The human tendency to impose order upon the world is perhaps nowhere more visible than in our architecture and design. We favor the square, the hexagon, and the cube—rigid structures that satisfy our need for predictability and structural stability. Yet, as Professor Alain Goriely of Gresham College masterfully argues in his recent lecture, this human obsession with sharp-edged geometry stands in stark contrast to the natural world. In a profound examination of the mathematics of tilings, Goriely posits that nature operates on an entirely different set of principles, favoring "soft" or irregular patterns that offer not just aesthetic beauty, but superior mechanical and optical properties. This shift in perspective—from the rigid to the organic—serves as a transformational framing of how we perceive the structures that govern our universe.
Goriely begins by demystifying the fundamental language of tiling: the arrangement of shapes to cover a plane or a three-dimensional space. By defining cells and nodes, he maps out the mathematical classification of regular tilings, such as the familiar grid of squares or the honeycomb of hexagons. However, he quickly pivots to the more complex, irregular reality of brickwork and random fractures. It is in this transition that he introduces the concept of "softening." In the world of traditional geometry, polygonal tiles are bound by the iron laws of dimension, requiring at least three corners in a two-dimensional plane to exist. Goriely’s research, however, demonstrates that when we replace straight edges and sharp corners with curved, organic shapes, the rules change entirely: a "soft" cell can exist with as few as two corners.
The implications of this shift become even more striking as Goriely moves into the third dimension. He dissects the limitations of the Platonic solids, noting that while shapes like the cube are mathematical marvels, they are often ill-suited for the dynamic requirements of biological growth. His exploration reveals a startling mathematical truth: while regular shapes struggle to fill space efficiently, a "soft" cell in three-dimensional space requires zero corners. This discovery is a triumph of intelligent curation, distilling complex topological proofs into a narrative that reveals why nature so frequently eschews the energy-expensive, sharp-cornered geometry that human designers find so comforting.
Related article - Uphorial Shopify
To understand why nature prefers the curve, one need only look at the architecture of the living world. Goriely identifies the Nautilus shell and the microscopic structures of biological cells as primary examples of soft tiling in action. These are not merely artistic choices by nature; they are functional solutions to the problems of growth, structural integrity, and environmental adaptation. The lecture connects these soft cells to the fascinating world of Minimal Surfaces—soap films and Triply Periodic Minimal Surfaces (TPMS) like the Schwartz P surface. These surfaces, which balance tension and area in perfect equilibrium, are not just mathematical curiosities; they are foundational to materials science and engineering, providing blueprints for the next generation of smart materials.
Perhaps the most compelling evidence for these principles was provided by a 2025 experiment conducted aboard the International Space Station. In the unique, zero-gravity environment of space, researchers successfully utilized liquid water to demonstrate the formation of a 3D soft cell. This experiment offered a rare, tangible window into the hidden patterns of reality, proving that without the intrusive forces of gravity and rigid physical constraints, water naturally seeks out these organic, curved architectures. It is a moment of strategic storytelling that bridges the gap between high-level mathematics and the visceral, observable reality of our cosmos. The cultural understanding fostered by this lecture is clear: our human-made world is defined by a desire for efficiency through rigidity, but the natural world is defined by a desire for elegance through fluidity. From the iridescent wings of butterflies to the resilient structures of weevils, nature consistently produces shapes that offer superior durability and light-manipulation capabilities. These organic structures do not require the energy-intensive manufacturing processes that characterize our own building techniques; instead, they are the result of biological processes optimized over millions of years of evolution.
In concluding his lecture, Goriely challenges his audience to reconsider their relationship with the shapes that define their environment. By moving beyond the binary of the regular versus the random, we begin to see the beauty of the soft cell not as an exception, but as a rule. His work invites us to embrace a new philosophy of design—one that learns from the hidden, organic geometry of the natural world. This is not just a lesson in mathematics; it is an invitation to look at the structures around us—the patterns in the fabric of a wing, the cellular architecture of a leaf, and the soap bubbles in a glass—with a renewed sense of emotional precision. The shift toward soft tiling represents a broader movement in contemporary science toward biomimicry and smart materials, where the goal is no longer to impose our will upon nature, but to align our innovations with the patterns nature has already perfected. By understanding the mathematics behind the curve, we gain the agency to design technologies that are not only more sustainable and efficient but also more fundamentally harmonious with the world they inhabit. Professor Goriely’s lecture serves as a reminder that the universe is not just a collection of rigid shapes waiting to be categorized; it is a dynamic, evolving tapestry of soft, interconnected patterns that we are only just beginning to truly understand.
