Acronym | ... |
Name | triangle triacontahexahedron |
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Vertex figures | [3^{4}], [3^{7}], [3^{9}] |
This non-convex deltahedron happens to be threefold antiprismatic and acoptic
(non-selfintersecting), but only is triform (has 3 different vertex types).
It can be thought of aligning patches of 6 triangles each (ships) alternatingly above / below the equator in a circle.
Note that all those ships are not inclined but align with the overall axis.
It also can be thought of as a digged and cut icosahedron.
©
Idea and animation are courteous to T. Dorozinski.
The tip-2-tip length of those ship patches can be calculated as the x(n,3) chord length of some n-gon (n to be evaluated and non-integral). The respective width of that ship patch would be twice the height of the according n-gonal pyramid. As the axial projection of this ship has to be a 60°/120° rhomb, those lengths are in ratio sqrt(3). This leads to the equation 16 T^{3} - 24 T^{2} - 3 T + 3 = 0 with solution T = sin(π/n)^{2} = 0.327896 and thus n = 5.152683. This now allows to calculate those measures of the ship, resulting in a length of 1.688417 and a width of 0.974808. By mere Pythagoras follows from the last measure that the distance of those 2 central points of the polyhedron will be 0.223046.
2 * * | 6 3 0 0 0 0 | 3 6 0 0 [3^{9}], central * 12 * | 1 0 1 1 1 0 | 1 1 1 1 [3^{4}], polar * * 6 | 0 1 0 2 2 2 | 0 2 1 4 [3^{7}], external tropal -------+----------------+---------- 1 1 0 | 12 * * * * * | 1 1 0 0 1 0 1 | * 6 * * * * | 0 2 0 0 0 2 0 | * * 6 * * * | 1 0 1 0 0 1 1 | * * * 12 * * | 0 1 0 1 along rim of external band 0 1 1 | * * * * 12 * | 0 0 1 1 across external band 0 0 2 | * * * * * 6 | 0 0 0 2 -------+----------------+---------- 1 2 0 | 2 0 1 0 0 0 | 6 * * * yellow {3} 1 1 1 | 1 1 0 1 0 0 | * 12 * * yellow {3} 0 2 1 | 0 0 1 0 2 0 | * * 6 * blue {3} 0 1 2 | 0 0 0 1 1 1 | * * * 12 blue {3}
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