The Minuet of Marching Drill — Homotopy Analysis Based on Topological Behavioral Science

Overview

This article is a satirical piece adapted from an essay of mine. Its creation was inspired by an article criticizing marching drill written by my classmate Bao Haimo — with its incisive perspective on deconstructing the marching drill scene, Bao’s article sparked extensive discussion in the class and served as the inspiration for this work. Centered on using mathematical concepts such as algebraic topology, category theory, and geometric group theory as playful carriers, this article conducts an interesting deconstruction and teasing of the campus marching drill scene. All content is for entertainment purposes only, does not possess any serious academic value, and does not constitute a formal evaluation or interpretation of relevant mathematical theories or educational scenarios. Readers may view it as a fun essay and need not examine it against the standards of academic rigor.

Introduction

The renowned social critic Mr. Bao Haimo once pointed out incisively in his article "Morning Exercise That Outrages Both Humans and Gods": "Morning exercise turns every student into a CS." This assertion is often superficially interpreted as a slang trick, but if we turn our attention to the profound field of algebraic topology, we will find that "CS" here actually refers precisely to "Configuration Space". This is not only a geometric metaphor, but also a profound social topological model. Its core is the mathematical deconstruction of "individual alienation" and "power discipline" in the marching drill scene. In the following, through the strict definition of mathematical concepts, axiom laying, and logical deduction, we will gradually disassemble the topological structure and algebraic laws behind marching drill, expose the essence of its formalism and the logic of power operation, and explain the in-depth academic value of Mr. Bao Haimo’s assertion.

I. Configuration Space and the Tyranny of Quotient Space

To rigorously describe the alienation process in marching drill where "individuals are erased and the collective is deified", it is necessary to first establish a complete mathematical formalization system, accurately define and deduce core topological and algebraic concepts, and clarify how the marching drill system achieves dual discipline of individuals’ bodies and wills through rigid topological rules, algebraic operations, and group actions. All mathematical constructions correspond to the reality of "the collective overriding the individual" and "discipline being superior to personality", deconstructing the absurdity of formalism with mathematical rigor.

(1) 2-Dimensional Compact Manifold with Boundary

The physical carrier of marching drill — the school playground — is the basic space for the entire topological analysis. Its mathematical modeling can be defined as a 2-dimensional compact metric space with boundary, denoted as $M$. From a topological perspective, the playground is compact, meaning that any open cover has a finite subcover, which corresponds to the reality that the playground has a limited area and marching drill activities are confined to a fixed area; at the same time, the playground has clear boundaries (such as runway edges, playground fences, etc.), so $M$ is a manifold with boundary, denoted by $\partial M$. The existence of the boundary limits the range of individual movement during marching drill, making it impossible to move beyond this boundary. From a metric perspective, a distance function $d: M \times M \to \mathbb{R}_{\geq 0}$ can be defined on $M$, satisfying the three axioms of a metric space: positivity (${d(x,y)=0}$ if and only if ${x=y}$), symmetry (${d(x,y)=d(y,x)}$), and triangle inequality (${d(x,z) \leq d(x,y)+d(y,z)}$). This distance function corresponds to the actual spatial distance between individuals on the playground, laying the foundation for the subsequent definition of the configuration space.

(2) Ordered Configuration Space and Fat Diagonal

Assume a class consists of $k=62$ individuals. Treat each student as an independent physical individual, and define the individual set of the class as an ordered set $N = \{1, 2, \dots, 62\}$, where each element corresponds to a student. The order reflects the uniqueness of individuals in a free state (such as differences in height, posture, and movement habits). In a free state, each student can move freely on the playground $M$, and their state space can initially be regarded as the $k$-fold product space of $M$, i.e., $M^k = M \times M \times \dots \times M$ (with $k$ copies of $M$). Each point $(x_1, x_2, \dots, x_k)$ in the product space corresponds to a configuration of individual positions, where $x_i \in M$ represents the position of the $i$-th student on the playground.

