MIES - Bridge Pavilion

The Zaragoza Bridge Pavilion by Zaha Hadid is organized around 4 main elements, or “pods”, that perform both as structural elements and as spatial enclosures. The Bridge Pavilion design is a result of detailed examination and research into the potential of a diamond shaped section which offers both structural and programming properties. As in the case of space-frame structures, a diamond section can efficiently distribute forces along a surface, whilst underneath the floor plate the resulting triangular pocket space can be used to run services.

The diamond section has also been extruded along a slightly curved path. The extrusion of this rhombus section along different paths has generated the four separate ‘pods’ of the Bridge Pavilion. The stacking and interlocking of these truss elements (the ‘pods’) satisfies two specific criteria: optimizing the structural system, and allowing for a natural differentiation of the interiors - where each ‘pod’ corresponds to a specific exhibition space. By intersecting the trusses/pods, they brace each other and loads are distributed across the four trusses instead of a singular main element, resulting in a reduction in size of load- bearing members.

Located above the main flood level, the Bridge Pavilion connects with each river bank via a smooth inclined terrain. Each pod is located on the same level, except one which is 1.5 meters above this main level and intersects with its adjacent pods. All but one of the pods include an upper floor, which hangs from the diamond section structure and provides views of the lower level.

MIES - Penthouse

The Penthouse Defined by Louvers and a Terminating Curve

Recently, I've posted some on parametric curves and their extrapolation in contemporary architectural practice.  This, I find, is a good example of their usage.  This penthouse, designed by Bentham Crouwel Architekten in The Netherlands, shows panache in detailing the rounded features toward the termination of the building.  The curves are used in places explicitly and implicitly (more about that in another post), but of vital concern is how the curves translate to exteriors and interiors.  First, a couple images to get some background:

The Opposing Side of the Penthouse; View Overlooking the Maas River

Aerial View of the Penthouse

The Penthouse floats a good three metres above the Las Palmas building, held aloft on thin steel columns. This two-story office volume is rounded off vertically at the head ends, in deliberate contrast with all other buildings on Wilhelmina Pier. The maritime mood projected by the Penthouse refers to its waterside location and to the history, inextricably interwoven with shipping, of the Kop van Zuid project on the south bank of the Maas River.

Explicit Curves Used to Demarcate the End Wall of the Penthouse

The Penthouse stands on twenty-three columns, with special attention to the feet to enhance the floating effect. The main core of Las Palmas stitches together old and new elements, lending stability to the whole. A large goods lift in this core ascends to a roof terrace, lying below the volume of the roof structure, and containing fourteen parking places. Above, the white volume opens up to the north and south with story-height butt-jointed glazing. Both directions offer an unimpeded view of the river, the Rijnhaven harbor basin and the shorelines of the city beyond. The Penthouse acts as an eye-catcher anchoring the refurbished Las Palmas in the skyline of Wilhelmina Pier.  It uses curves both explicitly and implicitly, to attract attention and to provide warmth and habitability.

Implicit Curves Used to Guide Louvers to Harness Light

AGS - Parametric Curves

One of the More Famous Curves, the Butterfly Curve, and its Parametric Function

Parametric curves are the representation of curvilinear extrapolations over the course of an interval.  Thus, the coordinates of a point "p" of a parametric curve "c" are expressed as functions of parameter, "t."  This means that a spatial curve c is represented by c(t) = ( x(t), y(t), z(t) ), where x(t), y(t) and z(t) are known as the coordinate functions.  Thereby, every parameter t is mapped to a curve point p(t).  Often, it is helpful to think about t as time, although t may not be time.  An interval t mapped over over the curve of three-dimensional space would look thusly:

Spatial Three-Dimensional Curve Showing Length of an Interval for Parameterization of t

By restricting the interval of a parameter, t, one obtains a subset of curve c (sometimes referred to as the curve segment).  A parametric curve defined by polynomial functions is called a polynomial curve.  The highest order of the parameter t in any of the three coordinate functions is called the degree of the polynomial curve.  So for a third order equation (or cubic root function), the degree of a polynomial curve would be three.  For a second order equation, the degree would be two and so on.

A Circle Represented as a Parametric Curve

Oftentimes, typical functions such as the equation of a circle, are represented as parametrics.  In the above photo, one can see that c(t) or p is represented by the variables x and y, which are dependent on the origin,  m(t), and the radius, n(t).  Rather than a unit circle, where cosines and sines determine a constant radius of 1 toward the perimeter, this circle is far more useful when delineating the most common curve used in architecture: namely, the circular curve.

AGS - Curves

Today I'm going to start blogging a little about something that continually fascinates me: architectural geometry and surfaces (which I'll label as AGS from now on).  Whether it's surface, texture, contours, shape or volume, geometry is such a vital part of architecture that without it designing, as an artform, ceases to exist.  As a structural engineer, I understand that stiffness in structures doesn't exactly correspond to stiffness in materials, and that by extrapolating the correct structure based on overall shape - and not volume - one introduces deviatoric stresses throughout a whole integrated structure.  These principal stresses, primarily, are the foundation of all dynamically-responsive structures.  Therefore, the mastering of geometry is crucial to an artist/architect/engineer's overall education.

To begin with, I'm going to elucidate a little on something sexy: curves.  First of all, there are curve tangents, curvature of curves, inflection points of curves, and so on.  Curves are generally many things too, but for the purposes here, they are only thing explicitly: they are profiles used to generate surfaces.

The discussion of curves leads to the study of surfaces in a natural way.  Mathematically, this happens because of two analytical approaches to curves: (1) tangent planes and (2) surface normals.  A tangent plane is a manifold that facilitates the generalization of vectors (such as a surface normal) from affine spaces to general manifolds (surfaces).

  The Deep Blue Vector is a Surface Normal, the Transparent Blue Rectangle a Tangent Plane

A curve can also be more easily considered as a connected one-dimensional series of points.  As can be seen in a hyperbola, these point series can consist of different parts or the branches of a curve.  All of these one-dimensional curves are called planar curves, naturally, in contrast to spatial curves (such as helixes).

Next I will write about parametric curves, which are the curves used most for kinematic equations and structural dynamic analysis.   (Note: curves are both artistic and mathematical.  I will explain what I believe to be the overlap later in another post.)