Question textWhen spheres are packed like these oranges and centres of adjacent spheres are connected by edges you arrive at a packing of space using two different types of regular polyhedra. a) Which polyhedra?Answer:Answer 1 Question 9[select: , tetrahedron:octahedron, dodecahedron:icosahedron, tetrahedron:cube, cube:octahedron]b) In a large chunk of such a packing the ratio of the number of the polyhedra of the two types is what?Answer:Answer 2 Question 9[select: , 3:2, 2:1, 3:1, 5:3]

Question

BlackTom AI · Accepted Answer

Question restatement: The problem describes spheres packed like oranges with centers connected by edges, yielding a space-filling arrangement using two different regular polyhedra. Part a asks which polyhedra are involved. Part b asks for the ratio of the numbers of the two polyhedron types in a large packing.

Part a options analysis:
- Option "tetrahedron:octahedron": This reflects the well-known tetrahedral-octahedral honeycomb that fills space when regular tetrahedra and regular octahedra share faces. In close-packed arrangements (FCC/HCP), the decomposition into space-filling cells naturally leads to tetrahedra and octahedra. This option aligns with standard geometric decomposition of such sphere packings.
- Other options mix inappropriate pairings (for example, dodecahedron or icosahedron do not tessellate space alongside tetrahedra in this packing context). A common misconception is to pair tetrahedra with cubes or other non-tessellating regular polyhedra, but regular cubes and regular tetrahedra do not form a uniform space-filling partition with just those shapes here.

Part b ratio analysis:
- The known tessellation of space by regular tetrahedra and regular octahedra (tetrahedral-octahedral honeycomb) yields a fixed proportion between the two types. In this packing, the number of octahedra is twice the number of tetrahedra, giving the ratio octahedra:tetrahedra = 2:1. This result arises from how each tetrahedron shares faces with octahedra in the tiling, enforcing a consistent count relationship throughout a large assembly.
- Among the provided options, the other ratios (3:2, 3:1, 5:3) do not match the established tessellation balance of tetrahedra and octahedra in the space-filling arrangement.

Putting the pieces together, the reasoning supports the interpretation that the two polyhedra involved are tetrahedra and octahedra, and the ratio of the numbers of the two types in a large packing is 2:1.

In summary, for part a, the pairing that fits the known space-filling pattern is tetrahedron:octahedron. For part b, the corresponding ratio is 2:1. The remaining choices fail to reflect either the correct tessellation partner or the correct global count relationship in the tetrahedral-octahedral honeycomb.

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