However, it should be noted that as physical entities, students have non-overlapping exclusivity, meaning that no two students can occupy the same spatial position. This physical constraint is transformed into a topological constraint, requiring the elimination of all configurations with overlapping individuals from the product space $M^k$. That is, define the "Fat Diagonal" $\Delta = \{(x_1, \dots, x_k) \in M^k \mid \exists i \neq j, x_i = x_j\}$, where $\Delta$ includes all configurations where at least two students are in the same position. Based on this, the ordered configuration space of the class in a free state is formally defined as:$\text{PConf}_k(M) = M^k \setminus \Delta$

The core significance of the ordered configuration space $\text{PConf}_k(M)$ is that it retains the uniqueness and order of individuals. Each configuration point corresponds to a position distribution with individual differences, reflecting the autonomy of individuals in a free state, which is also the mathematical mapping of individual states before marching drill.

(3) Unordered Configuration Space and Permutation Group

The essential feature of marching drill is to erase individual differences and emphasize collective unity. From the perspective of upper observers (such as school leaders and physical education teachers), the uniqueness of individual students is ignored, and only the overall geometric shape and neatness of the class are concerned. This cognition is transformed into the action of a permutation group (symmetric group) $\Sigma_k$ in mathematics. The permutation group $\Sigma_k$ is composed of all permutations of $k$ elements, and each element $\sigma \in \Sigma_k$ in the group corresponds to a permutation operation of individual positions, i.e., $\sigma: N \to N$, mapping the position of the $i$-th student to the position of the $\sigma(i)$-th student.

The permutation group $\Sigma_k$ acts freely on the ordered configuration space $\text{PConf}_k(M)$, and the action is defined as $\sigma \cdot (x_1, x_2, \dots, x_k) = (x_{\sigma(1)}, x_{\sigma(2)}, \dots, x_{\sigma(k)})$. The core meaning of this action is: if you swap positions with your deskmate, it is regarded as "the same point" in the macro topological structure of marching drill. All individual $x_i$ are forcibly quotiented out and collapsed into an orbit with only cardinal significance.

This construction process of the quotient space is precisely the topological nature of individual homogenization, and also the first layer of satirical meaning of "CS" mentioned by Mr. Bao. The construction of the quotient space is essentially an operation of "modulo permutation group action" on the ordered configuration space, classifying all configuration points that can be transformed into each other through permutation into the same equivalence class, i.e., the same orbit. This means that the uniqueness (order) of individuals is forcibly erased, and all students are collapsed into indistinguishable spatial elements. Regardless of an individual’s height, posture, or movement state, as long as their position can be swapped with other individuals through permutation, they are regarded as "indistinguishable individuals", which is highly consistent with the reality of "uniformity and erasure of personality" in marching drill.

II. From Fiber Bundles to Kan Extensions

The configuration space defined earlier mainly describes the static position distribution of marching drill, but marching drill is a dynamic process with periodicity, constraint, and other characteristics. The original text mentions $Conf(X) \cong \text{colim}(B\Sigma_n \to \text{Top})$. To deepen the understanding of dynamic discipline in marching drill, it is necessary to start from the perspectives of category theory and fiber bundle theory, analyze the dynamic constraint mechanism in the marching drill process, and reveal how power achieves continuous discipline of individual behaviors through "dynamic rules".

(1) Fiber Bundle Modeling of Marching Drill

Ideally, free running can be regarded as the free movement of individuals on the base space $M$, corresponding to a flat connection in topology, meaning that individual movement has no additional constraints and can move along any geodesic. However, marching drill is not free movement, but a dynamic process strictly constrained by rules such as "alignment", "synchronization", and "constant speed". This dynamic constraint can be accurately modeled through a Fiber Bundle, which is defined as follows:$F \xrightarrow{i} E \xrightarrow{\pi} B$

