Underlying Principles and Emerging Designs: Design Exercises Based on Magic Squares

  • Jin-Ho Park
Part of the KAIST Research Series book series (KAISTRS)


This article discusses how a simple and efficient design principle may be used to create a large collection of hybrid designs. Applying such principle in creating assembled arrays of designs can generate visually elusive final designs. This paper has three tasks. First, it examines the influence of the fundamental principle on the creation of new designs. Some notions and approaches of designers and architects are also reviewed. Second, this paper examines the basic properties of magic squares and employs its principles in the analysis and synthesis of designs. Third, this paper applies the principles of magic squares in creating new designs. Some examples of designs are also presented in this paper.


Magic sqaure Principle Grid Geometric pattern Parametric design 

1 Introduction

One may encounter various designs or patterns in any cultural artifact, especially artwork. Some designs seem to come out randomly, by chance, or as a capricious act of the designer’s will. These designs are very personal to the extent that they have been criticized for being arbitrary, ad hoc, or eccentric although nothing is wrong with such designs. By contrast, the sophisticated look of these designs have been praised by others. Given their complexity, the composition of such designs is difficult to analyze at first glance. However, these designs may be explained in a rational and methodical manner.

Some designers have developed their own styles and come up with designs that may be consistent or similar with their other projects. To achieve consistency in their designs, designers must develop their own design principles or rules. Styles with different designs may share the same design principles [1]. By following their own design principles or methods, designers have the power to invent new designs. Applying certain principles can guide these designers in creating new designs. Accordingly, some designers keep in mind the notion of consistency when creating subtle designs. These designers may also be challenged in creating completely new styles for each of their project.

By modifying their principles through their creative talent, some designers can efficiently generate a multitude of design variations. They must also leave some room for subjectivity according to their creative desires and tastes. Designers may even break the stylistic conventions on purpose to achieve a certain effect, to improve their focus, or to highlight a specific reason. For example, by mastering the symmetry fundamentals, designers can easily break the symmetry and create asymmetrical designs [2]. One does not have to follow particular principles or rules; a mere appreciation of the fundamental principles in arts and architecture will be extremely useful in developing unique design styles or comprehending other styles.

This paper discusses the influence of fundamental principles on the creation of new designs. Aside from reviewing the notions and approaches employed by designers and architects, this study also examines the basic properties of magic squares and employs its principles in the analysis and synthesis of designs. This study also applies the same principles in creating new designs, which will be presented in this paper.

2 Notion of Underlying Principes

The fundamental principles of designs imply the ways that designers array the motifs of design in their artwork. These principles also help designers understand the spatial forms of their designs from the initial planning to the design stages. On the one hand, these principles compose the basic framework for understanding complex designs. On the other hand, these principles offer inventive insights into the creation of innovative designs.

The significance of the fundamental principles in creating new designs must not be overemphasized. In his article, The Metamorphosis of Plants, Johann Wolfgang von Goethe proposed the notion of metamorphosis and pursued urform, an ideal form, that underlies all plant and animal forms [3]. This article also lays the foundation for the subsequent developments in arts and architecture. Goethe also coined the term, morphology, to delineate the study of natural forms.

Many researchers, including Ernst Haeckel, D’Archy Thompson, and Wilson Bentley, have tried to find the underlying principles in natural objects. Their approaches have greatly influenced the search for pure geometric forms and the underlying principles of spatial compositions in architecture with stripped-out ornamental elements. Such tendencies are most apparent in the works of Étienne–Louis Boullée, Claude–Nicolas Ledoux, and Jean–Nicolas–Louis Durand in France as well as those of Friedrich Gilly and Karl Friedrich Schinkel in Germany. Gottfried Semper also used the term, “tectonics,” to describe his theoretical perspective toward the fundamental principles of forms in nature and in art. He wrote, “Tectonics is an art that takes nature as a model—not nature’s concrete phenomena but the uniformity [Gesetzlichkeit] and the rules by which she exists and creates” [4]. He emphasized that appreciating the fundamental principles would help one generate various creative works through modification and transformation.

