Markdown environments, along with brief interpretative comments.







#### [1](). Basic Twisting Band (Simplified Shape)







```text

        ______

      /       \

     /         \

    /           \

   /             \

  /               \

  \               /

   \             /

    \           /

     \         /

      \_______/

       \     /

        \   /

         \_/

```



> *A twisted loop showing the Möbius strip’s single continuous surface with a half-twist.*







#### [2](). Stylized Loop with a 







```text

    +---------+

    |         |

    |         |

    |     +---|----+

    |     |        |

    +-----|--------+

          |

          |

```



> *A conceptual loop with a twisting line, indicating orientation reversal in the Möbius strip.*







###  [Möbius Strip](): Traversing the Sides of Subjectivity







[**Clinical Example**]():



A patient insistently oscillates between feeling “inside” (withdrawn, self-absorbed) and “outside” (dependent or seeking validation). Every time therapy seems to clarify a boundary—such as understanding a symptom as “mine” versus “the Other’s”—the patient’s experience slips to the opposite side, as if the distinction melts away.







- [**Implication**]():

The Möbius strip helps the analyst and patient see that psychic boundaries (self/Other, conscious/unconscious) are fundamentally twisted and continuous. Recognizing this can reduce frustration in analytic work and foster acceptance of ambiguity within identity and desire, making the analytic process less about achieving “closure” and more about navigating complexity.







#### [3](). Zigzag Path Representing Half-Twist







```text

Start ->———/\/\/\/\————< Back to Start

```



> *The zigzag symbolizes the half-twist in the strip before returning to the start point.*








## [Lacan’s Topological Figures Beyond the Möbius Strip]()







### [1](). Torus







[**The torus**]() is a surface shaped like a doughnut: a circle rotated around an axis outside itself.



Lacan uses the torus (toro in Portuguese) as an important topological figure in his psychoanalytic theory to represent the structure and dynamics of desire and subjectivity. The torus is a doughnut-shaped surface characterized by its ring form with a central hole, and it symbolizes continuity with a central void that is conceptually linked to Lacan’s notion of lack or "object a", the unattainable object of desire.
















### [Torus](): 







- [Key points about the torus in Lacan’s theory]():




  



 * Continuity and Repetition: The torus illustrates the cyclical, repetitive nature of desire and demand. Desire is not fixed but circulates like movement around the ring of the torus.







  * Central Hole as Lack: The hole in the middle of the torus symbolizes a fundamental void or lack that is the core of subjectivity and desire — this lack is what drives the subject’s continual movement around desire.







  * Relation to Demand and Desire: Lacan used the torus in his seminars, particularly in "Function and Field of Speech and Language" and "The Seminar on Identification," to explain the dynamic relationship between demand (what    is asked from the Other) and desire (what remains unfulfilled and elusive).

 

 


  

  * Topological Reversal: In psychoanalytic dynamics, the torus can be “turned inside out” or “reversed,” reflecting phenomena like identification and the shifts in psychic boundaries between inside and outside.



 


   

  * Incorporation in Clinical Theory: The torus helps conceptualize how the subject’s structure is organized around a dynamic of emptiness and repetition, which informs psychoanalytic treatment and interpretation.








[Below is a simple ASCII representation of the torus in Markdown format]():







```text

     ****

   *      *

  *        *

 *          *

 *          *

  *        *

   *      *

     ****

```



> *The torus is depicted as a circular ring, representing a continuous surface with a central hole. In three dimensions, it looks like a doughnut.*







#### [Lacanian Use]():



- [**Psyche and Repetition**](): The torus shows cyclical structures in the psyche—repetition, desire circling a central void (the unattainable object a). It embodies continuity, yet the hole at its center represents the structural lack at the heart of subjectivity.







####  [Torus](): The Circuit of Repetitive Desire







[**Clinical Example**]():

A patient repeatedly finds themselves in similar relational patterns—always circling back to the “same” dysfunctional relationship or self-defeating job scenario, despite conscious efforts to change.







- [**Implication**]():

The torus enables the analyst to conceptualize repetition compulsion not merely as a linear return of the same but as circulation around a structural hole: the *object a* (cause of desire). Analytically, interventions can focus less on content and more on how the repetitive pattern structures the patient’s world, potentially opening new routes of desire that recognize, rather than deny, lack.









### [2](). Klein Bottle







[**The Klein bottle**]() is a non-orientable surface with no inside or outside. It cannot be fully represented in three dimensions without self-intersection.

















### [Klein Bottle](): 







```text

    ___

   /   \

  /     \

 /       \

|         |

 \       /

  \_   _/

    | |

    | |

```



> *This shows a simplified bottle whose neck curves back into its side, passing "through" the surface (as possible only in four dimensions).*







### [Lacanian Use]():



- [**Beyond Distinction:**](): The Klein bottle challenges the division between inside and outside even more radically than the Möbius strip. For Lacan, it models the way language and subjectivity suffuse each other, erasing fixed boundaries between self and Other, conscious and unconscious.



 
 



#### [Klein Bottle](): Inside/Outside Dissolution and Psychotic Structure



 
 



[**Clinical Example**]():

A psychotic patient lacks clear boundaries between thoughts and reality—their inner speech “leaks” into the outside world in the form of hallucinations or delusions. Ordinary distinctions between “what I think” and “what happens” collapse.



