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.

Fractal Branching in the Victoria Amazonica

Much can be learned from the biological system of the Victoria Amazonica, also known as the Giant Water Lily. The plant has long generated curiosity about its delicate appearance yet impressive strength which is owed to its exceptional structural characteristics. An intricately webbed system of spines and ribs contributes to the success of the Victoria Amazonica, as it evenly distributes the weight of the plant while leaving pockets to store air and increase buoyancy. Once studied in detail, the branching pattern of the spines and ribs becomes apparent. Each Water Lily is unique, but all follow the same fractal branching pattern which can be defined by the following geometric sequence:

This branching system proves to be most effective for the Victoria Amazonica. Therefore, because of the efficiency of the system, its quick growth, and incredible strength, the plant can become somewhat of an invasive species. While the Lily Pad remains planar, in different forms, its system proves to be equally as strong. Below, the application of the system to different forms is tested.

Ribs are evenly dispersed between spines, lending to the plant’s equally distributed surface weight. In this example, the ribs are roughly spaced 8.3 cm from each other. This can be described with the linear expression: y=(1/8.3)x

Finite Element Method Mesh structurally analyzes how the surface of the Lily Pad is broken up into geometric cells. The FEM of the Pad can be tested for its movement under force and for its elasticity . These diagrams were generated with data from Nature-Inspired Fractal Geometry and Its Applications in Architectural Designs. Asayama, Riane, Sassone. 2014.

Naturally, the model without ribs (left model) was not as strong. However, it had more potential for movement and was still structurally impressive. Going forward, this was used as a base model for form studies.

As the above models proved to be structurally sound, they were scaled up to test their application to a larger model. At 1.5 m, the structure was not as stable because the spacing of the branches increased. Going forward, increased branching will be tested. Added weight will also be applied to see how it affects the stability.

Branching Structures

The inspiration for this body of work is the complex formations of tree branching. It is fascinating to comprehend the geometry of the branches and the reasoning for this, from aerodynamics to solar gain. The branching can be seen as fractal, with each branch migrating into two or more as it moves through space, also the thickness of the trunks is mathematically interesting, with Leonardo Da Vinci’s deduction of the sum of branch thickness moving up the tree, at a certain height is equal to the thickness of the initial tree trunk. This analysis has been taken forward throughout these studies, creating more structurally sound physical and digital models. Through studying these geometries the work aims to produce architecture which is thoughtfully designed, and can be adapted based on its environment.

Point Cloud Branching

Travelling through the point cloud the branching lines are created, although to produce a smoother geometry the initial branch lines are reformed with smooth curvature. Then, a pipe structure is applied to the branch lines, with varying thickness moving from top to bottom, as seen in the drawing above.

To form the point cloud branching system, first the path the branches aim to take must be formed, in the below case a cone shape was chosen to closely mimic the geometry naturally formed by the dragon-blood tree. Next a bounding box is created to inform the population of points. Once the points are established within the bounding box they can be joined to its nearest counter points, thus forming a point cloud.

Pyramid Branching


The voronoi geometry is offset, with the distance halved each iteration.


The outer most geometry is divided into points which meet on the offset geometry.


The process of meeting the points continues.


Finally the points are all joined at a central point.

To form the 3D printed models the below file was used, is it comprised of 10 pyramid like branching structures, perfectly joined. Therefore the 3D printed model to the right could form the below geometry, if all of its counterparts were also printed.

Curved Plywood Branching

Test 01

The initial test model using 0.8mm plywood utilises an initial layer of 8 segments. Once built the model had little tension, therefore needed string to create the dome like form. Although, even with the tension string the resulting form was flatter than anticipated.

Test 02

Proceeding the first test, the second aimed to produce a more substantial dome geometry. To do this 5 rather than 8 initial segments where used.
When the tension string was weaved the dome form was created, giving the model an ability to stand upright, as shown in the image above.

