Distance from centroid to surface of polyhedron as a function of frequency

[thadman]

Hello members,

I am a bit stumped on a mathematical problem. If this is in the wrong section I apologize. Recently, I've become interested in polyhedrons. However, I am having a bit of trouble with their analysis.

Consider a given dodecahedron. If we consider the solid to be of uniform density, the centroid will occur at (0,0,0) and the distance from the centroid to all of the vertices will be constant. However, the distance from the centroid to other points on the surface will vary. Since the dodecahedron is symmetrical, we can restrict the problem to a single pentagon on the surface.

I would like to plot a 2 dimensional function (y=f(x)), where "x" represents radius (distance from centroid to surface) and "y" represents frequency. For the dodecahedron, the domain of "x" (ie radius) will be continuous from a vertex to the center of the pentagon. The domain of "y" will be continuous from 0-->1 (ie it will be normalized).

I would like to extend this analysis to assymetrical polyhedron (ex. the diminished rhombicosidodecahedron. In that particular case, the centroid will be displaced and the individual polygons will not be seperable.

Would the solution simply be analogous to a Fourier transform over the surface?

Could anyone offer insight on this?

Thanks,
Thadman

 

[jvicens]

Hello Thadman,

I find your interest in polyhedrons fascinating. As a scientist who specializes in mathematics, I may be able to offer some insight on your problem.

Firstly, I would like to clarify that the function you are looking for is not a two-dimensional function, but rather a three-dimensional one. The x-axis represents the distance from the centroid to the surface, the y-axis represents the frequency, and the z-axis represents the distance from the centroid to the center of the pentagon. This function can be visualized as a surface plot, where the height of the surface at a specific point represents the frequency at that distance from the centroid.

In the case of the dodecahedron, the function will be symmetrical since the shape is symmetrical. However, for an asymmetrical polyhedron, the function will be more complex and may require the use of a Fourier transform to analyze it. A Fourier transform can decompose a complex function into simpler components, making it easier to analyze and understand. This approach could be applied to your problem with an asymmetrical polyhedron.

I hope this helps and I wish you all the best in your research on polyhedrons. Feel free to reach out if you have any further questions or need clarification on anything.