The 3-piece Twist Connection

In a single solid 3 square pieces connection with the pieces intersecting each other, one of them has to be able to rotate to release a crossing piece. The intersected cubic volume of the pieces can be divided in 8 units. 1 unit is a cube with each side length as a notch dept wich is half the width of the square piece. Each piece needs 2 units in axial direction to keep together. That leaves the designer 2 units to place. The rounded piece can only have a straight axle of (2) units. If the pieces crossing the rounded piece gets 1 unit each, either can be released depending of the twist direction. If one crossing piece gets the 2 units only the other crossing piece can be released with a twist in the right direction. And if the other crossing piece also is rounded, it can release the first rounded piece if twisted in the right direction.
And further; in a puzzle with adjacent 3-piece connections, when one connection is solved the next can be designed to use the extra space to slide apart and thus does not need any twisting piece.

A rotating piece needs tangential space around it to rotate. In a multi twist connection puzzle, e.g. a 3x4 pieces block burr, the four pieces are cornered and have no tangential space at the inner sides to rotate. The designer can get over this by also rounding the two inner corner edges of the twisting piece, or make the connection loose to give space. This is necessary when the rounding of the rotating axle is circular.

The Triangular Rounding

When you drill a hole, sometimes the drill with two edges starts to make a triangular shaped hole. This is when the drill's one edge is held momently in the move by the vibration or the material and the other edge swings around until it stops and the first edge swings around etc. If you measure the diameter of the triangular rounded hole it is the same as drill diameter in every direction. At the triangle edges the drill edges has stopped momently and the rounding in between is shaped by the opposite edge swinging.
If you draw straight lines around the triangular hole it will be a square figure with the same width as the drill.

Such triangular (the geometric figure is known as the Reuleaux Triangle) formed axle rotating in a square hole with the same width as the axle diameter, will all the time "fill" the hole.
So in the puzzle design, if you make the twist piece axle rounding triangular with radius 1 unit and align one of the edges with the nearest corner of the piece, during rotation the corner edges will slide along the adjacent pieces and the "tangential space" is taken on the other sides of the twist piece. But to be precise in the example, the twist piece corner edge nearest the rounding need to be broken a little to go smooth.

Conclusion: With triangular rounded twist pieces it would be possible to design a new range of multi twist burr type of wooden puzzles beside the well known slide burr type. Combinations of the two types are also very interesting, with or without internal holes.

Examples to study are the symmetrical Five-Star and Seven-Star burrs with internal holes, and the solid symmetrical Four-Block and Eight-Square and Nine-Block burr designs.

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