Test 03

The final test, aimed to decreased the percentage of material necessary to construct the model. This lead to cutting parts to make them thinner, causing breakage. Through the breakage, it was found that the dome like form could be created without the use of tension string.

Curved Paper Void Branching

Initially points are set using populate tool in grasshopper, then the points are plugged into the voronoi tool, giving the geometry of meeting circular entities.

From the voronoi geometry, the curves are filleted, then offset, at an increment doubling each time.

On the outer curve points are set equidistant apart, from these points a mid point is set along the inner curve, then creating a line to the closest point on the inner curve.

The task of joining the points at their midpoint, along the closest point on the adjacent curve, is repeated. Finally the points all join together in the centre of the final curve.

Once all the branches meet at the central point of the final curve the offset curves can be removed to reveal the branching geometry.

Finally, tracing over the straight branching geometry with the curve tool. Revealing petal like closed curves filling the voids between the straight branches.

Thread Voronoi Branching

Crown Shyness

Dragon Blood Tree Analysis

Asymptotic Grid Structure of a Triply Periodic Minimal Surface

Through extensive research into the construction of grid shells, as well as differential geometry, I present a design solution for a complex grid structure inspired by the highly symmetrical and optimised physical properties of a triply periodic minimal surface. The proposal implements the asymptotic design method of Eike Schling and his team at Technical University of Munich.

‘Minimal Matters’ utilises the several geometric benefits of an asymptotic curve network to optimise cost and fabrication. From differential geometry, it is determined asymptotic curves are not curved in the surface normal direction. As opposed to traditional gridshells, this means they can be formed from straight, planar strips perpendicular to the surface. In combination with 90° intersections that appear on all minimal surfaces (soap films) this method offers a simple and affordable construction method. Asymptotic curves have a vanishing normal curvature, and thus only exist on anticlastic surface-regions. 

Asymptotic curves can be plotted on any anticlastic surface using differential geometry.

On minimal surfaces, the deviation angle α is always 45 (due to the bisecting property of asymptotic curves and principle curvature lines). Both principle curvature networks and asymptotic curve networks consist of two families of curves that follow a direction field. The designer can only pick a starting point, but cannot alter their path.

(a) Planes of principle curvature are where the curvature takes its maximum and minimum values. They are always perpendicular, and intersect the tangent plane.

(b) Surface geometry at a generic point on a minimal surface. At any point there are two orthogonal principal directions (Blue), along which the curves on the surface are most convex and concave.
Their curvature is quantified by the inverse of the radii (R1 and R2) of circles fitted to the sectional curves along these directions. Exactly between these principal directions are the asymptotic directions (orange), along which the surface curves least.

(c) The direction and magnitude for these directions vary between points on a surface.

(d) Starting from point, lines can be drawn to connect points along the paths of principal and asymptotic directions on the respective surface.

Gyroid TPMS

The next step is to create the asymptotic curve network for the Gyroid minimal surface; chosen from my research into Triply Periodic Minimal Surfaces.

As the designer, I can merely pick a starting point on an anticlastic surface from which two asymptotic paths will originate. It is crucial to understand the behaviour of asymptotic curves and its dependency on the Gaussian curvature of the surface.

Through rotational symmetry, it is resolved to only require six unique strips for the complete grid structure (Seven including the repeated perimeter piece).

The node to node distance, measured along the asymptotic curves, is the only variable information needed to draw the flat and straight strips. They are then cut flat and bent and twisted into an asymptotic support structure.

Plywood Prototype: 600mm cubed

Eight fundamental units complete the cubit unit cell of a Gyroid surface. Due to the scale of the proposal, I have introduced two layers of lamellas. This is to ensure each layer is sufficiently slender to be easily bent and twisted into its target geometry, whilst providing enough stiffness to resist buckling under compression loads.

Minimal Matters’ aims to create an explorative, meditative and interactive experience for visitors. It is a strained grid shell utilising the geometrical benefits of an asymptotic curve network; digitally designed via algorithmic rules to minimise material, cost, and construction time.