Spectrum of the Laplacian on regular polyhedra

Figure 1.  Eigenfunctions on the cube

Figure 2.  Restriction of eigenfunctions on the cube to $ y = 0 $

Figure 3.  Eigenfunctions on the octahedron: (a) is obtained from (b) by rotating and dilating by $ \sqrt{3} $, which reduces the eigenvalue by a factor of $ \frac{1}{3} $

Figure 4.  Identified edges on the cube

Figure 5.  Identified edges on the tetrahedron

Figure 6.  Eigenfunctions on the tetrahedron

Figure 7.  Restriction of eigenfunctions on the tetrahedron to $ y = 0 $

Figure 8.  The torus covering

Figure 9.  Reflection symmetry

Figure 10.  Skew-symmetric (1-) reflection

Figure 11.  The hexagonal lattice

Figure 12.  A generic orbit ($ j > k > 0 $)

Figure 13.  Eigenvalues on the tetrahedron

Figure 14.  Counting function on the tetrahedron

Figure 15.  Identified edges on the octahedron

Figure 16.  Eigenfunctions on the octahedron

Figure 17.  Restriction of eigenfunctions on the octahedron to $ y = 0 $

Figure 18.  Types of reflections on the octahedron

Figure 19.  Symmetry properties of eigenfunctions on the octahedron

Figure 20.  Tetrahedron-type eigenfunctions

Figure 21.  Distribution of signs

Figure 22.  Counting function on the octahedron

Figure 23.  Eigenfunctions on the icosahedron

Figure 24.  Restriction of eigenfunctions on the icosahedron to $ y = 0 $

Figure 25.  Identified edges on the icosahedron

Figure 26.  Counting function on the icosahedron

Figure 27.  Reflections on the cube

Figure 28.  Symmetric properties of eigenfunctions on the cube

Figure 29.  Integer lattice

Figure 30.  Generic orbit

Figure 31.  Distribution of signs over the orbit on the cube

Figure 32.  Counting function on the cube