©

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 T³ - 24 T² - 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.

Incidence matrix

2  * * |  6 3 0  0  0 0 | 3  6 0  0	[39], central
* 12 * |  1 0 1  1  1 0 | 1  1 1  1	[34], polar
*  * 6 |  0 1 0  2  2 2 | 0  2 1  4	[37], 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}