Explore the fascinating connections between mathematics, art, and the human form in this 47-minute Oxford Mathematics Public Lecture. Delve into the language of curves, from the stability of oil rigs to gravitational lensing and life drawing. Discover how René Thom's Catastrophe Theory provides the key to understanding curved surfaces, and learn about folds, cusps, and swallowtails. Journey through diverse fields including physics, engineering, biology, and art, with a focus on the beauty of curves. Examine concepts like structural stability, geometric sequences, and energy surfaces, while appreciating their applications in natural phenomena and artistic expression. Gain insights into the Renaissance language of curves, wave nature, and the intriguing Whitney umbrella. Note: This lecture contains nudity in the context of life drawing.