The Nature of Gridshell Form Finding

Grids, shells, and how they, in conjunction with the study of the natural world, can help us develop increasingly complex structural geometry.

Foreword

This post is the third installment of sort of trilogy, after Shapes, Fractals, Time & the Dimensions they Belong to, and Developing Space-Filling Fractals. While it’s not important to have read either of those posts to follow this one, I do think it adds a certain level of depth and continuity.

Regarding my previous entries, it can be difficult to see how any of this has to do with architecture. In fact I know a few people who think studying fractals is pointless.

Admittedly I often struggle to explain to people what fractals are, let alone how they can influence the way buildings look. However, I believe that this post really sheds light on how these kinds of studies may directly influence and enhance our understanding (and perhaps even the future) of our built environment.

On a separate note, I heard that a member of the architectural academia said “forget biomimicry, it doesn’t work.”

Firstly, I’m pretty sure Frei Otto would be rolling over in his grave.

Secondly, if someone thinks that biomimicry is useless, it’s because they don’t really understand what biomimicry is. And I think the same can be said regarding the study of fractals. They are closely related fields of study, and I wholeheartedly believe they are fertile grounds for architectural marvels to come.

7.0 Introduction to Shells

As far as classification goes, shells generally fall under the category of two-dimensional shapes. They are defined by a curved surface, where the material is thin in the direction perpendicular to the surface. However, assigning a dimension to certain shells can be tricky, since it kinda depends on how zoomed in you are.

A strainer is a good example of this – a two-dimensional gridshell. But if you zoom in, it is comprised of a series of woven, one-dimensional wires. And if you zoom in even further, you see that each wire is of course comprised of a certain volume of metal.

This is a property shared with many fractals, where their dimension can appear different depending on the level of magnification. And while there’s an infinite variety of possible shells, they are (for the most part) categorizable.

7.1 – Single Curved Surfaces

Analytic geometry is created in relation to Cartesian planes, using mathematical equations and a coordinate systems. Synthetic geometry is essentially free-form geometry (that isn’t defined by coordinates or equations), with the use of a variety of curves called splines. The following shapes were created via Synthetic geometry, where we’re calling our splines ‘u’ and ‘v.’

A-Barrel-Vault
Uniclastic: Barrel Vault (Cylindrical paraboloid)

These curves highlight each dimension of the two-dimensional surface. In this case only one of the two ‘curves’ is actually curved, making this shape developable. This means that if, for example, it was made of paper, you could flatten it completely.

B-Conoid

Uniclastic: Conoid (Conical paraboloid)

In this case, one of them grows in length, but the other still remains straight. Since one of the dimensions remains straight, it’s still a single curved surface – capable of being flattened without changing the area. Singly curved surfaced may also be referred to as uniclastic or monoclastic.

7.2 – Double Curved Surfaces

These can be classified as synclastic or anticlastic, and are non-developable surfaces. If made of paper, you could not flatten them without tearing, folding or crumpling them.

C-Dome.gif
Synclastic: Dome (Elliptic paraboloid)

In this case, both curves happen to be identical, but what’s important is that both dimensions are curving in the same direction. In this orientation, the dome is also under compression everywhere.

The surface of the earth is double curved, synclastic – non-developable. “The surface of a sphere cannot be represented on a plane without distortion,” a topic explored by Michael Stevens: https://www.youtube.com/watch?v=2lR7s1Y6Zig

D-Saddle.gif
Anticlastic: Saddle (Hyperbolic paraboloid)
This one was formed by non-uniformly sweeping a convex parabola along a concave parabola. It’s internal structure will behave differently, depending on the curvature of the shell relative to the shape. Roof shells have compressive stresses along the convex curvature, and tensile stress along the concave curvature.
Pringle
Kellogg’s potato and wheat-based stackable snack
Here is an example of a beautiful marriage of tensile and compressive potato and wheat-based anticlastic forces. Although I hear that Pringle cans are diabolically heinous to recycle, so they are the enemy.
11 Tensile and Compressive behaviour of shells.jpg
Structural Behaviour of Basic Shells [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]

7.3 – Translation vs Revolution

In terms of synthetic geometry, there’s more than one approach to generating anticlastic curvature:
E-Hyperbolic-Paraboloid-Saddle.gif
Hyperbolic Paraboloid: Straight line sweep variation

This shape was achieved by sweeping a straight line over a straight path at one end, and another straight path at the other. This will work as long as both rails are not parallel. Although I find this shape perplexing; it’s double curvature that you can create with straight lines, yet non-developable, and I can’t explain it..

