A world from a sheet of paper - Tadashi Tokieda

The speaker begins by highlighting the importance of numbers in our lives, using examples like 3.14152 and the iconic James Bond number to illustrate how numbers can have different meanings. The focus then shifts to the concept of surface tension and its relevance to everyday phenomena like water droplets. The lecture's central theme revolves around the number 2.5, which is portrayed as a bridge between two and three dimensions, particularly in the context of paper and its interaction with the three-dimensional space. A magic trick involving a paper square and a circular coaster demonstrates the surprising properties of paper when it transitions between 2D and 3D, setting the stage for a deeper exploration into the intrinsic and extrinsic dimensions of objects.

The speaker delves into the historical Greek problem of angle trisection, which questions whether any angle can be divided into three equal parts using only a compass and straightedge. While traditional methods fail for certain angles, like 60 degrees, the art of origami offers a versatile solution. The process of folding a sheet of paper can achieve angle bisection and trisection through simple folds. The speaker then connects this concept to the broader mathematical principle of constructing regular polygons with a compass and straightedge, emphasizing the importance of prime numbers in this process. The narrative culminates in a demonstration of how origami can construct a regular pentagon, showcasing the power of paper folding in solving geometrical challenges.

The speaker explores the theorems behind origami, discussing how the creasing and flattening of paper can lead to the formation of various geometric patterns. It is shown that at any point on a crumpled sheet of paper, the number of creases meeting is always even, a principle that has significant mathematical implications. The speaker also presents a theorem related to the angles formed at the vertices of these creased patterns, stating that the sum of angles meeting at a point must satisfy certain conditions to be achievable through origami. This section reveals the intricate relationship between the physical act of folding and the mathematical principles governing these actions.

The speaker introduces a 'gift' in the form of a mathematical theorem related to the paper folding, emphasizing the joy of discovering such phenomena that cost nothing yet hold significant value. The discussion then shifts to the patterns that emerge when folding a sheet of paper, and how these patterns must adhere to specific mathematical conditions. The speaker illustrates this with a theorem about the sum of opposing angles in a crumpled pattern, which must equal each other. This theorem is proven through a step-by-step explanation, highlighting the interconnectedness of paper folding and geometric principles.

The speaker broadens the perspective to the universality of origami, discussing its application across various scales – from the smallest to the largest in the universe. Examples range from the creation of paper in Mino, Japan, to the formation of mountain ranges like the Andes due to tectonic plate movements. The speaker touches on the concept of buckling, a phenomenon where structures bend under compression, and its significance in the context of origami. The challenge of extracting science from the complexity of these natural and man-made phenomena is presented, inviting the audience to consider the deeper implications of origami in diverse fields.

The speaker explores the anisotropic properties of paper, demonstrating how paper can exhibit different behaviors depending on the direction in which it is manipulated. Through a series of experiments, the speaker shows that paper can be easily torn in one direction but not another, highlighting the microscopic structure's influence on the paper's macroscopic properties. The concept of anisotropy is further illustrated through the use of origami to create structures with unique directional flexibility and rigidity. The speaker concludes by suggesting that these principles can be applied in various practical applications, such as in the design of medical equipment and industrial components.

The speaker introduces the H-principle, a concept that suggests we can ask nature to solve complex geometrical problems, such as embedding a large object into a smaller space isometrically. This principle is exemplified through the construction of a momotanic carpet, a complex origami pattern that exhibits surprising elasticity and structural integrity. The speaker also discusses the work of a French team that used this principle to create a concrete model of a sphere that could be compressed into a smaller space without changing its surface length, demonstrating the practical applications of these geometrical concepts in engineering and design.

The speaker concludes the lecture by discussing the discovery of negative Poisson's ratio in random structures, such as crumpled paper. This finding challenges the traditional belief that Poisson's ratio, a measure of how materials respond laterally to being compressed or stretched, is always positive. The speaker demonstrates that by crumpling a sheet of paper, it is possible to achieve a negative Poisson's ratio, where the material expands in all directions when compressed. This discovery has significant implications for the design of materials with unique mechanical properties. The speaker reflects on the broader impact of scientific inquiry, suggesting that science is not confined to laboratories but is a part of the natural world and everyday life.