A note on the algebra of colors (and other
sense data)
Athanassios Tzouvaras
Dept. of Mathematics, Univ. of Thessaloniki,
541 24 Thessaloniki, Greece
e-mail:[email protected]
Abstract
Using the fact that all colors are generated from three basic ones,
which are independent, we can identify the set of colors to the structure R+3 with ordinary addition and scalar multiplication. Using some
standard facts from the geometry of R3 it is shown that: (a) If a color
is producible as a mixture of any number of other colors, then it is
producible by at most three of them. (b) For any k > 0, there are
k independent colors, i.e., none of them is producible as a mixture of
the rest.
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Some basic facts about colors
It is well known that the shades of all colors of the world are numerous (practically infinite), but all can be produced as mixtures of three basic ones: Red
(R), Green (G) and Blue (B). They correspond to certain radiation frequencies of the visible spectrum. A material substance receives red, green and
blue radiation and, depending on the energy states of its atoms, it absorbs
some them and emits the others. E.g. a red substance absorbs the green and
blue radiation, and emits the red one, while a yellow substance absorbs the
blue and emits the red and the green. So yellow is a derivative color and we
may say that, concerning radiations, yellow=red+green. The following table
contains the basic derivations from the basic colors:
1
color (frequency) of radiation
red green
blue
1
1
1
1
1
0
1
0
1
1
0
0
0
1
1
0
0
1
0
1
0
0
0
0
color of substance
white
yellow
reddish (magenta)
red
cyan
blue
green
black
(1 means emission, 0 means absorbtion)
Table 1
For example, the fifth row of the Table says that if a substance absorbs
all red radiation and emits all green and blue one, then its color will be cyan.
It follows that there are two complementary ways to look at colors: Either as forms of light radiation of certain frequencies, or as visual properties of
material objects, which are due to selective absorbtion and emission of light
radiation. The first system is usually referred to as additive system, and the
latter as subtractive system. The additive system is used in image processing,
TV screens etc, while the subtractive system is used in painting, printing,
drawing etc. Structurally the two approaches are almost identical. They
simply give rise to different basic colors and different generations. For example in the radiation approach, the basic colors are those we considered above,
i.e., the red, the green, and the blue (yellow is derivative: green+red), the
mixture of the three produces white (the well-known rainbow phenomenon
of the analysis of white light into its constituents ) and the absence of all
produces black. In contrast, in the absorbtion approach, the basic colors
are the red, the yellow and the blue (green is now derivative: yellow+blue)
the mixture of which produces black. White is not producible by the above
three, so we must add it as a fourth basic color. In this article we follow
the additive system, which is more fundamental and simpler, since no fourth
color is needed.
Let C be the set of all colors. Not only R, G, B generate all elements of
C, but moreover, none of the R, G, B can be produced as a combination of
the other two. Therefore R, G, B look like a basis of a vector space, where
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the elements of a basis exhibit these two characteristics: (a) They generate
the space and (b) they are linearly independent. But there are important
differences: Only nonnegative reals can be used in the combinations of the
basic colors, and only the proportions between these scalars are taken into
account, not the scalars themselves. As a result, the operation of “adding”
colors, i.e., mixing them up, is irreversible (that is, an ingredient of the
mixture cannot be regained by “subtracting” from the mixture the other
ingredients). Hence no group structure exists. C is just an additive semigroup
with a neutral element.
Let us come to our formalism. We use capital letters X, Y, Z, X1 · · · ranging over colors and lowercase letters a, b, c, k, . . . ranging over (nonnegative)
real numbers. Let R+ be the set of positive reals and R+
0 be the set of nonnegative reals. We assume two basic operations on C: An “addition” and a
multiplication by scalars. Namely for every X, Y ∈ C and every a ∈ R+ ,
X + Y and a · X, or just aX, are elements of C. Intuitively, + corresponds to
mixing, while the multiplication aX corresponds to “intensifying” (whenever
other colors are present).
Some obvious properties of + and · are accepted:
(1)
(2)
(3)
(4)
(5)
+ is commutative and associative.
a(bX) = (ab)X.
1 · X = X.
(a + b)X = aX + bX.
a(X + Y ) = aX + aY .
In terms of + and ·, the nontrivial derivations of Table 1 above are rewritten as follows:
white
= R+G+B
yellow = R+G+0V
= R+G
reddish = R+0G+B
= R+B
cyan
= 0R+ G+B
= G+B
black
= 0R+0G+0B =
We see that black is the absence of any color, and we shall denoted by 0,
since X + 0 = X for any X ∈ C. (Of course this holds only for radiations,
not for colors of substances, where black is heavily non-neutral.)
