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THE ART OF THREE-DIMENSIONAL DESIGN

HOW TO CREATE SPACE FIGURES

Dover Publications, Inc.

INTRODUCTION

Any object or geometric magnitude that has length, breadth, and thickness is three-dimensional. The three-dimensional or space object has the property of envelopment; that is, it may be incased in a rectangular prism whose faces will be tangent to the extreme limits of the space object.

There are many different types of three-dimensional objects. The differences for the most part are explained by the geometric properties that characterize each type.

We must realize that when we talk about geometric properties we talk about imaginary conditions. The objects and surfaces with which we deal in real life are our own approximations of the mathematical concepts. It is well to remember that the mathematical line or surface or object is one thing and its counterpart in reality is quite another thing.

From a practical point of view, the closer we try to make an object or surface or line conform to its mathematical equivalent the more exact we must be in both tooling and measuring. As a result the economic factor enters the picture. Strange as it may seem, the economics of the situation may have a very profound effect on the design. Any designer working in industry knows only too well the intimate relationship between design and economics and the challenging problems that must be overcome for a satisfactory solution. As an example, an industrial designer may be called upon to create a design which can be produced by casting or die-stamping. If the concern does not have a foundry and does not wish to parcel out its jobs, the designer is confronted with a problem different from the one which he would otherwise have if casting were feasible.

The question of how to construct the object and what materials to use is ever present when dealing with three-dimensional design. Essential as this is, this book is primarily concerned with an investigation into the properties of surfaces, their modification, adaptation, and transformation into new forms.

In the course of everyday living we see countless objects with the greatest variety of shapes and functions. As a matter of interest and curiosity, count the number of differently shaped objects that you see in your own home. Do the same when you are out-of-doors. I am quite sure that you will be astonished at the large number of space objects that you see, that you live with, that you use—each one in some mysterious way adding to the sum total of your ever-widening experience.

What are the basic shapes or surfaces? How can they best be described for the profitable use of the designer? How can they be combined and modified to serve the purposes of the designer? This book tries to answer these questions. We must bear in mind, however, that design involves much more than an understanding of the basic geometric properties of space objects.

It includes as well:

1. An awareness of, and knowledge concerning, the properties of various types of material.

2. The availability of tools for manufacture.

3. The availability of skilled craftsmen.

4. Knowledge of municipal and state codes which impose restrictions affecting design.

5. The ability to cultivate a reasonable attitude toward the ideas of others with whom it may be necessary to collaborate on a design project. (This is especially important in production and architectural design.)

6. A questioning outlook not easily satisfied by any ready-made solution.

7. A healthy disrespect for historic design. (This does not mean that we must deny the greatness of earlier forms in order to proceed with the solutions to our own problems.)

8. A healthy curiosity and a desire for continuing inquiry into the essential nature of source material.

Each designer has his own limited capacity for self-expression. At each step of his development he acquires new skills and new emotional outlooks which come from the subtle influences of the works of others and from his own continuing growth. He constantly calls on all of his resources to give fullest expression to his ideas. Only after he is adequately prepared can he bring into play the imagination, the technical skill, the daring that each design must have in order to do justice to its creator.

FUNDAMENTAL DEFINITIONS

1. A line A line is a geometric figure created by a point moving through successive positions.

Straight line The direction remains the same throughout the length of the line.

Broken line Each successive segment changes direction.

Curved line There is a constant change in direction. The direction at any point in the curve is described by the direction of the tangent line at that point.

2. Plane surface A plane surface is a geometric magnitude produced by the motion of a straight line. The moving straight line always passes through a fixed point and always intersects a fixed straight line.

3. Prismatic surface A prismatic surface is a combination of plane surfaces in which the successive planes change direction. It is produced by a straight line moving parallel to a fixed straight line and always touching a fixed broken line.

4. Cylindric surface A cylindric surface is generated by a straight line moving parallel to a fixed straight line and always touching a fixed curved line.

5. Pyramidal surface A pyramidal surface is a combination of plane surfaces having a common point. It is created by a straight line moving through a fixed point and always touching a fixed broken line.

6. Conical surface A conical surface is created by a straight line moving through a fixed point, always touching a fixed curved line.

7. Surface of revolution A surface of revolution is formed by the motion of a line about a straight line axis. Each point of the moving line generates a circle. The moving line may be straight, curved, or a combination of the two.

