The sculpture with the lines on it is included in Geometry and the Imagination as a photograph. So those lines may very well be the same ones arranged by Klein. --Dan P.S. The following passage from this document may be slightly relevant: < http://www.cs.rutgers.edu/~decarlo/readings/interrante-sg03c.pdf > (illustrations omitted): ----- 4.5: Parabolic curves — perceptually less-relevant shape feature lines Many people, and most notably the mathematician Felix Klein (as related by [Koenderink 1990]), have explored the possibility of enhancing the perception of a surface’s shape by marking the locations of the parabolic curves. Parabolic curves are the locus of points on a surface where one of the principal curvatures is zero and correspond to places where the surface is in some sense locally flat. The Gaussian curvature, which is equal to the product of the two principal curvatures, is zero at these parabolic points, which on generic surfaces separate the hyperbolic (or “saddle-shaped”) patches from the elliptic (convex or concave) regions. Although they are geometrically very meaningful [Koenderink and van Doorn 1982], arguments for the perceptual significance of these curves are not particularly strong. Various attempts, over the years, to gain deeper perceptual insights into the shape of a surface from the explicit representation of the zerocrossings of Gaussian curvature have been of only marginal success, if any at all. Figure 4.18, from [Hilbert and Cohn-Vossen 1952], illustrates one of the most famous instances of an attempt to use parabolic lines to reveal the features of a surface shape; in this case the possible aim was, reputedly, to better understand the “mathematical relations concerning artistic beauty” [Koenderink 1990]. Figure 4.19 illustrates a very similar attempt by [Bruce et al. 1993] to gain insight into the shape features of the human face from a surface partitioning by parabolic lines. Thirion [1994] described one disadvantage of decomposing a surface into patches using the parabolic lines: the problem of instability (a relatively small, relatively distant change in surface shape may have a relatively large effect on the shape of a parabolic line). Ponce and Brady [1987] alluded to another practical problem that arises when defining parabolic lines on a surface. While principal curvature values can be used to help differentiate the most significant ridge or valley features, it is not clear, from purely local information, how to successfully determine which zero-crossings of Gaussian curvature correspond to important patch boundaries and which merely indicate insignificant detail. The main difficulty I see with attempts to divide a surface into perceptually meaningful subunits based on zero-crossings of Gaussian curvature, however, is that this representation is just not very perceptually intuitive. The difficulty we have in estimating or even verifying the accuracy of parabolic line placement, confirmed by [Koenderink 1990] in his discussion of the lines on Apollo Belvedere, is evidence of this gap between mathematical and perceptual understanding. I do not mean to say, by all of this, that parabolic lines are not useful shape descriptors. Displays of Gaussian curvature properties have been shown to be of great practical use in computer aided surface design [Seidenberg et al. 1992] and in other computational geometry applications in which their values directly correspond to quantities of interest [Maekawa and Patrikalakis 1994]. It’s just that illustrating the locations of the parabolic lines on a surface does not appear to be a promising technique for conveying perceptually meaningful shape information in an intuitive and immediately understandable way. ----- On Apr 10, 2014, at 9:01 AM, Whitfield Diffie <whitfield.diffie@gmail.com> wrote:

Given that Klein died in 1925, I suspect it was doen by hand by an artist who felt the surface for the poins of zero curvature. What I would think would be interesting would be to get a three-dimensional digital representation of the statue; apply some more formal technique; and see how well the two agree.