**Optimizing dimensions in furniture design: A literature review**,”

*BioResources*19(3), 4727-4748.

#### Abstract

Wooden furniture design necessitates the integration of both technological requirements and aesthetic considerations. To guide designers in achieving this balance, this article explores how established design principles, such as proportions and preferred numerical sequences, can inform decision-making for both technological and aesthetic aspects. The goal is to demonstrate how these principles can be integrated with modern CAD tools. In reviewing the scientific literature, this study compiled and compared mathematical and non-mathematical models that support dimensional decision-making. These models included ancient canons (Egyptian, Greek, and Roman) alongside those of Leonardo da Vinci, Palladio, Dürer, Le Corbusier, Zeising, McCallum, and Brock. Additionally, the article examines numeral systems used in modern technology, such as Renard’s series and convenient numbers. It is proposed that designers should experiment with geometric design templates to achieve balanced proportions. All geometric design principles contribute to aesthetics, creativity and effectiveness in design. The literature identifies two groups of dimensional design templates: organic, inspired by the human body or the Fibonacci sequence, and inorganic, based on numerical order. It’s impossible to pinpoint a single “optimal algorithm” to support dimensional decisions in design. Specific geometric design principles serve as valuable tools, not the ultimate answer.

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**Optimizing Dimensions in Furniture Design: ****A Literature Review**

Anna Jasińska,^{a,}* Maciej Sydor,^{a} and Miloš Hitka ^{b}

Wooden furniture design necessitates the integration of both technological requirements and aesthetic considerations. To guide designers in achieving this balance, this article explores how established design principles, such as proportions and preferred numerical sequences, can inform decision-making for both technological and aesthetic aspects. The goal is to demonstrate how these principles can be integrated with modern CAD tools. In reviewing the scientific literature, this study compiled and compared mathematical and non-mathematical models that support dimensional decision-making. These models included ancient canons (Egyptian, Greek, and Roman) alongside those of Leonardo da Vinci, Palladio, Dürer, Le Corbusier, Zeising, McCallum, and Brock. Additionally, the article examines numeral systems used in modern technology, such as Renard’s series and convenient numbers. It is proposed that designers should experiment with geometric design templates to achieve balanced proportions. All geometric design principles contribute to aesthetics, creativity and effectiveness in design. The literature identifies two groups of dimensional design templates: organic, inspired by the human body or the Fibonacci sequence, and inorganic, based on numerical order. It’s impossible to pinpoint a single “optimal algorithm” to support dimensional decisions in design. Specific geometric design principles serve as valuable tools, not the ultimate answer.

*DOI: 10.15376/biores.19.3.4727-4748*

*Keywords: Decision making; Design; Preferred value systems; Aesthetic and technological evaluation; CAD objects*

*Contact information: a: Poznań University of Life Sciences, Faculty of Forestry and Wood Technology, Wojska Polskiego 28, 60-637 Poznań, Poland; b: Technical University in Zvolen, Department of Economics, Management and Business, T. G. Masaryka 24, 960 01 Zvolen, Slovakia;*

** Corresponding author: anna.jasinska@up.poznan.pl*

**INTRODUCTION**

Designers consider many dimensional product variants when designing spaces, buildings, and smaller forms, such as furniture. Design in a CAD environment provides additional ease of alternative testing and almost unlimited design possibilities. Analyzing alternative courses of action is an essential part of the decision-making process. Therefore, design is a creative process of constant search for new forms and assimilation of known solutions (Falessi *et al*. 2011). Design engineering literature identifies different techniques for making choices among alternatives but does not provide clear guidance on which technique is more appropriate than another and under what circumstances. The best possible decision-making technique probably does not yet exist, but some techniques are more prone to specific difficulties (Polikar 2006; Jasińska and Sydor 2021). According to Falessi *et al.* (2011), some decision-making techniques are more vulnerable to specific challenges than others. Ideally, architects could choose a technique based on the problems they want to avoid. However, lacking empirical evidence hinders understanding of which circumstances lead to particular difficulties with specific techniques. This makes it difficult to predict the challenges that might arise.

