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Published online by Cambridge University Press: 25 June 2025
The attractive pattern of 168 shaded and 168 unshaded triangles shown in Figure 1 has an interesting history. Since its discovery by Klein in 1878 (see [Klein 1879]), it has often been reproduced; a close cousin (Figure 2) inspired the badge of the 1978 International Congress of Mathematicians in Helsinki. This article considers its origins, which lie in the fields of nineteenth century geometry and the theory of equations.
In each of the 14 slices emanating from the center there are 12 shaded and 12 unshaded triangles, so there are 168 of each kind. The sides of each triangle are arcs of circles orthogonal to the boundary circles, or are diameters. The figure can be continued in this fashion to reach indefinitely close to the boundary, and it provides in this way a tessellation of the non-Euclidean plane. The unshaded tessellation is preserved by non-Euclidean reflection in any side of any triangle (i.e., by inversion) and so has the group of all such reflections as its symmetry group. The group generated by all products of pairs ofreflections is the symmetry group of the shaded figure.
Klein had been led to construct the figure because of its use in studying a certain polynomial equation (described at the end of this paper) for which the group permuting the roots is PSL(2; ℤ/7ℤ), sometimes known as G168 because of the number of its elements. Our first task, then, is to understand this group geometrically.
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