F-Hyperbolic-Paraboloid-Tower.gif
Ruled Surface & Surface of Revolution (Circular Hyperboloid)
The ruled surface was created by sliding a plane curve (a straight line) along another plane curve (a circle), while keeping the angle between them constant. The surfaces of revolution was simply made by revolving a plane curve around an axis. (Surface of translation also exist, and are similar to ruled surfaces, only the orientation of the curves is kept constant instead of the angle.)
 
Cylinder_-_hyperboloid_-_cone.gif
Hyperboloid Generation [Source:Wikipedia]

The hyperboloid has been a popular design choice for (especially nuclear cooling) towers. It has excellent tensile and compressive properties, and can be built with straight members. This makes it relatively cheap and easy to fabricate relative to it’s size and performance.

These towers are pretty cool acoustically as well: https://youtu.be/GXpItQpOISU?t=40s

 

8.0 Geodesic Curves

These are singly curved curves, although that does sound confusing. A simple way to understand what geodesic curves are, is to give them a width. As previously explored, we know that curves can inhabit, and fill, two-dimensional space. However, you can’t really observe the twists and turns of a shape that has no thickness.

Geodesic Curves - Ribon.jpg
Conic Plank Lines (Source: The Geometry of Bending)

A ribbon is essentially a straight line with thickness, and when used to follow the curvature of a surface (as seen above), the result is a plank line. The term ‘plank line’ can be defined as a line with an given width (like a plank of wood) that passes over a surface and does not curve in the tangential plane, and whose width is always tangential to the surface.

Since one-dimensional curves do have an orientation in digital modeling, geodesic curves can be described as the one-dimensional counterpart to plank lines, and can benefit from the same definition.

The University of Southern California published a paper exploring the topic further: http://papers.cumincad.org/data/works/att/f197.content.pdf

8.1 – Basic Grid Setup

For simplicity, here’s a basic grid set up on a flat plane:

G-Geocurves.gif
Basic geodesic curves on a plane

We start by defining two points anywhere along the edge of the surface. Then we find the geodesic curve that joins the pair. Of course it’s trivial in this case, since we’re dealing with a flat surface, but bear with me.

H-Geocurves.gif
Initial set of curves

We can keep adding pairs of points along the edge. In this case they’re kept evenly spaced and uncrossing for the sake of a cleaner grid.

I-Geocurves.gif
Addition of secondary set of curves

After that, it’s simply a matter of playing with density, as well as adding an additional set of antagonistic curves. For practicality, each set share the same set of base points.

J-Geocurves.gif
Grid with independent sets

He’s an example of a grid where each set has their own set of anchors. While this does show the flexibility of a grid, I think it’s far more advantageous for them to share the same base points.

8.2 – Basic Gridshells

The same principle is then applied to a series of surfaces with varied types of curvature.

K-Barrel
Uniclastic: Barrel Vault Geodesic Gridshell

First comes the shell (a barrel vault in this case), then comes the grid. The symmetrical nature of this surface translates to a pretty regular (and also symmetrical) gridshell. The use of geodesic curves means that these gridshells can be fabricated using completely straight material, that only necessitate single curvature.

L-Conoid
Uniclastic: Conoid Geodesic Gridshell

The same grid used on a conical surface starts to reveal gradual shifts in the geometry’s spacing. The curves always search for the path of least resistance in terms of bending.

