p. 1 — Introduction
张 语 芮
Definition 1.1
identity := "mathemati"
∀ challenges ∈ ℝ, ∃ solution ∈ mind
A high school student who quit competition math, rediscovered it through real analysis, and found that the perfect fifth is a continued fraction convergent.
§ 1 — About
I'm a high school student at Chongqing Weiming School, applying to U.S. universities in the Class of 2026. I transferred here this semester, seeking an environment that could support the mathematical depth I was hungry for — my previous school offered AP courses but not the kind of mentorship I needed for serious mathematical study.
My mathematical journey began with competition training in middle school, hit a wall of burnout, and was quietly rekindled by the elegance of AMC/AIME problems. Now I'm working through real analysis — ε-δ definitions, uniform convergence, Fourier series — and finding that abstraction isn't obscurantism. It's clarity.
Outside of math, I hold a Level 8 National Art Certificate in sketch drawing, and I'm fascinated by how mathematical structures appear in music, architecture, and the natural world.
// quick facts
§ 2 — Proof of a mathematician
"Six years ago I thought math was a race. Now I know it's a field."
Selected for the school math competition team. Trained for Gaolian, learned to love the camaraderie of problem-solving. Math felt alive.
n! = n × (n−1)!Friends left the team. Training became mechanical — endless drills, no joy. A teacher's warning that 'one CMO result determines your fate' extinguished the flame. I quit.
∅Discovered AMC/AIME. Their emphasis on flexible thinking over computational speed felt like breathing fresh air. A quiet realization: math competition can look different.
AMC → AIME → ?Started working with a Yau competition coach. Built the framework from scratch: ε-δ limits, uniform convergence, parametric integrals, Fourier series. Spent two weeks stuck on proving the equivalence of six completeness theorems — then broke through.
∀ε>0, ∃δ>0: |x−c|<δ ⟹ |f(x)−L|<εSelf-studied linear algebra and abstract algebra. Moved from computation to structure — groups, rings, fields, Sylow theorems. The shift from 'how to calculate' to 'why this structure exists' changed everything.
G/N ≅ HExploring number theory through Diophantine approximation and continued fractions. Discovered their connection to musical tuning systems — why the perfect fifth sounds harmonious is a question about rational approximations of log₂(3).
φ = [1; 1, 1, 1, ...] = (1+√5)/2"Mathematics is not a racetrack with a finish line. It is an open field — and curiosity is the only fuel that doesn't run out."
— from my application essay
§ 3 — Academic record
AP courses taken at previous school · self-directed study in real analysis & abstract algebra
§ 4 — Activities & projects
Math is the lens. Everything else is what I see through it.
Passed the national standardized sketch examination at the highest level. Nine years of drawing practice, from observation to abstraction.
Founding member of the school's Women in STEM initiative, creating space for female students to pursue rigorous science and mathematics.
Designed and implemented a kitchen waste composting system, studying microbial decomposition rates and soil nutrient recovery.
Building a 3D plant model database combining architectural modeling techniques with biological illustration. Bridges design, biology, and computational geometry.
Community service initiative redesigning public spaces for elderly accessibility. Applying landscape architecture principles to real-world social challenges.
Creating an illustrated calendar combining traditional Chinese ink painting and oil painting techniques, one artwork per solar term.
§ 5 — Explorations
// a question that started everything
My interest in number theory grew from Diophantine approximation: how well can we approximate irrational numbers with rationals? Continued fractions are the optimal tool.
Musical harmony is the same question. The octave is a 2:1 frequency ratio. The perfect fifth is 3:2. Consonant intervals correspond to simple integer ratios — the foundation of just intonation.
Equal temperament? It's the rational approximation of 2^(1/12) — making all intervals slightly impure so all keys are equally usable. The twelve-tone system is a continued fraction convergent.
consonance ≈ simplicity of ratio ≈ quality of approximation
Fascinated by how mathematical structures — tessellations, minimal surfaces, fractal geometries — manifest in built and natural environments.
future study directionNine years of drawing practice. Level 8 National Certificate. Art as a different language for describing structure and form.
Level 8 certifiedThe most interesting questions live at the borders: math × music, biology × geometry, architecture × topology.
the real subject§ 6 — Contact
Whether it's a hard problem, a beautiful proof, or a question about continued fractions and musical tuning — I'm always up for a conversation.
Chongqing Weiming School
Class of 2026 · Mathematics & Architecture
φ = [1; 1, 1, 1, ...] = (1+√5)/2