Given this shape:
Is it possible to divide the cyan area into 5 equal area shapes
- Each shape is the same
- Each shape has an edge touching the red square
- Each shape has an edge touching the outside.
- No diagonal lines.
Its reasonably easy to perform #2 and #3 as long as you violate #1
And I don't believe its possible to actually satisfy #1, and the very fact its contained within a square suggests #1 is not satisfiable given the presence of #2.
Though its just a bit of fun really =).
Context: Was simply devising a town plan for a guild oriented town for a minecraft world, and it became possible that we might want 5 guilds, and the fun of giving each guild a fair equal area region , in conjunction with all 5 guilds having a shared space in the middle.
Diagonals are unwanted as it makes dividing the land fairly and applying region controls onerous, as most things are rectangular, regions included.
So while you can roughly approximate a diagonal with sufficiently many rectangles, the less steps, the better.
If it is not possible
Please provide reasoning as to why not.
It would be interesting to see what sort of alternatives people can come up with, perhaps there is an optimal shape that results in all the shapes being highly similar geometrically, despite not being identical.