# Non-Euclidean

## Non-Euclidean Patternmaking

“Non-Euclidean Patternmaking” is a revolutionary new form of fashion patternmaking based on the mathematics of curved Non-Euclidean geometry which fundamentally changes the way we understand and practise fashion design. Developed during Liu's PhD research (2015), it addresses systemic problems in traditional patternmaking techniques due to their reliance on linear measurements, Euclidean geometry and subjective judgement. Built from the ground up on rigorous modern mathematical principles, it offers new strategies to bypass existing systemic problems, enabling greater accuracy, control, creative freedom and new possibilities.

“Non-Euclidean” geometry is the modern mathematics of curved surfaces. Developed in the 19th century it forced mathematicians to understand that curved surfaces have completely different rules and geometric properties to flat surfaces. Most of us are familiar with Euclidean geometry from high school, but never learn Non-Euclidean geometry unless we study mathematics or science degrees. Most fashion designers are not familiar with advanced Non-Euclidean geometry.

Conventional patternmaking techniques were developed using Euclidean geometry and linear measurements. Linear measurements have a limited ability to capture the three-dimensional curvature of the body. For example, three people may have the exact same linear body measurements, but completely different three-dimensional body shapes. This means the intervention of a skilled patternmaker is often required to guarantee an accurate fit.

Liu’s PhD research sought to understand the underlying geometry of fashion patternmaking. His research identified that conventional patternmaking techniques use Euclidean mathematics on curved Non-Euclidean surfaces. This incompatibility in mathematical rules is why they can never be completely accurate. Linear measurements are also incapable of recording the curvature of a surface which fundamentally limits their accuracy. The only way to know which geometric rules applied to a surface is to measure its curvature. Different parts of the body can have completely different geometric properties. Only by understanding Non-Euclidean geometry is it possible to accurately map the curved surfaces of the human body.

Parts of the body have different curvatures which have different geometric properties and mathematical rules.

Liu’s research used Non-Euclidean geometry to explain the systemic fitting problems in traditional fashion patternmaking systems. To solve this problem, he built a new patternmaking system from the ground up based on curved Non-Euclidean geometry. The challenge was to make the complex mathematics powerful enough to increase the accuracy of patternmaking, yet simple enough for a fashion patternmaker to understand. He also invented a new device called the “drape measure” that can measure the curvature of a surface and record it as an angle measurement which can be used in patternmaking.

An early prototype of the "drape measure" a new invention that changes shape to measure the curvature of a surface and improves the accuracy of patternmaking.

The drape measure recording the curvature of a spherical surface as an angle measurement.

The drape measure recording the curvature of a hyperbolic surface as an angle measurement.

Understanding fashion patternmaking from a modern geometric perspective allows us to fundamentally rethink many areas of fashion design. Fashion patternmaking is pervasive and this affects everyone from high fashion to fast fashion. A Non-Euclidean patternmaking system which has greater accuracy can improve the efficiency of fashion production. It allows us to rethink ready-to-wear sizing systems or more efficiently fit tailored clothes. A patternmaking system that bridges the gap between traditional techniques and modern science creates new possibilities for technology. It will be possible to build better 3D scanning algorithms that can create accurately fitted clothing.

Fashion education requires students to rote learn many techniques, while teaching them the underlying geometry of patternmaking rapidly accelerates their ability to master advanced techniques. It is also possible to develop fashion STEAM workshops that teach advanced mathematical principles such as calculus and geometry through fashion design. This is a great way of engaging high school students who do not respond to traditional mathematics education, especially young women.

Non-Euclidean geometry opens a new frontier for fashion design with exciting new possibilities...