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What is the meaning of aX? Although we follow the radiation approach
we can think of colors as visual impressions produced by substances, e.g.
paints. So if X represents a drop of a paint of color X, then 2X represents
two drops of this paint and it is clear that the visual impression caused by one
drop in no different (as such and not in contrast e.g. with the background)
from the impression caused by two drops. Therefore X and 2X, and in
generally aX for any a ∈ R+ , are equivalent, denoted X ∼ 2X ∼ aX.
On the other hand, mixing one drop of X-paint with one drop of Y -paint
produces a different impression than mixing four drops of X-paint with one
drop of Y -paint, that is X + Y 6∼ 4X + Y . Here the scalar 4 acts indeed as
an intensifier making the impression caused by X stronger compared to that
of Y . On the other hand it is clear that we should have
X + Y ∼ 4X + 4Y ∼ aX + aY ∼ a(X + Y ).
The above simply says that the outcome of color mixing depends only one
one thing: The quantitative analogy of the ingredients. To put it formally we
define an equivalence relation ∼ in the space C as follows.
Definition 1.1 For any X, Y ∈ C, let X ∼ Y iff Y = aX for some a ∈ R+ .
Clearly ∼ is an equivalence relation. Given X1 , . . . , Xn ∈ C, for any
a1 , . . . , an ≥ 0, the combination Σi ai Xi = a1 X1 + · · · + an Xn is a color. Let
P os(X1 , . . . , Xn ) = {Σi ai Xi : ai ≥ 0}
be the positive hull of X1 , · · · , Xn .
Lemma 1.2 Let X1 , . . . , Xn ∈ C be distinct, and let Y = Σi ai Xi and Z =
Σi bi Xi be two elements of hX1 , . . . , Xn i. Then Y ∼ Z iff for all i, j ≤ n,
ai =
6 0 ⇐⇒ bi 6= 0 and ai /aj = bi /bj .
Proof. Let the condition hold. Without loss of generality suppose all ai
are 6= 0. Then so are bi and for any two i, j, ai /aj = bi /bj , or ai /bi = aj /bj ,
for all i, j. If c = ai /bi , then Y = Σi cbi Xi = cΣi bi Xi = cY , that is Y ∼ cZ.
The converse is similar.
a
Note that ∼-equivalent colors are not substitutable in combinations. Namely,
X ∼ Y 6⇒ (X + Z) ∼ (Y + Z). In view of the equivalence ∼, given
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X1 , . . . , Xn , instead of the elements of the positive hull P s(X1 , . . . , Xn ) we
may consider only the combinations a1 X1 + · · · + an Xn such that ai ∈ R+
0
and a1 + · · · + an = 1, i.e. the convex hull of X1 , . . . , Xn ,
Conv(X1 , . . . , Xn ) = {Σi ai Xi : ai ≥ 0, Σi ai = 1}.
Indeed, obviously, Conv(X1 , . . . , Xn ) ⊆ P os(X1 , . . . , Xn ). Conversely if X =
a1 X1 + · · · + an Xn is an element of P os(X1 , . . . , Xn ) and a = Σi ai , then
a−1 X ∼ X and a−1 X ∈ Conv(X1 , . . . , Xn ).
To sum up: Using the colors {R, G, B} as a basis, the set C of colors can
be identified with the subset R+3 = {(a1 , a2 , a3 ) : ai ≥ 0} of nonnegative
triples of reals. Addition and scalar multiplication for colors is reduced to
the corresponding operations on R+3 , which is a semigroup with unit element
(0, 0, 0). Any notion concerning colors, such as convex hull, independence,
etc, is better studied in the abstract context of R+3 or R+d in general, so we
recall certain properties and facts of R+d .
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Notions from euclidean spaces
Let Rd be the d-dimensional euclidean space. Bared letters x, y, . . . denote
vectors, i.e., elements of Rd , while x, x1 , a, a1 . . . denote elements of R. For
the following definitions and facts the reader is referred e.g. in [2].
Given X = {x1 , . . . , xk } ⊆ Rd , we let
hXi = {Σi ai xi : ai ∈ R}
be the linear hull of X,
Af f (X) = {Σi ai xi : Σi ai = 1}
be the affine hull of X, and
Conv(X) = {Σi ai xi : ai ≥ 0 and Σi ai = 1}
be the convex hull of X.
