Reciprocal Fern Fronds

The fern is one of the basic examples of fractals. Fractals are infinitely complex patterns that are self-similar across different scales, created by repeating a simple process over and over in a loop. The Barnsley fern (Example here) shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas.

Fern Parameters

Due to the fractal nature of the fern fronds, the perimeter of the laser cutting took a long time. By simplifying this, I began joining fronds to each other and the large perimeter allowed for enough friction for the fronds to adhere to the adjacent one. I explored this through a series of 4 different frond types (X Axes on matrix below), angles of rotation (Y2, Y3) and distance between each leaf (Y4).

Reciprocal Testing

With the study of many different arrangements of fronds and distances between each leaf in the frond, I was then able to select those that slotted in to the adjacent ones best and began arranging them with more components.

Reciprocal Testing – Flat Component

The arching nature of each individual leaf meant the configuration was only stable once the fitting in of each component had passed the node of the arch. By flattening each component into rectangular members, the friction that allows the components to adhere to each other would be constant throughout the length of the individual part. This means they could now be placed more or less fitted in to the other component, as desired.

Reciprocal Testing – Large Component

I then scaled up the component and attempted to array these as done with the smaller components above. Each component measured 600 mm length-wise and consisted of 5 members (3 facing one way and 2 facing the other, with a gap between them matching the width of each member). They originated from a central “stem” and attached to this by using glue and nails as to allow for easy manufacturing.

Ferntastic Azolla

Simultaneously, I also became intrigued by a small aquatic fern called Azolla which I thought would be worth exploring too.

What is interesting about this little plant is that it holds the world record in biomass producer – doubling in size from 3-10 days. It is all thanks to its symbiotic relationship with the nitrogen fixing cyanobacterium, Anabaena. This superorganism provides a micro-climate in exchange for nitrate fertilizer.They remain together during the fern’s reproductive cycle. They also have a complimentary photosynthesis, using light from most of the visible spectrum.

Bending Lattice System

My initial studies stemmed from researching into Stellation. This, in simple terms, is the process of extending  polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions, to form a new figure. Through researching the application of this process, I came across the sculptures created by George Hart, as he has experimented with stellated geometries to which are subdivided to create mathematical interweaving structures.Stellation 1

My Research into the method and calculations of George Hart’s Mathematical Sculpture’s focused on the sculpture ‘Frabjous’. Through rigorous testing and model making I have understood the rules behind the complex form. This is based on the form of a stellated icosahedron, whose shape is contained within a dodecahedron.grey card model

Lines are drawn from one point, to a point mirrored at one edge of the face of the dodecahedron form – as shown in the diagram. This creates intersecting lines at each face as you can see from the diagrams below. Each dividing line has two intersection points, with symmetry at the center of the line. The sculpture aims to avoid the intersections of these lines by introducing a sine curve with the domain 0 to 2*pi. As you can see, each component is exactly the same – for this model, 30 components are used.

george hart diagram 1george hart diagram 2george hart diagram 3

`To simplify the construction of the sculpture, I extracted a build-able section which uses ten components in total. Two of these sections are then weaved together and joined up by a further ten single components to form the entire sculpture.Diagram Sequence of Researched SculptureOne Component ImageryGeometry 2

Following this research, I extracted the concept of avoiding the intersection and subdivided a cube with lines from each corner of the cube. These lines were then weaved around eachother using a sine curve with a domain of 0 to pi. I then mirrored the curves and rotated them to create an intertwining form.Avoiding Self Intersection 2

Another test was created with the same process, however subdividing a cube using the midpoint of each face. – This creates an octahedral geometry.Avoiding Self Intersection octahedron

Using this interweaving geometry, I have created different three dimensional arrays to create a spatial form. The concept of avoiding intersections naturally cause a structure to fail. To form a structurally efficient version of this geometry, I introduced the idea of a reciprocal structure, and allowed the beams to self support by resting on eachother. This did not create a structure strong enough to stand on, however through adding a cube whose dimensions are equal to the width of the beams, the structure became very strong.