The specific meanings of each space are as follows: the base space $B$ is taken as the time axis $S^1$ (circle), symbolizing the periodicity of marching drill — marching drill usually proceeds cyclically along the runway, and the movement process is repetitive, which is highly consistent with the topological properties of the circle (no starting point, no ending point, and cyclic repetition); the total space $E$ is defined as the actual movement trajectory space of students, and each point in $E$ corresponds to the position and movement state (speed, direction) of a student at a certain moment, i.e., the elements in $E$ can be expressed as $(t, x_i(t))$, where $t \in B=S^1$ is time, and $x_i(t) \in M$ is the position of the $i$-th student at time $t$; the fiber $F$ is the unordered configuration space $\text{Conf}_k(M)$ defined earlier, representing the position configuration of all students in the class at a fixed moment, and the fiber $i: F \to E$ is an embedding map, embedding the static configuration into the dynamic trajectory space; the projection $\pi: E \to B$ is defined as $\pi(t, x_i(t)) = t$, representing the projection of the dynamic trajectory onto the time axis, reflecting the time periodicity of marching drill.

The core significance of the fiber bundle is that it reveals the "local triviality and global constraint" of the dynamic process of marching drill: in any local neighborhood $U \subset B$ of the time axis $B$, the fiber bundle can be trivialized into $U \times F$, meaning that the student configuration at a local moment can be regarded as the product of time and static configuration; but overall, the fiber bundle is not a trivial product, because the marching drill rules require that the configurations at different moments must satisfy the "synchronization" constraint (such as maintaining a consistent front-back distance, synchronized arm swing frequency, etc.). This constraint gives the total space $E$ a non-trivial topological structure, corresponding to the requirement of "dynamic neatness" in real marching drill.

(2) Braid Group Action and Trivial Braid

During marching drill, the whistles and commands of physical education teachers (such as "Dress right!", "Align front and back!", "Synchronize arm swings!") are the direct embodiment of dynamic constraints. This constraint can be mathematically described through the representation of the Braid Group. The braid group $Br_k(M)$ is the fundamental group of the configuration space $\text{Conf}_k(M)$, i.e., $\pi_1(\text{Conf}_k(M)) \cong Br_k(M)$. Its core elements are "braids", corresponding to the state where the trajectories of $k$ individuals are intertwined when moving in space.

In a free movement state, individual trajectories can form non-trivial braids, i.e., the trajectories are intertwined, reflecting the autonomy of individual movement; but the marching drill rules require that all students’ actions and trajectories must be synchronized, and "trajectory entanglement" is not allowed. This requirement is transformed into a "trivial braid constraint" in mathematics — the blowing of whistles and the issuance of commands are essentially the application of a trivial representation of the braid group $Br_k(M)$ to the fiber $F$, forcing all generated braids to be trivial braids, i.e., individual trajectories are not intertwined and completely synchronized.

Specifically, "Dress right!" requires all students’ horizontal positions to satisfy the same constraint condition, corresponding to the trivialization of horizontal generators in the braid group; "Align front and back!" requires vertical positions to satisfy linear constraints, corresponding to the trivialization of vertical generators; "Synchronize arm swings!" requires the synchronization of individual movement states (speed, direction), corresponding to the trivialization of dynamic generators in the braid group. This trivial braid constraint is essentially power forcing the restriction of individual movement freedom through external signals such as "commands", realizing the unification of dynamic behaviors.

(3) Right Kan Extension of Behaviors

The colimit description mentioned earlier, $Conf(X) \cong \text{colim}(B\Sigma_n \to \text{Top})$, needs to be corrected in combination with the Kan extension in category theory to more accurately describe the compliance of individual behaviors. In marching drill, individual behaviors are not freely generated, but are shaped by rules imposed by the power system (school leaders, physical education teachers, head teachers). This shaping process can be modeled through functors and Kan extensions in category theory.

Define the category $\mathbf{Power}$ as the power category, whose objects are various marching drill rules (such as alignment rules, arm swing rules, command rules, etc.), and morphisms are the implication relationships between rules (such as "Dress right!" implies "horizontal distance constraint"); define the category $\mathbf{Metric}$ as the metric behavior category, whose objects are individual movement behavior patterns, and morphisms are the approximation relationships between behavior patterns. Define the power functor $R: \mathbf{Power} \to \mathbf{Metric}$, which maps each rule in $\mathbf{Power}$ to the corresponding individual behavior pattern in $\mathbf{Metric}$, i.e., rules determine individual behavior methods.

