10 Sections

10   Sections

1If the lines of the 31-zone star followed no pattern, it would be possible to form ${\displaystyle \frac{31\times 30}{2\times 1}}=465$ planeseach with a different orientation.

2 In our 31-zone system, the pairs of lines form only 121 different planesthis is because some of the pairs of lines lie on the same plane. Thus, our 31-zone star is singular.¹

3 In any one 31-zone star there are:

10.1   R Section

10.2   $S$ Section

22In the $S$section there is only one triangle possible:

10.3   $T$ Section

27In the $T$section there are two kinds of triangles possible:

10.4   $V$ Section

32In the $V$section there are equilateral triangles:

10.5   $X$ Section

37In the $X$section there is a rectangle:

10.6   $Y$ Section

42In the $Y$section there is a rectangle:

10.7   The Formation of Triangles

44The $R$, $S$, $T$, and $V$ sections contain triangles; the $X$ and $Y$ don’t. They couldn’t because they don’t have enough lines. Every triangle has sides running in three different directions, but the X and Y have only two directions.

45 If we have three zones in one plane, then we can form a triangle. If it is a structure with only certain lengths of structural members, then we must have them proportioned correctly so that they begin and end at the vertices of these particular kinds of triangles. This is a more difficult problem. The $A$ and $B$ lines are unique in that stars formed of only $A$ lines or B lines are non-singular. They define maximum numbers of planesmost uniformly distributed through spacewith minimum numbers of lines. But they cannot triangulate themselvesmixed $A$ and $B$ lines do form triangles as can be seen in the triangles within the $R$ sections. But it is the 15 $C$ lines that serve as the triangulators in our star. In the $T$ and $V$ sections they form triangles with themselves while in the $R$ and $S$ sections the very relationship between the lengths of the $A$ and $B$ lines was determined by choosing the same $C$ lines as a base for two different trianglesone with $A$ lines and one with $B$ lines.

10.8   A Star Problem

46An interesting problem for the designer would be to construct a star in which every plane defined in the system by two lines could be triangulated by other lines in the system. Or, to arrange lines in space, so there were never only two lines in one plane. This is impossibleas you add more lines to triangulate a plane, the new lines define new planes with old lines, and the star needs even more new lines to form triangles. That the $X$ and $Y$ sections in the thirty-one zone star are unable to triangulate themselves cannot be avoided. A proof of this assertion can be derived from the necessity of all convex polyhedra to have some faces with fewer than six sides.

47 We have shown all the triangles that can be formed with our system. We have not illustrated all the convex polygonsthose with an even number of sides are straight forward. The existence of irregular pentagons, septagons, nonagons and eleven sided figures has not been investigated.

48 There are a finite number of classes of such an angle similar convex polygons. If one does not insist that the polygons be convex, then there are infinite numbers of such polygons.

49 In three dimensions, the smallest convex polyhedron is the tetrahedron. A tetrahedron has 4 triangular sides. The stock of possible triangles to form tetrahedra is those we have shown in the $R$, $S$, $T$ and $V$ sections. These triangles must fit together along the proper planes to form tetrahedra within our system. For instance, in the $V$ sections there are equilateral triangles, but the dihedral angles between $V$ sections do not allow us to form a regular tetrahedron in our system.