M-Dome
Synclastic: Dome Geodesic Gridshell

This case illustrates the nature of geodesic curves quite well. The dome was free-formed with a relatively high degree of curvature. A small change in the location of each anchor point translates to a large change in curvature between them. Each curve looks for the shortest path between each pair (without leaving the surface), but only has access to single curvature.

N-Saddle
Anticlastic: Saddle Geodesic Gridshell

Structurally speaking, things get much more interesting with anticlastic curvature. As previously stated, each member will behave differently based on their relative curvature and orientation in relation to the surface. Depending on their location on a gridshell, plank lines can act partly in compression and partly in tension.

On another note:

While geodesic curves make it far more practical to fabricate shells, they are not a strict requirement. Using non-geodesic curves just means more time, money, and effort must go into the fabrication of each component. Furthermore, there’s no reason why you can’t use alternate grid patterns. In fact, you could use any pattern under the sun – any motif your heart desires (even tessellated puppies.)

6 - Alternate Grid
Alternate Gridshell Patterns [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]

Here are just a few of the endless possible pattern. They all have their advantages and disadvantages in terms of fabrication, as well as structural potential.

Biosphere Environment Museum - Canada
Biosphere Environment Museum – Canada

Gridshells with large amounts of triangulation, such as Buckminster Fuller’s geodesic spheres, typically perform incredibly well structurally. These structure are also highly efficient to manufacture, as their geometry is extremely repetitive.  

Centre Pompidou-Metz - France
Centre Pompidou-Metz – France

Gridshells with highly irregular geometry are far more challenging to fabricate. In this case, each and every piece had to be custom made to shape; I imagine it must have costed a lot of money, and been a logistical nightmare. Although it is an exceptionally stunning piece of architecture (and a magnificent feat of engineering.)

8.3 – Gridshell Construction

In our case, building these shells is simply a matter of converting the geodesic curves into planks lines.

O - Saddle 2
Hyperbolic Paraboloid: Straight Line Sweep Variation With Rotating Plank Line Grid

The whole point of using them in the first place is so that we can make them out of straight material that don’t necessitate double curvature. This example is rotating so the shape is easier to understand. It’s grid is also rotating to demonstrate the ease at which you can play with the geometry.

Hyperbolic-Paraboloid-Plank-Lines
Hyperbolic Paraboloid: Flattened Plank Lines With Junctions

This is what you get by taking those plank lines and laying them flat. In this case both sets are the same because the shell happens to the identicall when flipped. Being able to use straight material means far less labour and waste, which translates to faster, and or cheaper, fabrication.

An especially crucial aspect of gridshells is the bracing. Without support in the form of tension ties, cable ties, ring beams, anchors etc., many of these shells can lay flat. This in and of itself is pretty interesting and does lends itself to unique construction challenges and opportunities. This isn’t always the case though, since sometimes it’s the geometry of the joints holding the shape together (like the geodesic spheres.) Sometimes the member are pre-bent (like Pompidou-Metz.) Although pre-bending the timber kinda strikes me as cheating thought.. As if it’s not a genuine, bona fide gridshell.

Toledo-gridshell-20-Construction-process
Toledo Gridshell 2.0. Construction Process [source: Timber gridshells – Numerical simulation, design and construction of a full scale structure]

This is one of the original build method, where the gridshell is assembled flat, lifted into shape, then locked into place.

9.0 Form Finding

Having studied the basics makes exploring increasingly elaborate geometry more intuitive. In principal, most of the shells we’ve looked are known to perform well structurally, but there are strategies we can use to focus specifically on performance optimization.

9.0 – Minimal Surfaces

These are surfaces that are locally area-minimizing – surfaces that have the smallest possible area for a defined boundary. They necessarily have zero mean curvature, i.e. the sum of the principal curvatures at each point is zero. Soap bubbles are a great example of this phenomenon.

hyperbolic paraboloid soap bubble
Hyperbolic Paraboloid Soap Bubble [Source: Serfio Musmeci’s “Froms With No Name” and “Anti-Polyhedrons”]
Soap film inherently forms shapes with the least amount of area needed to occupy space – that minimize the amount of material needed to create an enclosure. Surface tension has physical properties that naturally relax the surface’s curvature.

