A reciprocal frame is a self-supported three-dimensional structure made up of three or more sloping rods, which form a closed circuit. The inner end of each rod rests on and is supported by its adjacent rod, gaining stability as the last rod is placed over the first one in a mutually supporting manner.

These rods form self-similar and highly symmetric patterns, capable of creating a vast architectural space as a narrative and aesthetic expression of the frame. The appearance of the entire structure is determined by the geometric parameters of each individual unit and the connections between the units.

*Precedent image*

Reciprocal frame (RF) principles have been around for many centuries, proving themselves versatile, efficient and resistant. They were present in the neolithic pit dwelling, the Eskimo tent, Indian tepee and the Hogan dwellings where mutually supporting beams form a rigid skeleton. The Hogan dwellings consist of a larger number of single RFs being supported by a larger diameter RF structure. Later development of the structural form can be seen in the timber floor grillages of larger medieval buildings where they were used for spanning spaces wider than the length of available beams.

*Eskimo tent*

Leonardo da Vinci explored two forms of reciprocal structure: a bridge and a dome. His work was commissioned by the Borgia family, with the purpose of designing light and strong structures which could be built and taken down quickly. This was to aid them in their constant quest for dominance over the Medici family in Renaissance Italy. The bridge would have been used for crossing rivers, and the dome could have functioned as a military camp.

*Leonardo da Vinci’s sketchbook*

Understanding the geometry of the reciprocal frame and the parameters that define it is essential in order to design and construct larger systems. The parameters that define RF units with regular polygonal and circular geometry are the following:

– n: number of beams;

– R: radius through the outer supports;

– r: radius through beam intersection points;

– H: vertical rise from the outer supports to the beam intersection points;

– h: vertical spacing of the centerlines of the beams at their intersection points;

– L: length of the beams on the slope;

– l: plan projection of the length of the beam.

Manipulating the length (L), height (H) and radius of the circumscribed circle of the three intersection points (r), the geometry of the structure changes as follows:

-increasing the length of the beams reduces the height of the entire structure;

-increasing the height of the RF structures reduces the span of the overall structure;

-increasing the radius of the circumscribed circle reduces the span of the overall structure.

Each RF member is subject to forces of compression, bending moments and shear forces as well as axial forces. The members transmit the vertical forces of their own weight and any imposed loads through compression in each member. These forces must be resisted at the perimeter supports. In addition, the lower part of the beam, between the outer support and the point where the beam is supporting the adjacent one, is in compression whereas tension forces will occur in the upper part of the beam.

*Rhino model*

Having investigated various morphologies through digital and especially physical modelling, I have started creating a dome-like structure which, through an irregular reciprocal unit, folds into a super-dome. Repeating the process, I arrived at a spiralling domical structure which I have then panelled, using the same reciprocal morphology. This lends a recursive effect to the entire structure.

*Progression of the structure in physical form*

If you haven’t come across it already, check out the work of dutch artist Rinus Roelofs, who has done a lot of experimentation with “reciprocal structures”: http://www.rinusroelofs.nl

Nice website, thanks Nselikoff!

Your reciprocal structures are very interesting!

We have been investigating reciprocal / weaving techniques in paper and bamboo in Italy

it would be interesting to compare notes…

Thanks Alison, your work looks fascinating. I am amazed how reciprocal patterns can lead into minimal surfaces!