13.1 Math and Art

The golden ratio, a mathematical concept of about 1.618, pops up everywhere in nature and art. It's in plant patterns, human body proportions, and even our faces. This ratio is often seen as pleasing to the eye.

The Fibonacci sequence, where each number is the sum of the two before it, is closely linked to the golden ratio. As you go further in the sequence, the ratio between adjacent numbers gets closer and closer to that magical 1.618.

The Golden Ratio and Fibonacci Numbers

Golden ratio in nature and anatomy

• Golden ratio of approximately 1.618 appears frequently in nature and is considered aesthetically pleasing
• Found in spiral arrangement of leaves and branches in plants (sunflower seed pattern)
• Observed in proportions of the human body
  • Ratio of total height to height of the navel close to golden ratio
  • Ratio of length of hand to length of forearm near golden ratio
• Seen in proportions of the human face
  • Ratio of distance between eyes to width of nose often close to golden ratio
  • Ratio of distance from top of head to chin compared to distance from lips to chin approximately equal to golden ratio
• Appears in sacred geometry, which explores geometric patterns found in nature and religious symbolism

Fibonacci ratios vs golden ratio

• Fibonacci sequence is series of numbers where each number is sum of two preceding ones (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...)
• As Fibonacci sequence progresses, ratio between adjacent numbers converges to golden ratio
  • 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615, 34/21 = 1.619, and so on
  • Further along sequence, closer ratio gets to golden ratio of approximately 1.618
• To calculate ratio between adjacent Fibonacci numbers:
  1. Divide any Fibonacci number by previous number in sequence
  2. $F_n / F_{n-1}$ approaches golden ratio as $n$ increases

Golden rectangles in art and architecture

• Golden rectangle is rectangle with side lengths in golden ratio of approximately 1:1.618
• Used in art and architecture for centuries due to aesthetically pleasing proportions
  • Parthenon in Athens, Greece (440 BC) incorporates golden rectangles in facade
    • Overall dimensions of structure and spacing of columns closely approximate golden rectangles
  • "Mona Lisa" by Leonardo da Vinci believed to have been composed using golden rectangles
    • Dimensions of painting itself and placement of key elements within composition align with golden rectangle proportions
  • United Nations Secretariat Building in New York City (1952) features golden rectangle facade
    • Building's proportions, with width-to-height ratio of approximately 1:1.618, exemplify use of golden ratio in modern architecture

Mathematical concepts in art

• Perspective: Mathematical technique used in visual arts to create the illusion of depth and space on a flat surface
• Tessellation: Repetitive pattern of geometric shapes that fit together without gaps or overlaps, often used in decorative art and architecture
• Fractals: Self-similar patterns that repeat at different scales, found in nature and used in computer-generated art
• Platonic solids: Five regular polyhedra (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) that have been used in art and design for their symmetry and aesthetic appeal