At the same time, define the compliance relationship $\iota: \mathbf{Power} \to \mathbf{Metric}$ as an embedding functor, reflecting the compliance of individuals with power rules — individual behavior patterns must be embedded in the behavior space specified by power rules. Based on this, the actual behavior pattern of individuals can be defined as the Right Kan Extension of the power functor $R$ along the compliance relationship $\iota$, denoted as:$CS_{behavior} = \text{Ran}_\iota R$

In category theory, the core significance of the Right Kan Extension is to approximate a specific structure through a Limit. This means that an individual’s behavior pattern is not actively generated, but is forced to converge to the "standard behavior" specified by the rules under the constraint of power rules. Specifically, for any student (corresponding to a point in $\text{Conf}_k(M)$), their behavior pattern must satisfy the local trivialization condition under the open cover $\{U_\alpha\}$ of their neighborhood, i.e., constantly calculate the $\epsilon$-distance between themselves and the previous student ($\epsilon$ is the minimum distance specified by the marching drill rules) to ensure that their behavior is consistent with adjacent individuals, thereby maintaining the continuity of the holomorphic section of the entire fiber bundle. The essence of this process is: you are not running actively, but solving the inverse limit of your own behavior in this rigorous, immutable commutative diagram constructed by power rules. You must always calculate the $\epsilon$-distance from the person in front of you to ensure the continuity of the holomorphic section.

III. Distance Metricization and Topological Alienation

Mr. Bao Haimo jokingly referred to CS as "Chusheng" (a homophone for "beast" in Chinese), which is not only a linguistic joke, but also a profound insight into the dehumanization phenomenon in marching drill. Behind it lies the metric properties of configuration spaces and the theory of homological stability, which can be used to accurately interpret the topological alienation essence referred to by "Chusheng" through mathematical tools.

(1) Potential Energy Function and Distance Constraint

In the unordered configuration space $\text{Conf}_k(M)$, any collision between two points ($x_i \to x_j$) corresponds to a boundary or singularity of the manifold. To avoid collisions (maintain formation), the system introduces a potential energy function $V: \text{Conf}_k(M) \to \mathbb{R}$. When the distance $|x_i - x_j| < \delta$, $V \to \infty$.

This repulsive potential energy is packaged as a moral imperative — "maintain formation".

However, a deeper irony lies in Homological Stability. The mathematician Arnol'd proved that the homology groups of configuration spaces stabilize as the number of particles $k$ tends to infinity. This maps to marching drill: when there are enough people on the playground, the life or death, emotions, or even whether an individual falls behind has no impact on the algebraic topological properties of the whole (such as the homology group $H_*(\text{Conf}_k(M))$). The disappearance of an individual (dropping out) does not change the Euler characteristic of the system.

(2) Homological Stability and Individual Alienation

Here, "humans" as subjects with independent will are completely alienated into pure elements that maintain the stability of the quotient topological structure. What Mr. Bao refers to as "Chusheng" actually refers to such**"dehumanized topological elements"**. This adaptation process is to forcibly transform one’s own originally rich high-dimensional soul into a low-dimensional CW-complex defined by rules through originally homotopy equivalent transformations.

IV. Geometric Group Theory of the Minuet

Finally, regarding the analogy of the "court minuet", we can introduce the perspective of Geometric Group Theory.

A court minuet is not a random Brownian motion; it is an isometric action of a discrete group $G_{minuet}$ on the dance floor $X$. This group is generated by a finite number of generators (stepping, bowing, rotating) and has extremely high rigidity.

The same is true for marching drill. The head teacher and physical education teacher define a set of generators $S = \{\text{Stand at attention}, \text{Swing arms}, \text{Shout slogans}\}$, and the generated group $\Gamma = \langle S \rangle$ is the vertex walk on the Cayley Graph.