00---Minimal-Surface-Model
Kangaroo2 Physics: Surface Tension Simulation

We can simulate surface tension by using a network of curves derived from a given shape. Applying varies material properties to the mesh results in a shape that can behaves like stretchy fabric or soap. Reducing the rest length of each of these curves (while keeping the edges anchored) makes them pull on all of their neighbours, resulting in a locally minimal surface.

Here are a few more examples of minimal surfaces you can generate using different frames (although I’d like stress that the possibilities are extremely infinite.) The first and last iterations may or may not count, depending on which of the many definitions of minimal surfaces you use, since they deal with pressure. You can read about it in much greater detail here: https://tinyurl.com/ya4jfqb2

Eden_Project_geodesic_domes_panorama.jpg
The Eden Project – United Kingdom

Here we have one of the most popular examples of minimal surface geometry in architecture. The shapes of these domes were derived from a series of studies using clustered soap bubbles. The result is a series of enormous shells built with an impressively small amount of material.

Triply periodic minimal surfaces are also a pretty cool thing (surfaces that have a crystalline structure – that tessellate in three dimensions):

Another powerful method of form finding has been to let gravity dictate the shapes of structures. In physics and geometry, catenary (derived from the Latin word for chain) curves are found by letting a chain, rope or cable, that has been anchored at both end, hang under its own weight. They look similar to parabolic curves, but perform differently.

00---Haning-Model
Kangaroo2 Physics: Catenary Model Simulation

A net shown here in magenta has been anchored by the corners, then draped under simulated gravity. This creates a network of hanging curves that, when converted into a surface, and mirrored, ultimately forms a catenary shell. This geometry can be used to generate a gridshell that performs exceptionally well under compression, as long as the edges are reinforced and the corners are braced.

While I would be remiss to not mention Antoni Gaudí on the subject of catenary structure, his work doesn’t particularly fall under the category of gridshells. Instead I will proceed to gawk over some of the stunning work by Frei Otto.

Of course his work explored a great deal more than just catenary structures, but he is revered for his beautiful work on gridshells. He, along with the Institute for Lightweight Structures, have truly been pioneers on the front of theoretical structural engineering.

9.3 – Biomimicry in Architecture

There are a few different terms that refer to this practice, including biomimetics, bionomics or bionics. In principle they are all more or less the same thing; the practical application of discoveries derived from the study of the natural world (i.e. anything that was not caused or made my humans.) In a way, this is the fundamental essence of the scientific method: to learn by observation.
Biomimicry-Bird-Plane
Example of Biomimicry

Frei Otto is a fine example of ecological literacy at its finest. A profound curiosity of the natural world greatly informed his understanding of structural technology. This was all nourished by countless inquisitive and playful investigations into the realm of physics and biology. He even wrote a series of books on the way that the morphology of bird skulls and spiderwebs could be applied to architecture called Biology and Building. His ‘IL‘ series also highlights a deep admiration of the natural world.

Of course he’s the not the only architect renown their fascination of the universe and its secrets; Buckminster Fuller and Antoni Gaudí were also strong proponents of biomimicry, although they probably didn’t use the term (nor is the term important.)

Gaudí’s studies of nature translated into his use of ruled geometrical forms such as hyperbolic paraboloids, hyperboloids, helicoids etc. He suggested that there is no better structure than the trunk of a tree, or a human skeleton. Forms in biology tend to be both exceedingly practical and exceptionally beautiful, and Gaudí spent much of his life discovering how to adapt the language of nature to the structural forms of architecture.

Fractals were also an undisputed recurring theme in his work. This is especially apparent in his most renown piece of work, the Sagrada Familia. The varying complexity of geometry, as well as the particular richness of detail, at different scales is a property uniquely shared with fractal nature.

