# The Impacts of Symmetry in Architecture and Urbanism: Toward a New Research Agenda

## Abstract

**:**

## 1. Introduction

## 2. Definition and Background

## 3. Classes of Symmetry

**reflectional**(Figure 1a and Figure 2a), in which one geometric configuration is reflected across an axis. For example, in many Classical buildings, the axis is the centreline of the building, and the façade as it appears on one side of the axis is exactly reproduced on the other in a matching progression (identical geometries moving to left as moving to right). It is common to say that one side is the “mirror image” of the other. In nature, a reflection in a body of water has a vertical reflectional symmetry with the scene above it, across the axis of the horizon.

**rotational**(Figure 1b and Figure 2b), in which a configuration is symmetrical as it rotates about a point. A perfect circle has a rotational symmetry about the centre point of the circle. Many structures in nature have rotational symmetry at least in part: planetary bodies, the iris of the human eye, an inflated balloon (except where its nozzle distorts the shape) and so on. In the case of the inflated balloon, the symmetry of its shape is a product of the symmetry of the pressures radiating outward from within the balloon.

**translational**symmetry (Figure 1c and Figure 2c) is a correspondence between different shapes that may not be otherwise symmetrical about an axis. For example, a repeated figure has a translational symmetry with itself. In architecture, one part of a repeated motif has a translational symmetry with another part. A child’s blue eyes can be said to express a translational symmetry with its parents’ identical blue eyes (and the child may have partial translational symmetries in other respects, e.g., shape of the face, etc.)

**scaling**symmetry (Figure 1d and Figure 2d) is similar to a translational symmetry, but instead of symmetry across positions, the symmetry occurs also across sizes. The most familiar example is the so-called fractal pattern, in which the same geometry is visible at multiple scales.

**Symmetry breaking,**though not strictly speaking a class of symmetry in its own right, is an important process of symmetry formation and transformation, and therefore it is included as an important element in this list. When existing symmetries are perturbed or “broken,” they often produce new symmetries at smaller scales, or new compound structures combining smaller symmetries and partial asymmetries (Figure 3). For example, a tubular structure, which is symmetrical about its axis, may break at some point along the axis, producing a new form of mirror symmetry. It is now understood that many classes of biological morphogenesis rely on broken symmetries to generate new compound symmetries, and importantly, complex body geometries as well as functionalities in organisms [20]. Symmetry breaking also plays a large role in recent theories of physics and cosmology [21,22]. It may be that symmetry, together with the process of its breaking and recombining into new complex forms, is a fundamental aspect of the evolving structure of the Universe at many different scales.

**information**symmetry is a symmetry that is not per se between geometric forms, but has some other more abstract correspondence. At its most fundamental level, mathematics is a representation of symmetries, or preserved correspondences, in the transformation of mathematical constructs [23]. The familiar “equal” sign in a formula expresses just such a symmetry. As forms of information, these constructs can therefore be described as a major class of “information symmetry.” One can also describe an “information symmetry” between the pattern of a DNA sequence and the complex protein that it generates, which in turn allows the “information” to be further transmitted, e.g., through inheritance. In the realm of human technology, one could describe an information symmetry between the pattern of grooves in a vinyl audio recording and the changes in air pressure at the microphone or speaker, corresponding to the sounds that are made or heard (Figure 4). Any language or code can be said to contain information symmetries as well, for example, the symmetries between a symbol and its referent, or a map and its territory. In the case of literature, information symmetries can take the form of analogies or metaphors. In the case of the built environment, they can take the form of literal signs or symbols.

**Compound**symmetries, which combine other kinds of symmetry into more complex forms, are what is experienced most commonly in natural and human environments. Many classic tile patterns and motifs combine reflectional and translational symmetry, in what is known as a glide reflection (Figure 5): a figure is repeated, then reflected across a glide axis [24]. The three basic plane symmetries, plus their 14 possible combinations, form 17 symmetry groups in two dimensions (called “wallpaper groups”). The familiar kaleidoscope pattern is also a compound symmetry in two dimensions (Figure 6).

