13.1 Math and Art
3 min read•june 18, 2024
The , 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 , 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
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How to Use the Golden Ratio in Design (with Examples) View original
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Top images from around the web for Golden ratio in nature and anatomy
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- Golden ratio of approximately 1.618 appears frequently in nature and is considered aesthetically pleasing
- Found in 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 , 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:
- Divide any Fibonacci number by previous number in sequence
- approaches golden ratio as increases
Golden rectangles in art and architecture
- 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
- in Athens, Greece (440 BC) incorporates in facade
- Overall dimensions of structure and spacing of columns closely approximate golden rectangles
- "" by believed to have been composed using golden rectangles
- Dimensions of painting itself and placement of key elements within composition align with golden rectangle proportions
- 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
- in Athens, Greece (440 BC) incorporates in facade
Mathematical concepts in art
- : Mathematical technique used in visual arts to create the illusion of depth and space on a flat surface
- : Repetitive pattern of geometric shapes that fit together without gaps or overlaps, often used in decorative art and architecture
- : Self-similar patterns that repeat at different scales, found in nature and used in computer-generated art
- : Five regular polyhedra (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) that have been used in art and design for their and aesthetic appeal
Key Terms to Review (18)
Aesthetics: Aesthetics refers to the branch of philosophy that deals with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It plays a crucial role in understanding how mathematical concepts can inspire artistic creations and influence visual experiences, bridging the gap between mathematical precision and artistic expression.
Fibonacci sequence: The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. This sequence is closely related to the golden ratio and appears frequently in nature, art, and architecture, showcasing the deep connection between mathematics and aesthetics.
Fractals: Fractals are intricate geometric shapes that can be split into parts, each of which is a reduced-scale copy of the whole. This self-similar property showcases the connection between mathematics and art, as they illustrate how complex patterns emerge from simple, repeated processes. Fractals highlight the beauty in mathematics, bridging the gap between numerical concepts and visual representation.
Golden ratio: The golden ratio is a mathematical constant approximately equal to 1.6180339887, often denoted by the Greek letter phi (Φ). This unique ratio arises when a line is divided into two parts, such that the ratio of the whole line to the longer part is equal to the ratio of the longer part to the shorter part. Its aesthetic appeal has made it a foundational element in art and architecture, linking mathematics with beauty and proportion.
Golden rectangle: A golden rectangle is a rectangular shape in which the ratio of the longer side to the shorter side is the golden ratio, approximately 1.618. This unique ratio has been admired for its aesthetic properties and is often found in nature, art, and architecture, contributing to visually pleasing compositions that capture the viewer's attention.
Golden rectangles: A golden rectangle is a rectangle whose side lengths are in the golden ratio, approximately 1.618:1. It is often considered aesthetically pleasing and appears frequently in art and architecture.
Leonardo da Vinci: Leonardo da Vinci was a Renaissance polymath known for his contributions to art, science, and engineering. His innovative approaches combined mathematical principles with artistic techniques, creating masterpieces like the 'Mona Lisa' and 'The Last Supper', which illustrate the intersection of math and art through proportions, perspective, and symmetry.
Mona Lisa: The Mona Lisa is a world-renowned painting created by the artist Leonardo da Vinci, believed to have been completed in the early 16th century. The artwork is famous for its sophisticated use of techniques such as sfumato, which creates a soft transition between colors and tones, contributing to the subject's enigmatic expression. It represents a fusion of art and mathematics through its composition, proportions, and use of perspective, showcasing how mathematical principles can enhance artistic beauty.
Parthenon: The Parthenon is an ancient temple on the Acropolis of Athens, dedicated to the goddess Athena, whom the people of Athens considered their patroness. This architectural marvel symbolizes the glory of Ancient Greece and embodies the ideals of harmony, proportion, and beauty that define Classical architecture.
Perspective: Perspective is a technique used in art and mathematics to create the illusion of depth and space on a flat surface. It involves representing objects in a way that mimics how they appear to the human eye, giving a sense of three-dimensionality. This concept plays a vital role in the way artists compose their works, making use of vanishing points and horizon lines to guide the viewer's eye.
Platonic Solids: Platonic solids are the five regular polyhedra that are convex, with congruent faces composed of regular polygons. These solids have equal edges, angles, and faces, representing a harmonious relationship between geometry and symmetry that has fascinated artists and mathematicians throughout history. Their unique properties and aesthetic appeal have made them a fundamental part of both mathematical theory and artistic representation.
Proportion: A proportion is an equation that states that two ratios are equal. It reflects the relationship between quantities and can be used to express how one quantity compares to another, whether through scaling, sharing, or finding parts of a whole. This concept connects to various mathematical applications, including rational numbers, where it helps understand comparisons, and visualizations, where proportions can illustrate data relationships.
Sacred geometry: Sacred geometry is the study of geometric shapes and proportions that have symbolic and spiritual significance, often reflecting the fundamental patterns of the universe. It connects mathematics to art, architecture, and nature, illustrating how geometric principles can be seen in the design of ancient structures, religious symbols, and natural formations. This relationship highlights the idea that geometry can serve as a bridge between the material and spiritual realms.
Spiral: A spiral is a curved geometric shape that winds around a central point, moving farther away from that point as it progresses. This unique shape can be found in various forms in both mathematics and art, serving as a fundamental concept that bridges the two fields. Spirals can be categorized into several types, including logarithmic spirals and Archimedean spirals, each with distinct mathematical properties and visual representations.
Symmetry: Symmetry refers to a balanced and proportionate similarity in the arrangement of parts on opposite sides of a dividing line or around a central point. It plays a crucial role in understanding patterns and designs, as it helps create harmony and visual appeal in various forms, including art and geometric shapes.
Tessellation: Tessellation refers to the covering of a plane with one or more geometric shapes, called tiles, without any overlaps or gaps. This concept is crucial in various fields such as mathematics and art, where it showcases patterns and symmetries, and emphasizes how shapes can fit together harmoniously. The study of tessellations reveals important properties of shapes and their relationships in two-dimensional spaces.
Tessellations: A tessellation is a pattern of shapes that fit together perfectly without any gaps or overlaps. These patterns can cover a plane infinitely using one or more geometric shapes.
United Nations Secretariat Building: The United Nations Secretariat Building is the headquarters of the UN, located in New York City, and serves as the central administrative office for the organization. It is a symbol of international cooperation and diplomacy, showcasing architectural innovation and urban design, reflecting the ideals of the United Nations through its modernist aesthetic and use of geometric shapes.