8. Spherical surface The spherical surface is a surface of revolution. It is formed by the rotating motion of a circle about one of its diameters.

9. Toroidal surface A toroidal surface is a surface of revolution created by the rotating motion of a curve about a straight line lying outside the curve. If the curve is a circle, the surface is called an annular torus.

10. Hypoid surface A hypoid surface is a surface of revolution formed by the motion of one straight line about a second straight line as the axis. (The two straight lines are non-parallel and non-intersecting. They are called skew lines.)

11. Hyperbolic paraboloid surface A hyperbolic paraboloid is a surface generated by a moving line that always touches two skew lines and is always parallel to a fixed plane.

12. Conoid surface The conoid is a surface created by a line moving so that it always touches a fixed straight line and a fixed curve and is always parallel to a fixed plane.

13. Helicoid surface The helicoid is a surface generated by a line moving so that it touches a helical curve and makes a constant angle with a fixed straight line axis of the helix.

14. Convolute surface The convolute is a surface formed by the tangents to a space curve.

15. Serpentine surface The serpentine is a surface formed by the successive positions of a sphere as the center of the sphere moves along a curved line.

THE PLANE SURFACE

The plane surface may be generated in a number of different ways. Two of the most easily understood are:

1. Assume any straight line and any point lying outside the straight line. A second straight line is now introduced and moved through an infinite number of positions always passing through the fixed point and always intersecting the fixed straight line. The sum of all the positions of the moving line is called a plane surface.

2. Assume two parallel straight lines. Introduce a third straight line intersecting the parallel lines. Move the third straight line parallel to its initial position, always touching the two fixed parallel straight lines. The sum of the infinite positions of the moving line is called a plane surface.

The concept of motion as expressed in the definition of the plane surface adds a dynamic quality to this geometric magnitude.

In all of the fundamental statements regarding the various types of surfaces considered in this book, motion of a line, whether straight or curved, plays an important role. In a sense, the successive positions of the moving line generator are like the skeleton of an animal figure. Many examples of contemporary art show the surfaces uncovered, so to speak, and the positions of the generating line exposed. This so-called exposure heightens the feeling of motion because it enables the eye to go from position to position. It is the change of position that induces a sense of motion.

The plane surface considered by itself is two-dimensional. We know that it is composed of an infinite number of straight lines. The plane surface may also be composed of an infinite number of plane curves, open and closed, and all sorts of combinations of straight and curved lines.

In fig. A, all of the lines, heavy as well as light, lie in the same plane.

All that we have to do to add the third dimension and produce a space figure is to take one of the infinite lines, or part of one, out of the plane. The combination of the plane surface and the line out of the plane surface constitutes a space figure having three dimensions.

The perspective drawing in fig. B shows the effect of moving several lines out of the plane.

Keeping in mind the general nature of a plane surface, we see that there are many apparently complex space figures which can be evolved from combinations of plane surfaces and elements taken from the plane surfaces.

The plane surface, and combinations of plane surfaces with elements from them, occur most frequently in man-made objects. The plane surface in nature is a rarity, at least in ordinary visual experience. For example, crystals are very common in nature, and their geometric faces are plane surfaces; but unless the crystals are very large, the faces cannot be seen.

BASIC SURFACES

PLATES

A. Surfaces composed of planes and singly ruled developable surfaces.

B. Non-developable surfaces. (Variation 6, the convolute, is an exception.) Variations ID and 3 show the internally tangent spheres.

PLATE A Types of surfaces.

All of the surfaces shown in Plate A are called ruled surfaces, i.e., surfaces generated by the motion of a straight line.

Figures 1, 2, and 3 are prismatic surfaces; Figures 4, 5, and 6 are pyramidal surfaces.

Each of the six surfaces is composed of planes for the faces. The number of faces is limitless.

Figure 1 is an oblique prism: The lateral edges do not make right angles with the plane of the base.

Figure 4 is an oblique pyramid: The axis—the line joining the apex and the center of the base—is not the same length as the altitude, which is the perpendicular line from the apex to the base.

The development or pattern accompanying each figure represents the surface unrolled. Only ruled surfaces can be developed without tearing or stretching the surface.

Figures 7, 8, and 9 represent cylindric surfaces; Figures 10, 11, and 12 represent conical surfaces.

Figures 7 and 10 are oblique for the same reasons given for Figures 1 and 4, above.