Unlike others, Louis Sullivan demonstrated how these principles could be elaborated and developed for the creation of new organic designs. On the first page of his book, A System of Architectural Ornament, Sullivan wrote, “Remember the Seed-Germ” [5]. He defined the notion of seed-germ as the universal beginning or fundamental principle from which a plant will grow. In other words, the seed-germ is a concept of biological growth where several complex designs are elaborated through organic or geometric means. In A System of Architectural Element, Sullivan demonstrated how designs evolved and transformed from principles. Several creative and idiosyncratic designs can be developed by grasping the fundamental principles at work. In fact, these principles have been transformed into integral parts of designs (Fig. 1).
Fig. 1

Sullivan’s generating process where simple polygons and their axes are used as underlying principles of final organic designs (from Louis H. Sullivan: A System of Architectural Ornament)

The most striking notion in architecture that emerged during the 20th century appears in the discussion of Louis I. Kahn regarding form and design. According to Kahn, form is an abstract notion, while design is an interpretation of the form. Moreover, the former has to do with eternal essence, while the latter relates to elaboration and manifestation [6].

Albeit their slight differences, abstract artists and architects have attempted to find the underlying rules or essential forces of forms. De Stijl architect Theo van Doesburg described how he simplified a cow into an abstract painting in Study for Composition (1917–18) and how he designed the stained glass window, Grote Pastorale (1922), where human figures were simplified into basic geometric components by arranging and juxtaposing rectangular and triangular elements with primary colors [7].

Pablo Picasso adopted the similar process. In 1945, Picasso introduced a novel abstraction approach in his drawings of a bull in lithograph [8]. He demonstrated how realistic objects could be transformed into abstract ones through a series of sketches. During this process, Picasso aimed to present the essential elements or presence of a bull through a successive analysis of its form. By reducing and simplifying its anatomy, Picasso sketched a few lines that represented the essential parts of forms. While van Doesburg simplified the cow with planes and colors, Picasso outlined the bull with lines. This whole process aims to find and express the absolute essence of the bull in a concise and abstract manner.

Several theoretical and practical perspectives have emerged to highlight the significance of the fundamental principles in the creation of new designs. Some of these perspectives search for the notion of design principles, while others encode such principles as guiding laws of their designs. When equipped with the necessary knowledge, designers have a higher degree of freedom to break the existing rules or set their own rules. Instead of following an array of seemingly arbitrary rules, designers may be better off by creating their own styles. This tradition continues to shape much contemporary scholarship of architecture like Christoper Alexander’s pattern language, Lionel March’s fundamentals of architectonics, and George Stiny’s shape grammar [9, 10, 11].

The following sections present analytic and constructive approaches that adopt the underlying principles of designs. These approaches are complimentary in the sense that the analytic approach seeks to reduce a substantial whole into parts and components to clarify how the parts are arrayed and to infer the principles and rules that comprise the whole, while the constructive approach seeks to achieve the opposite. Partial motifs are combined into a coherent whole according to certain principles and rules [12]. Some design examples are presented as proof of applications. Both of these approaches have also been applied in analyzing the designs.

3 Magic Square: An Underlying Principle of Spatial Designs

Although different computational tools may be applied in the analysis and synthesis of designs, we use the fundamental principles of magic squares as an example. Despite being a relatively modern trend, magic squares have been used by designers and architects to devise design solutions. The principles of magic squares have fascinated mathematicians for many years. This technique is believed to have originated from China before spreading across the world [13]. The principles of magic squares have various applications relating to divination, alchemy, cosmology, and astrology. Some scholars have used the magic square to create a pattern for the layout of royal cities in China. The application of this technique can also be found in Melancholia I by Albrecht Dürer [14] and in Sagrada Familia by Antoni Gaudi.

Mathematicians have developed variations of magic squares, such as alphamagic squares, panmagic squares, antimagic squares, Franklin squares, and Latin squares. This paper limits its focus on an elementary property of the mathematical structure of magic squares. A magic square is an arrangement of consecutive natural numbers. The sums of the numbers in each row, column, or diagonal of an n × n matrix are the same. Given that a magic square follows an n × n format, its size is described by the order of n. Accordingly, a 3 × 3 magic square is described as the order of three magic squares. The sum of each row, column, and diagonal of a basic magic square is calculated as [n(n2 + 1)]/2, while the middle number of an odd order magic square is calculated as (n2 + 1)/2. Accordingly, a 3 × 3 magic square contains every number from 1 to 9, thereby having a sum of 15 and a middle number of 5. When n = 4, 5, and 6, the sum becomes 34, 65, and 111, respectively.