## 4. Theories of Symmetry in History

“…a proper agreement between the members of the work itself, and relation between the different parts and the whole general scheme, in accordance with a certain part selected as standard. Thus, in the human body there is a kind of symmetrical harmony … and so it is with perfect buildings.”[29] (Book 2, Chapter 2)

“Beauty is a kind of concord and mutual interplay of the parts of a thing. This concord is realized in a particular number, proportion, and arrangement demanded by harmony, which is the fundamental principle of nature… There are three basic things which contain everything that we seek: number, what I have called proportion, and arrangement (numerus, finitio, collocatio). But besides these there is something else which originates from the linking and mutual relationship of these things, and which makes the surface of beauty glisten with a marvelous brilliance; this thing we call harmony (concinnitas).”[31] (Book 9, Section 5, pp. 337–340)

## 5. Modern Developments

## 6. The Role of Symmetry in Biology

## 7. The Role of Symmetry in Human Biology

## 8. Evidence for the Health Impacts of Symmetry and Its Absence

## 9. Applications of Symmetry to Built Environments: Art in Service to Life

- “Create many linked symmetries of different types as a response to activities on distinct scales (but don’t impose a global overall symmetry)…
- “Implement approximate spatial correlations using similarities at a distance and scaling symmetries (i.e., similarity under magnification)...” [56] (p. 52)

- Levels of Scale (similar figures at a range of scales)
- Strong Centers (prominent geometrical zones in between others)
- Boundaries (geometrical zones that bound others, e.g., centers)
- Alternating Repetition (patterns that repeat with some alternating variation)
- Positive Space (a geometric region that does not have excessively acute sub-regions)
- Good Shape (a geometric region that is coherent and interrelated)
- Local Symmetries (groups of regions that are internally symmetrical but may not be externally symmetrical)
- Deep Interlock and Ambiguity (patterns that inter-relate in complex ways)
- Contrast (adjacent figures that are starkly different from one another)
- Gradients (figures whose characteristics gradually transition)
- Roughness (figures with many small-scale asymmetrical characteristics)
- Echoes (figures that repeat some aspect from other figures)
- The Void (areas where few or no figures are present)
- Simplicity and Inner Calm (overall figures that are highly unified and harmonious)
- Not-separateness (connectedness of all figures to one another and to the viewer)

- Levels of Scale (scaling symmetries)
- Strong Centers (rotational, reflectional symmetries)
- Boundaries (rotational, reflectional symmetries)
- Alternating Repetition (compound symmetries)
- Positive Space (net convex symmetrical spaces)
- Good Shape (coherent symmetrical shapes)
- Local Symmetries (reflectional symmetries within symmetry breaking)
- Deep Interlock and Ambiguity (translational symmetries)
- Contrast (reflectional symmetries)
- Gradients (translational symmetries)
- Roughness (translational symmetries)
- Echoes (deep translational symmetries)
- The Void (symmetry void)
- Simplicity and Inner Calm (symmetry simplicity ratio)
- Not-separateness (ultimate symmetry, with symmetry breaking, of all things)

“…life, as I have defined it, is mathematical. By this I mean that it arises because of the mathematics of space itself. Since living structures arise primarily as symmetries and structures of symmetries, their presence and their density can, in principle, be calculated for any given configuration.”[55] (p. 469)

## 10. A Basic Distinction

- The immense structures of the physical world, with their vastly complex geometric properties;
- One’s innate experiences of these structures, which are conditioned by humans’ evolution as complex neurological organisms; and
- One’s own synthetic mental constructs, which sometimes interact with these physical symmetries and one’s experiences of them, and sometimes go into other directions, creating linguistic or artistic symbols, metaphors, allegories, artworks, and other constructions.

## 11. Tentative Conclusions and Hypotheses for a New Research Agenda

**Hypothesis 1.**

**Hypothesis 2.**

**Hypothesis 3.**

**Hypothesis 4.**

**Hypothesis 5.**

**Hypothesis 6.**

## 12. Discussion

## 13. Conclusions

- A clarified understanding of the ways that people are impacted by buildings and urban environments containing varying forms of symmetry under varying circumstances, and the lessons to draw about likely shared reactions and evaluations.
- A clarified understanding of the measurable impacts of symmetry on different aspects of human health and well-being, and the conditions under which these impacts occur.
- A clarified understanding of the role that symmetry might play in achieving other urban goals, including walkability, active public space, social capital, urban resilience, low-carbon living, and mitigation of loneliness.
- A better articulated understanding of the different forms and combinations of symmetry, including “deep symmetry” (as initially outlined herein), and how they combine and interact to produce human impacts.
- New methods for defining and measuring these forms of symmetry, building on the work of Alexander, Salingaros, and others, and providing useful diagnostic and design guidance tools.
- Articulation of the philosophical framework of “symmetric structuralism” described herein, in order to provide a comprehensible and useful model of the relationships between these issues.
- Finally, and perhaps most controversially—but supported by evidence—new normative and practical guidance for inclusion of these geometric characteristics in building design so as to enhance positive user impacts and attenuate negative ones, as part of responsible professional practice.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Examples of (

