The Hidden Math Behind Spirograph Art: How to Predict Your Patterns
Spirograph drawings may look beautifully unpredictable, but there’s a clear mathematical structure underneath every curve. Once you understand a few simple ideas—gear ratios, prime factors, and pen offsets—you can predict what your design will look like before you even start. This transforms Spirographing from a nostalgic activity into a precise and creative design process. 1.…
Understanding Transformers: An Intuitive Guide to AI’s Most Powerful Model
Introduction The rise of large language models (LLMs) like ChatGPT, Claude, and Gemini has transformed the AI landscape. But behind these innovations lies a single fundamental breakthrough: Transformers. Originally introduced in the groundbreaking 2017 paper Attention Is All You Need, Transformers have since revolutionized natural language processing (NLP), speech recognition, and even computer vision. But…
Short History of Idealism
“Idealism,” in its original sense, should actually be understood as “formalism.” (Incidentally, the term materialism—which is often used as the opposite of idealism—should more accurately be translated as hylomorphism or material formalism). In ancient Greek philosophy, form (or idea) signified the true essence of a thing—something that exists eternally, without birth or decay. Greek philosophers…
Emotional Resilience While Facing Audience
Most people’s self-confidence is built upon external validation. In reality, everyone you encounter is a reflection of yourself. There is a Buddhist saying: “When a Buddha sees a Buddha,” which means that if you truly understand this principle, those deeply rooted feelings of inferiority will disappear in an instant. Let’s first consider a common phenomenon:…
Beyond Ordinary Life
Can You Accept Living an Ordinary, Even Mediocre Life? Many people underestimate themselves. This statement isn’t intended to urge you to achieve great feats, amass wealth, or chase after fame. Rather, it’s to highlight how easily we diminish the vastness of our own being, eroding parts of our true selves in the process. The concepts…
How to Cultivate High-Quality Relationships
Why Are Kind-Hearted, Simple People Often Disliked? Human interactions are driven by three core motivations: These three motivations—affinity, achievement, and security—form the foundation of all social interactions. Unfortunately, individuals described as “kind-hearted, simple, and without ulterior motives” often fail to align with these three motivations, which is why they may be less favored in social…
Is Immortality a Kind of Torture?
Do not use a mortal brain to ponder the troubles of immortality. When you look at a cicada, its eyes can only see close distances, and its ears are deaf. How pitiful. But the cicada looks at you and thinks, unable to hear ultrasonic waves and unable to fly, how unfortunate you are. When you…
Why Leaders Eat Last: Building Cultures of Trust
Great leadership doesn’t come from authority or titles—it emerges from the ability to foster trust, empathy, and a shared sense of purpose. True leaders inspire us contributing to something bigger than ourselves, something with value that would last well beyond our own lifetimes. Simon Sinek’s insights in Leaders Eat Last challenge us to rethink how…
Nothing That Is Worth Knowing Can Be Taught
The statement attributed to Oscar Wilde invites a deeper exploration into the intrinsic nature of knowledge acquisition. It might appear confrontational to traditional education systems, yet a more nuanced interpretation reveals a profound truth: the limitations of formal teaching in encompassing the entirety of wisdom and worthwhile understanding. Formal education undoubtedly plays a pivotal role…
The Art of Induction and Deduction in Critical Thinking
Critical thinking, the cornerstone of rational inquiry and problem-solving, relies on the dynamic interplay between induction, deduction, and their modern counterpart, machine learning. These reasoning methods form the bedrock of logical analysis, enabling individuals—and increasingly, machines—to navigate complexities and arrive at informed conclusions. Induction, a reasoning method rooted in observation and empirical evidence, finds its…
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Natalie Feng 