An exploration of the simplest Hyperbolic Paraboloidic ‘saddle’ form has lead to the development of a modular system that combines the principles of the hypar (Hyperbolic Paraboloid) and elastic potential energy.

A hyperbolic paraboloid is an infinite doubly ruled surface in three dimensions with hyperbolic and parabolic cross-sections. It can be parametrized using the following equations:

Mathematical:   z = x2 – yor  x = y z

Parametric:   x(u,v)=u   y(u,v)=v   z(u,v)=uv

The physical manifestation of the above equations can be achieved by constructing a square and forcing the surface area to minimalise by introducing cross bracing that has shorter lengths than the  square edges.

A particular square hypar defined by b = n * √2 (b=boundary, n=initial geometry or ‘cross bracing’) thus constricting the four points to the corners of a cube leads to interesting tessellations in three dimensions.

Using a simple elastic lashing system to construct a hypar module binds all intersections together whilst allowing rotational movement. The rotational movement at any given intersection is proportionally distributed to all others. This combined with the elasticity of the joints means that the module has elastic potential energy (spring-like properties) therefore an array of many modules can adopt the same elastic properties.

The system can be scaled, shaped, locked and adapted to suit programmatic requirements.

## Minimal Surface – Hyperbolic Paraboloid Folding

Triply Periodic Minimal Surfaces

Minimal surface is an area minimizing surface whose mean curvature at any point is zero, and is often represented by the shapes of soap bubbles that span wire frames. Some minimal surfaces have crystalline structures that repeat themselves periodically in three dimensions. Many of these surfaces were discovered by Alan Schoen who analysed them in his technical report, ‘Infinite Periodic Minimal Surfaces without Self-Intersection‘, written in 1970. I first started researching the different types of triply periodic minimal surfaces to understand the rules behind their structures.

Folded Hyperbolic Paraboloid

Parametric Kerf Bending

I have also been investigating different types of lattice hinges or cutting patterns that could help fold a hyperbolic paraboloid from a rigid single sheet material.