Avoiding Self Intersection octahedron 3

Testing the component at a small scale required the design of a joint which allowed me to assemble these components together through interlocking elements. Each beam element slots into the joint; When two joints and two beams are connected together the curves naturally stay in place due to the angle cut into the joint. Three of these connected elements together form the component.

Diagraming the Joint

As mentioned previously, avoiding intersections create inefficient structures – For this small scale experimentation, the concept of Tensegrity was implemented. Tensegrity is a structural principle based on using isolated compression components within a net of continuous tension, allowing the compression members to not need to touch each other. This model was constructed using 1.5mm plywood which has been laser cut; the modularity of the system ensures minimal material wastage.

Construction Sequence of ModelModel Photographs

The three dimensional array of this geometry creates many interesting shapes and patterns when viewed from different angles – this is visible in the following video:





Kinetic systems

“…the main task is to unfreeze architecture- to make it a fluid, vibrating, changeable backdrop for the varied and constantly changing modes of life…”

Reciprocal systems can be used to create a wide variety of movable structures based on pin-joint assemblies, especially in planar form.

One of the most widely known reciprocal kinetic structures is the iris diaphragm which uses four or more elements hinged at their ends with pin joints to generate a sliding motion for opening and closing. The elements join one another at different points along their spans and these intermediate points of connection can be used to determine new kinematic behaviour.

              658px-Lens-iris   7a4b032b5d49095bf3d6465baa2ee078

Iris diaphragm with 6 and 8 elements

Jean Nouvel’s facade design for the Institut du Monde Arabe is based on the iris mechanism, with aluminium diaphragm panels employing squares, circles, stars and polygons to generate decorative patterns through rotation. This light-responsive south facing facade uses a photoelectric cell to adjust the admission of natural light by the opening and closing of the mobile diaphragm.


Institut du Monde Arabe by Jean Nouvel

Calatrava’s project for a restaurant in Zurich has some similarities with the principles of retractable reciprocal frames. The roof structure is composed of nine metal and glass tree-like elements 12m high. Each of the nine columns is mechanically operated and folds simultaneously with all the others to provide shelter for the restaurant underneath.



Model for a restaurant in Zurich by Santiago Calatrava

The idea of a retractable roof which operates similar to the iris of the camera lens was first patented in 1961 by Emilio Perez who proposed a dome built of 3D curved segments which retract. The segments twist simultaneously and create a circular opening at the top.

Patent of retractable dome

Patent for a retractable dome by Emilio Perez

However, issues such as cladding materials, the changing geometry due to the retraction, design details of the hinges, eliminating the danger of progressive collapse, drive mechanisms which will provide simultaneous reaction to the beams and the cladding as well as overall construction detailing have to be considered and developed.

Chuck Hoberman’s research in the field of mobile and folding structures can have a remarkable impact on the development of kinetic reciprocal structures. His unique approach in the field of transformable design has created created objects that simulate the behaviour of living organisms, fostering a dynamic relationship between structure and user.

The Iris Dome has a fixed perimeter with a centre retracting in a smooth radial motion. A lamella dome with a geometry of interlocking spirals, the structure is based on a Vierendeel grid which carries the load by bending action rather than by axial forces which makes it similar to a retractable reciprocal structure. The main difference however is that the segments which form the Iris Dome are an assembly of pairs of structural elements connected with hinges at their midpoints which move like scissors.

Scissor-like movement is the main generative force also for the Hoberman sphere. The unfolding structure resembles an expanding geodesic sphere which can reach a size up to five times larger than the initial one. It consists of six great circles, each made of 60 elements which fold and unfold in a scissor-like motion. There are also 60 nodes which give rigidity to the structure and prevent the circles from expanding further into elliptical shapes.