Note that there is an alternative definition of convex hull. Namely, given
x, y, the line segment [x, y] determined by x, y, is defined to be the set
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{ax + (1 − a)y : 0 ≤ a ≤ 1}. A set Y ⊆ Rd is said to be convex if for
any x, y ∈ Y , [x, y] ⊆ Y . Then the convex hull of X can be alternatively
defined as the intersection of all convex sets that include X. The equivalence
of the two definitions, proved in [2], p.15, is referred to as “Carathéodory’s
theorem”. In fact Carathéodory’s theorem says more.
Theorem 2.1 (Carathéodory) Let X be any subset of Rd . If x ∈ Con(X)
then there are at most d vectors xi ∈ X, i ≤ d, and ai ≥ 0 such that Σi ai = 1
and x = Σi ai xi .
Corollary 2.2 Let X1 , . . . , Xk be k colors of C. If X is a combination of
Xi , i ≤ k, then X is a combination of (at most) three of them.
Proof. By the remarks at the end of the last section, X can be taken to
be in the convex hull of X1 , . . . , Xk . Also C can be identified with R+3 , so
apply the previous theorem for d = 3.
a
Further let
P os(X) = {Σi ai xi : ai ≥ 0}
be the positive hull of X.
Definition 2.3 x1 , . . . , xk are said to be positively dependent if for some
i ≤ k, xi ∈ P os(X − {xi }).
Fact x1 , . . . , xk are positively dependent iff for some i ≤ k and some
a > 0, axi ∈ Conv(X − {xi }).
Proof. Let x1 , . . . , xk be positively dependent. Then for some i ≤ k, there
are aj ≥ 0 such that xi = Σj6=i aj xj . If a = Σj6=i aj , then a−1 xi = Σj6=i bj xj ,
where bj ≥ 0 and Σj6=i bj = 1, hence a−1 xi ∈ Conv(X − {xi }). Conversely, let
axi ∈ Conv(X − {xi }) for some i ≤ k and some a > 0. Then axi = Σj6=i bj xj ,
where bj ≥ 0 and Σj6=i bj = 1. Thus xi = Σj6=i a−1 bj xj , and since a−1 bj ≥ 0,
xi ∈ P os(X − {xi }).
a
By 2.2, if a color is generated by a combination of k other ones, then it is
generated by the combination of at most three of them. What about positive
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independence? Namely, how many positively independent colors can we have
in R+3 and more generally in R+d ? Contrary to what happens with ordinary
independence, if d ≥ 3, there can be arbitrarily many positively independent
vectors. For d = 3 this fact, as well as result 2.1, has an easy geometric
explanation.
Lemma 2.4 (a) Let d ≥ 3. For every k > 0, there are k positively independent vectors in In R+d .
(b) In contrast, at most two vectors of R+2 are positively independent.
Proof. (i) We show it for d = 3. The generalization to d > 3 is easy.
By Fact above it suffices to find a set X = {x1 , . . . , xk } of k vectors such
that for every i ≤ k, and every a > 0, axi ∈
/ Conv(X − {xi }). Choose
+3
k points in R , x1 , . . . , xk forming the vertices of a convex k-polygon. If
X = {x1 , . . . , xk }, then xi ∈
/ Conv(X − {xi }), Since all xi are on the same
plane, clearly, also for any a > 0, axi ∈
/ Conv(X − {xi }).
(b) For any three vectors x, y, x, of R+2 , if identified with their position
vectors, then one of them, say x, lies between the other two. Then obviously,
x is a positive combination of y and z.
a
Corollary 2.5 For every k > 0, there are k independent colors in C.
Lemma 2.6 If {x1 , . . . , xm }, {y 1 , . . . , y n } are positively independent and
Conv(x1 , . . . , xm ) = Conv(y 1 , . . . , y n ),
then {x1 , . . . , xm } = {y 1 , . . . , y n }.
Proof. Let S = Conv(x1 , . . . , xm ). Since {x1 , . . . , xm } is positively independent, {x1 , . . . , xm } is the set of vertices of S, i.e., of the points of
S which are not interior in any line segment contained in S. Since S =
Conv(y 1 , . . . , y n )), the same holds for the set {y 1 , . . . , y n }. So the two sets
are equal.
a
Corollary 2.7 Let X1 , . . . , Xn be independent colors. Then X1 , . . . , Xn are
the only colors generating P os(X1 , . . . , Xn ).
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References
[1] Colors on the web,
http://www.webwhirlers.com/colors/coloursphysics.asp
[2] B. Grünbaum, Convex polytopes, Interscience Publishers, Purs and Applied Mathematics, vol. 16, 1967.
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