For the regular kaleidocycles the order minimum is 8; nevertheless we can build a ring with 6 tetrahedra, but which cannot turn completely. 

regular kaleidocycle of order 8 
closed kaleidocycle (non regular) of order 6 may be cut using its symmetry plane (when closed) into two mirror image rightangled kaleidocycle 
rightangled kaleidocycle of order 6 ("Schatz cube" below) 
The "Paul Schatz cube" contains a kaleidocycle of order 6 one position of which sketches a cube which may be completed with two "bolts" (order 3 symmetry).
The volume of the kaleidocycle is one third of the volume of the cube. proof: the base of the six pyramids is a half equilateral triangle whose long side, twice the small side, is the side of the equilateral triangle, and the third is its height which is equal to the edge c of the cube; the height of these pyramids is equal to the short side of their base base, i.e. (1/2)×c/(√3/2) = c√3/3; the volume of the kaleidocycle is therefore 6×1/3×(1/2×c×c√3/3)×c√3/3 = c³/3. This "cube" is linked to a curious geometric object: the oloid, another discovery by Paul Schatz. 

The "Konrad Schneider cube" contains a kaleidocycle of order 8 one position of which also sketches a cube which may be completed with two "bolts" (order 4 symmetry).
 
The three pieces build the eversible cube (Umstülpwürfel in German) : we move from the cube (positive form) to the rhombic dodecahedron (negative form) with a cavity equal to the initial cube.
The net of the kaleidocycle is easy to draw starting with a strip of eight format A (a√2×2a) rectangles. The eight link edges (four of length a and four of length 2a) are drawn in magenta (the pairs of the small ones superimpose themselves). The dashed segments point out cuts. 
references: 
• the site by Jürgen Köller (special kaleidocycles, also in German)
• Umstülpkörper by Ellen Pawlowski (2005, in German) • see also eversible polyhedra 
more kaleidocycles: IsoAxis  kaleido 2  AniKA (Marcus Engel)
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