# Designing Color Symmetry in Stigmergic Art

## Abstract

**:**

## 1. Introduction

## 2. Color, Symmetry, and Color Symmetry

**ne**), north-west (

**nw**), south-east (

**se**), and south-west (

**sw**). Each section can be mapped into any other section by the elements of the symmetry group. The sections are separated in Figure 1a by white lines (which are not present in the original Kandinsky study). We assume that the image is composed of $n\times m$ pixels and each pixel ${p}_{i}$ has a color ${c}_{i}$ in a HSV color space: ${c}_{i}=({h}_{i},{s}_{i},{v}_{i})$. In the following discussion (and also in the experiments reported later) the focus is on the hue ${h}_{i}$, while the saturation ${s}_{i}$ and the value (brightness) ${v}_{i}$ remain constant; however, of course the same method can be applied to the whole HSV color space.

**ne**). The colors in this section remain unchanged. The colors of the pixels of the other sections are changed according to the rectangle’s symmetry group. For instance, the color permutation associated with the rotation is

**sw**is changed from ${h}_{i}$ to ${f}_{1}\left({h}_{i}\right)$. For the remaining sections

**nw**and

**se**, we may, likewise, define for the reflections

## 3. Stigmergy and Generative Art

## 4. Experiments and Visual Results

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Vasily Kandinsky (1866–1944): Farbstudie-Quadrate mit konzentrischen Ringen (Color study. Squares with concentric rings), 1913. (

**b**) Color symmetry using a color permutation associated with the symmetry group of the rectangle.

**Figure 2.**Symmetry and color symmetry on a color wheel with 12 slots. We denote the slots according to the digits of a clock face. Axes connect slots on the right-hand side of each slot. A rotation $\alpha $ rotates counter-clockwise with 30${}^{\circ}$, and a reflection $\beta $ reflects in the axis connecting 12 and 6 o’clock. (

**a**) The outer circle is the RYB (red–yellow–blue) color wheel. (

**b**) Color symmetry described by the color permutations of Equation (1) (outer circle) and Equation (2) (inner circle).

**Figure 3.**Four different stigmergic color-symmetric patterns with reminiscence of Kandinsky’s color study in Figure 1a. (

**a**) Vertical reflection imposing a color symmetry of circles and squares from primary and secondary colors to tertiary colors. (

**b**) Vertical reflection imposing a color symmetry of Archimedean spirals and squares from primary and secondary colors to tertiary colors. (

**c**) Color reversed version of (

**b**). (

**d**) Vertical reflection imposing a color symmetry of circles and squares which preserve primary colors and change secondary colors to tertiary colors.

**Figure 4.**Four different stigmergic color-symmetric patterns with isometric as well as color symmetry. (

**a**) Vertical reflection (

**b**) Archimedean spirals with color-symmetric counterpart obtained by a rotation about a random center point. (

**c**) Diagonal reflection (

**d**) Circle symmetry following a Fibonacci circle curve.

**Figure 5.**Johannes Vermeer (1632–1675): (

**a**) Girl with a Pearl Earring, 1665. (

**b**) View of Delft, 1661. The same Vermeer painting with color symmetry imposed by a color permutation. (

**c**) Triangular pattern. (

**d**) Chess board pattern with 16 elements.

**Figure 7.**Color symmetric art works, which can be seen as a quotation and homage to Vermeer’s Girl with a Pearl Earring.

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**MDPI and ACS Style**

Richter, H.
Designing Color Symmetry in Stigmergic Art. *Mathematics* **2021**, *9*, 1882.
https://doi.org/10.3390/math9161882

**AMA Style**

Richter H.
Designing Color Symmetry in Stigmergic Art. *Mathematics*. 2021; 9(16):1882.
https://doi.org/10.3390/math9161882

**Chicago/Turabian Style**

Richter, Hendrik.
2021. "Designing Color Symmetry in Stigmergic Art" *Mathematics* 9, no. 16: 1882.
https://doi.org/10.3390/math9161882