Lie Group Analysis of Scaling Law for Domino Toppling Wave Velocity

1. Introduction

The domino effect, the successive toppling of regularly spaced elements in a periodic array, is not merely an entertaining game but a profound physical phenomenon that models cascading energy transfer and sequential failures in both macroscopic structures and microscopic systems. The phenomenon is also widely used as a metaphor for social and systemic catastrophes, such as the cascading consequences of research misconduct. Historically, the dynamics of domino toppling have been approached through both analytical modeling and numerical simulation.

In 1983, McLachlan et al. [1] proposed a scaling law for the propagation speed v for dominoes with zero thickness: $v_{\mathrm{McLachlan}}=\sqrt{gh}f({\displaystyle \frac{\lambda }{h}})$, where g is the gravitational acceleration, h is the domino height, $\lambda$ is the domino spacing, and $f\left(x\right)$ is an undetermined function. Szirtes and Rozsa [2] utilized traditional Buckingham dimensional analysis to include domino thickness $\delta$, obtaining $v_{\mathrm{Szirtes}}=\sqrt{gh}f({\displaystyle \frac{\lambda }{h}},{\displaystyle \frac{\delta }{h}})$. Although these analytical strides provided foundational insights, the traditional Buckingham theorem assumes isotropic space, which contradicts the physical reality of dominoes where gravity acts vertically and propagation occurs horizontally.

To overcome this limitation, Sun [3] employed directed dimensional analysis to propose a universal scaling law for the propagation speed:

$$v=\lambda \sqrt{{\displaystyle \frac{g}{h}}}f\left({\displaystyle \frac{\delta }{\lambda }}\right)$$

(1)

By curve-fitting Stronge’s theoretical data [6], Sun [3] further deduced an explicit approximate power law $v\sim \sqrt{{\displaystyle \frac{\delta \lambda }{h}}g}$, revealing that the speed is proportional to the square root of separation and thickness, rather than directly proportional to $\sqrt{gh}$. Subsequently, the validity and applicability of this scaling law were extensively verified and expanded. Song, Guo, and Sun [4] investigated the domino toppling motion in curved paths (circular and S-shaped) using the finite-element analysis program ABAQUS. Their work proved that Sun’s scaling law [3] remains universally true for curved arrangements, although the coefficient C varies with the path curvature. They formulated an amendatory function ${\varphi }_{revise}$ to account for the curvature effect, demonstrating the robustness of the $\sqrt{\delta \lambda g/h}$ dependency across complex geometric configurations.

More recently, the focus has shifted towards the highly nonlinear dissipative mechanisms in domino toppling, particularly friction. Cantor and Wojtacki [5] conducted systematic discrete-element method simulations to study the effects of domino-domino friction (${\mu }_{dd}$) and domino-surface friction (${\mu }_{sd}$). Crucially, they adopted Sun’s scaling law [3] as the foundational framework and enriched it by incorporating frictional terms. Their proposed scaling law, $v=C\sqrt{swg/h}{(h/s)}^{\beta }/{(1+{\mu }_{dd})}^{\alpha }$, successfully produced a master curve over a wide range of parameters. They found that while the wavefront speed is significantly reduced by domino-domino friction, it is practically independent of domino-surface friction (except for triggering anomalies like vertical stacking or backward sliding).

Despite these significant advances, two fundamental issues remain obscure. First, the mathematical symmetry underlying the domino scaling law lacks a rigorous framework. Directed dimensional analysis corrects the physical intuition but lacks a rigorous mathematical formulation for symmetry. Second, the question of whether the exponent $\alpha \approx 1/2$ found in the approximate power law is a universal constant or merely an empirical approximation requires clarification. To bridge these gaps and provide a robust theoretical basis that can naturally accommodate future complexities, this paper introduces the Lie group method [7] to re-investigate the problem. The Lie group approach explicitly reveals the invariance and anisotropic symmetry of the physical system, offering deeper mechanical insights into the energy cascade and mathematically substantiating the phenomenological laws established in recent literature.

