Examples of connected sums of unicursal deltahedra

"Pizza" (or zero-based-index Plantri 7-1) is the smallest unicursal deltahedron that can derived as a connected sum of other unicursal deltahedra.

The smallest composite unicursal deltahedron (composite in the sense that it is a connected sum of two smaller unicursal deltahedra) is the shape we have been calling pizza, a V-shaped, symmetrical composition of two triangular bipyramids. Pankov and Tyc prove that the sum is unicursal, and in fact, in the the photo the shape has been made by wrap forming.

Two triangular bipyramids can be summed to make "pizza".

Though the triangular dihedron is considered a degenerate deltahedron, we make exception for it here because it is identity element in connected sums of deltahedra (and it is in fact unicursal.) Connected summation with the triangular dihedron does not change a thing, since removing one of its triangles leaves an exact replacement for the triangle being removed from the other deltahedron.

Apart from the triangular dihedron, the only unicursal deltahedron smaller than the trio of shapes with 7 vertices, shown in the figure below, is the triangular bipyramid (TBP). Since connected summation with a TBP adds a net of 5 - 3 = 2 vertices, we must look among the 9 vertex unicursal deltahedra for the next composites.

Clockwise from top: pizza, tricorn and spaceship.