= Seven-triangle Dirichlet transplantation
Take seven copies of a scalene <triangle>, with sides labelled $a,b,c$ opposite its three vertices, and reflect adjacent copies across their glued side. For the first <planar domain>, the glued pairs are $a:(2,3),(6,7)$, $b:(4,6),(5,7)$, $c:(1,5),(3,7)$; for the second they are $a:(1,3),(5,7)$, $b:(2,6),(3,7)$, $c:(4,5),(6,7)$. Both coloured trees have central tile $7$. For triangles sufficiently close to equilateral the interiors do not overlap. If its three angles are distinct and between $\pi/4$ and $\pi/2$, the only reentrant vertices are the original central vertices, with angles four times the corresponding triangle angles. They force any proposed <isometry> to preserve the labelled central triangle, which is incompatible with the different attachments on its arms.
For the spectral proof index the nonzero vectors and covectors of $\mathbb F_2^3$ by $1,\ldots,7$, using binary coordinates. The <Fano plane> incidence matrix is $B_{y,x}=\mathbf1_{y\cdot x=0}$. It satisfies $B^TB=2I+J$, so is invertible, and intertwines the point and line permutations of each of the three elementary transvections. Put $D=\operatorname{diag}(1,1,-1,1,-1,-1,1)$. The signed reflection matrices for the <Dirichlet boundary condition> are $-DP_sD$ and $-DQ_sD$, hence $C=DBD$ intertwines them. Its polar orthogonal factor gives a <unitary equivalence> of the <Dirichlet Laplacians>, preserving both the $L^2$ <inner product> and the Dirichlet energy of the piecewise functions. This produces a three-parameter family of noncongruent isospectral <planar domains>.
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