CYCLIC ARRANGEMENTS of ANGLES:
Dihedral angle = 90 – C5
Hip Kernel:
Standard arrangement
of Angles
R1
DD
T1
SS
T2
Dihedral angle = 90 – C5
Dihedral angle = SS
DD
90 – P2
90 – P2
P2
90 – R1
90 – SS
R1
90 – C5
90 – DD
Transformation 1:
Transformation 2:
This is the equivalent of
interchanging “Miter” and “Bevel”,
with their corresponding Dihedral
angles. The angle on the
“Compound face” (90 – P2)
maintains its original location.
There is no fundamental change or
transformation; the entire kernel is
positioned so that the face
containing angle SS lies on the
deck. This grouping relates the
complements of the constituent
angles.
TABLE of ANGLE ARRANGEMENTS:
Since the transformations could each be independently applied to the
original kernel, the process may be continued with successive kernels. For
example, it is possible to apply T1 first, then T2; or, T2, then T1, T2,
T1…etc., each application producing a different grouping. The cycle returns
to the initial arrangement upon the tenth operation.
All kernels thus transformed, and the equations relating their
angles, are equivalent. It is important to observe that this method differs
from the technique used to create the formulas in “Hierarchy of Developed
Kernels”, especially the Group 1 angles. Both sets of kernels use the same
angular values as starting points for their respective developments, but the
orientations of the triangles substituted in the locations of DD and SS are
completely different in each case. Thus, the “Hierarchy …” kernels generate
distinct types and values of angles.
Different angle groupings can occur “naturally”, a kernel extracted
from another kernel may possess a different ordering of angles than a kernel
extracted from the stick. One pattern matches a kernel extracted directly
from the stick at a compound joint, and is used as a basis for worksheet
programs to generate values for this type of joint.
This process is not restricted to the kernel of Common rafter angles; it
may be applied to any kernel. The table below lists the cyclic permutations,
beginning with the Standard position, and applying, in order, T2, then T1,
followed by T2, T1, … etc.
DD
90 – SS
P2
C5
R1
DD
90 – SS
P2
C5
R1
Table of Cyclic Arrangements of Angles
SS
R1
P2
90 – DD
P2
R1
90 – C5
90 – SS
R1
90 – P2
90 – SS
R1
90 – DD
90 – SS
C5
90 – R1
90 – SS
C5
90 – P2
DD
C5
SS
C5
DD
90 – R1
P2
DD
90 – C5
DD
P2
C5
C5
DD
DD
P2
P2
R1
R1
90 – SS
90 – SS