(1) Minuet: Rigid Behaviors in Group Theory

Students do not flow as a continuous medium, but jump as discrete points on a huge and outdated Cayley Graph lattice. This is a strictly limited "Penrose Tiling". Just as the preferences of nobles in the minuet define the metric tensor $g_{ij}$, the "norms" in marching drill define the geodesics allowed between students.

This rigid geometric structure is the mathematical expression of the "pedantry" described by Maupassant — it rejects changes in the curvature of the manifold, rejects the free growth of the tangent space, and only allows finite state machine-style cycles on specified lattice points.

(2) Marching Drill: Discrete Group and Cayley Graph

Morning exercise’s rigidity is exactly the same as that of the court minuet, and its mathematical modeling can be analogous to the isometric action of a discrete group on the playground space $M$. Define the set of marching drill rule generators $S = \{\text{Stand at attention}, \text{Swing arms}, \text{Shout slogans}, \text{March at a constant speed}\}$, where each generator corresponds to a basic action of marching drill, and the action standard of each generator (such as arm swing amplitude, step height, slogan frequency) is specified by the head teacher and physical education teacher, with fixity and unchangeability. The discrete group generated by the generator set $S$ is denoted as $\Gamma = \langle S \rangle$. Each element in the group $\Gamma$ corresponds to a combination of marching drill actions, all elements are generated by the generators through finite operations, and the group $\Gamma$ has extremely high rigidity, which cannot change the action standard through continuous transformation, corresponding to the insurmountability of marching drill rules.

From the perspective of geometric group theory, the group $\Gamma$ can be visualized through a Cayley Graph: the vertices of the Cayley Graph correspond to the elements of the group $\Gamma$ (marching drill action combinations), and the edges correspond to the actions of the generators (switching of basic actions). The behavior of students during marching drill is essentially a "vertex walk" on the lattice of the Cayley Graph — they can only switch from one vertex (action combination) to another along the edges (basic actions), and cannot move independently outside the lattice of the Cayley Graph.

Furthermore, the movement of students during marching drill is not the flow of a continuous medium, but jumping as discrete points on the lattice of the Cayley Graph, which corresponds to the "Penrose Tiling" in topology — the tiling pattern has periodicity, but there are subtle local constraints, showing a rigid neatness as a whole. The "norms" in marching drill (action standards, spacing requirements) essentially define the distribution of Cayley Graph lattice points, i.e., define the geodesics allowed between students, and students must move along these geodesics and cannot break the constraints. This rigid geometric structure is the mathematical expression of the "pedantry" described by Maupassant — it rejects changes in the curvature of the manifold (rejects individual action differences), rejects the free growth of the tangent space (rejects individual behavioral autonomy), and only allows individuals to perform finite state machine-style cyclic movements on specified lattice points, completely losing the flexibility and autonomy of behavior.

Conclusion

In summary, marching drill, as a collective behavior, is essentially a large-scale topological experiment under power discipline, which can be completely deconstructed through mathematical tools such as algebraic topology, category theory, and geometric group theory. The marching drill system maps students into indistinguishable points in the unordered configuration space $\text{Conf}_k(M)$, erases the order and uniqueness of individuals through permutation group actions, and achieves individual homogenization; through fiber bundle modeling and trivial braid constraints, it achieves continuous discipline of individual dynamic behaviors and deprives individuals of behavioral autonomy; through potential energy functions and homological stability, it alienates individuals into pure elements that maintain system topological stability and achieves individual dehumanization; through discrete group actions and Cayley Graph modeling, it limits individual behaviors within a rigid rule framework and achieves behavioral rigidity.

The "CS" referred to by Mr. Bao Haimo is not only the abbreviation of Configuration Space, but also can be interpreted as Canonic Slave, which accurately summarizes the alienated state of individuals in the marching drill system. The "macro neatness" pursued by marching drill is achieved at the cost of sacrificing individual freedom, erasing individual personality, and depriving individual autonomy. Just like the court minuet, it is an absurd projection of power aesthetics in low-dimensional space. The formalism and dehumanizing discipline hidden behind it are exposed without reservation in the rigorous mathematical deduction.