# 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

## The green line** shows one appearance of the equilateral triangle which appears at four stages of the inversion of the Schatz Cube. The large dots represent vertical hinges. Red dots indicate that the hinge lies above the plane of the horizontal hinges, which are purple, and blue dots indicate that the vertical hinge lies below the plane of the horizontal hinges.

## 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 2: The four triangular states of the Schatz Cube.

## Comparing FourFold Schatz and the Schatz Cube reveals that there are two further ways to link the elements of the Schatz Cube to obtain invertible models which lie within FourFold Schatz. These are shown in Figure 3.

# Figure 3: Two models which contain the elements of the Schatz Cube and are contained within FourFoldf Schatz. The model on the left is known as Inversis "Star Variation".

## Note that the green triangle appears again in each of these models, as it will do four times in a cycle of each model which is contained within FourFold Schatz.

## Remarkably, it is possible to obtain related models within FourFold Schatz. These differ from the Schatz Cube, however, in that although they consist of the same two elements, these elements are not present in equal numbers.

## When each element of FourFold Schatz is cut into four pieces to produce the Schatz Cube (indeed it produces four identical cubes) two different shapes - which are mirror images of each other - are created. The Schatz Cube contains three of each of the two different shapes, as do each of the models shown in Figure 3. If the elements are labelled "R" and "L", their order in the Schatz Cube is R L R L R L. In Inversis Star Variation the order is R R L L R L, while in the third model the order is R R R L L L.

## Two further models with the same elements, but not in equal numbers, can be found. These models - which are shown in Figure 4 - are mirror images of each other and contain respectively 5 R elements and 1 L element or 1 L element and 5 R elements.

# 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 5: Two characteristic models which are not contained within FourFold Schatz.

## The motion of these models differs essentially from the motion of Fourfold Schatz. Although a version of the triangular (in this case not equilateral) form occurs four times in each cycle, there are also four states in which four hinges are parallel to each other and perpendicular to the other two hinges.

## Figure 6 shows these four states during the inversion of the model on the left of Figure 5. In this diagram, the open red circle represents a vertical hinge (above the plane of the neighbouring horizontal hinges) which remains fixed throughout the cycle. The horizontal hinges (purple) have different thicknesses to represent that the hinge corresponding to the thinner line lies below the hinge corresponding to the thicker line. Only four of the green lines are visible as two of them are vertical, joning the two neighbouring hinges.

# 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