# Mathematical order of colors

## Fundamental Colors Indexing System

How many colors exist ? A color-sensitive observer is able to distinguish approximately 100.000 color shades but no human being can distinguish more than 1 million shades. Therefore, it does not make any sense to reckon with more than one million.

Küppers´ Fundamental Colors Indexing System refers to one million color sensations. Each sensation is identified by a 6-digit code, e.g. : " 25 75 50 ". Every code consists of three two-figure groups. Each group refers to one of the three sensory forces, in other words to one Fundamental Color (FuCo). The first group stands for the Fundamental Color Violet-blue (FuCo B), the second group for Fundamental Color Green (FuCo G), and the third one for Fundamental Color Orange-red (FuCo R)

The order of Fundamental Color Codes is not RGB but rather BGR. This is based on the fact that in scientific presentations of the spectrum short-wave rays are always on the left and long-wave rays on the right. We are used to read from left to right. Short-wave rays stimulate FuCo B and long-wave rays FuCo R. So the code order is BGR.

Kueppers divides each sensory force, i.e. each Fundamental Color into 99 tiny bits of energy using them as arithmetical units. He calls these units " sensation quanta ". The minimum is 00 and the maximum 99 producing 100 degrees of sensation. As a result the following Fundamental Colors Codes apply to the 8 Basic Colors:

 FuCo B FuCo G FuCo R Basic Color 00 00 00 = S 99 00 00 = B (V) 00 99 00 = G 00 00 99 = R (O) 99 99 00 = C 99 00 99 = M 00 99 99 = Y 99 99 99 = W Each Fundamental Color Code (FuCoCode) defines a possible color sensation and with it a geometrical spot within the Rhombohedral Color Space which will be explained later. FuCoCode " 25 75 50 " indicates that 25 sensation quanta of FuCo B plus 75 of FuCO G and 50 of FuCo R have been stimulated. It is possible to deduce from this FuCoCode all the other connections and relations concerning that specific shade of color.

From a Fundamental Color Code one can determine the Code of the Basic Colors composing a particular shade. It indicates the parts (quanta) of the eight extreme sense impressions of a color sensation in the sensory mechanism. It is equally possible to determine the mixing formula of an Integrated Mixture (IntMi), i.e. the subsets of individual Basic Colors required for a mixture with opaque dyes.

In addition, the Fundamental Color Code helps to calculate The Four Distinctive Esthetic Features of Colors.

## Basic Colors Indexing System

Equal potentials of all three Fundamental Colors produce the achromatic color sensation White (W). Equal potentials of two Fundamental Colors generate the chromatic color sensations Yellow (Y), Magenta-red (M), and Cyan-blue (C). Potentials of only one Fundamental Color produce values of the chromatic color sensations Violet-blue (B), Green (G), and Orange-red (R). And the difference between the greatest available and the greatest possible (99) potentials represents the value of the achromatic color sensation Black (K). In consequence, the following calculation can be made :

 Fundamental Colors B G R Basic Colors 25 25 25 00 25 25 00 25 00 00 00 00 ------------- 25 75 50 (FuCoCode) = W 25 = Y 25 = G 25 = S 24 ------------- = 99 set parts

The described method makes it possible to deduce from each Fundamental Color Code (FuCoCode) the Basic Colors subsets that are needed for a mixing with opaque color material. We are always speaking of 99 set parts representing the mathematical set 1 or 100% because it always concerns a single opaque ink layer where no Basic Color helps to fill in the differential values. In an Integrated Mixture (IntMi), the first step is mixing and then the mixture is applied as a single opaque ink layer.

A Fundamental Color Code may contain maximum 4 Basic Color subsets which together are composing a single color shade. The above example demonstrates this clearly. If there are four subsets of Basic Color, the achromatic Basic Colors W and K are always involved to produce the achromatic value. The chromatic value is then the result of two vicinal chromatic Basic Colors. Naturally, a Fundamental Color Code can also be composed of 3, or 2, or possibly only 1 Basic Color subset. The latter is the case if it is a pure Basic Color.

### Calculating the mixture components

If we want to stimulate the visual organ with an Additive Mixture to produce the color sensation " 25 75 50 " the intensity of the three chromatic lights must be such that FuCo B generates 25 sensation quanta , FuCo G 75, and FuCo R 50.

We remember that in a Subtractive Mixture (SubMi) the translucent ink layer Y modulates FuCo B by absorbing short-wave radiation. The translucent ink layer M controls FuCo G by absorbing the medium-wave radiation, and translucent ink layer C generates FuCo R by absorbing long-wave radiation. In other words, the filtering layer Y must absorb enough so that the passing radiation produces 25 sensation quanta of FuCo B. The filtering layer M has to absorb an appropriate amount of radiation so that the transmitted radiation generates 75 sensation quanta of FuCo G. And the absorbing power of filtering layer C must be such that the passing radiation produces 50 sensation quanta of FuCo R.

