= Solution
We construct the two domains explicitly from seven congruent <triangles>. Let a scalene reference <triangle> have sides $a,b,c$ opposite its three labelled vertices. Choose its three side lengths in a small open neighbourhood of $(1,1,1)$, with a fixed strict ordering. Thus the three angles are distinct and lie between $\pi/4$ and $\pi/2$. Each copy keeps these side labels.
For the first assembly glue the following tile pairs, reflecting across the indicated side:
$$
\begin{array}{c|c|c}
\text{side}&\Omega_P&\Omega_L\\ \hline
a&(2,3),(6,7)&(1,3),(5,7)\\
b&(4,6),(5,7)&(2,6),(3,7)\\
c&(1,5),(3,7)&(4,5),(6,7)
\end{array}
$$
Each coloured adjacency graph is a tree with central tile $7$ and three two-tile arms. Place tile $7$ in the plane and obtain every other tile by the prescribed reflections. At the equilateral reference shape the copies have disjoint interiors and form a polygonal disk; the same holds throughout a sufficiently small neighbourhood, since the only contacts between tiles are the prescribed edges and their incident vertices. In particular each assembly is a bounded connected <planar domain>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-20-drums.png]
{title=Seven-triangle point and line assemblies with matching Dirichlet spectra}
To prove equality of spectra, index tiles by the seven nonzero <vectors> $x\in\mathbb F_2^3$, using $x=(x_1,x_2,x_3)$ with label $x_1+2x_2+4x_3$. Let
$$
s_a=I+E_{12},\qquad s_b=I+E_{23},\qquad s_c=I+E_{31}.
$$
These <involutions> act on points by $x\mapsto s_sx$ and on nonzero <covectors> by $y\mapsto s_s^{-T}y=s_s^Ty$. Their swaps are exactly the two columns of the table. Write $P_s,Q_s$ for the corresponding permutation <matrices>. The <Fano plane> incidence <matrix>
$$
B_{y,x}=\mathbf1_{y\cdot x=0}
$$
satisfies $BP_s=Q_sB$, since incidence is preserved by a linear transformation and its inverse transpose. Each point lies on three lines, while two different points have one common line. Therefore
$$
B^TB=2I+J,
$$
where $J$ is the all-ones <matrix>, proving that $B$ is invertible.
For the <Dirichlet boundary condition>, a glued side has a positive swap and an exposed side has diagonal entry $-1$. Both trees have the same bipartition, with sign <matrix>
$$
D=\operatorname{diag}(1,1,-1,1,-1,-1,1).
$$
Their signed reflection <matrices> are $M_s=-DP_sD$ and $N_s=-DQ_sD$. On a glued edge its endpoint signs are opposite, so the off-diagonal entries are $+1$; on an exposed edge the entry is $-1$. Consequently the <invertible matrix> $C=DBD$ satisfies
$$
CM_s=N_sC\qquad(s=a,b,c).
$$
We spell out why this gives <isospectrality>. Pull each function back from the seven tiles to the reference <triangle>. Its seven boundary values along side $s$ form a <vector> $u_s$ satisfying $M_su_s=u_s$: this imposes equal values at glued edges and zero values at exposed edges. Outward <normal derivatives> of a smooth <eigenfunction> satisfy $M_sv_s=-v_s$, giving opposite <normal derivatives> on glued edges. The identities above preserve both conditions under $u\mapsto Cu$. Since $C$ is constant, it also commutes with the <scalar> <Laplacian> on each <triangle>.
One can include all weak <eigenfunctions> without any corner regularity issue by <orthogonalization of a transplantation matrix>. The <matrix> $C^TC$ commutes with the symmetric $M_s$, so $O=C(C^TC)^{-1/2}$ is orthogonal and still intertwines $M_s,N_s$. On the disjoint tiles, $O$ preserves the $L^2$ norm and the sum of the <Dirichlet energies>; it maps the <trace>-matching form domain $H_0^1(\Omega_P)$ bijectively to $H_0^1(\Omega_L)$. Thus it is a <unitary equivalence> of the <Dirichlet Laplacians>, proving equality of every <eigenvalue> with its multiplicity.
Finally, the only reentrant vertices of either polygon are the three vertices of tile $7$, with interior angles four times the corresponding reference-<triangle> angles. All other boundary angles are less than $\pi$. These three distinct angles force any hypothetical <isometry> to map each labelled central vertex to its counterpart. An <isometry> between connected open <planar domains> is a restriction of a rigid motion: its derivative is orthogonal, and preservation of the flat <Levi-Civita connection> makes that derivative constant. After aligning the two central <triangles>, this rigid motion fixes three noncollinear points and is the identity. But the two domains do not coincide: the tile beyond side $a$ of the central <triangle> has its next attachment along $b$ in the point assembly and along $c$ in the line assembly. These occupy different open regions. This contradiction proves nonisometry.
Therefore \b[the three freely varying side lengths give a three-parameter family of nonisometric Dirichlet-isospectral bounded <planar domains>]. The parameters include scale; no congruence normalization removes it.
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