Scherk's Minimal Surface

01: System

Scherk’s Minimal Surface - Visual Diary, Research & Experiments-

Andreea - Laura Nica | University of Westminster | DS10 | 2016

CONTENTS

I SCHERK’S MINIMAL SURFACE

II SYSTEM CATALOGUE

Minimal Surfaces - Frei Otto Experiments

Building a Scherk’s Surface

Tensile Model

Minimal Surfaces in Nature

Iterations 1 (surpassing limits)

Kerf Cuts

Skin Concept - Stretched Model

Main Iterations/ Rules

Unrolling Surfaces

3D Printed Models

Mould fabrication

Iterations 2

Digital fabrication explorations

Scherk’s Minimal Surface Ways of Representation Properties Types of Scherk’s Surface

III FABRICATION EXPLORATIONS

Iterations 3 Iterations 4

IV

APPENDIX Minimal Surfaces in Euclidian Space (Math Theory) Voronoi Diagrams - Burnt Skin concept

I | Minimal Surfaces In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having a mean curvature of zero.

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.

Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.

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I | Minimal Surfaces - Frei Otto’s Experiments Frei Otto is a revolutionary architect and structural engineer. He is renowned for his development and use of ultra-modern and super-light tent-like structures, and for his innovative use of new materials.

For Frei Otto, experimentation with models and maquettes was a fundamental part of his work. In 1961, he began to conduct a series of experiments with soap bubbles. His experiments centered on suspending soap film and dropping a looped string into it to form a perfect circle. By then trying to pull the string out a minimal surface was created.

Another experiment were the famous ‘Wool Experiments’ that started at the beginning of the 1990s.

The experiments were influenced by Gaudi’s catenary chain models used to create the Sagrada Familia.

These experiments were meant to create strategies for calculating two-dimensional city infrstructure, as well as three-dimensional canellous bone structures. These are considered analog computational and form finding.

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I | Approximation of Minimal Surfaces in Nature

Living creatures

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Bone Structure

Degenerated skin cells

I | Minimal Surfaces in Architecture

Bend wood

Curved Wooden structures

‘Fabric’ Installation

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I | Initial Experiments The initial experiments were inteded to facilitate the understanding of minimal surfaces and help decide on a type that would be further developed. The concept and research on the burnt skin cells was continued by looking at surfaces that would imitate their behaviour.

The Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz.

Schoen named this surface ‘diamond’ because it has two intertwined congruent labyrinths, each having the shape of an inflated tubular version of the diamond bond structure.

2 Edge simulations

3 Edge simulations

Schwarz ‘D’ Surface - diagram

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4 Edge simulations

Digital Simulations

I | Tensile Fabric Model - ‘D- Schwarz’ Minimal Surface

Metal Frame 1m x 1m x 1m , Black lycra and Black thread Model

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I | 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.

∞

A²∂

1 b∂ b∂ cos ∂ -3

10

-2

∂

-i

1

-1

2

3

x 1

Singly periodic surface -1

A²∂

A³∂

sin(x) · sin(z) = sin(y),

for θ in [0,2), and r in (0,1).

A¹∂ 0

An implicit formula for the Scherk tower is:

x = ln((1+r²+2rcosθ)/(1+r²-2rcosθ)) y = ((1+r²-2rsinθ)/(1+r²+2rsinθ)) z = 2tan-1[(2r²sin(2θ))/(r-1)]

W¹(∂)

2

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.

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

Scherk’s second surface can be written parametrically as:

-1

i

y

W²(∂)

-2 W³(∂)

W³(∂)

intersection with z=0 intersection with z=� asymptote planes A ∂i

Doubly periodic surface

I | Ways of Representation The theory of translation surfaces is always one of interesting topics in Euclidean space. (see appendix). Translation surfaces have been onvestigated from various viewpoints by many differential geometers: L. Verstraelen, J.Walrave and S.Yaprak have investigated minimal translation surfaces in n-dimensional Euclidean spaces.

A surface that can be generated from two space curves by translating either one of them parallel to itself in such a way that each of its points describes a curve that is a translation of the other curve.

Gauss curvature of a translation surface generating by space curves in zero if and only if at least one of generator curves is an asiymptotic line of surface.

