Distributional Logic

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Distributional LogicThe Material Economy of Turtle Shells

Lawrence Lynch

Distributional LogicThe Material Economy of Turtle Shells

ContentsOverall FormSegmentationGrowthDigital Modeling

Initial ConceptVoronoi and GrowthLandscape GrowthUnit DevelopmentPlan DevelopmentRoof DevelopmentDigital FabricationFinal DesignReferences

Material Economy of Turtle Shells

House-Scapes: Ecologies of Dwelling

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Material Economy of Turtle Shells

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The most efficient shape in terms of material economy is the sphere. It provides the minimum surface area for the maximum internal volume. However, like many other natural forms turtle shells are not quite spheres, but are more “egg-shaped” as this does little to compromise their efficiency. This shape therefore maximises the internal volume for the turtle to grow whilst minimising the material investment. The dual curvature enhances the strength for a given thickness and thus allows the turtle to have an incredibly thin yet strong shell. The shell is so strong in fact that very little internal structure is required.

Overall Form

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“double curvature gives greater strength and stiffness for a given investment of mate-rial. Maximization of internal volume for a given surface compounds the material economy-for that nothing beats a sphere, and a slight egg-shapedness (spheroidicity) doesn’t make

things much worse…” (Vogel 2003:440-441)

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Segmentation

After studying a wide range of turtle shells I realised that all were segmented in exactly the same way. They all shared the same pattern of three central hexagons, surrounded by ten pentagons. The reason for this is that a regular hexagon is the most ef-ficient interlocking shape as it has the lowest perimeter for the area it encloses. The turtle therefore wants as many of its seg-ments to be hexagons so as to minimise the joints between segments which weaken the shell. However perfect hexagons do not interlock on a curved surface and so the shapes vary. The pentagons are required to create a straight edge around the shell.

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I constructed the following nets as an exploration into how varying segment shape affects curvature.

REGULAR

IRREGULAR

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Growth

A turtle’s shell grows from a number of points. The centres of growth are in the middle of each segment. As the segments grow they push up against one another forcing themselves into geometric shapes. If each segment grows at an equal rate the joins will be equidistant from the two nearest centres of growth. (They will be their perpendicular bisectors.) The pattern generated is therefore solely dependent on the distribution of the centres of growth. This principle is described by the voronoi algorithm. It is also found in many other natural systems such as bubbles and scales.

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I made the following model to demonstrate this idea of certain geometries materialising due to forms pushing up against one another. It was constructed by filling balloons with plaster and then tightly packing them into a box. Once set the balloons were removed and the pieces reassembled.

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Digital Modeling

To create a 3D digital model I first traced an image of an existing turtle shell, which I then simplified by straightening some of the edges. Finally I applied a gradient to each segment which, when entered into 3ds Max could be reinterpreted as the con-tours of the shell.

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These models were an experiment into the effects of modifying the curvature of the shell and the contours of the segments.

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House-Scapes: Ecologies of Dwelling

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Initial Concept

This is a conceptual image representing my initial thoughts on how to use voronoi as a tool for dividing up space in plan. It demonstrates the idea of overlapping different voronoi patterns to create an organic plan which fills the entire site. Each layer represents a different function corresponding to its scale.

SITE BUILDINGS LANDSCAPELAYOUT

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The voronoi algorithm describes how natural cellular systems grow and push against each other, saving material by sharing walls. The pattern is determined by the array of points. The lines themselves are always equidistant from the two nearest points as they are the perpendicular bisectors of the lines linking the points.

Voronoi and Growth

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Beginning from specific points, each plant grows, allowing people to walk between them. As the plants spread, they push up against one another, causing the pathways to become more defined. Therefore both the natural spreading of plants and move-ment of people naturally generates a voronoi landscape. Simultaneously the lines which generate the voronoi pattern (blue lines) can be utilised as an irrigation system as they link the plants via their centres of growth.

Landscape Growth

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Unit Development

By arraying the points regularly a hexagonal voronoi plan is created, thus minimising the perimeter for a given area. A variety of room sizes is created by imagining two layers of hexagons (one for units and one for rooms) and then offsetting one layer. The perimeter of each housing unit is the generator (blue lines) which creates the smaller hexagons (rooms). The diagram on the right shows the three different types of housing unit.

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This system of units allows for a degree of flexibility as different units can be joined or separated if required. For example if a couple were to move into a single unit, eventually they may have children and so require extra space, but then need less once they have left.

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Plan Development

I first established the layout of where buildings would go by creating a voronoi pattern whose generator points marked the surrounding buildings. This meant that my buildings would be equidistant from their surroundings. I then created a regular hexagonal voronoi pattern along these lines, the remainder being a random voronoi. I then simply selected regular hexagons to be housing units and the random voronoi pattern as the landscaping.

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As turtle shells achieve material economy by employing a dual curvature which maximises their strength, I too wanted to utilise this as a means of reducing the thickness of my roofs. My solution came when I discovered that I could use sand to generate my voronoi based plan. The form of each roof is based on the same principle as catenary arches. After the sand has fallen through the holes, the remaining form is left because it is able to support itself against gravity. If this form is then inverted, it creates the most natural self-supporting dome which fits a hexagonal plan and accounts for the diameter of the hole in the centre. This model was created by mapping the points which generate the voronoi pattern used to create my plan and allowing the sand to fall through the holes.

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Roof Development

In order to work out the precise shape of my roofs I created a 1:50 model which cuts through the form, allowing me to view the two sections necessary to describe it (one between opposite corners and the other between the midpoints of opposite sides).

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Because these two sections are the maxima and minima of the domes curvature I was then able to calculate the form in-be-tween. Having constructed the wire frame I was then able to convert it into a surface with a given thickness in 3DS Max.

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Digital Fabrication

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Having made a digital 3D mode l then took horizontal sections throughout it, which could then be cut out and stacked on top of one another. The model produced is therefore both an accurate reconstruction of the form created by the sand and a 1:50 scale model of my roof (the hole in the centre to let light into a courtyard below).

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Final DesignSection: Housing Units

33Unit Elevation 1:50

Elevation: Housing Units

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Plan: Housing Units

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Plan: Complete Scheme

36 Axonometric 1:200

Axonometric

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Perspectives

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References Biology of sea turtles Vol. 2. / edited by Peter L. Lutz, John A. Musick, Jean-ette Wyneken

Turtles of the world, Schildkroten der Welt . Vol. 1, Africa, Europe and West-ern Asia , Band 1, Afrika, Europa und Westasien / Holger Vetter

http://photo-dict.faqs.org/phrase/691/soap-bubble.html

http://www.tradenote.net/trade29-products/Fruit-87/

http://sjconnor.wordpress.com/2007/01/

http://www.pollsb.com/polls/p9487-cutest_mobile_home

http://www.asknature.org/strategy/0b182d72f993909bbcef484b5f3a5ff9

http://pterosaurheresies.wordpress.com/category/triassic

http://www.jonathans.me.uk/indexcgi?section=main&pic=tortoise_shell

http://www.flickr.com/photos/judyjowers/5164117422/

http://stoneplus.cst.cmich.edu/hematite.htm

http://www.eco-divers.com/galleries/v/caryyanny/02.jpg.html

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