Antoni Gaudí and his legacy are unquestionably one of a kind, but I don’t think this is a coincidence. I believe the reality is that it is exceptionally difficult to peruse biomimicry, and especially fractal geometry, in a meaningful way in relation to architecture. For this reason there is an abundance of superficial appropriation of organic, and mathematical, structures without a fundamental understanding of their function. At its very worst, an architect’s approach comes down to: ‘I’ll say I got the structure from an animal. Everyone will buy one because of the romance of it.”

That being said, modern day engineers and architects continue to push this envelope, granted with varying levels of success. Although I believe that there is a certain level of inevitability when it comes to how architecture is influenced by natural forms. It has been said that, the more efficient structures and systems become, the more they resemble ones found in nature.

Euclid, the father of geometry, believed that nature itself was the physical manifestation of mathematical law. While this may seems like quite a striking statement, what is significant about it is the relationship between mathematics and the natural world. I like to think that this statement speaks less about the nature of the world and more about the nature of mathematics – that math is our way of expressing how the universe operates, or at least our attempt to do so. After all, Carl Sagan famously suggested that, in the event of extra terrestrial contact, we might use various universal principles and facts of mathematics and science to communicate.

Da Vinci Codex

‘Da Vinci Codex’ is a latticed sculptural piece which creates unique poetics of morphology that merge structure and movement. It transgresses the artificial boundary between art, science and technology, casting seemingly established analogies in a new light while inviting visitors to rethink the relationship between form, geometry and construction. Linear and curved scissor elements form a series of recursive cubes which speak of infinity and the complexity of our world. It denotes a recognizable metaphor of ‘object-within-similar-object’ that appears in the design of many other natural and crafted objects. The precision of the cubic form reflects the organised chaos of our universe. Poignant patterns inspired by a study into the scissor movement of the cube elements are perforated into the triangulated parts of the Codex.

Da Vinci Codex 1

Da Vinci Codex 2

As they expand and collapse, the triangles form unique and intricate shadows which highlight the transitional quality of human life and emotions, changing from a state of happiness to sadness, from calm to anger, from life to death. The structure provides shelter from the heat of the sun while entertaining its guests with opportunities to engage with the structure. A deployment mechanism inspired by study into Leonardo da Vinci’s machinery sketches found in his Codex Atlanticus is actuated by a series of gears situated at the base of the structure, which are set into motion by a pedal system powered by visitors. As burners interact with the piece, they contemplate a fascinating and spectacular change of light and decor. ‘Da Vinci Codex’ stands as a piece of event architecture, a spatial construct where movement is a transformational creative force.

The visitors interact with the piece by powering one of the four pedal systems connected to the deploying mechanism. As they pedal, the burners witness a captivating movement: the synchronised expanding and collapsing of the three cubes which cast intricate shadows and stimulate a sense of play. The visitors can also step inside the cubes and experience a series of ‘in-between’ spaces before reaching the central volume and enjoying a level of protection from the wind and sun. The highly abstract aesthetic of the ‘da Vinci Codex’ is meant to affect the community with a spirit of experimentation and encourage each and every burner to question preconceived ideas, beliefs or desires.

Da Vinci Codex

Da Vinci Codex3Da Vinci Codex2

The size of each member has been carefully considered not only to allow structural integrity but also to respect the proportions of the human body. Each face of the cube moves in a synchronised manner. The relationship between the size of each face and proportions of the human body has been inspired by da Vinci’s Vitruvian man.

BM open cubeBM night render

Deployable structures

A deployable structure includes an enclosed mechanical linkage capable of transformation between expanded and collapsed configurations while maintaining its shape.

These types of structures have the advantage of creating versatile, modulated spaces, with easy and fast assembly which generate benefits such as adaptability, flexibility and space transformation.

