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Figure 2. The dihedral group D3, which is the symmetry group of an equilateral triangle, or the outerautomorphism group of the real roots of the Borcherds–Kac–Moody algebra (the group of symmetry modthe Weyl group), is generated by an order two element corresponding to a reflection and an order threeelement corresponding to the 120◦ rotation.
Figure 3. A plane (directions|X, α) = 0 of orthogonality to a positive real root α always intersects the hyperboloidX|2 = 1, or equivalently the upper-half plane or the Poincar´e disk
Figure 4. (i) The wall of marginal stability for the two-centered solution with charges (projected onto the two-dimensional slice of moduli space equipped with a natural hyperbolic metric, andmapped to the Poincar´e disk and the upper-half plane
Figure 5. Tessellation of the Poincar´e disk using the groupto the three “mirrors”, namely the three sides of the regular triangle in the middle
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