Several methodical accounts for constructing odd and even order magic squares have been presented in the literature. Figure 2 shows a sample order of 3, 4, and 5 magic squares (Fig. 2).
Fig. 2

The order of 3, 4, and 5 magic squares

Given that magic squares are presented as a square grid, the symmetry group of the square is used to transform the arrangement of magic squares. The symmetry group of the square comprises eight distinguishable operations, namely, four reflections and four quarter-turns. Accordingly, any magic square can produce eight distinct squares through rotation and reflection.

Claude Bragdon and Richard Paul Lohse discussed the active usage of magic squares in arts and architecture. Influenced by the organic architecture of Louis Sullivan and Frank Lloyd Wright, the architect Bragdon actively applied magic squares to create geometric patterns for artwork and to provide “a source of formal beauty.” He argued, “There is nothing strained or illogical in this, for beauty is ever the fine flower of order and necessity, and in magic squares order and necessity predominantly rule” [15]. Bragdon followed a relatively simple logic when creating the patterns in his designs; starting from a number, he connects the centers of squares in a numerical sequence. The line paths, which he called “magic paths,” connect the consecutive numbers in each square to form a pattern. By tracing the numbers in their consecutive order, Bragdon creates a beautiful motif for an ornament. He also proposed other novel uses of magic squares for generating characteristic designs [16] (Fig. 3).
Fig. 3

Bragdon’s system of connecting the centers of magic squares in a numerical sequence in the case of order 3, 4, 5, and 6 magic squares (drawn by the author)

Bragdon elaborated and enumerated a large spectrum of patterns following his own logic. He also created several patterns for architectural applications, such as ornamental panels, brickwork, gate patterns, and ceiling designs [17]. Despite following a simple process, the final designs of Bragon are very intriguing in the sense that their underlying principles cannot be identified at a quick glance. Above all, these designs may not be immediately apparent to the viewer. However, upon close analysis, one can find that these designs are highly superimposed, intricately planned, or based on reason.

Unlike the use of linear paths with magic squares in the work of Bragdon, Lohse created several gridiron paintings with colors. Many viewers have focused on the symmetries, proportions, and color sequences of these paintings, while others have focused on their underlying systems. Upon closer observation, one can study how Lohse composed the grids and permuted colors on the canvas.

Figure 4 illustrates how the spatial layouts of each painting of Lohse are designed. Most paintings of Lohse are based on a grid structure that comprises proportional or gradually incremented subdivisions that form the basis of a systematic approach for constructing a layout. Lohse systematized the groups of colors by arranging them in a rotational progressive sequence. These colors are arranged in such a way that only one color occupies a grid unit by means of sequential principles. The sizes of grids and colors progressively and proportionally increase or decrease clockwise or counter-clockwise. The geometric design of the grid and the succession of colors form the principles that determine the expression of colors and forms of the entire painting.
Fig. 4

Lohse’s three paintings where proportions and symmetries are used as instrumental devices for the spatial compositions (redrawn by the author)

The choice of colors follows a clear pattern; Lohse based his choices and arrangements of colors not on intuitive or impromptu but on certain principles. Therefore, the color and form in his paintings are closely bound to strict design principles. For example, in his famous painting, ‘Nine Vertical Systematic Color Sequences with Horizontally and Vertically Increasing Density’, Lohse followed a particular sequence of color choices and positions (Fig. 5).
Fig. 5

Lohse’s paintings nine vertical systematic color sequences with horizontally and vertically increasing density, and a latin square (redrawn by the author)

Based on the underlying modular grid of a square, each of the nine colors in the painting is distinctively used for each row and column, that is, the same color does not appear twice in the same row and column. This method has introduced a unique type of magic square that was called a “Latin square” by Leonhard Euler. In this square, each number or alphabet can only be used once to fill an n × n grid. Each row, column, and diagonal uses each number or alphabet. Lohse repetitively applied a similar methodical approach in many of his paintings. His methodical approaches have been well illustrated in the working sketches and drawings presented in his book [18].

4 Parametric Designs

Parametric design is a rule-based approach that deals the algorithmic and geometric properties of a design with constraints and rules. Therefore, the strategic transformation of components is applied to explain the design of complex compositions. This approach allows for an unusual control over the designs in such a way that the aesthetic qualities of the final design is described as complex yet structurally organized.

This section provides examples of constructive designs where motifs can be arranged, transformed, and superimposed according to the underlying principles and rules. Some methodical approaches are applied and manipulated as generative methods for creating new designs. Through these approaches, the development of various designs requires only a few abstracting steps and several sequences.

