using congruent tiles
Definition
We say a shape S is called n-convex if n congruent copies of S can be assembled into one convex shape.
We say a shape S 'has spectrum (n,...,m)' if S is k-convex for k=n,...m, but not k-convex for any other k.
'Trivial' and 'non-trivial' solutions:
Sectors of a circle have spectra of the form {1,2,3, . . . n}
if the sectors cannot form a complete circle, and the form {1,2,3, . .
. n,2n}
or {1,2,3, . . . n,2n+1} if they do.
Therefore, a 50 degree sector of a circle has spectrum {1,2,3}.
A 360/n degree sector of a circle dented so that the pieces fit
together into a circle,
has spectrum {n}.
We will call such radial arrangements around one fixed point 'trivial'
solutions
for a given spectrum because they are straightforward and easy to find.
We call a shape S with spectrum {n,...,m) a 'non-trivial' solution for
this spectrum
if there is a k in {n,...,m) such that no convex shape can be achieved
by
arranging k copies of S in a 'sector-like' mannner, i.e. by radially
arranging all k copies
with the same vertex around one fixed point.
The problem of convex tilings was originally dicussed by Erich Friedman (in April 1999), see this link.
However, Erich
- gives only one solution per spectrum
- does not mention whether other solutions exist and how easy they are to find,
- whether other solutions can be chosen to be of a specific type (polyamond etc)
This of course leaves many questions unanswered.
Table of solution:
Table covers all finite spectra plus the special case
N-{1} .
The table only contains non-trivial solutions (except for n=2, n=3).
