Category Archives: Program

Programmatic Considerations

-The diagram [above] depicts circulation and relationships linking various programmatic entities; [public] living space, kitchen/dining, patio, [private] bathroom, bedroom. Each programmatic typology becomes an attractor, where particles emitted from a point of entry gravitate towards.

-Wikipedia article on Attractor:

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.
A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor. The trajectory may be periodic or chaotic or of any other type.

A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamic system specify its behavior over any given short period of time. To determine the system’s behavior for a longer period, it is necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers.
Dynamical systems in the physical world tend to be dissipative: if it were not for some driving force, the motion would cease. (Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to combine to kill out initial transients and settle the system into its typical behavior. This one part of the phase space of the dynamical system corresponding to the typical behavior is the attracting section or attractee.
Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.
For example, the damped pendulum has two invariant points: the point x0 of minimum height and the point x1 of maximum height. The point x0 is also a limit set, as trajectories converge to it; the point x1 is not a limit set. Because of the dissipation, the point x0 is also an attractor. If there were no dissipation, x0 would not be an attractor.

Fixed point:
A fixed point is a point of a function that does not change under some transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under the whole series of transformation. In general there would not be such a point, but there may be one. The final state that a dynamical system evolves towards, such as the final states of a falling pebble, a damped pendulum, or the water in a glass corresponds to a fixed point of the evolution function, and will occur at the attractor, but the two concepts are not equivalent. A marble rolling around in a basin may have a fixed point in phase space even if it doesn’t in physical space. Once it has lost momentum and settled into the bottom of the bowl it then has a fixed point in physical space, phase space, and is located at the attractor for that system.