We use a rhythmic grid instead of other conventional regular grids. Lionel March categorized a series of grids with “three linear elements A, B, C, which had distinct integer dimensions with a < b < c [19]. The grids were designed to accept permutations of these elements.” In this exercise, we use the permutation of the 3, 4 and 5 linear elements that produce a polyrhythmic grid, 3 + 1 + 1 + 2 + 1 + 1 + 3. Given that the grid has seven intervals, a 7 × 7 magic square is used as an arraying tool to array a motif. Afterward, each motif is placed on the center, corner, or side of each square grid. During this process, one must keep in mind that the type of accuracy and regularity that connects each number in sequence is an important aspect of the spatial composition of the designs [20].

We begin by placing the simplest possible motif on a grid to explain the increasing complexity that is captured in the parametric designs. The ordered structure of the whole design is revealed. Following Bragdon, we place the motif on the center of the grids in a numerical sequence. By applying the principle without any transformation in the square grid, the resulting configurations design becomes too apparent (Fig. 6).
Fig. 6

Left: A polyrhythmic grid. Middle: A 7 × 7 magic square. Right: A simple motif is arranged without any transformation in the grid

Several other parametric rules can be applied in this process. For example, we apply symmetry operations, particularly rotations, where transformations and displacements of motifs are performed such that a motif can be simply displaced in a certain position within the grid according to the symmetric operations [21]. Instead of placing a motif on the center of the grid, a motif can be displaced on each corner of the grid according to the symmetric operations. Therefore, a motif can be rotated and placed while moving according to the sequence of numbers. Assume that we follow a set of rules where we place a motif on a corner of a point of a departure grid [22, 23]. Afterward, we move the motif clockwise before placing it on the next corner of the next grid according to the sequence of numbers while being rotated by 90°. In this process, a change in a single movement of the motif will change the entire design (Fig. 7).
Fig. 7

Sequential positions for adding a motif to the corner of the grid

Such consecutive deployment of a motif will generate an entirely unexpected design. Upon removing the grid, the resulting pattern presents a dynamic image with an unrecognizable design (Fig. 8).
Fig. 8

A square dot is placed on the grid according to the magic square. When the grid is removed, the resulting design is highly complex and dynamic

The above rule illustrates the possibilities for generating several irregular patterns. By slightly changing the rules, a huge array of different designs can be generated. For example, a motif can be scaled up or down during its placement on the side of the grid. Figure 8 presents a design where a motif is placed on each side of the grid and then stretched or diminished according to the length of the grid. This rule increases the complexity of the final clustered design (Fig. 9).
Fig. 9

A pattern where a motif is constantly transformed when moving according to the consecutive numbers of the magic square

Following the same principles, the symmetrical or asymmetric motifs can be clustered together. An asymmetric motif increases the complexity of the final design. When larger motifs are used and placed on the grid, the motifs may become overlapped. Therefore, we apply a rule to reduce the number of overlapping parts, and highly complex aggregated patterns are generated as a result (Figs. 10 and  11).
Fig. 10

Designs where the symmetric (left) and asymmetric motifs (right) are aggregated together

Fig. 11

An acrylic painting using a pattern generated in Fig. 10 (painted by the author)

A simple set of parametric rules is iteratively applied to generate complex forms. The results of such application are significantly more complex than expected. The parametric rules provide designers with a tool for controlling complex aggregates. By slightly changing the rules, the number of emerging combinatory possibilities is increased. Although the generated final designs are unexpected, their principles can be detected.

5 Summary

This paper discusses the underlying principles of spatial designs based on magic squares. The basic properties of magic squares are delineated, and some designs of Loshe and Bragdon are presented as examples. Along with the symmetry operations, some complex designs have been generated to illustrate the unique application of these principles. The resulting designs are too complex that their rules and principles are very obscure. Yet, the motifs in the final designs are all tied together to create a cohesive whole. Using magic squares and symmetric operations as underlying principles can facilitate the analysis and construction of complex designs.

Once a series of sequences of operations are developed, further rules are setup and used to create more sophistigated designs. Instead of using a simple rhythmic grid of the same unit, we use a highly complex polyrhythmic grid that we derive from the permutation of three distinctive linear elements. More complex designs can be made by slightly changing grid patterns as well.

This paper presents a novel method to generate a variety of different designs. The results of this study can be further extended and utilized in design practices. In particular, the method can be adopted from small to large scale architectural components such as windows, carpets, wallpapers, housing clusters and complex building blocks.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of ArchitectureInha UniversityIncheonKorea

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