004 LectureDM2_2011

TRANSFORMATION DM2

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Lecture 4

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TRANSFORMATIONS This series of lectures will focus on the many types of transformations that are used in constructing the proper geometry for specific architectural applications. + Planar Transformations will be explored. + Tilings - Regular and Semi Regular tessellations will be examined and how they transition into building components. + Motion, Sweeping, and Shape Evolution + Skinning

PLANAR TRANSFORMATIONS Basic Transformations are congruence transformations (translation, rotation, reflection) which preserve all lengths and angles occurring on an object.

Slightly more general are similarity transformations, which still preserve angles but multiply all distances by the same factor.

Shear transformations preserve the area of the transformed objects.

Scaling transformations provide even more freedom for shape modification (but it’s still linear)

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M.C.ESCHER Escher illustrates a perfect way various types of transformations are created. We study his work to learn about tiling’s which can be used to create sophisticated facades or surface tilings.

He used his knowledge of the properties of planar transformations to generate nontrivial tessellations.

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SAN FRANCISCO RAILWAY STATION Located on the BART transit line.

Ceramic hexagon tiled wall.

Form features a hexagon shape blended into a spherical form.

Completed in 1973

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SPANISH PAVILION BY FOA FOA - FOREIGN OFFICE ARCHITECTS. 2005 World Exposition Aichi, Japan FOA created this seemingly irregular hexagonal tiling by simply shifting some of the internal vertices of groups of 8 tiles. By coloring the tiles in various colors, this further breaks up the under laying regular pattern.

Constructed of slip cast glazed ceramic.

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FEDERATION SQUARE - LAB ARCHITECTS Only three cladding materials; sandstone, zinc (perforated and solid) and glass have been used within a modular basis established by the triangular pinwheel grid This fractal incremental system uses a single triangles, whose proportion is maintained across the single tile shape, the panel composed of five tiles and the construction module of the mega-panel composed of five panels.

The unique quality of the pinwheel grid lies in the possibility of surface figuration and framing shapes to be independent from the grid’s smallest component unit, the triangle.

This grid allows the building facades to be treated in a continuous changing and visually dynamic way, instead of being traditionally composed as a regularly repeating flat surface.

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TRANSLATION, ROTATION AND REFLECTION OF A PLANE A transformation is a general term for four specific ways to manipulate the shape of a point, a line, or shape. The original shape of the object is called the pre-image and the final shape and position of the object is the image under the transformation.

Types of transformations in math • Translation • Reflection • Rotation Translation – a translation is defined by a translation vector t, which specifies the direction and the magnitude of the translation.

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TRANSLATION, ROTATION AND REFLECTION OF A PLANE A simple example that is not as trivial as the square and the hexagon is the chair family of polynomial. Chairs are not symmetric as the square and the regular hexagon are, and still they manage to tile the plane in the same simple way, just by translating them in two different directions.

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TRANSLATION, ROTATION AND REFLECTION OF A PLANE Rotation – We define a rotation by a fixed point c, the center of rotation, and the rotational angle p The order in which you perform transformations can affect the final result. Consider, for example, translating and rotating an image. If you perform the transformations in this order, you end up with a rotated model translated, for example, down the Xaxis, as shown

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TILINGS

We have now looked at how planar congruence’s transformations as effective tools for positioning objects in the plane.

They are very useful in creating regular tessellations and tiling’s.

The art of designing tiling’s and patterns has a long history and is therefore well developed

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REGULAR AND SEMI-REGULAR TESSELLATIONS A tessellation is a way of filling the entire plane with congruent shapes without overlaps or gaps.

The term tiling is sometimes used to describe a special tessellation of the plane using planar polygons.

There are 14 types classes of convex pentagonal tilings known and three classes of tilings with irregular hexagonal tiles.

Quadrangular Tessellation

There exist only three regular tessellations due to the fact that the vertex angle of the tiles must be a divisor of 360 degrees, Therefore, we only have regular tessellations with regular triangles, squares and hexagons.

Triangular tessellation

Irregular Tilings

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SEMI REGULAR TESSELATIONS

Tessellations that use two or more different regular polygons, we add the rule that every vertex must have exactly the same configuration.

This means that every vertex there has to be the same number and the same sequence of congruent regular polygons.

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EXAMPLES IN ARCHITECTURAL TILINGS SKYLER TIBITS - SJET a built project which demonstrates a system of flat panel tessellation derived from complex surfaces to enable ease in constructability. Each panel’s uniqueness is afforded by the efficiency of digital fabrication while coded parametric relationships allow an emergent structural efficiency.

Recently the development of planar quadrilateral meshes has become a strong interest in the architectural community due to their potential ease for constructing complex surfaces. The project responds to this problem and proposes a method for flat panelization of free form surfaces which provides large scale, efficient and economic construction from flat sheet material.

Tesselion was constructed using: -189 panels (7 rows of 27 panels)

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EXAMPLES IN ARCHITECTURAL TILINGS DECOI - HYPOSURFACE It was proposed as a dynamically reconfigurable surface capable of real-time responsiveness to events in the theatre, such that movement or sound can create actual deformation of the architectural surface. Effectively Aegis is a dynamically reconfigurable screen where the calculating speed of the computer is deployed to a matrix of actuators ( 896 pneumatic pistons ) that drive a ‘deep’ elastic surface. The implicit suggestion is one of a physically responsive architecture where the building develops an electronic central nervous system, the surfaces responding instinctively to any digital input (sound, movement, Internet, etc).

-CNC routed 1/16” white aluminum -1 week of assembly

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