Fig. 5

A visual depiction of simplices and cycles present in \(G_2\). Left: Four trivial cycles filled by individual 2-simplices: \([1 \quad 2 \quad 3]\), \([3 \quad 4 \quad 5]\), \([1 \quad 5 \quad 6]\), and \([ \quad 1 \quad 3 \quad 5]\). Center Left: A non-trivial, but null-homotopic cycle, 1, 2, 3, 5, 1 filled in by two 2-simplices \([1 \quad 2 \quad 3]\) and \([1 \quad 3 \quad 5]\). Center Right: All eight 2-simplices represented as the faces of a regular octahedron. Right: The closed surface of \(G_2\) is filled in by four 3-simplices \([1 \quad 2 \quad 3 \quad 6], [1 \quad 3 \quad 5 \quad 6], [3 \quad 4 \quad 5 \quad 6], [2 \quad 3 \quad 4 \quad 6]\)