I'm wondering whether it is possible to split an octagon into a finite number of quadrilaterals, such that the result is symmetric from all 8 directions (sides or points). There is one condition — any new points that are created must be on the inside of the octagon (i.e. no new points on the edges of the octagon).
The figure below illustrates 4 of my attempts. Most of them (the $1^{st}$, $2^{nd}$ and $4^{th}$) are not symmetric from all 8 directions, and are therefore no valid solution. The $3^{rd}$ one is symmetric, but creates points on the edges.
I'm curious how one could prove that there is (or isn't) a solution to such a problem?
I'm assuming that you want rotational symmetry of degree 8. (If what you want is reflection along the 4 different axis, then let me know.) Such a configuration is not possible.
Recall that the angles in a octagon sum to $180^\circ \times 6$, and the angles in a quadrilateral sum to $180^\circ \times 2 $.
Since we can only add internal points, note that adding a point would increase the sum of angles by $360^\circ$. If we add a point that is not in the center, then by rotational symmetry, we must add all 8 symmetric points.
By double counting the angles, this shows that we must have $8k+3$ or $8k+4$ quadrilaterals.
However, by rotational symmetry of the figure, there must be $8l$ of $8l+1$ quadrilaterals.