Hoberman’s piece emerged in part from working with NASA on their deployable structures programme: ‘rather than constructing a structure in space, you unfold a structure in space’.

The Burning Seed

The Burning Seed
The Burning Seed

Mother nature protects the most vulnerable, the weak, the young, the very seeds of growth. At the heart of nature is the dispersal of its seeds. Burrs which catch a lift, dandelion seeds that ride on the wind and those that are catapulted, but perhaps the most enterprising of all are the seeds that are distributed far and wide…

The Burning Seed has arrived in the desert having fallen far from its original host. Inspired by the intricate geometry of the Flower of Life, the impetus behind the design is rooted in nature. The structure’s spiky exterior belies its beauty within – a space of refuge, protection from the prevailing winds and shelter from the intensity of the sun’s rays. The design of The Burning Seed has been driven by its interactivity. In daytime, people climb between the structure’s wooden thorns and look through to its centre at those who have managed to penetrate its hostile shell. Inside they bask in the smooth, spherical interior, interacting with the harmony of the structure and one another. At night the spectacle is truly realized as a flurry of fibre optic lights protrude from the thorns, adorning the structure in a wash of blue, purple and green light.

The Burning Seed Model - Scale 1:4
The Burning Seed Model – Scale 1:4

The structure consists of 37 star shaped plywood components which interlace to form the lightweight dome. The legs of the components are bent outwards from the centre and join to create the thorns of its exterior. The structure is reciprocal, with each component supporting the neighbouring pieces, there is no requirement for a central support. This results in a homogeneous internal space with a beauty derived from its inherent highly symmetrical patterns. The components have differently sized circular holes though their centres. This creates a porous structure and a patterned light effect to be cast on the desert floor. The size of these openings reflects the geometry of the ‘Fruit of Life’ symbol, a derivative of the Flower of Life. It is said to be the blueprint of the universe, containing the basis for the design of every atom, molecular structure, life form, and everything in existence.

Construction Diagrams
Construction Diagrams


An optional flame system for The Burning Seed allows fire to intermittently burst from the tips of the copper from six spikes, illuminating the Burning Seed in orange glow. This feature can be incorporated into the design following discussion with the Burning Man Festival Organisers.

The Burning Seed
The Burning Seed

Lotus Hypars

Lotus Hypars – A study of hyperbolic bamboo structures

The Lotus Hypars symbolise the “Caravansary” trading centre. The structure is assembled as the centre for exchange after journeying across land and water to a resting point, Burning Man. Hammocks offer a space for the festivals unique style of trading to be discussed and carried out. The tangible nature of the Lotus also creates a playfulness in an otherwise formal system of resources exchange. The lightweight structure evolves from the horizontal lines of the desert and forms a hyperbolic shelter. The user can inhabit not only underneath the structure, but also the petal shaped hammocks. Here, individuals can exchange stories, supplies and treasures.


In Buddhism, the Lotus flower is symbolic of fortune. It grows in muddy water, and it is this environment that gives forth the flower’s first and most literal meaning: rising and blooming above the murk to achieve enlightenment. The Lotus Hypar story has evolved from the same principles. In the harsh desert environment, man can create beauty. The folded geometries are playfully excited by human participation. A twist, a fold and a push.


The structure is assembled using bamboo sticks that are arranged in a reciprocal formation. These canes are then bound using high strength elastic bands. This allows for the flat cells to twist and take on new shapes. The Lotus Hypar is formed by a repetitive series of folds and the result forms petals. These are symbolic of the Lotus flower. The cells are covered with a white semi-elastic membrane that adds to the strength of the structure and the petal geometries become more visible. These are also the hammocks that can be inhabited by the Burning Man users.


In order to test the structural performance of the proposal, I constructed a series of 1:1 scale models. This was done using 6m and 3m bamboo canes (35mm diameter). By testing a small segment of the full proposal, it is easier to determine the success of the final proposal.

LH.4 LH.5