Charles Hoberman pioneered a type of deployable structure based on curved scissor pairs as seen in his 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 loop assemblies (or great circles), each made of 60 elements which fold and unfold in a scissor-like motion. Portfolio 2.jpg

Hoberman Sphere by Charles Hoberman

A loop assembly is formed of at least three scissors-pairs, at least two of the pairs comprising two identical rigid angulated strut elements, each having a central and two terminal pivot points with centres which do not lie in a straight line, each strut being pivotally joined to the other of its pair by their central pivot points. The terminal pivot points of each of the scissors-pairs are pivotally joined to the terminal pivot points of the adjacent pair such that both scissors-pairs lie essentially in the same plane.Portfolio 22

Regular curved scissor-pairs in motion

When this loop is folded and unfolded certain critical angles are constant and unchanging. These unchanging angles allow for the overall geometry of structure to remain constant as it expands or collapses.Portfolio 23

Regular and irregular curved scissor-pairs in motion

The above diagrams show a closed loop-assembly of irregular scissors pairs where each scissors-pair is pivotally joined by its two pairs of terminal pivot points to the terminal pivot points of its two adjacent scissors-pairs. This loop-assembly is an approximation of a polygon in the sense that the distances between adjacent central pivot points are equal to the corresponding lengths of the sides of the polygon. Further, the angles between the lines joining adjacent central pivot points with other similarly formed lines in the assembly are equal to the corresponding angles in the polygon.

The beams forming scissor-pairs can be of almost any shape, providing that the three connection points form a triangle. The angle of the apex would dictate the number of scissor-pairs that can be linked together to form a closed loop.Portfolio 28.jpg

Scissor-pairs of varying morphologies

My physical experiments started with materials that would allow a degree of bending and torsion in order to test the limits of the system. Using polypropylene for the angular beams and metal screws for the joints, I created these playful models that bend as they expand and contract.Portfolio 214.jpg

Later I started using MDF for the beams as well as joints and noticed that a degree of bending was present in the expanded state of the larger circle.Portfolio 215.jpg

After using curved scissor pairs of the same angle to form closed linkages, I decided to combine two types of scissors and vary the proportion between the elements to achieve a loop which would offer the highest ratio between the expanded and contracted state.Portfolio 216.jpg

900 curved scissors loops

Portfolio 217.jpg

900 curved scissors with linear scissors loops

The above diagrams show a combination of 900 curved scissors with linear (1800) scissors to form rectangles that expand and contract. The length of the 900 beam was gradually increased  and by measuring the diagonals  of the most expanded and most contracted forms, I obtained the following ratios for the three rectangles:

R1 = 0.87

R2 = 0.67

R3 = 0.64

By keeping the curved scissor with the best ratio, I created three more rectangles, this time by varying the length of the linear beam. The following ratios were obtained:

R1 = 0.64

R2 = 0.59

R3 = 0.67Portfolio 218.jpg

900 curved scissors with linear scissors loops

I then took the linkage with the best ratio of 0.59 and rotated it 900 to form a cube which expands and contracts.Portfolio 219.jpgPortfolio 220.jpg

Combined linkage cubes

The change of state from open to closed is visually attractive and could have the potential of creating spaces that are transitional.Portfolio 223If more linear scissors are placed between the 900 scissors, a better contraction ratio is obtained.Portfolio 222

Combined linkage cubes with two linear scissors

Auxetics

Auxetics are materials that have a negative Poisson’s Ratio. When stretched they become thicker perpendicular to the applied force, from our own experiences when a material is stretched we expect the material to not only become longer but also thinner. Auxetics behave in a different way because of there internal structure.

Poisson’s ratio (v) of a material is the ratio of the lateral contractile strain to the longitudinal tensile strain for materials undergoing tension in the longitudinal direction. It shows how much a material becomes thinner when stretched, therefore most materials have a positive Poisson’s ratio.

 

The images below show modules of four structures that have Auxetic behavior. The images show the change in state of the structure as they undergo tension in the longitudinal direction.

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This video shows an interesting application of an auxetic structure with inflatables by Fergal Coulter. http://fergalcoulter.eu/

 

 

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Final Render 01

Scherk’s Minimal Surface

In mathematics, a Scherk surface (named after Heinrich Scherk in 1834) is an example of a minimal surface. A minimal surface is a surface that locally minimizes its area (or having a mean curvature of zero). The classical minimal surfaces of H.F. Scherk were initially an attempt to solve Gergonne’s problem, a boundary value problem in the cube.