In general, Scherk Minimal surfaces can be represented through: a. Weiserstrass Representation (Every regular minimal surface in RÂł has a local isothermal parametric representation) b. Planar harmonic mappings

t=1

t = 1/2

t=0

t=1

t = 1/2

t=0

Diagrams of concentric circles under f and corresponding minimal surfaces for various values of t

Translation surface generated by two helices

Scherk Surface is one of minimal tranlation surfaces

Translation surface generated by two helices

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I | Types of Scherk’s Minimal Surface

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Scherk’s first surface (doubly periodic)

Scherk’s first surface (doubly periodic)

Scherk’s first surface (doubly periodic)

Scherk’s second surface (singly periodic)

Scherk’s second surface (Saddle Tower)

Twisted Sherk’s second surface

The sheared (Karcher-)Scherk surface

The sheared (Karcher-)Scherk surface

Scherk’s second surface (Saddle Tower)

Scherk’s with 3 saddle branches

Scherk’s with 5 saddle brances

The doubly periodic Scherk surface with handles

The Meeks-Rosenberg surfaces

Scherk’s with 4 saddle branches

The doubly periodic Scherk surface with handles The doubly periodic Scherk surface not

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I | Scherk’s Surface Properties 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. 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.

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It contains an infinite number of straight vertical lines. 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.

By sliding the parameter for the associated family, it allows to see the whole associated family of Scherk’s minimal surfaces: 0 corresponds to Scherk’s doubly periodic surface,1 to the singly periodic Saddle Tower.

The values inbetween show how the first surface can be deformed isometrically into the second, passing through a family of minimal surfaces. When the parameter is 0 or 1, several periods of the surfaces are shown.

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I I| Building a Scherk’s Surface 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. 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.

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Define 2 range of domains with resolution set as nr. of steps. Evaluate with function of Scherk Surface.

Build a mesh plane and deconstruct the mesh.

Reconstruct the mesh (points from flat place and function points)

3D proximity and join curves.

Complete Scherk Surface

Copy, mirror and flip.

Create a box and trip the edges

Create a box and trip the edges

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I I| Iterations 1 (surpassing the limits) TOP VIEW

r: o to pi*2 rez.: 3 r2: 0 to 0.9

PERSPECTIVE VIEW

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r: o to pi*2 rez.: 5 r2: 0 to 0.9

r: o to pi*2 rez.: 7 r2: 0 to 0.9

r: o to pi*2 rez.: 9 r2: 0 to 0.9

r: o to pi*2 rez.: 15 r2: 0 to 0.9

r: o to pi*2 rez.: 30 r2: 0 to 0.9

TOP VIEW

r: o to pi*2 rez.: 80 r2: 0 to 0.5

r: o to pi*2 rez.: 80 r2: 0 to 0.8

r: o to pi*2 rez.: 80 r2: 0 to 1

r: o to pi*2 rez.: 80 r2: 0 to 5

r: o to pi*2 rez.: 80 r2: 0 to 9

PERSPECTIVE VIEW

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II | Main iterations - Rules 1. The main module of Scherk Second Surface, results from the basic parametric function.

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number of saddle branches - 2 number of holes/ stories - 1 strech x - 1 strech y - 1 strech z - 1

2

number of saddle branches - 2 number of holes/ stories - 3 strech x - 1 strech y - 1 strech z - 1

3

number of saddle branches - 2 number of holes/ stories - 3 turn around axis- 180ยบ strech x - 1 strech y - 1 strech z - 1

4

number of saddle branches - 2 number of holes/ stories - 3 flange- 2.6 turn around axis- 180ยบ bend towards axis - 360ยบ strech x - 1 strech y - 1 strech z - 1

1

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2. By copying, mirror it and flipping it, the surface becomes a Saddle Tower.

2

3. By rotating the saddle branches around its own axis, the surface becomes a twisted Saddle Tower.

3

4. And finally if the ends of the Saddle branches are connected by twisting 360ยบ, the desired surface closes.

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II| 3D Printed Models - Main Iterations

Iteration 1

Iteration 2

Iteration 3

Iteration 4

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II | Iterations 2 #- number of saddle branches n- number of holes/stories t- overall axial twist a- turn around axis b- bend towards axis x- stretch x y- stretch y z- stretch z

1

2 1

#- 2 n- 1 x- 1 y- 1 z- 1

2

#- 4 n- 2 x- 1 y- 1 z- 1

7

#- 2 n- 2 x- 1 y- 1 z- 1

3

#- 5 n- 2 x- 1 y- 1 z- 1

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#- 2 n- 3 x- 1 y- 1 z- 1

4

#- 2 n- 4 x- 1 y- 1 z- 1

5

#- 6 n- 2 x- 1 y- 1 z- 1

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#- 8 n- 3 x- 1 y- 1 z- 1

10 #- 2

14 #- 2

15 #- 2

#- 3 n- 2 x- 1 y- 1 z- 1

3 6

6

11 #- 2

n- 3 t- 15ยบ x- 1 y- 1 z- 1

12 #- 2

n- 3 t- 30ยบ x- 1 y- 1 z- 1

13 #- 2

n- 3 t- 45ยบ x- 1 y- 1 z- 1

n- 3 t- 90ยบ x- 1 y- 1 z- 1

n- 6 x- 1 y- 1 z- 1

n- 3 t- 180ยบ x- 1 y- 1 z- 1

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II| Iterations 2 - Plans

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II | Iterations 3 #- number of saddle branches n- number of holes/stories t- overall axial twist a- turn around axis b- bend towards axis x- stretch x y- stretch y z- stretch z

