From paper plate circles to scrap paper to scrap cardboard the circle pattern does not change. Using different materials folded to the circle pattern shows the adaptability and strength of structural organization and the limitations of different materials. Pattern and material form together reveal possibilities not possible in separation.

Below) are six views of a tetrahedron model using two pieces of scrap cardboard. Each piece is folded and scored as show in previous blogs (Order Without Boundary and Order Without Boundary II: Geocoding ). Only the folds necessary for the tetrahedron net as it appears in the circle have been folded. The measure for both folding is the same, the shape of the cardboard and placement of folding are different. Given those two variables every time a piece is folded the same way it will have a different proportional reconfiguration. This individual tetrahedron system can not be duplicated unless the configuration of paper is exactly the same as the position for folding, then the variables become consistently the same for all pieces. Then there would be a formula, not a discovery.

The surface that extends beyond the folded tetrahedron often makes it difficult to see the form of the tetrahedron or to even identify the pattern. Because of the differences in the original flat shapes it might even look arbitrary. Each piece is folded to form half a tetrahedron, two triangles each, then joined at right angles to each other forming a full tetrahedron. There is an inclosed "solid" tetrahedron that is central to holding the system together. You might be able to spot it by finding the small equilateral triangles.

Below) are multiple views of a truncated tetrahedron using four pieces of scrap cardboard. There are more creases added to the pattern in order to form the hexagon unit, one of the properties of the truncated tetrahedron (the folded 4-frequency diameter grid.) Each piece is different in configuration and placement of folds, but the measure and folding process remains consistently the same.

It is difficult to discern any symmetry or underlying pattern of organization by looking the diversity of different views. Each scrap is a different shape where each creased grid is located differently on each piece breaking the symmetry of anything recognizable. With the exception of one view above showing a vertical line of symmetry all other views seem random. In both models above, the formed pattern is not obvious by looking at the entire system. The form is an expression of two variables, the configuration of surface used and placement of the folded grid. All aspects are subject to the consistent organization of the circle pattern and sub-pattern of tetrahedron and truncated tetrahedron. The possibilities of variations appear to be countless depending on combinations of the two variables with the two constants. This reflects what we see with any two points when touched together and creased that will always generate two more points at the intersection of the crease and perimeter.

There are no folded line segments; each crease goes all the way across from edge-to-edge; as observed in folding the circle; the complete pattern for any shape. This provides a consistency of pattern and proportion throughout the surface allowing for a certain amount of reconfiguration beyond second level patterns as they are formed. To fold a line segment without connection to the perimeter is leaving information out, thus creating missing information.

The strength of these models lies with the structural nature of pattern and the rigidity of the material. When made from paper they are more fragile and appear less strong because of more flexibility in the extended surface area. Every material will have its effect on how we perceive the pattern and organization of order that seems to be primary in a forming process.