The term ‘minimal surface’ is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, minimal surface of revolution, Saddle Towers etc.).

Scherk's Surface Soap experiments

Scherk’s minimal surface arises from the solution to a differential equation that describes a minimal monge patch (a patch that maps [u, v] to [u, v, f(u, v)]). The full surface is obtained by putting a large number the small units next to each other in a chessboard pattern. The plots were made by plotting the implicit definition of the surface.

An implicit formula for the Scherk tower is:

sin(x) · sin(z) = sin(y),

where x, y and z denote the usual coordinates of R3.

Scherk’s second surface can be written parametrically as:

x = ln((1+r²+2rcosθ)/(1+r²-2rcosθ))

y = ((1+r²-2rsinθ)/(1+r²+2rsinθ)) 

z = 2tan-1[(2r²sin(2θ))/(r-1)]      

for θ in [0,2), and r in (0,1).

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Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic. They were the third non-trivial examples of minimal surfaces (the first two were the catenoid and helicoid). The two surfaces are conjugates of each other.

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Scherk’s first surface

Scherk’s first surface is asymptotic to two infinite families of parallel planes, orthogonal to each other, that meet near z = 0 in a checkerboard pattern of bridging arches. It contains an infinite number of straight vertical lines.

Scherk’s second surface

Scherk’s second surface looks globally like two orthogonal planes whose intersection consists of a sequence of tunnels in alternating directions. Its intersections with horizontal planes consists of alternating hyperbolas.

Other types are:

  1. The doubly periodic Scherk surface
  2. The Karcher-Scherk surface
  3. The sheared (Karcher-)Scherk surface
  4. The doubly periodic Scherk surface with handles
  5. The Meeks-Rosenberg surfaces

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Scherk’s surface can have many iterations, according to the number of saddle branches, number of holes, turn around the axis and bends towards the axis. Some of the design iterations and adaptations of the system are presented below:

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Scherk’s Surface can be adapted to several design possibilities, with multiple ways of fabrication. Interlocked slices using laser cut plywood sheets, folded planes of metal or CNC stacked wooden slices. With its versatile and flexible form it is adaptable to any interior space as an installation or temporary furniture.

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Moiré Patterns

Dutch Invertuals – ‘Cohesion’

Moiré patterns are superimposed secondary patterns created when two static surface patterns are overlaid one on top of the other. By displacing or rotating one or both patterns a new visual pattern becomes visible separate to the geometry of the first two. This moiré effect is created in the eye of the viewer, disparate from the shapes formed by the individual patterns themselves.

morie-patterns

A moiré pattern generated by overlapping two identical patterns of concentric circles

Associated mathematical formulas can be used to determine the size and spacing of inferred moiré patterns from a series of regularly spaced overlaid patterns. The beauty of the moiré effect is the illusion of movement created through completely static overlays. This forms a naturally interactive experience for the participant, giving over control to the superimposed pattern through visual movement and rotation.

Physical Moiré experiments

The video above illustrates the moiré effect in two dimensions by overlaying static linear and concentric patterns, printed on acetate, and manipulating their motion and rotation in order to create a new visual pattern.

IMG_1434

Concentric and Linear patterns, printed on acetate overlays

This effect is not restricted to two dimensional patterns but can also be applied in three dimensions. These spatial patterns then utilise the motion of the viewer in order to manipulate the moiré effect. The video below illustrates how three dimensional sculpted elements, set on separate spatial planes can form a visual pattern and take advantage of simple motions by the viewer.

Scale model of the facade for Brisbane Girls School Creative Learning Centre – M3 Architects

The two primary resultant effects from the physical experiment above illustrate the potential of moiré to create alternate visual patterns and to generate the illusion of movement. These were then applied digitally to create an animation that controls these aspects to create a recognisable representation of motion to the viewer, as opposed to an abstract pattern.

