Previous page of Issue

A Structure of Diagrams (Part 4) A Three Dimensional Problem

SERIES: A Structure of Diagrams
AUTHOR: Jim Schofield


This rather long series of papers are actually the contents of a complete book on Diagrammatic Realisation. Indeed, by the time the whole series has been delivered a couple of years will have elapsed. Nonetheless, as the individual papers each has a clear and separate purpose, they don’t have to be seen as part of the Whole. The range of diagrams delivered in this book is vast and perhaps sometimes quite surprising. Apart from principles of design, such as need-to-know, redundancy and multipurpose forms, there are many diagrams useful in quite unusual areas. The more obviously essential and functional diagrams are those developed for use with electronic circuits, but they range far and wide via hierarchical forms used in complex situations all the way to Music and Dance. These latter forms were developed to aid in both Composition and Choreography as well as in Teaching. A series of highly structured, and carefully organised, diagrams are presented that the author used in the design and construction of a music Synthesizer/Sequencer.
In quite a different area, there are many diagrams connected with research into Tilings (tessellations), to demonstrate how these can be both analysed and created, as well as revealing ideas such as Families of Tessellations.
Perhaps the most surprising are a series of animated diagrams using simple colour cycling which can deliver otherwise impossibly complex forms for analysis and development. A whole section is dedicated to to Music, and particularly to open tunings for the guitar, but also including new notation systems for finger picking styles of playing. Finally a whole range of music aid diagrams are introduced for use in both Composition and Improvisation.

One particular chapter shows how quite difficult 3D problems in the stacking of comples, re-entrant strands to fill space were both tackled and solved. The Geography of Dance works and their dual use to also include access to multimedia resources is also fully
described. Another area connected with Dance shows the design of special Parametered , animated figures to use superimposed upon successive frames of a dance video to capture movement for future analysis and use..
Finally among these special aids, is one for superimposing sequences of position on top of a moving dance video to communicate the real dynamics of movement, via precursor and subsequent positions. Rudolf Laban’s 26 orientations in space around an individual dancer which he used both in his famed Dance Notation and in his many Scales and exercises was captured into a polyhedral aid – called a Laban Pure Form, which was produced in both a series of desktop models and even a remarkable step-inside form, which proved very effective in many alternative uses. Philosophically useful diagrams used in explaining the role of Abstractions in human Thinking are also
included, as are a series of abstract mathematical diagrams which could be used in creating useable models of aspects of Reality. Finally a chapeter considers how diagrams can help with problems of Cosmology – particularly in the modelling of the actual “progress” of the Big Bang. Even maps developed for use in directing marketing have a chapter of their own....


1.This is an example of using diagrams to actually solve a problem, rather than merely illustrating an already understood solution. Here, a three dimensional, re-entrant and infinite “strand” (The Soma Strand), was known to tessellate to completely fill Space. But the crucial questions, “In how many different ways could it do this?”

2.Both model-making and 3D graphics packages did not help, for the first was too difficult to complete in sufficient numbers of units, and with adequate accuracy, and the second was simply not equipped for research, but only for presentation. After a great deal of effort both methods were abandoned, and an alternative was sought.

3.The author therefore constructed his own tool, by turning a simple 2D package into an “isometric” 3D system, by allowing only three directions in any drawings. Both x an d y were handled as normal, while z was implemented at a 45o angle. It worked.

4.As long as all constructions were built from the rear, so that all those forward overlapped those behind, the system could deliver a usable method: the user could correctly interpret what was happening in 3D.

5.The “grid-locking” feature of the package was modified to the unit size of the cubes from which the Soma Strand was constructed, and the “brush” facility allowed subsections of the image to be replicated wherever necessary.. It soon became possible to use the modified package effectively to both visualise and solve quite difficult 3D problems.

6.To conceptually construct a comprehensive list of possible tessellation attitudes of one strand with another, I first constructed all possible relations of a single cube with another. There turned out to be 13 of these (plus their mirror images), and these gave 12 face-to-face tessellations, 8 vertex-to-vertex tessellations, and 6 edge-to-edge tessellations.

7.But they were NOT all available for the full strand tessellations. The strands were constructed into flat “racks”and only 5 of the above forms were possible. The when attempting to form these into space filling “stacks” only 2 forms worked.

8.It is hard to see how any other system could have so clearly revealed what was possible..

Read Paper (PDF)

Left click to open in browser window, right click to download.

Previous paper in series
Next paper in series