Quadrays and Volumeby Kirby Urner 

For example, as we learned in An Introduction to Quadrays, the edge between (1,0,0,0) and (0,1,0,0) has a length and direction given by: (1,0,0,0)  (0,1,0,0) = (2,0,1,1). We could also write this difference as (0,1,0,0)  (1,0,0,0) which gives (0,2,1,1), a vector pointing in the opposite direction. However, since this is paper about volume, it is a vector's length, and not its positive versus negative directionality, which proves relevant in our calculations. 

For convenience, we can store the four vectors in a matrix, where each row represents a vertex. We can then feed a special case matrix to our volume function, a method defined within our generic tetrahedron class. Our class method for operating on such a matrix comes from Gerald de Jong. Although aware that Leonhard Euler had come up with an expression for the tetrahedron's volume, given only its six edge lengths as input, Gerald resolved to derive such an expression independently. Gerald's elegant result, which relates the algebraic terms to open and closed triangles, and opposite edge pairs, made its first public appearance in the form of a Java applet, subsequently modified to accommodate the synergetics approach, which takes our regular "home base" quadray tetrahedron as its unit of volume. 



Since the Amod is part of the concentric hierarchy definition in synergetics, we will add these new coordinates to allpoints and the new edge connections to our shapes table, and confirm a volume of 1/24. SHAPEID ID1 ID2  A1 ORG AF A1 ORG T1A A1 ORG AE A1 AE AF A1 T1A AF A1 T1A AE AE = midedge AF = midface ORG = (0,0,0,0) T1A = (1,0,0,0) AE = (T1A + T1B)/2 AF = (T1A + T1B + T1C)/3 


Picture four turtles starting from (0,0,0,0). A clock starts ticking off time intervals, and at each tick, the turtles each get to choose one of the 12 Cset directions. Turtle1, for example, might hop (2,0,1,1) in the first interval, and (2,1,0,1) in the second, bringing it to (4,1,1,2). This could go on for a long time, say n ticks. We can symbolize where Turtle1 ends up using the expression below, where the Greek letter sigma simply means the sum of all n hops, where each hop is a random selection from the Cset. 

One consequence of the above is that any shape assembled from various ivmTets will likewise have a whole number volume. This gives us a lot of freedom to define complicated components which nevertheless have simply expressed volumetric relationships. The vertices of fivefold symmetric shapes do not align with IVM sphere centers however, and cannot be modularized using whole numbers of A and B modules. Instead we make use of the Tmodule, 1/120th of the rhombic triacontahedron, to map this territory. David Koski's explorations have done the most to disclose the Tmod's ability to serve as the primary module in the world of fivefold symmetric shapes. Koski takes a Tmod and scales it up and down by a constant increment, providing a range of sizes, and shows how he can precisely fill a pentagonal dodecahedron, icosahedron, and many other familiar shapes using whole number combinations of these differently sized Tmodular "measuring cups". He even shows how the Tmod itself recursively selfdivides into an endless series of smaller and smaller Tmods  or a finite series plus a "remainder tet" terminator. For further reading:
Synergetics on the
Web 