Mass and Void

Space can be either full or empty

A void is an empty volume, while a mass is a filled volume.  

Voids and masses can be referred to as:

Mass/Void Interaction

When a void occurs, the space between two solids it is passive.

When it occurs as a removal, or subtraction, from a solid, the void is active.

Penetration of space can be regarded as empty space entering into a solid form.

Penetration is a deep incursion of space into a mass, where as concavity refers to slight indentations.

If space completely passes through a solid form it is a perforation.

Isamu Noguchi

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Subtraction and Addition

Carving is an example of a subtractive process where sections of simple mass are removed (or voided) to create a more complex mass.

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Subtractions should create distinctively shaped voids that interact visually with the mass to create strong positive/negative relationships and open up the space of a form.

Additive processes build form up and out into space.

Addition suggests growth.

Surface and Volume

A closed surface defines volume

The surface may be curved, like an egg, faceted, like a cube or a combination of both, like a cylinder.

Volumes enclosed by continuously curving surfaces are integrate volumes.

Integrate volumes have no sharp corners, straight line, or flat planes

All organic volumes are integrate volumes

Volumes  enclosed by faceted surfaces are called polyhedrons

Enclosing Volume

A cube constructed from six square plates of steel will read as a solid even though it is an empty volume.

The most common method of fabricating closed volumes is pattern construction

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Geometric Volumes



Polyhedrons are geometric solids whose faces are polygons. The prefix poly means "many" and hedron means "face".  A face (or facet) of a solid is by definition a flat plane. A polygon is a plane figure demarcated by straight sides. As part of a polyhedron the sides of the polygon become the edges of the polyhedron and the corners of the polygons meet to become vertices of a polyhedron.


Types of Polygons


Geometers name polygons according to the number of sides enclosing them. Triangles have three sides and quadrilaterals have four sides. Beginning with five sided figures polygons are named according to the Greek name for the number of sides: pentagons, hexagons, heptagons and octagon, for example, feature 5, 6,7 and 8 sides respectively.  

The sides and corner angles of regular polygons are all equal. Most polygons tend to be convex, meaning that all of their angles point outward, but should a corner turn inward, the polygon is labeled as concave.  


Types of Polyhedrons


The simplest and most common polyhedrons used in constructive modeling are prisms, pyramids and the truncation, or frustum, of a pyramid.




These three categories of polyhedrons constitute the vast majority of constructive modeling and are the subject of this tutorial. A particularly elegant, but less common, family of polyhedrons is the spherical polyhedrons. These tend to have many, many faces deployed around the center as if on a ball.

Spherical Polyhedra

Polyhedrons whose vertices fit evenly into a sphere

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Form in Nature

Biomorphic Form

Jean Arp

Rather than depict a particular form of life, Arp sought to express a vital force to which he believed inhabits all life.

In order to do so, he shaped marble into a sleek, continuous surface that appeared to have expanded or contracted from an inner force.

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Generated Form


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Wendell Castle

- Wendell Castle, 1996

John Cederquist