2. Lie Group Analysis of Domino Toppling

According to the Lie group theory of differential equations and invariants [7], a physical law $F(v,h,\delta ,\lambda ,g)=0$ must be invariant under the scaling group that preserves the physical dimensions. Unlike traditional isotropic scaling, domino toppling possesses distinct directional properties. We define a 3-parameter anisotropic scaling group $G({ϵ}_1,{ϵ}_2,{ϵ}_3)$ corresponding to the horizontal length $L_x$, vertical length $L_z$, and time T:

$$x\to {ϵ}_{1}x,z\to {ϵ}_{2}z,t\to {ϵ}_{3}t$$

(2)

To rigorously verify the scaling law, we utilize two distinct mathematical formulations of the Lie group method: the finite group transformation approach and the infinitesimal generator approach.

2.1. Formulation A: Finite Group Transformation

Under the anisotropic scaling group, the physical variables transform as:

• Speed v (horizontal propagation): $v^{\prime }={ϵ}_1{ϵ}_3^{-1}v$
• Height h (vertical dimension): $h^{\prime }={ϵ}_{2}h$
• Thickness $\delta$ (horizontal dimension): ${\delta }^{\prime }={ϵ}_1\delta$
• Separation $\lambda$ (horizontal dimension): ${\lambda }^{\prime }={ϵ}_1\lambda$
• Gravity g (vertical acceleration): $g^{\prime }={ϵ}_2{ϵ}_3^{-2}g$

By the fundamental theorem of Lie groups, the physical law must be expressible solely in terms of the absolute invariants of the group G. Since we have 5 variables and 3 independent parameters, there exist $5-3=2$ absolute invariants. To find the first invariant ${\mathsf{\Pi }}_1$ involving v, we eliminate ${ϵ}_2$ and ${ϵ}_3$. From the transformation of g and h:

$${\displaystyle \frac{g^{\prime }}{h^{\prime }}}={ϵ}_3^{-2}{\displaystyle \frac{g}{h}}\Rightarrow {ϵ}_3=\sqrt{{\displaystyle \frac{g^{\prime }h}{g{h}^{\prime }}}}$$

(3)

Substituting ${ϵ}_3$ into the transformation of v:

$$v^{\prime }={ϵ}_1\sqrt{{\displaystyle \frac{g{h}^{\prime }}{g^{\prime }h}}}v\Rightarrow {\displaystyle \frac{v^{\prime }}{\sqrt{g^{\prime }/h^{\prime }}}}={ϵ}_1{\displaystyle \frac{v}{\sqrt{g/h}}}$$

(4)

Let $V=v/\sqrt{g/h}$, which transforms as $V^{\prime }={ϵ}_{1}V$. To eliminate ${ϵ}_1$, we use the transformation of $\lambda$: ${\lambda }^{\prime }={ϵ}_1\lambda$. Thus, the first absolute invariant is:

$${\mathsf{\Pi }}_1={\displaystyle \frac{V}{\lambda }}={\displaystyle \frac{v}{\lambda }}\sqrt{{\displaystyle \frac{h}{g}}}$$

(5)

To find the second invariant ${\mathsf{\Pi }}_2$, we observe that both $\delta$ and $\lambda$ scale with ${ϵ}_1$. Their ratio naturally eliminates ${ϵ}_1$:

$${\mathsf{\Pi }}_2={\displaystyle \frac{\delta }{\lambda }}$$

(6)

By the principle of group invariance [7], the physical relation takes the form $\mathsf{\Phi }({\mathsf{\Pi }}_1,{\mathsf{\Pi }}_2)=0$, or equivalently ${\mathsf{\Pi }}_1=f\left({\mathsf{\Pi }}_2\right)$. Substituting the invariants yields the universal scaling law:

$$v=\lambda \sqrt{{\displaystyle \frac{g}{h}}}f\left({\displaystyle \frac{\delta }{\lambda }}\right)$$

(7)

2.2. Formulation B: Infinitesimal Generator Method

To provide a complementary and rigorous verification, we utilize the infinitesimal generator approach. The variables transform under the group parameters $a_1,a_2,a_3$ as $v^{\prime }=e^{a_1-a_3}v$, $h^{\prime }=e^{a_2}h$, ${\delta }^{\prime }=e^{a_1}\delta$, ${\lambda }^{\prime }=e^{a_1}\lambda$, and $g^{\prime }=e^{a_2-2a_3}g$. The generators corresponding to these scaling parameters are constructed based on the exponents of the finite transformations:

$$\begin{array}{cc}\hfill X_1& =v{\displaystyle \frac{\partial }{\partial v}}+\delta {\displaystyle \frac{\partial }{\partial \delta }}+\lambda {\displaystyle \frac{\partial }{\partial \lambda }}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill X_2& =h{\displaystyle \frac{\partial }{\partial h}}+g{\displaystyle \frac{\partial }{\partial g}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill X_3& =-v{\displaystyle \frac{\partial }{\partial v}}-2g{\displaystyle \frac{\partial }{\partial g}}\hfill \end{array}$$

(10)

An absolute invariant I must satisfy $X_i\left(I\right)=0$ for all i. First, we solve $X_1\left(I\right)=0$. The characteristic equation is ${\displaystyle \frac{dv}{v}}={\displaystyle \frac{d\delta }{\delta }}={\displaystyle \frac{d\lambda }{\lambda }}$, yielding the intermediate invariants:

$$I_1={\displaystyle \frac{v}{\lambda }},I_2={\displaystyle \frac{\delta }{\lambda }},I_3=h,I_4=g$$

(11)

Next, we apply $X_3$ to these intermediate invariants. Since $X_3$ only involves v and g, we compute:

$$X_3\left(I_1\right)=-I_1,X_3\left(I_2\right)=0,X_3\left(I_3\right)=0,X_3\left(I_4\right)=-2I_4$$

(12)

Solving $X_3\left(J\right)=0$ for the variables $(I_1,I_4)$ using the characteristic equation ${\displaystyle \frac{d{I}_1}{-I_1}}={\displaystyle \frac{d{I}_4}{-2I_4}}$, we obtain the invariant ${\displaystyle \frac{I_1^2}{I_4}}={\displaystyle \frac{{(v/\lambda )}^2}{g}}$. The variables $I_2$ and $I_3$ remain invariant. Thus, the invariants are updated to:

$$J_1={\displaystyle \frac{{(v/\lambda )}^2}{g}},J_2={\displaystyle \frac{\delta }{\lambda }},J_3=h$$

(13)

Finally, we apply $X_2$ to these invariants:

$$X_2\left(J_1\right)=-J_1,X_2\left(J_2\right)=0,X_2\left(J_3\right)=J_3$$

(14)

Solving $X_2\left(K\right)=0$ for the variables $(J_1,J_3)$ using the characteristic equation ${\displaystyle \frac{d{J}_1}{-J_1}}={\displaystyle \frac{d{J}_3}{J_3}}$, we obtain the invariant $J_1{J}_3={\displaystyle \frac{v^{2}h}{{\lambda }^{2}g}}$. The variable $J_2$ remains invariant. Therefore, the two absolute invariants of the system are:

$${\mathsf{\Pi }}_1={\displaystyle \frac{v}{\lambda }}\sqrt{{\displaystyle \frac{h}{g}}},{\mathsf{\Pi }}_2={\displaystyle \frac{\delta }{\lambda }}$$

(15)

This rigorously confirms the results obtained from Formulation A, mathematically substantiating the directed dimensional analysis performed in [3] and demonstrating that the scaling law is fundamentally a manifestation of the system’s invariance under anisotropic spatio-temporal scaling.

3. Approximation of Scaling Law and Data Validation

To obtain explicit information, we must determine the function $f\left(x\right)$. From the geometry of dominoes, toppling becomes possible only if $\lambda <\sqrt{h^2-{\delta }^2}$. If the domino separation is much smaller than its height, we can assume $f\left(x\right)$ obeys a power law: $f({\displaystyle \frac{\delta }{\lambda }})\approx C{({\displaystyle \frac{\delta }{\lambda }})}^{\alpha }$. The approximate scaling law becomes:

$$v\approx C\lambda {\left({\displaystyle \frac{g}{h}}\right)}^{1/2}{\left({\displaystyle \frac{\delta }{\lambda }}\right)}^{\alpha }=C\sqrt{{\displaystyle \frac{\delta \lambda }{h}}g}{\left({\displaystyle \frac{\delta }{\lambda }}\right)}^{\alpha -1/2}$$

(16)

We determine C and $\alpha$ using Stronge’s theoretical data [6] for dominoes with $h=41.78$ mm and $\delta =7.58$ mm. Linear regression of $ln\left(v\right)$ vs $ln(\delta /\lambda )$ yields the exponent $\alpha \approx 1/2$ and the coefficient $C\approx 3.488$. Substituting these into Eq. (16) gives:

$$v_{Stronge}\approx 3.488\sqrt{{\displaystyle \frac{\delta \lambda }{h}}g}$$

(17)

This explicit scaling law shows that the domino propagation speed is proportional to the square root of the domino separation $\lambda$, thickness $\delta$, and inversely proportional to the square root of the domino height h.

4. In-Depth Physical Mechanism and Modern Perspectives

The Lie group anisotropic scaling not only derives the formula but also decouples the physical dimensions, allowing for a profound understanding of the energy transfer cascade, inertial dynamics, and the role of dissipation in domino toppling.

4.1. Effective Dynamical Length and Potential Energy Cascade

The approximate law $v\sim \sqrt{{\displaystyle \frac{\delta \lambda }{h}}g}$ can be rewritten as $v\sim \sqrt{g{L}_{eff}}$, where $L_{eff}={\displaystyle \frac{\delta \lambda }{h}}$ acts as an effective dynamical length. The physical mechanism of domino toppling is a cascade release of gravitational potential energy. When a domino rotates and strikes the next, the collision angle is approximately ${\theta }_c\approx \lambda /h$. Because the domino has a finite thickness $\delta$, this thickness provides a horizontal geometric offset at the moment of impact. Consequently, the center of mass of the striking domino undergoes an effective vertical drop $\Delta h_{eff}$ proportional to the thickness times the impact angle: $\Delta h_{eff}\sim \delta ·{\theta }_c\sim {\displaystyle \frac{\delta \lambda }{h}}$. The toppling wave speed is therefore equivalent to the free-fall speed of an object dropping from this effective height $L_{eff}$.

This effective length perspective elucidates the findings in [4]: in curved paths, the primary geometry ($\delta ,\lambda ,h$) dictates the energy drop per domino, hence the scaling $\sqrt{\delta \lambda g/h}$ persists. However, the curved arrangement introduces an additional rotational constraint that modifies the efficiency of the energy transfer, which phenomenologically manifests as a curvature-dependent coefficient C or the amendatory function ${\varphi }_{revise}$ proposed by Song et al. [4].

4.2. Dynamic Decoupling of Inertia and Driving Force

The Lie group separation of $L_x$ and $L_z$ physically represents the dynamic decoupling between the driving force and the spatial transmission:

• The driving source lies entirely in the vertical $L_z$ dimension, dependent on gravity g and height h. This dictates the intrinsic time scale of the system, $\tau \sim \sqrt{h/g}$, akin to a compound pendulum.
• The spatial transmission occurs in the horizontal $L_x$ dimension, dependent on $\lambda$ and $\delta$.

The propagation speed is the ratio of the effective spatial step to the intrinsic time scale, $v\sim L_{x,eff}/\tau$. Because the collision geometry couples thickness and separation, the effective step is $\sqrt{\delta \lambda }$, leading to $v\sim \sqrt{\delta \lambda }/\sqrt{h/g}$.

4.3. Orthogonal Decoupling of Width and Mass

To understand the influence of domino width w, we introduce the cross-section area $A=h\delta$ and expand our Lie group to include the transverse dimension $L_y$. The group becomes $G({ϵ}_1,{ϵ}_2,{ϵ}_3,{ϵ}_4)$ scaling $x,z,t,y$ respectively. The variables transform as: $v^{\prime }={ϵ}_1{ϵ}_3^{-1}v$, $A^{\prime }={ϵ}_1{ϵ}_{2}A$, ${\lambda }^{\prime }={ϵ}_1\lambda$, $g^{\prime }={ϵ}_2{ϵ}_3^{-2}g$, and $w^{\prime }={ϵ}_{4}w$. Since no other variable contains the $L_y$ dimension, the invariant condition forces the exponent of ${ϵ}_4$ to be zero. This mathematical orthogonal decoupling implies that width w does not enter the dynamical invariants. Mechanistically, domino toppling is a 2D planar motion. The driving torque is $\tau \propto mgh$, while the moment of inertia is $I\propto m{h}^2$. In the rotational equation of motion $I\ddot{\theta }=\tau$, the mass m exactly cancels out. Thus, the speed remains independent of mass and width, relying solely on the geometry of the cross-section profile $A=h\delta$.

5. On the Nature of the Exponent $\mathit{\alpha }$: Symmetry and Symmetry Breaking

In the approximate scaling law $v\sim \sqrt{\delta \lambda g/h}$ proposed by Sun [3], the exponent $\alpha$ in the function $f(\delta /\lambda )\approx C{(\delta /\lambda )}^{\alpha }$ was found to be approximately $1/2$. A critical question arises: Is $\alpha =1/2$ a universal value derived from physical principles, or is it an approximation? A deeper perspective on this exponent can be understood through the lens of symmetry and symmetry breaking.

In the ideal, frictionless limit, the domino toppling process exhibits a complete spatio-temporal scaling symmetry where the energy cascade is purely governed by geometry. The exponent $\alpha =1/2$ can be regarded as a manifestation of this complete scaling symmetry. In this idealized regime, the spatial and temporal scales decouple perfectly, analogous to free-fall dynamics under conservative forces where the kinetic energy is exactly half the potential energy drop, naturally giving rise to the square root dependence (exponent $1/2$). The perfect symmetry of the collision geometry without energy loss ensures that the invariant function $f(\delta /\lambda )$ reduces precisely to a power law with $\alpha =1/2$.

However, the introduction of friction fundamentally breaks the time-reversal symmetry of the ideal conservative system and introduces dissipative scales into the dynamics. From the viewpoint of symmetry breaking, friction perturbs the ideal scaling symmetry. The presence of domino-domino friction ${\mu }_{dd}$ and domino-surface friction ${\mu }_{sd}$ introduces characteristic lengths and energy dissipation rates that do not scale isotropically with the geometric parameters. As a result, the exact symmetry governing the exponent $\alpha =1/2$ is broken, causing $\alpha$ to deviate from $1/2$ and fluctuate in its vicinity.

This symmetry-breaking interpretation perfectly aligns with the findings of Cantor and Wojtacki [5]. They demonstrated that the simple power law $v\sim \sqrt{\lambda }$ is insufficient for a wide range of parameters, as the wavefront speed depends nonlinearly on friction ${\mu }_{dd}$ and requires a correction term ${(h/s)}^{\beta }$ with $\beta \approx 0.7$. This correction modifies the effective exponent of separation s (which is $\lambda$) from $0.5$ to $0.5-\beta =-0.2$. In the context of symmetry breaking, the correction term ${(h/s)}^{\beta }$ and the frictional modification ${(1+{\mu }_{dd})}^{-\alpha }$ are precisely the mathematical manifestations of the broken scaling symmetry. The dissipative effects force the system away from the ideal symmetric state ($\alpha =1/2$), resulting in the observed fluctuations and modifications of the exponent.

Therefore, $\alpha =1/2$ is not merely an empirical approximation, but the signature of an ideal symmetric state governed by perfect energy conservation and geometric scaling. Friction breaks this symmetry, leading to the observed fluctuations and modifications of the exponent. The Lie group scaling law provides the general framework of the symmetric state, while the deviations capture the symmetry-breaking physics introduced by dissipation.

6. Conclusions

In conclusion, by applying the Lie group method of invariants [7], we have rigorously re-derived the universal scaling law for domino toppling propagation speed, $v=\lambda \sqrt{{\displaystyle \frac{g}{h}}}f({\displaystyle \frac{\delta }{\lambda }})$, through both the finite group transformation and the infinitesimal generator approach. These rigorous derivations overcome the isotropic limitations of traditional dimensional analysis and mathematically substantiate the directed dimensional analysis [3]. Data fitting yields the explicit approximate law $v\sim \sqrt{{\displaystyle \frac{\delta \lambda }{h}}g}$. Furthermore, the Lie group decoupling of dimensions provides profound physical insights: the propagation speed is governed by an effective dynamical length $L_{eff}=\delta \lambda /h$ characterizing the cascade release of gravitational potential energy during collision; the independence of speed on mass and width arises from the exact cancellation of mass in torque-inertia dynamics and the orthogonal decoupling of the transverse dimension in the scaling group. Regarding the exponent $\alpha \approx 1/2$, we provide a novel physical interpretation: it is the signature of a complete scaling symmetry in the ideal, frictionless limit. The introduction of friction breaks this symmetry, causing $\alpha$ to fluctuate around $1/2$. This symmetry-breaking perspective unifies the ideal power law with dissipative modifications, offering a deeper theoretical understanding of energy cascades in multi-body dynamics.