Juel's dissection uses the inscribed sphere while Gerling's dissection uses the circumscribed sphere. The vertices of a piece of Juel's are two vertices of the tetrahedron, the center of the inscribed sphere and its orthogonal projection on a face (the point of tangency). Jessen combined two such pieces into a symmetrical hexahedron with triangular faces (Frederickson 2002, 230-232).

Reference

[1] G. N. Frederickson, Dissections: Plane & Fancy, New York: Cambridge University Press, 1997, 2002 p. 22.