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#- 2 n- 3 b- 30º x- 1 y- 1 z- 1

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#- 2 n- 2 height- 1.5 flange- 2.9 x- 1.2 y- 1.3 z- 0.2

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#- 6 n- 13 height- 1.5 flange- 2.6 t- 60º a- 30º b- 360º x- 1.1 y- 1.1 z- 0.9

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#- 2 n- 3 b- 60º x- 1 y- 1 z- 1

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#- 3 n- 2 height- 1.5 flange- 2.9 x- 1.2 y- 1.3 z- 0.2

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#- 3 n- 13 height- 1.5 flange- 2.6 t- 360º a- 190º b- 360º x- 1.1 y- 1.1 z- 0.9

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#- 2 n- 3 b- 90º x- 1 y- 1 z- 1

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#- 7 n- 5 height- 1.5 flange- 0.5 x- 1.2 y- 1.3 z- 0.2

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#- 2 n- 13 height- 1.5 flange- 2.6 t- 360º a- 190º b- 360º x- 1 y- 1 z- 1

29 #- 4

n- 3 b 180º x- 1 y- 1 z- 1 #- 7 n- 5 t- 60º height- 1.5 flange- 2.9 x- 1.2 y- 0.9 z- 0.4 n- 5 height- 1.5 flange- 2.6 t- 360º a- 0º b- 360º x- 1.2 y- 4.1 z- 1

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#- 2 n- 3 b- 360º x- 1 y- 1 z- 1 #- 4 n- 5 t- 60º height- 1.5 flange- 2.9 x- 1.2 y- 0.8 z- 0.4 #- 4 n- 16 height- 1.5 flange- 1.3 t- 90º a- 0º b- 360º x- 1.6 y- 3.9 z- 1.6

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II| Iterations 3 - Plans

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II | Iterations 4 #- number of saddle branches n- number of holes/stories t- overall axial twist a- turn around axis b- bend towards axis x- stretch x y- stretch y z- stretch z

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#- 3 n- 2 x- 3 y- 4 z- 1

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#- 3 n- 2 height- 1.5 flange- 5 x- 3.8 y- 4.1 z- 5

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#- 3 n- 4 height- 1.5 flange- 0.7 a- 95º b- 180º x- 3.8 y- 1.5 z- 0.5

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#- 3 n- 2 x- 3 y- 4 z- 2 a- 55º b- 180º

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#- 1 n- 5 flange- 0.7 x- 2.8 y- 2.7 z- 4.2 t- 270º

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#- 6 n- 4 flange- 0.7 a- 95º b- 180º x- 4 y- 2.3 z- 4.8

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#- 1 n- 2 b- 90º x- 2.8 y- 2.6 z- 4.2 t- 210º

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#- 2 n- 5 flange- 0.7 x- 2.8 y- 2.9 z- 5 t- 270

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#- 6 n- 4 flange- 0.7 a- 95º b- 180º x- 1.4 y- 2.3 z- 4.8

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#- 7 n- 3 x- 2.8 y- 2.7 z- 4.2 t- 210º v- 0.10

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#- 5 n- 8 t- 30º flange- 0.7 x- 1.2 y- 1.5 z- 5 a- 60º

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#- 4 n- 4 flange- 1.3 a- 55º b- 360º x- 4 y- 2.3 z- 4.3

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#- 7 n- 5 x- 2.8 y- 2.7 z- 4.2 t- 210º #- 3 n- 4 flange- 0.7 x- 3.8 y- 1.3 z- 4.8 b- 180º a-95º #- 5 n- 10 height- 0.4 flange- 0.9 a- 55º b- 360º x- 4 y- 2.3 z- 4.3

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II| Iterations 4 - Plans

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III | Fabrication Explorations - Tensile Model 2 The second experiments with fabric were inteded to try and understand the construction of the Scherk Saddle Tower. By decomposing the geometry in grasshopper, the form was reduced to initial directory lines that could be inscribed into a paralelogram. The main lines were represented by drawing with thread and then fabric was streched inbetween the main points to form the base. The model was not completed, but showed another fabrication method.

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III | Mould Fabrication

Basic Scherk Surface

CNC fabrication

Framework fabrication

-Creating a Scherk Surface using the function in Grasshopper; -Combine and clean the mesh; -Weld the vertices; -Obtain the naked vertices; -Obtain the points; -Connect the points;

First method of fabrication would be CNC milling fabrication of the negative shape of the surface.

