You also have scaling symmetry. Look at that:
x^3y+y3^z+z^3x = 0
Can there be an equation more simple than that? It describes „The Riemann Surface of Klein with 168 Automorphisms”. And yet this simple equation, when analyzed, gives rise to beautiful 168 triangles representing "the fundamental domain".
The method of drawing is extremely simple. ;)
It is described in a paper "The Riemann Surface of Klein with 168 Automorphisms"
by
HARRY E. RAUCH1 AND J. LEWITTES
Research partially sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. AF-69-1641
...
….We call attention to an incorrect answer to (iii) by Hurwitz ([7], p. 159, criticized in [1]) and an abortive attempt on (i), (ii), and (iii) by Poincare in [16], p. 130, all noticed after the completion of our work.2
2. Klein's surface
Klein originally obtained his surface S in the form of the upper half-plane identified under the principal congruence subgroup of level seven, Gamma(7), of the modular group Gamma. In this form it is necessary to compactify the fundamental domain at its cusps. Klein's group then appears as Gamma/Gamma(7), which is simple and of order 168.
We need, however, another representation given by Klein, one which we recognize today as the unit circle uniformization of S. In the unit circle draw the vertical diameter L1 and another diameter L3 making an angle of Pi/7 with L1 and going down to the right. In the lower semicircle draw the arc L2 of the circle which is orthogonal to the unit circle and to L1 and which meets L3 at the angle Pi/3. Let t be the non-Euclidean triangle enclosed by L1,L2,L3 and let Ru R2i R3 be the non-Euclidean reflections in L1,L2,L3respectively. R1,R2,R3 generate a non-Euclidean crystallographic group, which we denote by (2, 3, 7)', with t as a fundamental domain. The images of t under (2, 3, 7)' are a set of non-Euclidean triangles each of which is congruent or symmetric to t according as the group element which maps t on it has an even or odd number of letters as a word in R1 R2i R3. These triangles form a non-Euclidean plastering or tesselation of the interior of the unit circle. The union of t and its image under R2 is a fundamental domain (with suitable conventions about edges) for the triangle group (2, 3, 7), which is the group generated by ….. A convenient fundamental domain for N is the circular arc (non-Euclidean) 14-gon Delta shown in Fig. 1. It will be noticed that t appears as the unshaded triangle immediately below and to the right of P0. There are 168 unshaded triangles, which are the images of t under Gamma168 in Delta and 168 shaded triangles, which are the images of t under anticonformal elements of (2, 3, 7)'. ….
And here, attached, is my own rendering:
