# **Symmetry, Structure, and Dynamics**

### **The 2025 Joint Mathematics & Physics Freshman Poster Symposium**

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## **Overview**

Welcome to the **2025 Joint Freshman Poster Symposium**, a collaborative academic showcase featuring the final projects of first-year students from the Departments of Mathematics and Physics. This event marks a significant milestone in the students' academic journey, representing the synthesis of their inaugural semester's theoretical learning and exploratory research.

This year's collection features **14 poster presentations** that highlight the intrinsic dialogue between the two disciplines. The **Mathematics Section** focuses on the rigorous construction of logical structures. Students here have distilled complex concepts—from the algebraic axioms of vector spaces to the geometric nuances of complex analysis—into clear, definition-focused presentations. These works emphasize the beauty of formal definitions and the power of mathematical language.

Complementing this, the **Physics Section** demonstrates the application of these languages to the physical world. Through computational modeling, theoretical derivation, and experimental observation, the physics cohort explores oscillatory systems, wave mechanics, and orbital dynamics. These projects reveal how mathematical foundations, such as Euler's formula and differential equations, serve as the essential framework for understanding the mechanisms of the universe.

Together, these presentations illustrate a shared academic pursuit: the quest to describe the natural world through precision, structure, and inquiry.

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## **Table of Contents**

### **Part I: Undergraduate Mathematics**

*(Abstracts generated by Gemini from poster files)*

1. **Linear Map** (S.-Y. Yang, Y.-S. Hung, Y.-W. Yu, L.-Y. Sung)
2. **Vector Space** (Daniel Lin, Lucas Tan, Tyler Z.-H. Yu, Desmond Y.-F. Luo)
3. **The Hidden Mathematics Behind Bitcoin: Weierstrass Elliptic Curve** (Zih-chen Fang, Pin-Cen Chen, Po-Jui Huang, Cheng-Hung Liu)
4. **Equivalence Relation** (Ray Huang, Lucas Lu, Ben Han)
5. **Determinant** (Claire Tsai, Stella Chou, An-yi Xiao, Melody Yu)
6. **Euler's Formula** (Pin-Hsun Liao, Chih-Ching Wang, Kai-Wen Wang, Hao Zhang)
7. **Derivative** (Yi-An Wang, Yu-Chen Chang)

### **Part II: Physics (Introduction to Theoretical Physics)**

8. **Group 1: Double Pendulum** (Kevin, Jack, Chris)
9. **Group 2: Foucault Pendulum** (Tim, Jashua, Ryan)
10. **Group 3: Coupled Oscillation and Beats** (Dylan, Harvey, Ricky)
11. **Group 4: Properties Between Wave and Medium** (Wendy, Ryan, Sally)
12. **Group 5: Damped and Driven Oscillations** (Caden, Clara, Ben)
13. **Group 6: Lissajous Figures** (Michelle, Van, Shakir)
14. **Group 7: Euler's Formula as a Language of Phase Evolution** (Jimmy, Neil, Quanta)

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## **Part I: Undergraduate Mathematics Poster Presentations**

*(Note: All abstracts in this section were generated by Gemini based on the review of poster files)*

### **1. Linear Map**
**Authors:** S.-Y. Yang, Y.-S. Hung, Y.-W. Yu, L.-Y. Sung

This poster defines a linear map $T: V \to W$ as a function that satisfies two core properties: **additivity** ($T(\mathbf{u}+\mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$) and **homogeneity** ($T(c\mathbf{v}) = cT(\mathbf{v})$). Linear maps model simple transformations that keep lines straight and the origin fixed. The poster uses **rotation** as a primary example of a linear map and highlights **translation** as a key counterexample, as it violates the essential property of **Invariance of the Origin** ($T(0) \neq 0$). [Generated by Gemini from Poster]

### **2. Vector Space**

**Authors:** Daniel Lin, Lucas Tan, Tyler Z.-H. Yu, Desmond Y.-F. Luo

This presentation outlines the formal **definition of a vector space** as a set $V$ over a field with operations (addition and scalar multiplication) that satisfy eight axioms (commutativity, associativity, additive identity/inverse, multiplicative identity, and distributivity). It provides a standard example: **Polynomials of degree ** $\leq 2$ ($P_2$) and an incorrect example: Polynomials of degree exactly 2, which fails closure. The poster also proves several properties derived from the axioms (e.g., uniqueness of identity/inverse, $0\mathbf{v} = \mathbf{0}$), and explores a less familiar example using a **Finite scalar system** ($\mathbb{Z}_p$). [Generated by Gemini from Poster]

### **3. The Hidden Mathematics Behind Bitcoin: Weierstrass Elliptic Curve**

**Authors:** Zih-chen Fang, Pin-Cen Chen, Po-Jui Huang, Cheng-Hung Liu

This work delves into the mathematics underpinning Elliptic Curve Cryptography (ECC), which is essential to Bitcoin's security. It defines a **Weierstrass elliptic curve** as a non-singular curve of genus 1. It details the **Group Law** on an elliptic curve, establishing that its points form a well-defined Abelian group under a geometric point addition operation (closure, identity, inverse, commutativity, associativity). An example of point addition on the curve $y^2 = x^3 - x + 1$ is shown, and the poster mentions visualization scripts developed by the authors. [Generated by Gemini from Poster]

### **4. Equivalence Relation**

**Authors:** Ray Huang, Lucas Lu, Ben Han

This poster introduces the formal concept of an equivalence relation $\sim$ on a set $X$. A binary relation is an equivalence relation if it is:

* **Reflexive** ($a \sim a$)
* **Symmetric** (If $a \sim b$, then $b \sim a$)
* **Transitive** (If $a \sim b$ and $b \sim c$, then $a \sim c$)

The main motivation is to formalize the idea of treating different things as "the same" when they share essential properties, which leads to the concept of **Equivalence Classes** and a **partition** of the set. Real-life examples include "Friend Circles" and "Pairs of Shoes". [Generated by Gemini from Poster]

### **5. Determinant**

**Authors:** Claire Tsai, Stella Chou, An-yi Xiao, Melody Yu

This work explores the definition and properties of the matrix determinant ($\det(A)$ or $|A|$). It presents both the **Formal Definition** (sum over permutations) and the **Equivalent Axiomatic Definition** (multilinearity, alternating property, normalization). Examples are provided to illustrate both methods on a $4 \times 4$ matrix, yielding a determinant of -2. Finally, it notes a practical application: using determinants to determine **Left and Right Turns in GPS Trajectories**. [Generated by Gemini from Poster]

### **6. Euler's Formula**

**Authors:** Pin-Hsun Liao, Chih-Ching Wang, Kai-Wen Wang, Hao Zhang

This poster introduces Euler's formula, $e^{i\theta} = \cos \theta + i \sin \theta$, a fundamental identity in complex analysis. It highlights its special case, **Euler's Identity** ($e^{i\pi} + 1 = 0$), which connects the five fundamental constants: $e, i, \pi, 1, 0$. The poster provides an idea for a proof by defining the exponential on the complex field and using De Moivre's formula. It also explores the geometric interpretation (a rotation by angle $\theta$ in the complex plane) and practical applications in engineering, specifically in modeling the forced vibration of a **launch rocket** and in signal **Navigation and Filtering**. [Generated by Gemini from Poster]

### **7. Derivative**

**Authors:** Yi-An Wang, Yu-Chen Chang

The poster defines the derivative as the instantaneous rate of change or the slope of the tangent line. Its key components include the **Average Rate of Change** (slope of the secant line) and the **Limit** as $h \to 0$, which transforms the average rate into instantaneous change. The geometric meaning of the derivative $f'(a)$ as the slope of the tangent line at $x=a$ is highlighted, along with interpretations: $f'(a) > 0$ (increasing), $f'(a) < 0$ (decreasing), and $f'(a) = 0$ (local max/min). It also lists the main properties: constant rule, linearity, product rule, quotient rule, and chain rule. [Generated by Gemini from Poster]

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## **Part II: Physics Final Poster Presentations**

### **Group 1: Double Pendulum**

**Authors:** 黃振峯 (Kevin), 朱俊諺 (Jack), 李宇翔 (Chris)

In daily life, we often encounter systems that resemble a double pendulum, such as trees oscillating in the wind, butterfly knives, and nunchaku. The double pendulum is a classic example in physics for studying nonlinear dynamics and chaotic behavior. In this project, we aim to investigate the relationship between the length ratio of the two connected pendulums and how easily the system oscillates. The main question of this study is how changing the ratio between the lengths of the upper and lower pendulums affects the motion, stability, and energy transfer of the system. Certain length ratios may allow oscillations to occur more smoothly, while others may lead to rapid chaotic motion. Understanding this relationship can help explain why some real-life tools and natural systems are designed with particular proportions. To explore this problem, we will use Newtonian mechanics, which provides a clear and intuitive framework based on forces, masses, and accelerations. Compared to more abstract approaches such as Lagrangian mechanics, the principle of Newtonian method is easier to understand. Using Python, we will model the equations of motion for a double pendulum and simulate its behavior under different length ratios while keeping other parameters constant. By analyzing angular displacement, velocity, and energy over time, we can identify which length ratios produce stable oscillations and which lead to chaotic motion. This study not only deepens our understanding of double pendulum dynamics but also demonstrates how computational tools can be used to analyze complex physical systems found in everyday life.