 
 



- [**Implication**]():

The Klein bottle dramatizes this radical dissolution of boundaries. Clinically, this perspective warns against insisting on rigid ego boundaries or interpretation “from the outside.” Instead, the analyst might aim to support the emergence of some degree of distance or separation that helps the patient locate themselves in relation to language and reality.









### [3](). Borromean Knot







The [**Borromean knot (or rings)**]() consists of three interlinked rings, no two of which are directly linked; removing one unlinks the rest.

















### [Borromean Knot]():







```text

    O    O

     \  /

      \/

      /\

     /  \

    O----O

```



> *Each circle is linked with the other two, forming a structure where all are interdependent, yet no pair is directly connected.*







### [Lacanian Use]():



- [**Registers of the Psyche:**](): The Borromean knot models the Real, the Symbolic, and the Imaginary—Lacan’s three primary registers. The subject’s coherence depends on the way these link together; if any are cut, psychic coherence collapses. It’s central in Lacan’s late work to show the topology of psychic structure and the impact of trauma or symptom.







#### [Borromean Knot](): Psychotic Breaks and Knotting Symptoms







[**Clinical Example**]():

A patient who previously functions well suddenly experiences a psychotic break following trauma, with symptoms such as delusions, hallucinations, or disintegration of meaning. There is a sense of unraveling or falling apart that the patient cannot control.







- [**Implication**]():

The Borromean knot models how the Real, Symbolic, and Imaginary need to be linked for psychic stability. A “break in the knot” after trauma explains the onset of florid symptoms. Analytically, the goal may be less about restoring the previous state and more about helping the patient re-knot these registers—sometimes by supporting the formation of a stabilizing symptom (“sinthome”), such as a creative activity, relationship, or belief.








## [Summary Table](): Lacan’s Main Topological Figures







| Topological Figure | ASCII/Markdown Example | Psychoanalytic Meaning |

| :-- | :-- | :-- |

| [Möbius Strip]()   | `see above` | Inside/outside continuity, paradoxical subjectivity             |

| [Torus]()          | `see above` | Repetition, the circuit of desire around lack (object a)        |

| [Klein Bottle]()   | `see above` | Radical erasure of inner/outer, entwined subject/Other          |

| [Borromean Knot]() | `see above` | Interlocking of Real, Symbolic, Imaginary (fundamental linkage) |







### [Explanations]():







- **Topology** allows Lacan to formalize paradoxical structures—such as lack, twisting, and interconnectedness—that define the unconscious, language, and the clinic.

- Each figure represents not just a metaphor but a dynamic, formalized mode of how subjectivity and desire operate, especially where simple spatial metaphors break down.








## [Resources and References]()



- **Heins, R., et al. (2025).** *Formalizing Lacanian psychoanalysis through the free energy principle: computational simulations of desire, the Borromean knot, and the Other.* Frontiers in Psychology. [DOI:10.3389/fpsyg.2025.1574650](https://doi.org/10.3389/fpsyg.2025.1574650)



- **Romanowicz, T., \& Moncayo, J. (2015).** *Going beyond Castration in the Graph of Desire.* The Letter, Issue 58.



- **Stanford Encyclopedia of Philosophy:** [Lacan](https://plato.stanford.edu/entries/lacan/)



- **TikZ Lacanian mathemes repo:** [gjoncas / Lacan-Mathemes](https://github.com/gjoncas/Lacan-Mathemes)



- **Seminar Videos:** Search Dr. Dan Collins’ Lacan lectures on YouTube







### [Links]()



- 1: https://arxiv.org/pdf/2309.06707.pdf



- 2: https://pubmed.ncbi.nlm.nih.gov/40547582/



- 3: https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2025.1574650/full



- 4: https://pmc.ncbi.nlm.nih.gov/articles/PMC12180394/



- 5: https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2025.1574650/xml



- 6: https://www.bohrium.com/paper-details/an-active-inference-model-of-lacanian-psychoanalysis/909142426050887756-108563



- 7: https://www.bohrium.com/paper-details/collective-behavior-from-surprise-minimization/987893228252430340-107588








## [Contributing]()



Contributions welcome from psychoanalysts, AI researchers, linguists, and data scientists! Please:



- Suggest new simulations or visualizations

- Extend symbolic formalisms

- Develop educational materials or case studies



Submit issues or pull requests via GitHub.








## 💌 [Let the data flow... Ping Me !](mailto:fabicampanari@proton.me)



#### [Contact and Support]()



- For notebook files, detailed tutorials, or enhanced visualizations, please reach out.

- Interested in Python notebooks simulating these dynamics or advanced Humanistic AI models? Just ask!







#### 
  🛸๋ My Contacts [Hub](https://linktr.ee/fabianacampanari)







### 
 







  ────────────── 🔭⋆ ──────────────



 ➣➢➤ Back to Top 







#



##### 
Copyright 2025 Mindful-AI-Assistants. Code released under the  [MIT license.](https://github.com/Mindful-AI-Assistants/lacan-psychoanalysis-math-graphs/blob/28d9178584b831679dec129fb0aa040203ce0e9e/LICENSE.md)