Mathematics in Art and Design LESSON 4

Geometry

framework for structure, symmetry, transformations, and spatial reasoning.

Euclidean geometry

lines, angles, polygons, circles

Symmetry

reflection, rotation, translation

Tessellations

Tessellations

Proportion and scaling

ratios, similar figures, enlargement

balance, visual stability, aesthetic harmony

These geometric ideas ensure?

Golden Ratio (φ ≈ 1.618).

Strongly studied by mathematicians and widely applied by artists, architects, and designers.

Examples of Golden Ratio

• Greek temples

• Renaissance paintings

• Modern photography composition

• Product design (smartphones, logos)

Perspective

one of the clearest intersections between mathematics and visual art.

Basic Perspective Concepts

• Horizon line

• Vanishing point(s)

• Orthogonal lines (lines that converge to a vanishing point)

• Foreshortening

One-Point Perspective

Used for roads, hallways, railway tracks, All depth lines converge to one point

Two-Point Perspective

Used for buildings and urban scenes,
Shows left and right depth using two vanishing points

Three-Point Perspective

Adds height/depth
Used for skyscrapers or dynamic angles (top view / worm’s eye view)

Op Art (Optical Art)

• geometric patterns

• symmetry and transformations

• contrast in color and line width

• visual illusions based on wave patterns, grids, and distortions

• often modeled using trigonometric functions and geometric sequences

Optical Art Artist

Victor Vasarely, Bridget Riley 

Minimalism

Characterized by:

• simple geometric shapes

• clean lines

• repetition

• emphasis on order and structure

• “less is more” philosophy

Math influences ______ through:

• geometric reduction

• linearity

• emphasis on form rather than decoration

Geometry and Coordinate Systems

Digital images use:

• Cartesian coordinates

• vectors

• matrices

Applications include:

• scaling

• rotation

• transformations

• projections (used in 3D rendering)

 Pixel Geometry

A digital image is a matrix of:

• tiny squares (pixels)

• arranged in rows and columns

• each pixel having coordinates and values

Thus, digital art is fundamentally discrete geometry.

3D Graphics and Modeling

Rely on

• geometry of polyhedra

• functions for shading and lighting

• matrix transformations for rotation, translation, projection

• algorithms for rendering surfaces

Used in:

• video games

• animation

• simulations

• architectural visualization