**a**) reflectional, (

**b**) rotational, (

**c**) translational and (

**d**) scaling symmetries in natural forms (tiger face, sun corona, mother and baby ducks, and fern leaves). Left image: S Taheri via Wikimedia Commons. Other images in public domain.

**Figure 2.**Examples of (

**a**) reflectional, (

**b**) rotational, (

**c**) translational and (

**d**) scaling symmetries in human architectural forms (Classical façade, stained glass window, archway tile motif, repeating arch shapes at different sizes). Images: Ryan Kaldari (left) and Thomas Ledi (second from left) via Wikimedia Commons. Other images in public domain.

**Figure 3.**An example of symmetry breaking in a drop of milk as it collides with a thin sheet of milk, shown in a famous series of photos by Harold Edgerton. The new structure is not a disordered mess, but in fact exhibits new forms of symmetry at smaller scales. Such processes are thought to be fundamental to the generation of structure in biological and natural systems. Images: courtesy of the Edgerton Digital Collections.

**Figure 4.**Two examples of information symmetries, one in human technologies—the encoding of audio information on a vinyl record (

**a**), and one in biological systems—the genetic code for the synthesis of proteins and other complex molecules in an organism (

**b**). Images: MIK81 (left), TBraunstein (center) via Wikimedia Commons. Image on the right in public domain.

**Figure 5.**A simple glide reflection (

**a**), and a more complex “wallpaper group” (

**b**). A figure is repeated, then reflected again along a glide axis. This process may be repeated to produce larger groups. Images: Kelvinsong and Martin von Gagern via Wikimedia Commons.

**Figure 6.**A view through a kaleidoscope, which seems to exhibit rotational symmetry (

**a**) In fact, it exhibits a compound form of reflectional symmetries (

**b**) created by mirrors within the kaleidoscope tube, which together form multiple reflectional planes (blue lines on the right). When aggregated together, they appear as rotational symmetry, also compounded together with many translational symmetries as well. This example also cleverly makes use of the reflectional sub-symmetries in the shell patterns, also producing scalar symmetries. Such patterns are widely regarded as beautiful. Images: public domain.

**Figure 7.**Most environments exhibit compound symmetries, including reflectional, rotational, translational, scaling, and broken symmetries. Humans are adept at perceiving these symmetries and the order that they manifest. Image: Sebastien Gabriel via Unsplash.

**Figure 8.**Fractal patterns generated by remarkably simple recursive mathematical formulas. Images: public domain.

**Figure 9.**The human face incorporates many classes of symmetry in compound and interrelated forms, as can be seen in the markings at right. Reflectional symmetry occurs across the dotted lines, rotational symmetry occurs in the circles or arcs around the points, translational symmetry occurs in the repeated curve patterns, and scalar symmetry occurs in the repetition of patterns at smaller scales, e.g., hairline-eyebrow-eyelash, etc. Finally, information symmetry occurs between the subject and the viewer’s instinctive ability to read biological signals of health (e.g., in clear skin, etc.) Images: the author.

**Figure 10.**Two urban environments in London, UK. The environment on the left includes literal natural forms as well as their properties, notably reflectional, scaling, translational and scaling symmetries. The environment on the right has few such characteristics. Images: the author.

**Figure 11.**Alexander’s “fifteen properties” of geometric order in human and natural environments. Illustration: the author.

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Mehaffy, M.W. The Impacts of Symmetry in Architecture and Urbanism: Toward a New Research Agenda. *Buildings* **2020**, *10*, 249.
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Mehaffy MW. The Impacts of Symmetry in Architecture and Urbanism: Toward a New Research Agenda. *Buildings*. 2020; 10(12):249.
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Mehaffy, Michael W. 2020. "The Impacts of Symmetry in Architecture and Urbanism: Toward a New Research Agenda" *Buildings* 10, no. 12: 249.
https://doi.org/10.3390/buildings10120249