When its diagonals are vertical and horizontal, however, the square exists in a balanced state of equilibrium. Surface first refers to any figure having only two dimensions, such as a flat plane. The term, however, can also allude to a curved two-dimensional locus of points defining the boundary of a three-dimensional solid. There is a special class of the latter that can be generated from the geometric family of curves and straight lines. Depending on the curve, a cylindrical surface may be circular, elliptic, or parabolic.
Because of its straight line geometry, a cylindrical surface can be regarded as being either a translational or a ruled surface.
Because of its straight line geometry, a ruled surface is generally easier to form and construct than a rotational or translational surface. Parabolas are plane curves generated by a moving point that remains equidistant from a fixed line and a fixed point not on the line. Hyperbolas are plane curves formed by the intersection of a right circular cone with a plane that cuts both halves of the cone. It can thus be considered to be both a translational and a ruled surface. If the edges of a saddle surface are not supported, beam behavior may also be present.
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The type of structural system that can best take advantage of this doubly curved geometry is the shell structure—a thin, plate structure, usually of reinforced concrete, which is shaped to transmit applied forces by compressive, tensile, and shear stresses acting in the plane of the curved surface.
The structure consists of a radial arrangement of eight hyperbolic paraboloid segments. Like shell structures, gridshells rely on their double curvature geometry for their strength but are constructed of a grid or lattice, usually of wood or steel. Gridshells are capable of being formed into irregular curved surfaces, relying on computer modeling programs for their structural analysis and optimization and sometimes their fabrication and assembly as well. See also pages — for a related discussion of diagrids.
Symmetrical curved surfaces, such as domes and barrel vaults, are inherently stable. Asymmetrical curved surfaces, on the other hand, can be more vigorous and expressive in nature. Their shapes change dramatically as we view them from different perspectives. It is for this reason that these are beautiful forms, the most beautiful forms.
Circles generate spheres and cylinders; triangles generate cones and pyramids; squares generate cubes. Sphere A solid generated by the revolution of a semicircle about its diameter, whose surface is at all points equidistant from the center. A sphere is a centralized and highly concentrated form. Like the circle from which it is generated, it is self-centering and normally stable in its environment. It can be inclined toward a rotary motion when placed on a sloping plane. From any viewpoint, it retains its circular shape.
Cylinder A solid generated by the revolution of a rectangle about one of its sides. A cylinder is centralized about the axis passing through the centers of its two circular faces. Along this axis, it can be easily extended.
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The cylinder is stable if it rests on one of its circular faces; it becomes unstable when its central axis is inclined from the vertical. Like the cylinder, the cone is a highly stable form when resting on its circular base, and unstable when its vertical axis is tipped or overturned.
It can also rest on its apex in a precarious state of balance. Pyramid A polyhedron having a polygonal base and triangular faces meeting at a common point or vertex. The pyramid has properties similar to those of the cone.
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Because all of its surfaces are flat planes, however, the pyramid can rest in a stable manner on any of its faces. While the cone is a soft form, the pyramid is relatively hard and angular. Cube A prismatic solid bounded by six equal square sides, the angle between any two adjacent faces being a right angle. Because of the equality of its dimensions, the cube is a static form that lacks apparent movement or direction. It is a stable form except when it stands on one of its edges or corners. Even though its angular profile is affected by our point of view, the cube remains a highly recognizable form.
They are generally stable in nature and symmetrical about one or more axes. The sphere, cylinder, cone, cube, and pyramid are prime examples of regular forms. Forms can retain their regularity even when transformed dimensionally or by the addition or subtraction of elements. From our experiences with similar forms, we can construct a mental model of the original whole even when a fragment is missing or another part is added.
Irregular forms are those whose parts are dissimilar in nature and related to one another in an inconsistent manner. They are generally asymmetrical and more dynamic than regular forms. They can be regular forms from which irregular elements have been subtracted or result from an irregular composition of regular forms.
Since we deal with both solid masses and spatial voids in architecture, regular forms can be contained within irregular forms. In a similar manner, irregular forms can be enclosed by regular forms. Note how the diagrid pattern becomes more dense in areas where moment stresses are higher. Dimensional Transformation A form can be transformed by altering one or more of its dimensions and still retain its identity as a member of a family of forms.
A cube, for example, can be transformed into similar prismatic forms through discrete changes in height, width, or length. It can be compressed into a planar form or be stretched out into a linear one. Subtractive Transformation A form can be transformed by subtracting a portion of its volume. Depending on the extent of the subtractive process, the form can retain its initial identity or be transformed into a form of another family. For example, a cube can retain its identity as a cube even though a portion of it is removed, or be transformed into a series of regular polyhedrons that begin to approximate a sphere.
Additive Transformation A form can be transformed by the addition of elements to its volume. The nature of the additive process and the number and relative sizes of the elements being attached determine whether the identity of the initial form is altered or retained. A pyramid can be transformed by altering the dimensions of the base, modifying the height of the apex, or tilting the normally vertical axis. A cube can be transformed into similar prismatic forms by shortening or elongating its height, width, or depth.
Carlo, Project, 17th century, Francesco Borromini St. If any of the primary solids is partially hidden from our view, we tend to complete its form and visualize it as if it were whole because the mind fills in what the eyes do not see. In a similar manner, when regular forms have fragments missing from their volumes, they retain their formal identities if we perceive them as incomplete wholes.
We refer to these mutilated forms as subtractive forms. Because they are easily recognizable, simple geometric forms, such as the primary solids, adapt readily to subtractive treatment.
These forms will retain their formal identities if portions of their volumes are removed without deteriorating their edges, corners, and overall profile. Ambiguity regarding the original identity of a form will result if the portion removed from its volume erodes its edges and drastically alters its profile. In this series of figures, at what point does the square shape with a corner portion removed become an L- shaped configuration of two rectangular planes?
Khasneh al Faroun, Petra, 1st century A. The basic possibilities for grouping two or more forms are by: Spatial Tension This type of relationship relies on the close proximity of the forms or their sharing of a common visual trait, such as shape, color, or material. Edge-to-Edge Contact In this type of relationship, the forms share a common edge and can pivot about that edge.
Face-to-Face Contact This type of relationship requires that the two forms have corresponding planar surfaces which are parallel to each other. The forms need not share any visual traits. For us to perceive additive groupings as unified compositions of A number of secondary forms clustered about a form—as figures in our visual field—the combining elements must dominant, central parent-form be related to one another in a coherent manner.
These diagrams categorize additive forms according to the nature of the relationships that exist among the component forms as well as their overall configurations. This outline of formal organizations should be compared with a parallel discussion of spatial organizations in Chapter 4. Linear Form A series of forms arranged sequentially in a row Radial Form A composition of linear forms extending outward from a central form in a radial manner Clustered Form A collection of forms grouped together by proximity or the sharing of a common visual trait Grid Form A set of modular forms related and Lingaraja Temple, Bhubaneshwar, India, c.
Because of their inherent centrality, these forms share the self-centering properties of the point and circle. They are ideal as freestanding structures isolated within their context, dominating a point in space, or occupying the center of a defined field.
They can embody sacred or honorific places, or commemorate significant persons or events. In the latter case, the series of forms may be either repetitive or dissimilar in nature and organized by a separate and distinct element such as a wall or path.