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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Gender Difference

As part of international woman’s day I’m exploring differences between males and females in relation to the built environment in order to inform my final project. It only takes two minutes to complete and will directly influence the design progression.

https://sites.google.com/site/genderpreference/

Some examples of questions found in the survey can be found below:

tetrachromacy

material preferencetimber shapes proximity

Many Thanks

Image : Jan Gehl, How to Study Public Life, http://www.blogadilla.com/2008/06/08/are-you-a-tetrachromat/

Reflection

Lorna Jackson_Reflection_Interior Visual

Reflection presents this years burners with an intimate setting in which to share their inner most confessions, secrets and tales – With the option to do so both openly with other burners face to face, or retain the mystery of their identity by sharing with a complete stranger through the pavilions semi private screen. Reflection embodies the theme ‘Carnival of Mirrors’ in a variety of manners:- the geometry of the pavilion not only mirrors itself in its own form, but also incorporates a reflective surface within its interior spaces. The reflective physicality of the pavilion beautifully juxtaposes its function, by giving its burners a physical platform with which to cogitate their innermost thoughts and feelings, and share these with others. The pavilion is created as a result of rigorous testing of origami in order to create a single Spiralhedron which is then mirrored through along all axis.
Lorna Jackson_Reflection_Plan ElevLorna Jackson_Reflection_sections

Based upon a geometric origami principle which outlines the rules for the triangular subdivision of a 2-dimensional shape and assigns mountain and valleys creases to each subsequent subdivision the Spiralhedron has been optimised through both digital and physical testing. Reflection takes an abstract approach to this years theme, the pavilion’s form manifests itself as a result of mirroring this singular Spiralhedron in the X,Y and Z axis, which in turn creates its enclosing plywood form. In order to create the semi-private confessional screen, the panels incorporate a pattern, providing both the function of privacy, but also narrating the origins of the pavilions final form.

Lorna Jackson_Reflection_Meta Diagram_PNGLorna Jackson_Reflection_constructionLorna Jackson_Reflection_Large Model_thin

The principles of Burning Man are carefully considered, by providing an interactive base for participation that is never fully accomplished without the burners involvement. By sharing their stories, burners create a unique experience manifested through the ideals of trust and sharing, which facilitates a special bond between the burners. Upon its burning at the end of the festival, ‘Reflection’ becomes a resting place for the confessions, secrets and stories of its burners, allowing new bonds to be formed.Lorna Jackson_Reflection_detailLorna Jackson_Reflection_Small Modelthin

Construction

Due to form being created through the act of mirroring the entire pavilion will be made of 9 unique laser cut panels which will be bolted together with both metal hinges and 90 degrees and wooden brackets at 135 degrees.

Lorna Jackson_Reflection_Daytime Visual

Dimensions

Constrained by the size of a plywood sheet each individual Spiralhedron is made of two sheets of plywood (requiring 16 in total). Made of eight spiralhedrons ‘Reflection’ has a footprint of 3.5metres*3.5metres with a maximum height of 3.5m creating a footprint equal to that of the height of the pavilion.

Website : ATLV/Education

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[Image: A screen grab from the homepage of ATLV/Education].

ATLV/Education is a learning platform where a lot of resources for tutorials that would be a major help for beginner and intermediate Grasshopper and Rhinoceros users. ATLV is actually an acronym for Architectural Technology Laboratory Venture, a computational design firm based in Los Angeles. The firm explores the frontier of computational design technology through design practice and research in contemporary architecture and spatial design.This computational design firm is founded in 2012 by Satoru Sugihara, with a mission ‘We make what we want to make with technology. This is our responsibility to society. ‘. He is currently a faculty member at Southern California Institute of Architecture teaching scripting for computational design. He has over 5 years of experience as a computational designer at Morphosis Architects as well as over 16 years of experience in computer programming. He holds Master’s degree in Architecture from University of California Los Angeles and another Master’s degree in Computer Science from Tokyo Institute of Technology.ATLV has been focussing in challenging area of design through new technologies and design process. Innovations in technology help in solving design problems in new perspectives and also broaden the design possibilities.

ATLV/Education is a very direct tutorial website and gives out clear step-by-step instructions for beginners . Diagrams and topics are displayed coherently, started from very fundamental and basic topics to a much complex processes , complete with file examples and pdf .

Screen Shot 2014-12-04 at 5.37.49 AM[Image: A screen grab from the website of ATLV/Education].

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[Image: A screen grab from the homepage of ATLV.org].

Read more at the ATLV/Education  and do check out the ATLV.org website for more information about the firm.

