Recursive 3D Tiling of Rhombic Dodecahedra

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In summary, the conversation discusses the concept of rhombic dodecahedra, which can be tiled indefinitely to create a new volume. The question is raised whether this process can be repeated with 13 of these shapes to create a fractal surface. However, it is mentioned that this may not be possible due to the hollow spaces that would be left.
  • #1
Ventrella
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I have a question about rhombic dodecahedra - polyhedra which can be tiled indefinitely, and which correspond to the closest packing of spheres. A single rhombic dodecahedron can be surrounded by 12 rhombic dodecahedra, making a tiling of 13 (having no gaps or holes or cracks). This creates a new volume which can be described as the union of 13 rhombic dodecahedra.

My question is this: Can I then take 13 of these shapes and tile them in the same fashion, as if they were each rhombic dodecahedra? And can this process be repeated indefinitely? If so, the boundary would approximate a fractal surface. I have an intuition that this would be true, but I'm not sure how to verify this (other than building models and coming up with a conjecture).

Thanks!
-j
 
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  • #2
I don't think you can. They leave 14 hollows, yes? Do the same with 13 balls; they do not stack to a ball.
 

What is "Recursive 3D Tiling of Rhombic Dodecahedra"?

"Recursive 3D Tiling of Rhombic Dodecahedra" is a mathematical concept that involves creating a three-dimensional structure by repeating the process of tiling rhombic dodecahedra in a recursive manner.

Why is "Recursive 3D Tiling of Rhombic Dodecahedra" important?

This concept has practical applications in fields such as architecture, computer graphics, and crystallography. It also has significance in mathematical research and can lead to the discovery of new geometric principles.

How is "Recursive 3D Tiling of Rhombic Dodecahedra" different from other tiling methods?

The process of tiling rhombic dodecahedra recursively results in a unique and complex three-dimensional structure that cannot be achieved through other tiling methods.

What are the properties of the resulting structure in "Recursive 3D Tiling of Rhombic Dodecahedra"?

The resulting structure is self-similar, meaning that the same pattern appears at different scales. It also has a high degree of symmetry, with icosahedral symmetry being the most common.

How can "Recursive 3D Tiling of Rhombic Dodecahedra" be applied in real life?

This concept can be used in the design of architectural structures, such as pavilions and domes. It can also be used in computer graphics to create visually interesting and complex objects. In addition, it has potential applications in nanotechnology and crystallography for creating new materials with unique properties.

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