What Are the Kissing Numbers for Different Sized Spheres in 3D?

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SUMMARY

The discussion centers on the concept of kissing numbers for spheres in three-dimensional space. It is established that when all spheres are of equal size, specifically unit spheres, a maximum of 12 smaller spheres can kiss a central sphere. The inquiry extends to scenarios where the outer spheres are smaller, specifically by fractions such as 1/4 or 1/2, prompting a search for existing research on non-uniform kissing numbers. The relationship between kissing numbers and the geometry of platonic solids, particularly the dodecahedron, is highlighted as a key factor influencing these configurations.

PREREQUISITES
  • Understanding of kissing numbers in geometry
  • Familiarity with three-dimensional shapes and their properties
  • Knowledge of platonic solids, particularly the dodecahedron
  • Basic principles of sphere packing
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  • Research the mathematical derivation of kissing numbers for non-uniform spheres
  • Explore sphere packing theories and their applications in higher dimensions
  • Investigate the relationship between kissing numbers and geometric configurations
  • Study the implications of varying sphere sizes on packing density and arrangement
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Mathematicians, physicists, and anyone interested in geometric packing problems, particularly those studying sphere arrangements and kissing numbers in three-dimensional space.

jal
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I am looking for info on kissing numbers.
If all the spheres are of one unit then there are 12 spheres in 3d that can kiss the center sphere.
Question:
If the outer spheres are smaller by say 1/4, 1/2 etc. what would be the kissing numbers?
I assume that this has already been done but my search has not found anything.
jal
 
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The reason it works out nice for spheres of equal size is because the dodecahedron has 12 faces. I think the answer is going to be ugly for sphere kissings that don't correspond to platonic solids.
 

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