We can fit other objects into circulant loops and lines. Kneemo tells me to think about the Fano star diagram. The star has $9$ vertices as shown, with lines of three vertices. With the unmarked central triangle $(j_1, j_2, j_3)$, there are many triangles in the diagram. Choosing $2$ separated sets of three vertices, we can form the off diagonal $1$-circulant positions of a $3 \times 3$ matrix. The $2$-circulant entries also form lines in the diagram.
Alternatively, we could have filled the matrix $1$-circulants with the three obvious triangles: small $j_i$, large $J_i$ and $I$. This can be done so that the $2$-circulants fit the remaining three long lines.
6 years ago