Exploring the Densest Space-Filling Polyhedra Lattice

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In summary, the cubic close packing (or face centered cubic lattice) and the hexagonal close packed (abab) structure are the same and considered to be the densest 3D lattice structure. The efficiency of this structure is still an open problem, with various upper bounds being calculated but no definitive proof yet. There are also two types of symmetric hexagonal close packed structures, but only the abab structure is equivalent to the FCC.
  • #1
Eppur si muove
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What is the densiest space-filling polyhedra lattice?My first guess would be the cubic one, but i am not sure.
 
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  • #2
I think it would be hexagonal close packed (tetrahedral).
 
  • #3
Can one prove that it is tetrahedral and possibly find the dimensions that would give the maximum density?
 
  • #4
Yes, you can calculate the density for the various lattices and compare them. It's a fun geometry problem!
 
  • #5
i believe it's called "cubic closest" packing, that is the best around. there is no proof that that is in fact, the best arrangement. this is an open problem. i forget what the effiiciency is, let's say, arbitrarily, it's 74% (i think that's close). it has been proven that (again, this is completely arbitrary, made up by me from loose memory) 80% is the best one can do. then again, something like 79.4% efficiency. the upper bounds are closing in, but as of yet, no proof.

EDIT: I"m a moron. nowhere did you ask about spheres. i replied about spheres. I will leave this anyways.
 
  • #6
The cubic close packing (or face centered cubic lattice) is the same as the hexagonal close packed (abab) structure...and is the densest 3D lattice structure. To see that the FCC is the same as the HCP, simply look at a set of parallel {111} planes in the FCC.

Trancefishy : You're not a moron. When the structure of a lattice "point" is unmentioned, it's more than reasonable to assume it is spherical.

Additional Note : There are two kinds of symmetric hexagonal close packed structures : "abab" and "abcabc". Both are equally dense, but only the abab structure is the same as the FCC.
 
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1. What is a densest space-filling polyhedra lattice?

A densest space-filling polyhedra lattice is a three-dimensional arrangement of polyhedral shapes that fills space as efficiently as possible, with little to no empty space between the shapes.

2. How are these lattices used in scientific research?

Densest space-filling polyhedra lattices are used in various fields of science, such as material science, physics, and mathematics, to study the properties and behavior of different materials and structures.

3. What are some real-life applications of these lattices?

These lattices have practical applications in fields such as architecture, engineering, and biology, where efficient space-filling structures are needed. They can also be used to create stronger and more lightweight materials for construction and transportation.

4. How are these lattices studied and analyzed?

Scientists use computer simulations and mathematical models to study and analyze the properties of densest space-filling polyhedra lattices. They also conduct physical experiments to test the strength and stability of these structures.

5. Is there a limit to the complexity of these lattices?

Yes, there is a limit to the complexity of these lattices as the number of polyhedral shapes increases. However, scientists are constantly exploring new ways to push these limits and discover more intricate and efficient space-filling structures.

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