7 Critical Constants

7.1 Six Zone

2 auto

\begin{array}{l} \frac{\bar{𝐵𝐷}}{\bar{𝐴𝐵}} & = 1.0514622 \\ \frac{\bar{𝐴𝐶}}{\bar{𝐴𝐵}} & = 1.7013016 \\ \cos 𝜃 & = 0.4472136 \\ \sin 𝜃 & = 0.8944272 \\ \tan 𝜃 & = 2.0000000 \\ \sin \frac{𝜃}{2} & = 0.5257310 \\ \cos \frac{𝜃}{2} & = 0.8506507 \\ \tan \frac{𝜃}{2} & = 0.6180339 \end{array}

7.2 Ten Zone

4 auto

\begin{array}{l} \frac{\bar{𝐵𝐷}}{\bar{𝐴𝐵}} & = 0.7136441 \\ \frac{\bar{𝐴𝐶}}{\bar{𝐴𝐵}} & = 1.8683446 \\ \cos ϕ & = 0.7453559 \\ \sin ϕ & = 0.6666666 \\ \tan ϕ & = 0.8944272 \\ \sin \frac{ϕ}{2} & = 0.3568220 \\ \cos \frac{ϕ}{2} & = 0.9341723 \\ \tan \frac{ϕ}{2} & = 0.3819660 \end{array}

7.3 Ten Zone

6 auto

\begin{array}{l} \frac{\bar{𝐵𝐷}}{\bar{𝐴𝐵}} & = 1.1547005 \\ \frac{\bar{𝐴𝐶}}{\bar{𝐴𝐵}} & = 1.6329931 \\ \cos X & = 0.3333333 \\ \sin X & = 0.9428090 \\ \tan X & = 2.8284271 \\ \sin \frac{X}{2} & = 0.8164965 \\ \cos \frac{X}{2} & = 0.5773502 \\ \tan \frac{X}{2} & = 0.7071067 \end{array}

7 $A$ line radii of an icosahedron with edge $= C$ . The pair of radii outline one end of a golden diamond.

10 $B$ line radii of dodecahedron with edge $= C$ . One pair outlines one end of the skinny diamond; another pair outlines an end of a maraldi diamond.

\begin{array}{l} A & = 0.9510565 C = \cos 18 ° C \\ A & = 1.0981855 B \\ B & = 0.8660254 C = \cos 30 ° C \\ B & = 0.9105930 A \\ C & = 1.0514622 A \\ C & = 1.1547005 B \end{array}

Table 7.1: Proportions Of

A

B

, and

C

Lines

7.4 The Appearance of the $C T^{n}$ Series in the Dimensions of the Triacontahedron and the Enneacontahedron

14Both these patterns can be found as sub patterns of the five-fold symmetry patterns of page.

16 All edges are $A$ linesone $A$ line is perpendicular to the plane of page.

18 All edges are $B$ lines $A$ line is perpendicular to the plane of page.

7 Critical Constants