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I'd like to construct a graph that approximates a sphere in 3-space, but I'm placed under the following constraints:

(1) - I am only allowed to use a construction block, $v_i$, consisting of a single vertex with $N$ edges of uniform length $L$.

(2) - The edges must exhibit rotational symmetry around any $v_i$ similar to the petals of a flower. The inner angle between any two edges across some $v_i$, $\theta(i)$, may be selected as desired, but it must hold for all possible pairs of edges for that instance of the building block. For practical reasons, I'll restrict the number of possible instances of $v_i$, and equivalently values of $\theta(i)$, to two or three.

(3) - When joining two vertices together, any two edges may be connected, but no bending is allowed. In other words, I should be able to draw a straight line between vertices across any two connected edges. As such, the distance between vertices across two connected edges should be $2L$.

Provided these constraints, for what values of $N$ (i.e. number of edges around $v$), and values of $\theta(i)$, is it possible to construct a closed surface (a sphere for example) from the building block $v$, and what are the properties of the closed surface? How many copies of $v$ are required?

Any and all feedback is appreciated!


Update: Instead of saying "approximates a sphere" I should say - "generates a closed surface". Eccentric and deformed spheres are certainly reasonable. Also (and with apologies to André Henriques!), I'm going to reformulate the question to allow for more than one interior bending angle $\theta(i)$ (two or at most three to hopefully keep the question reasonable).

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Look, who says what is allowed, what symmetries must be exhibited? –  Will Jagy Jul 16 '11 at 4:41
    
@Will Jagy, I added a further constraint that all edge lengths should be equal. Shouldn't there be a fixed value for theta(interior) that allows the formation of a N-gon approximating a sphere without gaps? –  FireAndCoffee Jul 16 '11 at 5:55
    
@Will, and thanks for reading my question! –  FireAndCoffee Jul 16 '11 at 5:57

1 Answer 1

up vote 2 down vote accepted

I think that your constraints are very restrictive, and that the only possibilities are the 5 Platonic solids: tetrahedron, octahedron, cube, icosahedron, and dodecahedron.

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@André Henriques, wouldn't a truncated icosahedron also fit within the constraints? –  FireAndCoffee Jul 19 '11 at 8:06
    
If fails "The edges must exhibit rotational symmetry around any $v_i$". –  André Henriques Jul 19 '11 at 11:22

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