For overcoming the design process complexity, an adaptive approach to the key meta-decision of how to make design decisions is necessary (Harman 2007). Many types of decisions include selections, acceptances, evaluations, and constructions. Each type, in turn, is assigned an approach that the decision-maker should follow (Acosta 2010). Two fundamental approaches are indicated: normative (rational) and naturalistic (Zannier *et al*. 2007). Depending on the approach taken, *e.g.*, decision matrices or SWOT analysis are performed (in the normative approach), or it is based on unconscious emotions by constructing stories, mental simulations, and various scenarios (in the naturalistic approach) (Jonassen 2012). Regardless of the approach, personal intuition is also a decisive factor; however, the primary decision points can be identified to help control decisions in the design procedure. These points are: project preparation, preliminary design, executive design, concretization, and materialization of the idea (Hsu and Chen 1996; Lee 2014).

Furniture designers, architects, and engineers in various fields of technology are obliged to make many decisions at the early design stage. In particular, the first two groups of designers focus special attention on making appropriate dimensional decisions (Yao and Carlson 2003). These considerations are multidirectional and concern, among others, technological feasibility, functionality, durability, and aesthetics (Thompson and Davis 1988; Suandi *et al*. 2022). From a technical perspective, the right dimensioning is crucial for ensuring structural integrity and low manufacturing costs. Simultaneously, the aesthetics of wooden furniture is impacted by dimension choices. Harmonious proportions, balanced dimensions, and thoughtful scaling are crucial in creating visually appealing and aesthetically pleasing designs. The synergy between engineering reasons and aesthetic considerations in dimension choices is essential for crafting acceptable wooden furniture for customers. In normative terms, the path of choices is seemingly simple; the design should be adapted to human dimensions and ensure the rational use of materials and appropriate dimensions of parts based on the functionality, strength requirements, and technological and economic aspects of product suitability. Using a more naturalistic approach, one can notice a tendency to choose certain dimension values without giving specific arguments but only relying on intuition and generally accepted standards for defining objects as aesthetic (Azizi *et al*. 2016). Assessing aesthetics seems complicated; however, research is being undertaken to indicate the relationship between the design decisions and the achieved aesthetic qualities of the end-product. Figure 1 shows exemplary variables that could function as guidelines for evaluating the aesthetics of projects.

Currently, CAD tools provide significant support in the design process. The software has complex modeling features that support various geometry types. Creating NURBS (Non-Uniform Rational B-splines) curves and surfaces is standard, offering possibilities for modeling organic shapes. Thanks to the mathematical model based on B-Spline curves, high precision is achieved, which is essential in applications requiring accurate measurements and rendering. Their widespread use is facilitated by standardization, which facilitates data exchange between different programs and work environments. Performing simulations and analyses or automating repetitive tasks using scripts and macros facilitate the design, production, and lifecycle management process (Hu *et al*. 2000). When describing modern CAD solutions, it is worth mentioning the increasingly popular method of generative design, based on the use of algorithms and parameters for automatic generation and optimization of projects. In this approach, the designer initiates the idea, and the resulting concepts result from artificial intelligence algorithms and cloud computing power. This allows for the generation of hundreds or even thousands of different solutions (Khan and Awan 2018). Even with significant advances in computer design techniques, a solution for fully supporting designers in making dimensional decisions has yet to be found.

**Fig. 1.** Scheme of aesthetic evaluation of furniture products (own elaboration)

This article aims to review canons, preferred numerical sequences, and proportions that support the decision-making process by the designer, who is the initiator of creative actions, possible to support in subsequent stages by artificial intelligence and tools for generating curves, shapes, and planes. Considering all these issues, the research questions can be formulated. Can one universal template for decision-making regarding dimensions by the designer be identified? What design assumptions should be adopted to ensure that the end-product has high aesthetic quality? Are objects with a known, harmonious, and rhythmic structure more attractive and perceived as beautiful?

**DIMENSIONAL DESIGN INSPIRED BY NATURE**

When looking for the optimal method of making dimensional decisions in design, the ideal proportions of the human body can be taken as inspiration. Over the centuries, canons, *i.e.*, systems of proportions, were born, and their summary is shown in Table 1.