### **Group 2: Foucault Pendulum**

**Authors:** 李俊廷 (Tim), 廖宥熙 (Jashua), 呂睿軒 (Ryan)

This study investigates the physical mechanisms and geometric principles behind the Foucault pendulum as a fundamental demonstration of Earth's rotation. Originally showcased in 1851, the device's operational effectiveness relies on an exceptionally long wire and a heavy bob to ensure a stable, slow swing that persists by providing sufficient inertia to overcome external friction. While the pendulum's plane of oscillation maintains its orientation in inertial space due to the absence of external torque, it appears to rotate relative to the ground because of the planet's rotation. To precisely quantify this precession rate across different latitudes, the research employs a Tangent Cone Model, where the daily trajectory at latitude $\theta$ is represented by a cone with a base radius of $R \cos \theta$ and a slant height of $R \cot \theta$. By unfolding this tangent cone into a planar sector, the analysis reveals a distinct "angular deficit" where the resulting sector angle $\alpha$ directly corresponds to the cumulative daily precession angle, $\Delta \phi = 360^{\circ} \sin \theta$. This geometric derivation confirms the precession period formula $T = 24hr / \sin \theta$. Numerical simulations demonstrate that while the precession period is 24 hours at the North Pole, it extends to 62.5 hours at lower latitudes such as Kaohsiung ($22.6^{\circ}$), resulting in a daily shift of only $138.3^{\circ}$. Ultimately, the study illustrates that the precession period approaches infinity at the equator and decreases toward the poles, visually proving how Earth's curvature directly dictates the magnitude of the pendulum's daily rotation through a perfect synthesis of geometric theory and physical observation.

### **Group 3: Coupled Oscillation and Beats**

**Authors:** 吳威達 (Dylan), 顏翰廷 (Harvey), 許少薰 (Ricky)

Coupled pendulums are classic systems where individual oscillators influence one another through shared interaction, producing motion that is complex yet governed by clear physical principles. Imagine two clocks placed on the same table and started at different times: although they initially move independently, their motions gradually influence one another and may transiently synchronize, illustrating the phenomenon of coupled oscillation. This behavior is not merely a mechanical curiosity; it reflects fundamental ideas that appear across physics, from molecular vibrations to analogues in quantum mechanics. This project focuses on a theoretical investigation of a two-pendulum coupled system, supported by a simple experimental demonstration using metronomes placed on a movable tray. The primary objectives are to examine how energy is exchanged between oscillators, identify stable and unstable motion patterns, and introduce the concept of normal modes. Beyond classical mechanics, the project aims to draw conceptual parallels between coupled pendulums and quantum systems, highlighting similarities between normal modes and energy eigenstates, as well as between mode superposition and quantum superposition. The theoretical framework is based on Newtonian mechanics, analyzing the forces acting on each pendulum and the shared movable base to construct the coupled equations of motion. Normal mode analysis is used to predict characteristic motion patterns, including in-phase and out-of-phase oscillations. Experimentally, metronomes on a sliding tray are used as a qualitative aid to visualize coupling effects and transient synchronization, while acknowledging practical limitations such as friction and device instability. Rather than emphasizing final quantitative results, this study aims to demonstrate how collective behavior emerges from simple interactions and how classical coupled systems can serve as intuitive models for more abstract quantum concepts. The project highlights the educational value of coupled pendulums as a bridge between classical dynamics and modern physics.

### **Group 4: Properties Between Wave and Medium**

**Authors:** 陳文靖 (Wendy), 陳彥呈 (Ryan), 彭苡萱 (Sally)

Waves are a mechanism for energy transfer. Under the "small-amplitude" (linear) approximation, most waves can be described by $\frac{\partial^{2}u}{{\partial t}^{2}}=v^{2}\frac{\partial^{2}u}{\partial x^{2}}$. While the mathematical form remains the same, the physical origin of the wave speed c varies significantly across different media. We have three main purposes: to derive the wave speed formulas for string waves, sound waves, and water waves (both shallow and deep); to compare the structural similarities of these formulas and identify the universal physical factors; and to specifically investigate why density is a critical factor in string. We utilize different branches of classical physics for each wave type:

* **String Waves:** Analyzed using Newton's Second Law $(F=ma)$ on a tiny string segment, applying small-angle approximations.
* **Sound Waves:** Derived from fluid dynamics, using the Bulk Modulus (B) to relate pressure changes to volume changes, combined with the continuity equation.
* **Water Waves:** Using a moving reference frame together with mass conservation and Bernoulli's principle for shallow water, and solving Laplace's Equation with specific boundary conditions to find the dispersion relation for deep water.

From the derivations, density acts as inertial resistance for string and sound waves. Higher density makes the medium respond more slowly, reducing wave speed. In conclusion, although the wave equation looks the same, the meaning of c depends on the medium. Thinking in terms of "restoring force vs. inertia" helps us understand and predict wave behavior in different situations.

