-------- Original Message -------- Subject: Re: [math-fun] In[569]:= vertexSolidAngles["RhombicTriacontahedron"] // FullSimplify //Tally // tim Date: 2017-12-20 13:40 From: Fred Lunnon <fred.lunnon@gmail.com> << Question 2: How many vertices, edges, and faces has the Stella Octangula? >> I'll buy it: why isn't the answer just 8, 12, 8 (two tetrahedra) ? WFL Mathematica agrees: In[603]:= PolyhedronData["StellaOctangula", "Vertices"] // Length Out[603]= 8 In[604]:= PolyhedronData["StellaOctangula", "Edges"] Out[604]= {{1, 3}, {1, 6}, {1, 7}, {2, 4}, {2, 5}, {2, 8}, {3, 6}, {3, 7}, {4, 5}, {4, 8}, {5, 8}, {6, 7}} In[592]:= PolyhedronData["StellaOctangula", "Faces"] Out[592]= {{4, 5, 8}, {5, 4, 2}, {8, 2, 4}, {2, 8, 5}, {6, 7, 3}, {7, 6, 1}, {3, 1, 6}, {1, 3, 7}} But that has interior boundaries. If it's really two tetrahedra, why isn't its volume twice the tetrahedron's? What do we call the union of an octahedron and 8 tetrahedra? https://en.wikipedia.org/wiki/Stellated_octahedron equates the two. But they're different. And what are the V and E counts of the stellate? (F=24.) --rwg On 12/20/17, Bill Gosper <billgosper@gmail.com> wrote:
During evaluation of In[569]:= 8.60579 secs, 2 components
Out[569]= {{π, 12}, {3π/5, 20}}
I.e., 12 vertices have solid angle π. True or false: Four RTs with disjoint interiors can intersect at a point and fill a neighborhood of it. Can five with disjoint interiors intersect at one point? Six?? 6×⅗π < 4π.
Question 2: How many vertices, edges, and faces has the Stella Octangula? --rwg
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