**Table 1. **Groups Supporting Dimensional Decision-Making

Only the Greek civilization departed from the conventional, mechanical canon and focused on the actual dimensions of the human body over their definition in the 5^{th} century BCE. An ancient Greek sculptor, Policlet, worked in this way and set himself to accurately determine the proportions of the human body, striving for an ideal. According to his guidelines, the head should be 1/7 of the total body height, the hand and face – 1/10, and the foot – 1/6. Policlet based his canon of proportions on the parts of the human body, *i.e.*, organic elements. Policlet had successors who also tried to establish ideal proportions, including the Greek sculptor Lysippos who developed a proportional system where the head served as a module, with the body’s height being eight times its size. The successive canons of the Roman architect Vitruvius (1st century B.C.) with eight heads, Dürer with 7, 8, 9, and 10 heads, *etc*. share the arithmetic mode with integers or fractions. In 1490, Leonardo da Vinci defined the canon of the human body in the famous drawing “Vitruvian Man” and stated that only the ideal proportion of the human body allows the figures to be inscribed in the indicated positions in a circle and a square. A breakthrough in the perception of the human body and the proportions of organic elements was the discovery of Adolf Zeising in the mid-nineteenth century. He stated that there is a law of proportionality that dominates Nature and governs the proportions of the body of every human being, male or female, of any age, race, and at any point in life. Based on the measurement results of about two thousand bodies, he concluded that the ideal proportions of the human body, such as the ratio of the length of the arms to the length of the head or the length of the face to the length of the whole body, can be expressed using the golden number (φ = 1.618)

The ideal body proportions can be calculated in various ways, *e.g*., using the McCallum formula to determine the ideal body weight using the Brock method. McCallum, for example, talks about the need for the trunk and legs to be the same length. He believes the chest should exceed the pelvis size (about 10 to 9). The chest and waist should be correlated in the proportions of 4 to 3, and the arms separated to the sides should be a person’s height. The same parameters were once introduced into the “Vitruvian man” phenomenon. Currently, scientists are trying to analyze different parts of the human body. The indicators are calculated for, for example, arms and forearms, fingers, and hands, *etc*. (Fig. 2). Undertaking detailed analyses, certain discrepancies are often found as to the correspondence of, *e.g*., the ratio of bones in the human hand; however, the conclusion is always a correlation with the Fibonacci sequence or the golden spiral (Park *et al*. 2003). The defining numbers are related by the terms of a sequence, where the first term is 0, the second is 1, and each subsequent one is the sum of two. The first 10 numbers are thus as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Based on this sequence, Fibonacci squares are constructed with sides that match the length of each term (Fig. 2)

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**Fig. 2. **A human hand inscribed in the Fibonacci squares and the golden spiral as a confirmation of the relationship between the Fibonacci sequence and the center of rotation of the joints of the hand (based on Park *et al*. 2003)

**THE GOLDEN RATIO IN NATURE, ARCHITECTURE, AND ART**

Interestingly, the ratio of consecutive Fibonacci numbers approaches the Golden Ratio (*φ*) of approximately 1.618 as the sequence progresses (Falcón and Plaza 2007). A graphical representation of this relationship is, for example, the golden ratio, the golden spiral, and the golden figure. These relationships can be applied to human work products and naturally occurring organic elements or phenomena. An example is the shape of sunflower or daisy flowers, as well as the number of petals in plants, the total number of which corresponds to one of the numbers of the Fibonacci sequence (*e.g*. 1 petal – calla lily, 2 – milkweed, 3 – lilac, 5 – ranunculus, 8 – delphinium, 13 – marigold, 21 – aster) (Fig. 3.) (Omotehinwa and Ramon 2013).

**Fig. 3.** The relationship between the sum of petals of individual flowers and numbers from the Fibonacci sequence ((a) White calla (1 petal), (b) Euphorbia (2 petals), (c) Trillium (3 petals), (d) *Aquilegia* (5 petals), ( e) Bloodroot (8 petals), (f) Black-eyed Susan (13 petals), (g) Shasta Daisy (21 petals) (based on Omotehinwa and Ramon 2013 and Canva Pro)

The seemingly random arrangement of branches on a tree trunk actually follows a hidden order – the Fibonacci sequence. Similarly, the spirals of galaxies also develop according to the golden ratio. In the human body, you can also find many dependencies remaining with the golden ratio, as already mentioned above (Cohen 2014). By reviewing achievements in the field of biomimetics, one can observe numerous structures inspired by biological systems, which serve as inspiration for problem-solving by understanding the mechanisms of biological systems. Imitating biological structures and functions significantly impacts the development of more efficient, sustainable, and innovative solutions (Kadri 2015).

