| Summary: | This paper is part of a series that describes the Fibonacci icosagrid quasicrystal (FIG) and its relation to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>8</mn></msub></semantics></math></inline-formula> root lattice. The FIG was originally constructed to represent the intersection points of an icosahedrally symmetric collection of planar grids in three dimensions, with the grid spacing of each following a Fibonacci chain. It was found to be closely related to a five-fold compound of 3D sections taken from the 4D Elser–Sloane quasicrystal (ESQC), which is derived via a cut-and-project process from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>8</mn></msub></semantics></math></inline-formula>. More recently, a direct cut-and-project from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>8</mn></msub></semantics></math></inline-formula> has been found which yields the FIG (presented in another paper of this series). The present paper focuses not on the full quasicrystal, but on the relationship between the root polytope of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>8</mn></msub></semantics></math></inline-formula> (Gosset’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mn>4</mn><mn>21</mn></msub></semantics></math></inline-formula> polytope) and the core polyhedron generated in the FIG, a compound of 20 tetrahedra referred to simply as a 20-Group. In particular, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>3</mn></msub></semantics></math></inline-formula> symmetry of the FIG can be seen as a five-fold or “golden” composition of tetrahedral symmetry (referring to the characteristic appearance of the golden ratio). This is shown to mirror a connection between tetrahedral and five-fold symmetries present in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mn>4</mn><mn>21</mn></msub></semantics></math></inline-formula>. Indeed, the rotations that connect tetrahedra contained within the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mn>4</mn><mn>21</mn></msub></semantics></math></inline-formula> are shown to induce, in a certain natural way, the tetrahedron orientations in the 20-Group.
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