Digital testing of the moiré effect in animation

The above digital animation illustrates the rotational movement of a circle through the movement of a linear overlay, created with the two static images below:

Moire Overlay   The linear moiré overlay       Resultant Shape

The moiré underlay  creating the circular motion 

This moiré underlay is created through a series of rules defined by the size of the overlay and the direction, factor and type of movement (linear or rotational). The diagram below explores the rules associated with this specific type of moiré animation.

Brief 01_Moire System Analysis_Linear Animation-page-001

Rules for defining a moiré ‘underlay’ for linear animation

Whilst primarily a visual effect it is the ability to translate spatially which gives the moiré effect the potential to be applied in a design context, particularly given it’s interactive nature and the reliance on the involvement of participants in order to reveal it’s true beauty.

The video below takes this concept to the extreme, exploring the effects of imagining matter as nothing more than multi-dimensional moiré patterns……

Moiré – Julias Horsthuis

Updated Research:

Video illustrating various physical moiré experiments

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Rules for defining moiré patterns in linear gratings

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Mathematical rules for defining moiré patterns of rotation

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Physical model for experimenting with moiré rotation patterns

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Results from the physical model using sin curves & square gratings

Moiré patterns can be ‘programmed’ using a certain mathematical formula. If two variables are known; the base layer and the desired moiré pattern (in this instance a sin curve) the resultant reveal layer can be determined, allowing moiré patterns to be programmed to any shape.

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Digital tests and physical proofs of programming moiré

Moiré patterns work in both ‘positive’ and ‘negative’ constructions. Positive moiré can be classed as additive, constructing patterns consisting of lines to create the effect. Negative moiré conversely removes elements of material (in this instance circles from card) to create patterns when held at a distance. The bottom row of images shows the most successful variables for discerning negative moiré patterns.

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Negative moiré, set-up & physical experiment

The above experiment was digitally reproduced, modelling its negative space in order to understand how the variables of distance affect the reception of pattern.

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Digital experiments with distance variables

In order to move from the plane into a spatial exploration of the moiré effect, sin curve gratings were mapped onto the faces of a cube, at varied rotations. The effect is a spatial understanding of moiré patterns when the various faces of the cube overlap. The moiré effect can be created by two distinct methods; a movement by the user, distorting the areas of overlap and the movement of the cube itself, visually shifting patterns.

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Physical model exploring moiré patterns in three dimensions

Curved Crease Folding

The history of curved crease folding goes back to as early as the Bauhaus, where a student had scored circular creases onto a paper in order to study its materiality. When a circular surface is folded along concentric rings, the resultant form bends on itself and forms a paraboloid in order to make up for the loss in circumference. Initial investigation involved the replication of such system and multiplying the modules which are then interlocked into each other to create various origami sculptures.

Circular Modules

Circular Modules

The system is then digitally simulated in order to extract the parameters which may affect the resultant geometry of the surface. With a combination of Kangaroo Physics, Hinge Forces and Springs, the digital simulation is created which allows anchor points to be placed, thus dragging for surface into various forms. Tests are carried out on different surfaces, including a closed circle of equal concentric rings, a closed circle of increasing concentric rings as well as an open circular strip with concentric rings. With an increasing fold angle, the bend angle increases.

System Exploration

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System Exploration

System Exploration

Upon cutting the closed circle, the surface becomes an open ended circular strip. The constraints that follow a closed surface no longer presents itself, thus allowing the strip to bend freely – although the principles of the system still applies. With increasing fold angles, the strip bends at greater angle. Having this revelation, different open ended strips are then tested against different parameters to extract the system further.

Parameters

Parameters

Parameters

Parameters

Parameters

In parallel to the research of curved crease folding is the investigation into the probability of transferring the system onto a more rigid, larger material, such as plywood. Here lattice hinge / kerf folds are employed, allowing the plywood to bend in a similar manner to card and paper. The final patterns for the hinges are a result of rigorous testing through trial and error. By repeating the modules we begin to see that, due to the folds, plywood can be as flexible as card.

Lattice

Lattice