In architecture and art created even before the golden ratio was identified, there are also many examples using the golden ratio. An example is the proportions of the Greek Pantheon or the Egyptian pyramids, built thousands of years ago. For example, the golden ratio in the Eiffel Tower or Notre Dame Cathedral can be observed. Works created based on this principle are widely recognized as extremely attractive and harmonic. Examples include Leonardo Da Vinci’s Mona Lisa, The Last Supper, The Birth of Venus, and the marble sculpture Venus de Milo.

Today, there are many applications of the golden ratio rules. Fibonacci numbers are found in architecture, graphics, fractals, electronics (Intel processors), botany, poetry, and music (Conti and Paoletti 2021). Document formats such as ID cards, driving licenses, and bank cards are golden rectangles. The layout of many websites is also closely related to the golden ratio and the golden spiral. Similarly, logotype designers use golden sequences in their work. It can also be used in furniture design or small architecture products. Even movements in the financial markets can be explained using a precise rule linked to Fibonacci numbers (Conti and Paoletti 2021). Examples of furniture designs and logos designed using the golden ratio are presented in Table 2.

**Table 2. **Examples of Furniture Designs and Logos Designed Using the Golden Ratio

Graph theory is a widely discussed topic in mathematics (Tutte 2001; Gross *et al*. 2012). Scientists are testing various applications of graphs in chemistry, biology, anthropology, and social sciences and their connections with various areas of mathematics. Fibonacci numbers are of particular interest in this area. Various graphs and structures are defined and tested later (Hsu 1993; Knopfmacher *et al*. 2007). An example would be graphs whose sequence of degrees consists of “n” consecutive Fibonacci numbers (Yurttas Gunes *et al.* 2020). Another example of this consideration is testing Fibonacci cubes used to create hypercube algorithms (Zhang *et al*. 2009). Fibonacci cubes are exactly resonance charts of the fibonaccenes that appear in chemical graph theory, and resonance charts reflect the structure of their ideal matches (Klavžar and Žigert 2005). Klavžar and Žigert (2005) showed that Fibonacci cubes are Z-transform graphs (resonance plots) of fibonaccens, *i.e*., zig-zag hexagonal chains (Fig. 4). The term for hexagonal and square systems were independently introduced by other researchers (Klavžar and Žigert 2002; Fournier 2003; Zhang *et al*. 2009).

**Fig. 4.** a) Fibonaccena, (b) linear square chain, and (c) Fibonacci cube Γ5 (based on Zhang* et al*. 2009)

**PROPORTIONS AND PREFERRED SHAPES**

Proportions play a vital role in architecture because different feelings are associated with them and depend on how a person perceives the surrounding space. Not only in the design of spaces but also in the design of objects, such as furniture, proportions influence the user’s perception. The proportions affect work efficiency by creating positive or negative stimuli in the human brain. The beauty of nature lies in the fact that each object has attractive proportions concerning its surroundings. Massive or light proportions are generated by changing various building elements’ size, mass, or volume. Depending on the environment, these proportions can promote feelings of strength or weakness, stability or instability, openness or closeness, and fear or security.

A sufficiently lit room can give the impression of openness, but the height of the roof should also be proportional to its length and width. The optimal ratio of room dimensions to ensure the impression of openness was proposed by Andrea Palladio (1508-1580) in Four Books of Architecture, published in 1570 (Palladio 1955). In his book, he suggests that the dimensions of a room can be determined by arithmetic mean, geometric mean, or harmonic mean (Fig. 5) (Constant 1993; Rybczyński 2003; Sharma *et al*. 2012). Describing the dimensions of the room with the symbols a, b, and c, in arithmetic terms, the following relationship is obtained: , in geometric terms: , and in harmonic terms: .

**Fig. 5.** Dimensions of the room according to Andrea Palladio (based on Sharma *et al*. 2012)

The renowned Italian architect Andrea Palladio, who lived during the 16^{th} century, is considered one of the most influential architects in Western architectural history. Andrea Palladio proposed a collection of architectural principles and measurements that, according to him, give the impression of order, beauty, and harmony in the rooms (Table 3).

**Table 3. **A Set of Figures and Proportions by Andrea Palladio (Sharma* et al*. 2012)

The influence of Palladio’s theoretical work on proportions and geometry is evident in the consistent use of preferred figures and ratios throughout his built projects. A prime illustration of Palladio’s preferred figures and proportions can be found in the villa plans depicted in Fig. 6.