# Proposal of a New Tool for 3D Pattern Generation

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## Abstract

The behavior of sound in an enclosed space is very complex. This behavior is closely related to boundaries, material properties and with the form of the space they are in. In this paper, the image source method as a method for modeling the sound field in enclosed spaces is re-visited and patterns of image sources and their relationships with the geometry of enclosing space is studied by using polygons/polyhedrons in 2D and 3D. Symmetries of these image source points, identification of possible polygons enabling simple geometrical shape formations and knowing the resulting geometrical pattern, is used as an input to an acoustic model, reducing the complexity and computational time. Visualization of such complex patterns provides a solid tool to grasp complex relationships. Therefore transcoding the invisible relationships of acoustics into geometric patterns of mathematics provides a valuable means to improve the cognition of sound phenomena and to extract information as design input.

## Keywords

Sound Source Sound Pressure Level Image Source Sound Field Acoustic Model## Introduction

In this paper, the ISM is re-visited and the patterns of image sources and their relationship with the geometry of the enclosing space have been studied by the use of regular and irregular polygons (polyhedrons) in 2D and 3D. Influenced by the symmetries of image source points in space, the identification of possible polygons (polyhedrons) permits us to model the reflection patterns as just simple geometrical shape formations. Knowing the resulting geometrical pattern, data is used as an input to the acoustic model, reducing the complexity and thus the computational time required by ISM.

The paper also aims to employ these geometrical patterns to visualize the behavior of the sound source in an enclosed space and the relationship between the source position and the form of space. The authors believe that the visualization of such complex patterns provides a solid tool to grasp complex relations. Therefore transcoding the invisible relations of acoustics into the geometric patterns of mathematics provides a valuable means to improve the cognition of sound phenomena.

## Image Sources Method

## Visualization of Sound Fields in Enclosed Spaces

However, efforts such as the Chladni plate, which can be grouped under the term cymatics studies, try to visualize the sound source not the sound field. In order to visualize the sound field within an enclosure, one may be influenced by the similarity between optics and acoustics.

As a result, the following study and the results shall also be perceived as an attempt to visualize the sound field by using an optics analogy and the ISM for more complex geometries than rectangular rooms.

## Identification of Patterns Generated by the Image Sources

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.

*R*

_{ n }×

*σ*

_{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.

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.

## Projection of the Proposal to 3-Dimensional Geometries

The proposal, in which the results on 2D enclosures are inspected above, is also applied to 3D spaces to investigate the correspondence of a potential room acoustics case and the proposed model.

## Computational Cost

*i*is the order of reflections to be calculated (Kuttruff 2001). So, the computational cost of ISM is an exponential function and is expected to increase dramatically as the number of reflections increase.

Comparison of conventional ISM and the proposed method in terms of computational cost with respect to the number of faces and number of edges

Name of platonic solid | Number of faces | Number of edges of a face | Number of points generated (up to 5 order reflections) | Computational time (conventional ISM) (s) | Computational time (proposed method) (s) |
---|---|---|---|---|---|

Tetrahedron | 4 | 3 | 388 | 0.4212 | 0.0131 |

Cube | 6 | 4 | 2190 | 0.7807 | 0.0139 |

Octahedron | 8 | 3 | 9448 | 1.5131 | 0.0146 |

Dodecahedron | 12 | 5 | 81872 | 15.9954 | 0.0167 |

Curve fitting results and error values

Name of the method | Equation | SSE | R-square | Adjusted R-square | RMSE |
---|---|---|---|---|---|

Conventional ISM | \(0.01853*e^{{0.5633*n}}\) | 0.1453 | 0.9992 | 0.9987 | 0.2679 |

Proposed method | \(0.00045*n + 0.0122\) | 6 × 10 | 0.9916 | 0.9874 | 0.0001732 |

## Conclusion

The tendencies of the human brain to search for patterns in nature as a way of simplifying complex problems inspired the authors of the present study. In the quest for these patterns basic geometrical transformations and uniform and semi uniform polygons and polyhedrons are employed to transcode acoustics to mathematics. Congealing the relationship between the sound source and the space that the source is in, as the geometric patterns in 2D and 3D can be considered as a demystification of the sound phenomena.

The attempt at mapping what is aural to visual provided significant data to map what is visual to reduce the computational cost of the ISM algorithm. As a result, two objectives of this research are accomplished: using an optics analogy for visualization as a tool for understanding sound phenomena in enclosed spaces, and reduction of the computational cost of ISM by using symmetrical relations among image sources.

In this study, the physical realm of acoustics and the abstract world of mathematics are brought together with the use of simple symmetry relationships inherited in polygons (polyhedrons) to reduce computational cost and complexity of the model. It is shown that instead of calculating all the reflections, calculation of the image sources in the major symmetry space reduces the computational time and overcomes the major drawback of the ISM for symmetrical spaces as shown in the examples. Further studies of these patterns and relationships, enabling modeling of the relationship between the space and the sound source, promises exploration in more intricate spatial forms as well.

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