A story
In my childhood memories, the lanterns in my hometown always fascinated me – circular palace lanterns, polyhedral colorful lanterns. How did my ancestors achieve the magical transformation from flat to three-dimensional through simple bamboo strips and paper? As a child, I was often confused: was there anything hidden behind these beautiful shapes?
Three decades slipped by, and as I returned to the art of paper folding, that long-forgotten question suddenly became clear. It turns out that the structural principles of those traditional lanterns align with geometric calculations—how a flat sheet of paper can be perfectly transformed into a three-dimensional structure with specific curvature. This process contains profound mathematical essence: straight creases correspond to developable surfaces, while curved folds achieve their shapes. From rectangular paper-based to porcelain like smooth curved surfaces, every fold undergoes intricate geometric transformations.
This perfect fusion of craft and mathematics not only showcases the ingenuity of folk artisans but also remarkably anticipated the evolution of modern computational origami techniques. The craftsmanship behind traditional lantern-making constitutes humanity’s earliest exploration and practice of spatial geometry—a profound legacy that has continuously inspired the development of structural design approaches.
Reflecting…
A question lingered unresolved for over three decades—until I encountered modern origami theory and finally grasped its profound mathematical wisdom. This long-delayed revelation compels me to ponder: In our relentless pursuit of so-called progress, have we overlooked something profoundly valuable? As masters worldwide have long pioneered innovations in paper folding, shouldn’t we pause and reflect? This isn’t merely an academic gap; it speaks to the neglect of foundational mathematical research. It’s time we reignite our appreciation for the enduring brilliance of mathematics!
As an archetype of curved-crease origami art, Origami Blue-and-White Porcelain transforms flat paper into exquisite three-dimensional forms through precise calculated creases.
Crease algorithm design
In the art of origami creation, I have always adhered to the concept of “subtractive aesthetics” – shaping forms by accurately hiding excess parts.
This principle is particularly typical in the structure of traditional Chinese circular lanterns: When the lantern ’ s uniformly arranged petal-like segments are unfolded into a planar sheet, the absent portions represent exactly those strategically eliminated excess regions.
This foundational geometric principle can be extended to designing curved vessels (e.g., circular lanterns, porcelain), with its computational workflow comprising three key steps:
- Unit Segment Width Determination
→ Based on the target model’s diameter. - Structural Length Calculation
→ Computed using the Pythagorean theorem. - Curved Petal Optimization
→ Derives optimal curvature for each unit segment.
Ultimately, the algorithm generates precise crease distribution diagrams, providing scientific mathematical guidelines for vessel fabrication.
Constructing universal geometric features
Applicable to rotatable vessels, papercraft lamps/lanterns, circular heritage architectures, and pagoda-style structures.
Engineering mathematics
This method overcomes the planar limitations of conventional origami techniques, offering an alternative computational approach for developable surface design with potential applications in industrial design and architectural mechanics.
Example: Bulbous porcelain vessels (Dulu)
We observe that exquisite ceramics and traditional circular lanterns alike are formed by rotational assembly of curved crease units. This exemplifies the core principle of origami engineering: a material optimization process where precise folding conceals redundant sections to achieve visually streamlined forms.
Example: Origami double-gourd vase (Song-style porcelain)
This artifact demonstrates uniform curved crease unit distribution in its structure. The waist contour is formed through rotational constriction, showcasing exceptional craftsmanship in curvature control. Notably, its spiral patterning precision surpasses comparable paper-folding implementations.
Wrong fold, it’s in another direction
These gear-like structures represent unanticipated structural outcomes during folding operations. The creative process requires continuous cognitive switching between 2D crease patterns and 3D topological realization, presenting significant neural adaptation challenges. Particularly without physical reference models, the spatial imagination ability of the brain often goes through a debugging stage, and these “wrong” works are considered a good thing from a certain perspective.
This is the mark of my encounter with mathematics. The folding and unfolding of paper is like a pair of hands revealing the science behind it. I returned to the old me, learning to discover the beauty of mathematics, exploring the mysteries of origami, and leaving my own thinking trajectory in this interlaced field.