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Please see the original problem specification (which Joseph O'Rourke was responding to in his answer) below.

Motivation -

I'm interested in a particular case of the problem where one wants to pack equiradius spheres around a central sphere s.t. every sphere has the same kissing number (i.e. contacts with its nearest-neighbors) and its nearest-neighbors are arranged in a rotationally symmetric fashion. In the problem specification (below), the centerpoint of each sphere represents a vertex, and edges represent kissing contacts between nearest-neighbors.

It troubles me that I have no intuitive understanding for why the set of solutions to this problem should be restricted to the platonic solids, and I'd really like a better appreciation for why this is so. Intuitively it feels like there should be some set of solutions involving more vertices/spheres than the dodedecahedron or icosahedron allows.


I'd like to decompose a sphere (say, or radius $R_s$) into a fully connected graph where:

(1) - The degree of each vertex is fixed at $N$,

(2) - Each edge in the decomposition graph has a constant bending angle $\theta$ and fixed length $L$,

(3) - The $N$ edges around each vertex, $v_i$, are arranged in a rotationally symmetric pattern,

(4) - The vertices $v_i$ lie on the sphere one is attempting to approximate.

As a function of the bending angle/curvature of each edge $\theta$, the uniform length of each edge $L$, and the number of edges around each vertex $N$, when can I generate a fully-connected graph that approximates the closed surface of the sphere, and how many vertices will it contain?

Again, any and all feedback is greatly appreciated!

Note - This is similar to my earlier question (which disallowed curved edges): "Constructing a graph that approximates a sphere using rotationally symmetric building blocks with equal numbers of edges". I now realize (thanks to André Henriques) that the requirement of having a vertex-transitive/isohedral graph, with fixed-length straight edges and rotational symmetry around vertices, should restrict one to the five platonic solids (though I'm not entirely sure if the rotational symmetry constraint matters in this case).

Update - I have now removed the constraint that the graph is vertex-transitive.

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Can you give a more precise definition of "decompose" here? If not, what kind of properties do you want your "decomposition graph" to have? –  j.c. Jul 17 '11 at 23:32
    
As I interpret it, (1) is essentially topological - you are searching for 1-skeleta of polyhedra all of whose vertices have the same degree. You can extract a list of candidates from en.wikipedia.org/wiki/… after taking the dual. From that list, you will find only a finite number of convex candidates (required by (4), I think), and I think you will be able to argue from your remaining constraints that you're again left with the Platonic solids. –  j.c. Jul 18 '11 at 5:08
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@F&C: As this is now the 3rd list of criteria you've offered, it may be more direct to explain the application that your criteria are intended to satisfy. –  Joseph O'Rourke Jul 18 '11 at 12:15
    
@Joseph O'Rourke, unfortunately I don't have a particular application in mind, but I have added a motivation section which directly addresses the problem I am having trouble understanding. I appreciate the answer you provided, and I hope this makes my question more fair. –  FireAndCoffee Jul 19 '11 at 4:43
    
FAC, you ought to try math.stackexchange.com We are not fond of tutoring. –  Will Jagy Jul 19 '11 at 4:50

1 Answer 1

up vote 2 down vote accepted

I assume you are aware that the vertex-transitive graphs represent a relatively constrained list? See this description. I fear that adding your conditions (2,3,4) on top of vertex-transitivity will again confine you to the skeletons of the Platonic solids.

Are you familiar with Fuller's geodesic domes? They fail to satisfy your criteria, but they get perhaps as close as is feasible, and may serve, depending upon your application:
      Geodesic Dome


After seeing your addendum on motivation, perhaps you should investigate the literature on packing disks on a sphere. For example, you might explore Neil Sloane's web pages on "Nice arrangements of points on a sphere in various dimensions" which address the problem of

placing $n$ points on a sphere in $d$ dimensions so as to maximize the minimal distance (or equivalently the minimal angle) between them.

For example, here is the best packing known for $n=50$, due to Székely (1974):
                     50 points on sphere
                     Image © Hugo Pfoertner, 2001

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@Joseph O'Rourke, I always thought there would be classes of vertex-transitive geodesics, or situations where one could achieve vertex transitivity via pruning? –  FireAndCoffee Jul 18 '11 at 4:50
    
@Joseph O'Rourke, I have now removed the requirement that the graph be vertex transitive. Can we construct a geodesic from a single building block? –  FireAndCoffee Jul 18 '11 at 4:56
    
@Joseph O'Rourke, I just wanted to say that I appreciate the time you spent answering my question (and the nice graphics!). Yes, I think you're right that my best bet is using putatively optimal arrangements of discs on spheres. –  FireAndCoffee Jul 19 '11 at 18:01

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