Mathematizing Human Relationships

　In architecture, “models” are essential tools for communicating ideas. The construction methods and materials used vary depending on what needs to be conveyed, such as design or structure. Based on this analogy, it seems possible to create mathematical models using mathematical formulas as the ‘material’ to convey the quantification of space. In fact, this approach is known as “mathematical modeling” and is widely utilized across a variety of fields.

　In the Yudai Honma Laboratory, we use mathematical modeling as a tool to quantify architecture and urban environments. Through daily discussions, we aim to understand their characteristics and translate these insights into better planning. Let me give you an example. The figure above shows an example of mathematical modeling used to optimize a floor plan. In the design of buildings such as offices and commercial facilities, floor plans are of critical importance. While it is essential to meet environmental constraints—such as light and ventilation—as well as equipment and structural requirements, the ideal plan is one that facilitates the flow of people and fosters a lively atmosphere as much as possible. In essence, this is nothing less than the task of determining the “location” and “size” of rooms that will stimulate the flow of people. From this perspective, by expressing an existing floor plan (left) mathematically (center), it becomes possible to recalculate it to maximize internal traffic flow (right).

　In the field of planning, where there is a growing need to engage more closely with “people” and “behavior” in the context of architecture and urban environments, it is difficult to say that mathematical discourse has been fully utilized. One reason for this may be that diversity is key when considering people and behavior, which ultimately ties into the difficulty of directly addressing the diversity of values.

　That said, it would be premature to conclude that mathematical reasoning has no place in architecture and urban planning. Facility sizing theory, based on queueing theory, stands as the prime example of the effectiveness of quantitative analysis in architectural planning. Since the time when facility sizing theory was first studied, both theoretical knowledge and computer technology have made tremendous strides. Consequently, the scope of applications where mathematical modeling proves effective in architecture and urban planning is likely expanding.

　One mathematical model that captures the essence of “human dynamics”—the diverse behaviors of diverse people—is the quadratic equation, which is essentially a form of multiplication. In other words, it considers the synergistic effect (x_i × x_j) resulting from the connection between oneself (x_i) and others (x_j). By using quadratic equations, we can explore phenomena such as the locations where people tend to linger while viewing art in a museum (Fig. – Left) and the clustering of populations in different regions resulting from job transfers (Fig. – Right) in a more natural way.

　There is a growing consensus that a sustainable society and a multigenerational collaborative society represent the ideal model for the 21st century. Given that individual buildings come together to form towns, cities, and ultimately nations, it seems increasingly important to engage in seamless discussions that transcend the confines of terms like “architecture” or “city” and span various scales of analysis.

　What impact does the construction of a single building have on a city? Or, what kind of buildings does a city hope to see built on a given site? As one of the research laboratories focused on architectural and urban planning, we are exploring the essence of “Connecting Architecture and the City.”

　If we assume that the purpose of urban planning is to establish a direction for the built environment, then the purpose of architectural planning can be interpreted as enhancing the quality of that environment so that this direction can be achieved as smoothly as possible. Space has its limits in terms of capacity. Movement within space entails a burden of distance. I am exploring approaches to architecture and urban design that take these structural realities of space into account.