Source : ATLV.org

System Development: Spidron

First developed in 1979 by Dániel Erdély the Spidron is created by recursively dividing a 2-dimensional hexagon into triangles, forming a pattern that consists of one equilateral followed by one isosceles triangle. The resulting form is of six Spidron legs that, when folded along their edges, deform to create a 3-dimensional Spidron.

Spidron Nest

Spidron System_Parametrics_Lorna Jackson

Initial investigations into the Spidron system using paper resulted in irregular shapes that could not be predicted, and therefore replicated precisely. Progressing onto using rigid materials allowed the system to be broken down into six components, removing unnecessary triangulated fold lines, and developing latch folded Spidron that is precisely the same as that formed parametrically.

Spidron System_Three SPidrons_Lorna Jackson

This relationship between parametric and physical tests of component based Spidrons in both regular and irregular hexagons, as well as various other equal-sided shapes, has enabled the development of large scale models concluding thus far in a 1:2 scale version being built which will continue to be developed as a pavilion for submission to the Burning Man festival.

In parallel there has been an investigation into the system at a smaller scale allowing for the Spidron nest to be made as one component. In order to achieve the 3-dimensional Spidron form lattice hinges, also known as kerf folds, have been employed. Rigorous testing into the best cutting pattern have resulted in a straight line cutting pattern that allows for bending on multiple axis at once.

Developing this smaller scale system for submission to Buro Happold the intention is to create an arrayed system that is a conglomeration of both regular and irregular spidrons with varying depths and apertures that are able to integrate various display models etc. within.

Thursday 12th December 2013

We just finished our last tutorials of the first term! Congratulations to all the students for the great three months and looking forward to the remaining two terms.

Students completed both briefs (brief01:systems and brief2A:festival) and are starting the case studies of events as part of our last brief (brief2B:realise).

Here are couple pictures of the projects we have seen during the last tutorials. Where do you suggest building the structures over the summer?

Merry Christmas & best wishes for the New Year!!

John Konings's towering gridshell.
John Konings’s towering gridshell.
John Konings's towering gridshell.
John Konings’s towering gridshell.
John Konings's towering gridshell.
John Konings’s towering gridshell.
Andres Jippa's 3D prints, driven by Chaos theory's strange attractors.
Andres Jippa’s 3D prints, driven by Chaos theory’s strange attractors.
Andres Jippa's 3D prints, driven by Chaos theory's strange attractors.
Andres Jippa’s 3D prints, driven by Chaos theory’s strange attractors.
Andres Jippa's 3D prints, driven by Chaos theory's strange attractors.
Andres Jippa’s 3D prints, driven by Chaos theory’s strange attractors.
Andres Jippa's 3D prints, driven by Chaos theory's strange attractors.
Andres Jippa’s 3D prints, driven by Chaos theory’s strange attractors.
Andres Jippa's 3D prints, driven by Chaos theory's strange attractors.
Andres Jippa’s 3D prints, driven by Chaos theory’s strange attractors.
Andres Jippa's 3D prints, driven by Chaos theory's strange attractors. Construction Component.
Andres Jippa’s 3D prints, driven by Chaos theory’s strange attractors. Construction Component.
Henry Turner's Curved Intersecting Plywood Wave Structure
Henry Turner’s Curved Intersecting Plywood Wave Structure
Ieva Ciocyte's Flame Tower made of Intersecting plywood components
Ieva Ciocyte’s Flame Tower made of Intersecting plywood components
Sarah Shuttleworth's Moebius Strips made of Steel Stars.
Sarah Shuttleworth’s Moebius Strips made of Steel Stars.
William Garforth-Bless' Bamboo Hammock Amphitheatre
William Garforth-Bless’ Bamboo Hammock Amphitheatre
William Garforth-Bless' Bamboo Hammock Amphitheatre
William Garforth-Bless’ Bamboo Hammock Amphitheatre

31/10/2013 Cross-Crit 1

Here are couple images of our first Cross-Crit which concludes Brief01:Systems and marks the beginning of our brief2A:Festival. Students will now chose a festival of their choice and use their design systems to submit a proposal for it. Thank you Michael Clarke, Kester Rattenbury and Andrew Yau for the great crit today!

Natasha Coutts' stacking spiky shingles system
Natasha Coutts’ stacking spiky shingles system
Joe Leach's soft reciprocal system
Joe Leach’s soft reciprocal system
Andrei Jippa's RepRap 3D prints.
Andrei Jippa’s RepRap 3D prints.
John Harding's woven lattice
John Koning’s woven lattice