Interior design and mathematics

Summary: Mathematics is involved in the layouts and color schemes of interior design.

Interior design is a career that combines mathematics and art and is the art and craft of making living spaces that bring positive emotional and aesthetic effects to the inhabitants. Mathematics has long been connected to interior design. One of the 10 books of Marcus Vitruvius Pollio’s mathematical work, De Architectura, is focused on interior decoration. These books heavily influenced the Western scientific, engineering, and mathematical developments in the 2,000 years since they were written. Mathematician Jamshid al-Kashi approximated the surface area of a muqarnas, a decoration made of flat and curved polygons that covers building joints, while the Art Deco design movement of the 1920s and 1930s relied on geometric forms.

Computations to estimate the amount of materials and their cost, such as the area of a surface that will be covered in fabric, tile, or paint, underlie interior design. The International Mathematics & Design Association was founded in 1998. It publishes a journal focusing on areas such as computer-aided design, computational geometry, mathematical modeling, visualization, and system media design.

Design Principles and Elements

Lists of overarching design principles tend to sound very mathematical. Figure 1 shows a widely accepted example, with mathematical elements listed next to each principle.

The principles are achieved through the combination of design elements. With the extensive use of software and digital media for home decorating, expressing design elements in formulas, graphs, tables, and other mathematical representations has become commonplace. A typical list of elements includes the following: line, shape, direction, size, texture, color, and value.

Figure 1.

Shape and Logistics

A circle has the maximum possible area to perimeter ratio of all two-dimensional shapes. This characteristic is the reason circles were adopted as elements of living space structures whenever exposure needs to be minimized. Houseboats may have circular windows to minimize leak danger. In noisy cities or harsh natural environments, architects opt for circular shapes of houses, or their key parts, to minimize the contact with the outside. This design was used in many cultures, such as in traditional Mongol yurts, Celtic roundhouses, Lakota tipis, and Lesotho’s mokhoros.

City and road building dictated cuboid houses for several reasons. Many tools and materials make it easy to mass-produce rectangular structures, such as boards, bricks, and panels. Also, circles do not tessellate (tessellation occurs when a repeated shape covers a plane without any gaps or overlaps), making it impossible to build circular houses adjacent to one another, as is done in cities. While hexagons tessellate, the edge of a block of hexagonal houses is not straight, making it problematic to build roads. Another tessellating floor shape candidate, a triangle, has sharp corners that are inconvenient to use and psychologically problematic.

Order in Complexity: Tessellations and Fractals

Repeating patterns satisfy the design principles of balance, rhythm, and unity. Since they are practical to make and use with a variety of simple tools, they are used in home decorating in all human cultures. Tessellations appear in mosaics, on all parts of buildings, and in designs of rugs, coverings, and wall hangings. Traditional designs often combine beauty, cultural and spiritual meanings, and utility through modular units.

The mathematics known to artisans is still being formally described. This process of rediscovery and formal mathematical description is called “ethnomathematics.” For example, medieval Islamic mosaics masters described very complex symmetric, never-repeating patterns, made of standard polygons. These patterns were rediscovered by mathematicians in the 1970s and named “quasicrystalline” and “Penrose tilings.” Many traditional African villages are laid out to form fractals, with the village shape repeated in the house clusters, houses, and interiors of each house.

These villages were first mathematically described in the 1990s. Mandalas—appearing in several cultural traditions, such as Hinduism, and in work of modern artists, such as M.C. Escher—exhibit elements of projective geometry, as their tessellating shapes shrink toward the edge of the circle. Once the underlying mathematical principles of complex patterns are understood, software can be programmed for further experimentation and discovery. Complex computer-generated patterns, often incorporating ancient artisan traditions, are now ubiquitous in home decorating materials such as wallpaper, flooring, textiles, and tiles.

Color Models, Circles, and Schemes

Colors are defined by spectral wavelengths; for example, the wavelengths of reds are approximately 630–730 nanometers. Color models define colors as additions or subtractions of primary colors and are used to pinpoint precise colors for decorating projects, often using software. The additive model known as “red, green, blue” (RGB) can be physically implemented in overlapping lighting with different colors. The subtractive model known as “cyan, magenta, yellow, key black” (CMYK) can be implemented by mixing pigments and is used in color printing, including wallpaper, yarn, and fabric dyes, as well as in mixing household paints. A color wheel is a traditional artist and designer infographic used to visualize color models. The wheel has the primary colors positioned at three equidistant points around it and color mixes between the primaries, with the position signifying the ratio of the mix.

A color scheme is a combination of two or more colors that work well together. Home decorators use special terms to describe colors, with each term having mathematical meaning in color models. The terms for describing colors include “warm,” “cool,” “hue,” “intensity,” “contrast,” and “tone.” On the other hand, colors can be described metaphorically, which is used more frequently in consumer-oriented product names such as “Light in the Leaves” or “Chilled Chardonnay.”

Bibliography

Cook, Tony, and Robin Prater. ABC’s of Architectural and Interior Design Drafting with an Introduction to AutoCAD 2000. Upper Saddle River, NJ: Prentice Hall, 2000.

Dold-Samplonius, Yvonne. “Practical Arabic Mathematics: Measuring the Muqarnas by al-Kashi.” Centaurus 35 (1992).

International Mathematics & Design Association. “M & D.” http://www.maydi.org.ar/index‗eng.html.

Lidwell, William, Kritina Holden, and Jill Butler. Universal Principles of Design: 125 Ways to Enhance Usability, Influence Perception, Increase Appeal, Make Better Design Decisions, and Teach Through Design. Rev. ed. Beverly, MA: Rockport Publishers, 2010.

“Modules and Monographs in Undergraduate Mathematics and Its Applications Project” (1980). http://www.eric.ed.gov/PDFS/ED218125.pdf.