ISM is based on reflection with respect to the axis defined by the boundary surfaces. Hence, it is expected to achieve highly symmetrical patterns while working with symmetrical boundaries. Yet, although the rules employed by ISM are simplistic and solid, these symmetrical patterns can be used to reduce computational time, and even to predict new case results without generating all image sources but employing necessary transformation functions to the patterns achieved.
For the sake of simplicity, the first test cases are constructed in 2D with regular polygons. The source, which the image sources will be generated from, is located at the centroid of the polygons and symmetrical relations are inspected to construct a base line. The image sources are treated as separate groups with respect to their order of reflection and are displayed with color codes of red, green, blue, yellow and black as the order of reflection increases, and the image sources of a particular image order are connected in a radial order to increase traceability of the symmetrical patterns. After the construction of the baseline, two relationships of pattern and boundary surfaces and original source are inspected as the relationship between patterns and the radius of the polygons and the relationship between the location of the original source and patterns.
The symmetrical patterns, which are achieved by generating image sources from a source located at the centroid of regular polygons with the number of edges from 3 to 9, are shown below.
As can be seen from the Figs. 5, 6 and 7, two types of lines (black and red) are dividing the space in which the patterns are generated. Patterns can be decoded with two types of symmetrical relations as rotation and reflection. Each of the patterns has rotational symmetry with the order of n, which is also the number of edges of the polygons they are generated from. These symmetrical parts are the polylines residing between two black lines dividing the space. In addition, each of these parts has a reflection relation with respect to the red line dividing the part in half. Hence, once we obtain 1/2n part of the pattern, we are able to obtain the full pattern by following the symmetrical operations as: R
1. This function reduces the computational cost of ISM by an order of 1/2n where n is the number of edges of the polygon which is the boundary of the space concerned.
Secondly, the relationship between the radius of the polygon and the pattern is inspected. It is observed that exactly the same pattern is obtained as expected as the radius of the polygon is changed. The radius of the polygon is linearly proportional with the distance of the image sources to the origin even though the generation of image sources has more complex relations. As the patterns obtained by scaling the polygon are exactly the same, the resultant patterns are not shown.
The third aspect within the scope of this research is the relationship between the pattern and the position of the original source. In this operation, the source location is changed firstly in x directly and then in both the x and y directions. Dislocation in only the y direction is excluded as the patterns have rotational symmetry and the resultant pattern will have the same behavior with the dislocation of the source in only the x direction. The resultant patterns obtained by dislocating the source are shown in Figs. 8, 9, 10, 11, 12 and 13.
It is observed that the pattern achieved in the case of the source being at the center of the concerned polygons forms the base of patterns achieved by dislocating the source. Patterns for dislocated sources have an additional relation resulting in the stretching and scaling of the patterns of centered sources. As the transformation function cannot be explained with symmetrical relations only, research of this function is left for future studies.