Question

A builder had been asked to lay an octagonal patio using square paving slabs. To do this, he cuts some of the bricks in half down the diagonal. Each paving slab is 1m wide. What will the perimeter of the patio be

Answer 1

Assuming you want an expression for the possible total area of the patio: if "completed" would be a rectangle of dimensions n, m: each must be >=3 to allow for octagonal corners. But each of the 4 corners must be missing, that's diagonals comprising (1/2)a^2, (1/2)b^2, (1/2)c^2, and (1/2)d^2, where a,b,c, and d must be variously limited and co-limited so as to allow at least 1 linear side of the original rectangle to be exposed. So A = (n*m)-(1/2)a^2-(1/2)b^2-(1/2)c^2-(1/2)d^2 as an expression.
Now, imagine replicating your possible (potentially irregular) octagons onto a plane and juxtaposing them so as to create a paved network. What geometric properties might such a network have? You now have a miniscule idea what nature does with silicate networked minerals, except that takes place in 3-D, with tetrahedra of SiO4 .