Since both the cylindric and conical figures are ruled surfaces, they may be unrolled to form patterns or developments.

Note: Every developable surface is a ruled surface but not every ruled surface is a developable surface. For example, both the hypoid and hyperbolic paraboloid surfaces are ruled surfaces, but they cannot be developed without stretching, tearing, or somehow deforming the surface, making it impossible to restore the surface to its original shape.

PLATE B Ruled surfaces and non-ruled surfaces.

Figures 1A, 1B, 1C, and 1D represent the same figure, the hypoid or hyperboloid.

The extreme contour shows the characteristic hyperbolic curve. It is a ruled surface, as shown in Figure 1A. In general, the hypoid surface can be generated by the rotation of one straight line (the generator) about a second straight line (the axis of rotation). The common perpendicular distance between the two lines must be constant and, furthermore, the two lines must be skew. (Skew lines are lines which do not determine a plane surface.)

In Figure 1A, successive positions of the moving generator are shown. Every point on the generator moves in the path of a circle and the plane of the circle is perpendicular to the axis (the fixed line).

Figure 1B shows the same hypoid generated by the rotation of a meridian section about the axis. (Any plane containing the axis of a surface of revolution is called a meridian section.)

Figure 1C shows the parallel circles, each of which represents the path of a point, as the hyperbolic line on which the point is situated rotates about the axis.

Figure 1D shows the same hypoid with a series of internally tangent spheres. The envelope of the spheres forms the hyperboloid surface.

There are other ways in which the hypoid surface may be generated. This is true for many of the surfaces that are mentioned, but a complete discussion of the analytic and descriptive properties of the various figures belongs in books on analytic and descriptive geometry and is beyond the scope of this work.

Figure 2 is called a conoid. The conoid is a ruled non-developable surface generated by the motion of a straight line always touching a fixed curved line and a fixed straight line (the fixed lines not in the same plane) and always parallel to a fixed plane surface.

Figure 3 is called a serpentine. The serpentine surface is the envelope of an infinite number of spheres whose centers are on a curve and whose diameters are constant. The serpentine is not a ruled surface, hence not developable.

Figure 4 is a hyperbolic paraboloid. This is a non-developable ruled surface generated by the motion of a straight line always touching two straight skew lines and always parallel to a plane surface.

Figure 5 is the right helicoid. It is a non-developable surface generated by a straight line touching two co-axial helices and at right angles to the axis.

Figure 7 is the oblique helicoid. It is a non-developable surface generated by a straight line touching two co-axial helices and making a constant angle other than a right angle with the axis. (Strictly speaking, the angle that the generator makes with the axis need not be constant. However, this fact does not affect the basic nature of the surface.)

Figure 6 is a convolute. This is a developable surface generated by a straight line moving so that it is always tangent to a space curve. In the illustration, the space curve is a helix (the curve that winds around a cylinder).

THE PRISM AND PRISMATIC SURFACES

PLATES

A. Surfaces whose faces are varied figures of three or more sides.

B. Sheet and wire-combining forms.

C. The dynamic diagonal in the skeletal wire prism.

D. The prismatic module—combinations varying in size and shape.

E. Plane surface angular variants from the square prism (tile design).

F. Further variants including cylindric surfaces (tile design).

G. Variations in over-all shape through rearrangement of prismatic parts, keeping the total volume constant.

Michelangelo once said that all the sculptor had to do was to remove the excess material to reveal the figure that is hidden in the marble block. The concept of removing or subtracting from the whole can be applied to any fundamental solid. It is obvious that the figure or figures which will reveal themselves are infinite in their variety and await only the imagination and technical skill of the creator to bring them to life.

I am using the term "figure" in its most inclusive sense. It embraces the simplest geometric form and the most complex combination of animal and human forms.

I have confined myself in the following illustrations to geometric combinations because:

1. Simple geometric figures are easiest to imagine and construct.

2. They are the easiest to control and describe.

In the illustrations marked D, page 17, the process of addition is used to bring a three-dimensional object into being. We can see that the line which is caused by the moving point "A," as it changes both direction and length, develops a space form. The general character of this form is controlled by the extent to which the moving point "A" is permitted to move.

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Excerpted from THE ART OF THREE-DIMENSIONAL DESIGN by LOUIS WOLCHONOK. Copyright © 1959 Louis Wolchonok. Excerpted by permission of Dover Publications, Inc..
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