# Models Contained Within FourFold Schatz.

## FourFold Schatz is the most complete model with six elements when the angle between successive hinges is 90 degrees. It is easy to see that four complete Schatz Cubes are contained within FourFold Schatz, The following diagram is a schematic representation of the Schatz Cube. # Figure 1: Schatz Cube

## As clarification, the following diagram shows all four appearances of the equilateral triangle during a complete cycle of inversion. The red and blue dots distinguish between the otherwise identical outlines. # Figure 4: The two models with unequal numbers of R and L elements which are contained within FourFold Schatz.

## There are many other models comprised of these elements which do not lie within Fourfold Schatz. Two are shown in Figure 5. The absence of a complete green equilateral triangle demonstrates that these models are not contained within FourFold Schatz. # Figure 6: Four states during the inversion of the model on the left in Figure 5 in which four higes are parallel to each other and perpendicular to the other two hinges.

## There are many other models which consist of the elements of the Schatz Cube. No attempt has been made to ascertain their number. Many such models which are - in principle - invertible will not permit full inversion because their hinges are too long. The hinges of the Schatz Cube are the longest hingest which permit inversion of this cube. Many of the models above could have longer hinges. Many other possible models would require shorter hinges.

** For the mathematician. The green lines represent the common normal of two adjacent hinges.

The necessary existence of the green equilateral triangle permits an easy proof that the five models listed here exhaust all models within FourFold Schatz.

The angle between successive hinges is always 90 degrees. Thus, as one steps around the triangle each hinge is twisted 90 degrees clockwise or anti-clockwise relative to its predecessor. The sixth step will only permit closure of the model if its final hinge (or axis) of this step corresponds to the initial hinge (or axis) of the first step. Thus the the sum of all rotations between adjacent axes must be 0, 360 or -360 degrees.

This is only possible if the elements consist of 3 R and 3 L elements (sum of rotations is 0 degrees), 5 L and 1 R elements (sum of rotations is 360 degrees) or 5 R and 1 L elements (sum of rotations id -360 degrees).

The only essentially distinct models which satisfy these conditions are:-

1.1) RLRLRL - Schatz Cube             1.2) RRLLRL - Star Variation             1.3) RRRLLL - Figure 3

2.1) RRRRRL - Figure 4                    2.2) RLLLLL - Figure 4