Polyhedra

fig. 9

Beyond three points, the next stage is about the volumes. Among these will be considered the  simpler and more regular: the polyhedra of Archimedes. In the part of this article we will not retain than the first three.

The simplest is the tetrahedron. It includes four vertices. His rendered is shown in Figure 9.

 

 

 

 

fig. 10

The octahedron has six vertices. It is composed of eight triangles equilateral all identical. His rendering is presented figure 10.

 

 

 

 

 

fig. 11

The hexahedron, that is to say the cube, includes eight vertices. (Figure 11)

The cube can be broken down into two tetrahedra. In Figure 11a the group of four colors of left is the first tetrahedron and the right group the second. It can also be broken down into six squares corresponding to each of his faces. (Figure 1 2)

 

 

fig. 12

With examples such as the octahedron or cube and their multiple possible decompositions we understand the profound richness of a geometric approach to color in a three-dimensional space. The possibilities of composition and of arrangement are endless while keeping the rigorous coherence of the given theme.

 The objects we saw in this first part constitute in effect, in our point of view,  color themes in that they are independent of each other and intrinsically coherent.

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A point

What can we do with a color point in a 3D space?

As such it corresponds to a unique color. How to do something with it ?

One point, if he does not have dimension, nevertheless has the quality of being an attractor, a vanishing point, a center of gravity, a destination.

It seems somewhat interesting to consider it as the center of a sphere and consider the colors located on the surface of this one. More the chosen radius will be smaller and more these colors will be close to its center. 

1a & 1b

For example, for red color we get the colors of Figure 1 a. All these colors are on the envelope of the sphere whose center is the red of the horizontal bar below. They are therefore located at a same distance from this color.

This approach illustrates the fact that with a geometric approach of colors we consider that relations between they are based on distances separating them or bringing them together according to whether they are large or small. The random arrangement of colors of Figure 1,b composed with fifty color squares, seems homogeneous because the colors used have a single common feature: to be at the same distance from a particular red.

The richness of the spatial approach of three-dimensional colors manifests now. Instead of thinking in terms of monochromatic or analog chord with a color wheel and then to look for variations of value and saturation that appear to agree, the use of Euclidian distance makes it possible to obtain directly colors that are related in their three dimensions .

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Geometrical approach of the color

At the beginning of the 20th century, the hypothesis of the existence of three receivers of colors in the visual apparatus was validated. The colors could then be spotted by three numbers and be represented as points in the geometric space. The coordinate system used is a orthonormal whose three axes match the colors red, green and blue primaries defined on the basis of concepts physical and photometry.

So far the needs of the industry and trade have required the setting up of tools and methods of measuring differences between colors perceived at the psychometric level. As an example a manufacturer of car that orders several tons of paint of a given color wishes to be able to check if the product received matches well to the requested characteristics. And when the economic and commercial stakes get bigger the human vision and its subjectivity quickly become insufficient. It then becomes necessary to be able to measure quantitatively the differences in perceived colors.  

Briefly the tools and color measurement methods perceived allow to do two things:

– determine the characteristics of a color regardless of the light with which one looks at it. This possibility has a very practical interest because it allows the appreciation of characteristics of a color in different contexts and places when a standard lighting source is used.

– locate this color in an absolute space of perceptual rendering. The term perceptual means that we refers to a space that is the transformation based on the appreciation of a sample statistically representative of human observers, of colors differences in a physical colorimetric and photometric space upstream of the human visual apparatus. The interest of this possibility is that we can deduce the location of the color in different spaces corresponding to materials that have been calibrated from absolute space. The expression perceptual rendering means that differences in colors measured in this space correspond to the differences perceived by the human vision.

This space, or rather these spaces should we say because many have been developed in response to different needs, are three-dimensional just like the RGB space from which they come. We will refer in this article to CIELAB space or CIE L * a * b * 1 976 which is used for surface  colors. The three axes of this system are:

– L * for lightness. Black is at origin 0 and white is worth 1 00.

– a * for the yellow blue axis centered around the origin. Blue corresponds to the negative values and yellow with positive values.

– b * for the centered red green axis around the origin. Negative values for the green and the positive values for the red.

Because the absolute spaces of perceptual rendering comply with our vision and are three-dimensional they are radically different from the common representation of the color scheme based on a two-dimensional object, the color wheel.

As an architect, I’m not indifferent to the representation of something so beautiful that color in a three-dimensional  space comparable to the architectural space. This article presents examples simple application of Euclidean geometry in CIELAB space. Simple application because the aesthetic prejudice that prevails in this approach is that what is simple is beautiful.

The purpose of this approach is to contribute to the enrichment of a color vocabulary limited since a long time by a practice based on the color wheel and the hue contrast.  In a second part will be developed the important notion of variation on a theme of colors.

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