### **Group 5: Damped and Driven Oscillations**

**Authors:** 郭天綸 (Caden), 李心彤 (Clara), 賴智詮 (Ben)

Tall buildings are subject to vibrations induced by external forces such as wind and earthquakes. When the frequency of these forces approaches the natural frequency of a structure, resonance can occur, leading to large oscillation amplitudes that may cause structural fatigue and discomfort for occupants. To reduce these effects, modern skyscrapers often employ damping mechanisms, such as the tuned mass damper installed in Taipei 101. In this project, the dynamic behavior of Taipei 101 was analyzed using the theory of damped and driven oscillations. The building was approximated as a large-scale elastic system represented by a mass-spring-damper model. The effective mass was assumed to be concentrated at the building's center of mass, while the overall structural stiffness was represented by an effective spring constant. Energy dissipation was modeled using a viscous damping term, and external forces were treated as time-dependent periodic driving forces. The equation of motion, $m\ddot{x}+c\dot{x}+kx=F(t)$, was derived using Newton's second law and was used to analyze the system's response under different driving frequencies and damping conditions. Analytical solutions and numerical simulations were employed to investigate resonance behavior and steady-state oscillations. The results showed that resonance occurred when the driving frequency approached the natural frequency of the system, resulting in a significant increase in oscillation amplitude. Furthermore, increasing the damping coefficient was found to effectively reduce the steady-state amplitude and suppress resonant motion. This study demonstrates that the complex vibrational behavior of a tall building such as Taipei 101 can be reasonably understood using a simplified damped and driven oscillator model. The results highlighted the critical role of damping in vibration control and provided a clear physical framework for understanding how oscillation theory applies to real-world structural engineering problems.

### **Group 6: Lissajous Figures**

**Authors:** 郭姵妏 (Michelle), 戴亞格 (Van), 艾夏軍 (Shakir)

Lissajous figures are patterns formed by combining two perpendicular oscillations, forming curves that range from simple ellipses to intricate multi-lobed loops. They arise from the interplay of frequency and phase, described mathematically by the parametric equations: $x(t)=sin(\omega_x t+\delta x)$, $y(t)=sin(\omega_y t+\delta y)$. The ratio of the two frequencies determines the shape and symmetry of the figure. Closed and repeating patterns appear when the ratio is rational, while more complex, non-repeating shapes emerge from irrational ratios. Phase angles also change the orientation and symmetry of figures formed even if the ratio of the frequencies is the same, but it can be kept constant unlike the frequencies. Lissajous figures are not only visually captivating but also highly practical: they can be used to measure unknown frequencies, determine phase differences, and even tune musical instruments relative to standard pitches such as $A4=440$ Hz. By counting the curve's intersections with the horizontal and vertical axes, the frequency ratio can be quickly inferred. Lissajous figures transform abstract vibrations into visible, elegant patterns, revealing the hidden beauty of oscillatory motion and offering a powerful bridge between mathematics, physics, and art. Their combination of scientific insight and visual elegance makes them an ideal subject for exploration and demonstration.

### **Group 7: Euler's Formula as a Language of Phase Evolution for Planar Motion**

**Authors:** 劉宥均 (Jimmy), 程昕 (Neil), 谷懷仁 (Quanta)

Planar orbital motion under a central gravitational force possesses an intrinsic rotational structure that is naturally captured by Euler's formula. In this work, two-dimensional motion is expressed in complex form as $z(t)=r(t)e^{i\theta(t)}$, where the exponential factor encodes the instantaneous orientation, or phase, of the trajectory in the orbital plane. This representation treats planar motion as the combined evolution of radial distance and phase, unifying geometry and dynamics within a single mathematical object. Differentiation of the complex position using standard calculus directly yields expressions for velocity and acceleration in which radial and tangential components emerge transparently, with multiplication by the imaginary unit corresponding to a ninety-degree rotation in the plane. When Newtonian gravity is introduced as a purely radial acceleration, the complex formulation reveals a clear structural constraint: the imaginary component of the acceleration must vanish. This condition leads immediately to conservation of $r^{2}\dot{\theta}$ which corresponds to constant areal velocity and constitutes Kepler's second law. In this framework, Kepler's 2nd law is interpreted as a statement about constrained phase evolution rather than as an independent geometric rule. Circular orbits arise as a special case of uniform phase evolution with constant radius, while non-circular orbits correspond to coupled radial and phase dynamics governed by the same underlying structure. This approach does not modify Newtonian mechanics or introduce additional physical assumptions. Instead, it provides a compact and conceptually transparent language for planar orbital motion in which rotational structure, conservation laws, and orbital behavior emerge directly from the mathematics of Euler's formula. By emphasizing phase evolution as the organizing principle of planar motion, this formulation clarifies the geometric origin of Keplerian dynamics and highlights the natural role of complex exponentials in describing gravitational orbits.