By extruding the lines created by the connections of the naked vertices, the result is a extruded framing with the main guides of the geometry.

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III | Wood Fabrication - Kerf Cuts

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III | Unrolling a Scherk’s Surface In order to produce and test the first paper model of the Scherk’s Surface, the ‘UnrollSrf’ command was used to flatten the Saddle Tower. By identifying the two longest edges and isolate the isocurves from the isocurves from the surface, I manage to create developable surfaces.

The command woks only for surfaces or polysurfaces with curvature in one direction to a planar surface.

After lofting it into grasshopper, I managed to smach the surfaces and obtain a 2D file proper for laser cutting and fabrication.

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4

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III | Interlocked Slices A way of defining the mesh is to divide the model into a number of slicing counts and interlocks between them. Despite being a pretty quick formula, to solve the model, the process takes long time to fi in right order all the slices.

A1 laser cut beds

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III | Folded Planes Another way of defining the mesh of the geometry is to subdivide it into folded planes.

In this case, separate connections would had been necessarily and quite hard to fabricate.

A1 laser cut bed

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III | Stacked Slices - Laser cut Fabrication The final fabrication method was to recreate the geometry by stacking thin layer on top of each other. The process was quick, but without precise scoring of the position, the model can not be precise.

A1 laser cut beds

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III | Stacked Slices

Scherk Surface mesh 360° degrees iteration

Trimmed Scherk Surface

Sliced Scherk Surface

Exploded axonometry of sliced Scherk Surface

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III | Staking Slices - CNC Fabrication A way of defining the mesh is to divide the model into a number of slicing counts and interlocks between them. Despite being a pretty quick formula, to solve the model, the process takes long time to fi in right order all the slices.

1m 0.63m 0.41m

1m

1m

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0.75m

0.70m

0.47m

0.49m

CNC Machining

CNC Machining

Laser Cutting

18 mm MDF sheet 2440 x 1220 mm bed size £18 x 11 sheets = £198 app.: 8 h 30 min

9 mm MDF sheet 2440 x 1220 mm bed size £12 x 3 sheets = £36 app.: 2 h

6 mm MDF sheet 600 x 400 mm bed size £1 x 33 sheets = £33 app.: 16 h 30 min

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III | Prototype Sc. 1:1

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III | Prototype Sc. 1:1

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IV | Appendix - Other experiments

-Weaverbird boundary; -joined curves; -Rhino isolate the edges;

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-Weaverbird boundary; -joined curves; -add horizontal planes; -creted a centre box; -cull the boxes;

-Weaverbird boundary; -joined curves; -add horizontal planes; -creted a centre box; -cull the boxes; -variations on lengths of brick;

IV | Appendix - Other experiments

-Weaverbird Sierpinsky subdivision; -Thickening the mesh;

-Weaverbird Cumulation;

Stellate/

-Weaverbird Sierpinsky’s Triangles Subdivision;

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IV | First System - Voronoi Diagram VORONOI DIAGRAM The Voronoi diagram, named after the mathematician Georgii Voronoi, is a decomposition of a metric space according to proximity criteria.

SITE POINTS

VORONOI EDGE

CONVEX HULL VORONOI VERTEX

Given a set of n points S (S1, S2, ... , Sn), called sites, the Voronoi diagram for S is the decomposition of the bidimentional space which associates a region V(Si) called Voronoi cell, to each point of S, so that all the points of V(Si) are closed to Si than any other points of S. The Voronoi diagram has practical applications in different fields, from physics to city planning (territorial division based on distances from a specific centre).

Basic Properties: a. The Voronoi diagram is a planar graph where evry vertex is of degree 3; b. The Voronoi diagram has a linear complexity; c. The dual graph forms a special kind of triangulation called the Delaunay Triangulation; d. Sites in the unbounded voronoi cells correspond to the vertices on the Convex Hull;

Typical Structure

Main Geometrical relations

In geometry, the Weaire–Phelan structure is a complex 3-dimensional Delaunay Triangulation structure representing an idealised foam of equal-sized bubbles.

Above there is a simulation of the evolution of the foam over time.

Basic Principles FOAM/ SOAP BUBBLES

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In geometry, the Weaire–Phelan structure is a complex 3-dimensional structure representing an idealised foam of equal-sized bubbles. Above A plant cell division algorithm based onofcell andfoam ellipse-fitting. there is a simulation thebiomechanics evolution of the over time.

PLANT SOAP CELLSBUBBLES FOAM/

The human skin is formed by several layers of cells. This is called the dermis (support+elasticity), hypodermis A plant cell division algorithm based on cell biomechanics and ellipse-fitting. (insulation+support).

BURNT HUMAN SKINCELLS PLANT

IV | First System - Voronoi Diagram

ns

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Andreea - Laura Nica | University of Westminster | DS10 | 2016