Spherical packing of a tetrahedon

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The largest sphere that fits in the interstitial site between four spheres arranged in a tetrahedron is approximately 0.414 times the diameter of the original spheres. This ratio is derived from the concept of the "Kissing Number," which indicates that 12 identical spheres can surround a central sphere without overlapping. The center of the interstitial sphere is located at the centroid of the tetrahedron formed by the four spheres. Understanding this arrangement is crucial in fields like crystallography and materials science, as it aids in the study of sphere packing and structural formations. The discussion highlights the mathematical principles underlying these geometric configurations.
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If I put four spheres together to form a tetrahedon, what is the largest sphere that would fit in the interstitial site between them?
 
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openminded said:
If I put four spheres together to form a tetrahedon, what is the largest sphere that would fit in the interstitial site between them?
What are your thoughts about how to approach this problem? Can you describe the location of the center of the interstitial sphere in relation to the centers of the other four spheres?

https://www.physicsforums.com/showthread.php?t=4825
 


The largest sphere that would fit in the interstitial site between four spheres arranged in a tetrahedron shape is approximately 0.414 times the diameter of the original spheres. This is known as the "Kissing Number" and is a well-known concept in mathematics and geometry. It represents the maximum number of identical spheres that can be arranged around a central sphere, each touching the central sphere and its adjacent neighbors without overlapping. In the case of a tetrahedron, the Kissing Number is 12, meaning that 12 spheres can fit around one central sphere without overlapping. This results in a ratio of approximately 0.414 between the diameter of the central sphere and the diameter of the surrounding spheres. This concept is important in understanding the packing and arrangement of spheres in various structures and is often used in fields such as crystallography and materials science.
 

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