 99 99 99 25 75 50 -------------- 74 24 49 Y  M  C

The difference between FuCoCode " 99 99 99 " (white light) and the FuCoCode of our color shade " 25 75 50 " indicates the values for the three translucent ink layers required in a SubMi .

Consequently, there are necessary 74 set parts in filtering layer Y, 24 in layer M, and 49 in layer C so as to stimulate the visual organ to generate the color sensation " 25 75 50 ".

If we want to obtain the color sensation " 25 75 50 " with an Integrated Mixture (IntMi), i.e. by mixing opaque color material, the following partial quantities are needed based on the above calculation: 25 set parts of achromatic Basic Color W, 25 set parts of chromatic Basic Color Y, 25 set parts of chromatic Basic Color G, and 24 set parts of achromatic Basic Color K. Mixing comes first and then the opaque ink layer is applied as a single layer. This single opaque ink layer represents always the mathematical set 1, i.e. 100 %. Therefore, IntMi consists always of 99 set parts irrespective of whether four, three, two, or only one Basic Color is involved in composing the color shade.

This demonstrates that the Basic Color Indexing Systems derives from the Fundamental Color Indexing System.

## The Four Distinctive Esthetic Features of Colors

We just learned that it is possible to deduce from Fundamental Color Codes the Basic Color Codes of a particular shade. And from those we can determine the distinctive esthetic features, also called Color Impressions or Quality Parameters, namely Achromaticity Type, Chromaticity Type, Achromaticity or Chromaticity Degree, and Brightness.

The following result is obtained from FuCoCode " 71 56 14 " :

 Fundamental Colors B G R Basic Colors 14 14 14 42 42 00 15 00 00 00 00 00 ------------- 71 56 14 = W 14 = C 42 = B 15 = K 28 ------------- = 99 set parts

### Achromaticity Type

The achromatic portion in " 71 56 14 " is composed of the subsets W 14 and K 28. The mixture ratio between W and K is 1 : 2. This ratio determines the Achromaticity Type, in this case dark gray which results from mixing one part of W with two parts of K. It was Kueppers who established this new parameter in the Theory of Color.

### Chromaticity Type

The chromatic portion is composed of the two subsets C 42 and B15. The mixture ratio between C and B is almost 3 : 1 determining the Chromaticity Type. In this case we have a Type of Chromaticity somewhere between Cyan-blue (C) and Violet-blue (B) which results from mixing slightly less than 3 parts of C with one part of B .

### Chromaticity or Achromaticity Degree

Adding the two achromatic subsets, the total is 42. The two chromatic subsets make together 57. The mixture ratio between the two joins of sets determines the extent of chromaticity or achromaticity of the color shade. The extent of chromaticity of a shade is called Chromaticity Degree and the extent of achromaticity is called Achromaticity Degree. Both together always make a total of 99 set parts or 100%. It is about reciprocal values as it were the two sides of a medal, namely one and the same color shade. Therefore, in our example " 71 56 14 " the Degree of Chromaticity is 57%, the Degree of Achromaticity 42 % (for easier calculation we set equal 99% and 100%). The above illustration demonstrates these aspects graphically. Above square (A) represents the four individual Basic Color subsets W, K, C, and B which make up the color shade. In square (B) we see the subsets W and K already united into one achromatic join of sets. The quantitative proportion between the two determines the Type of Achromaticity. Square (C) shows the subsets C and B united into one chromatic join of sets. The quantitative proportion between these two determines the Type of Chromaticity. Now, the mixing ratio between the chromatic and achromatic subsets defines the Degree of Chromaticity or Achromaticity, respectively. Finally, above square (D) demonstrates that all 4 subsets are perceived as one single shade of color, namely color sensation " 71 56 14 ".

### Brightness

The fourth distinctive esthetic feature is the Brightness of a color shade. This parameter is asymmetrically arranged within the color space. And it must be like this as the brightness of the 8 Basic Colors is absolutely different one from the other. Brightness of a color shade is the result of the inherent brightness of each individual Basic Colors involved in composing the shade and their quantitative proportion within the mixture.

In the domain of color exist, contrary to previous teaching and writing, not only the three esthetic characteristics of hue, saturation, and brightness but rather four distinctive esthetic features which in Kueppers' Theory of Color are clearly and unambiguously defined as Chromaticity Type, Achromaticity Type, Chromaticity or Achromaticity Degree, and Brightness.