Thursday, May 16, 2019

The Mathematics of Flower Arrangement

I have just returned from the Philadelphia Flower Show. The theme was “flower power” and it featured music and visual references to the 1960s. Because I learned most of the math I know in the 1960s, I searched restlessly for something that would resonate with that memory. (Yes, contrary to popular belief, I both lived through the 1960s and remember them.)  Eventually I found a curiously mathematical-sounding reference in a description of flower arrangements: the Hogarth curve.  A spray of flowers was described as a Hogarth curve if the dominant elements comprised or suggested an s-shaped form. 

To the consternation of my companion, I immediately searching for Hogarth curves on my phone. There was no mathematics--the curves had been used to describe the aesthetics in drawing, painting, and flower arrangement, and their name derived from the “line of beauty” extolled by William Hogarth in his 1753 book, The Analysis of Beauty. Hogarth was an 18th-century English painter and writer. He (and many subsequent flower arrangers) saw s-shaped curves as signifying liveliness and activity that cannot be found with straight lines or other curves. I found no mathematical or scientific discussion of Hogarth curves, so I will speculate on an interpretation now.

There is one special point on an s-shaped curve called the point of inflection.  It is in the middle of the curve, and is the point where the curvature changes from negative to positive.  In other words, it is a point of zero curvature.  Imagine looking at an s-shaped curve drawn on a flat sheet of paper, and noting (with a very sharp pencil) the point of inflection. Now change your point of view (e.g., slant the paper away from you) and re-image the figure with a pinhole camera.  Surprisingly, the point of inflection of the new image is exactly at the image of the point you noted with the pencil.  The point of inflection is an invariant of the perspective projection.

This invariance is a mathematical property of projective geometry (which includes perspective). A projective transformation sends any straight line into another straight line. It is more general than a linear transformation because parallel lines when mapped may converge: the meeting point is called a vanishing point.  When you look at an s-shaped curve, there is only one point of that curve whose neighborhood is a straight line, and that is the point of inflection.  Three points on the curve can be made as collinear as desired by moving them closer and closer to the point of inflection. Since the neighborhood of the inflection point is a straight line segment, the segment will stay at the same part of the s-curve when that curve is projected to another planar image.

What does this have to do with Hogarth’s praise for s-shaped curves in art? It might be that the inflection point is anchoring a moving viewer to a distinct point in the picture, relative to which all else is swirling around. By being a view-independent feature, it may be an effect opposite to the elusive smile of the Mona Lisa, which (as explained by Margaret Livingstone at Munsell2018) is a low-spatial-frequency rendering that is noticeable only in peripheral vision, which sees with much lower resolution than our fovea (center of vision). 

In any event, this tiny vestige of math in the middle of flower arrangements may allow me to revisit the 1960s with a sense of entitlement to its memories.

Michael H. Brill

Tuesday, November 27, 2018

Visualizing Four-Dimensional Colorimetry

This essay is a return to the now-popular topic of tetrachromacy (four-color vision [1]), but with a geometric flavor that responds to a challenge by Jan Koenderink:

 “I hold the view that it is not possible to understand human color vision completely without having appreciated the tetrachromatic (or polychromatic of any order) embedding. I consider it to be somewhat of a scandal that the literature has so little to offer there.” [2, p. 200]

Since the publication of Koenderink’s book, as if on cue, researchers confirmed one example of a human tetrachromat [3,4].  So it is timely to start to answer Koenderink’s challenge, comparing trichromacy with tetrachromacy and including some observations that he himself made.

In three color dimensions (e.g., CIE space), it is common to project out one of the dimensions, to produce a 2D chromaticity space.  The physical-light domain in this space is a planar region delimited by the spectrum locus and the line of purples: i.e., the familiar horseshoe diagram.  Inside the diagram, let’s denote a white point W and an arbitrary color A.  If you draw a line from A through W, any points encountered thereafter are complementary to A.  If you draw the line in the other direction from W through A, you will eventually meet the spectrum locus or the line of purples. If the meeting is with the spectrum locus, the meeting point is called A’s dominant wavelength. If the meeting is with the line of purples, then A has no dominant wavelength.

In four color dimensions, the chromaticity domain has 3 dimensions—still accessible to our spatial visualization, perhaps with some difficulty. The spectrum locus is still a curve, but it is a space curve that spans all three chromaticity dimensions---like a bent wire hanger.  Now imagine the wire hanger shrink-wrapped by a plastic sheet.* Every point within the shrink-wrap can represent a physical light. Now, as in CIE space, denote a white point W and an arbitrary point A within the shrink-wrap.  Again you can draw a line between A and W.  Extending on the W side, the points are legitimate complements of A: they add in certain proportions to give W. Because of the abundance of shrink-wrap area relative to wire-hanger area (theoretically zero), it will come as no surprise that the line will likely end at a shrink-wrap point and not at a wire-hanger point: light A probably has no spectral complement.  Extending the line on the A side, one similarly encounters shrink wrap and not wire hanger. That denies the existence of a dominant wavelength for the vast majority of colors. The selection of possible outcomes is the same for 3D as for 4D colorimetry, but the odds for each outcome are staggeringly different.

You can also use the picture of wire hanger and shrink wrap in understanding the optimal color reflectances in tetrachromatic color spaces.

In trichromacy, it has been shown many times that the optimal reflectances (on the exterior of the object-color solid in tristimulus space) have values 1 or 0 at each wavelength, with at most two transitions between 0 and 1.  To derive the number of transitions [5] one can notionally slice the chromaticity space with a line and define the optimal reflectance transitions as the points on the spectrum locus that were impinged by the slice.  The optimality follows from the argument that 1’s inhabit all of the curve’s wavelengths on one side of the slicing line and 0’s occupy the other side---you can’t do better than that.  In trichromatic space, by the way, the number of transitions is two, if the spectrum locus is convex (which it mostly is).

One can use the same trick in tetrachromatic space, but now one is slicing a 3D chromaticity space with a plane. The number of intersections of the slicing plane with the spectrum locus (wire hanger) is the number of 1-0 transition wavelengths.   It now remains to find the tetrachromat’s analogue to convexity of the spectrum locus and use it to minimize the maximum number of crossovers.   One could begin by asserting that the spectrum-locus curve must span the three dimensions of the chromaticity space, and posit as an axiom that no plane can cross the spectrum locus more than 3 times. 

In performing this exercise, I am realizing that, whereas one can use either chromaticity or tristimulus space to visualize basic colorimetry for trichromats, the chromaticity domain is essential for visualizing tetrachromatic relations.
*I intend shrink-wrap as a metaphor for the boundary of the 3D convex hull of the spectrum locus. Real shrink-wrap will sometimes incur concavity, so my metaphor is imperfect.
[1] Jennings C. All the colors we cannot see: tetrachromacy in humans. ISCC News Issue 482 (Spring 2018), pp. 13-14.

[2] Koenderink J. Color for the Sciences. Cambridge, MA: MIT Press, 2010, Section 5.10.3.

[3] Jordan G, Deeb S, Bosten J, Mollon JD. The dimensionality of color vision in carriers of anomalous trichromacy. J of Vision 10 (2010), p. 12.

[4] Jameson KA, Winkler AD, Goldfarb K. Art, interpersonal comparisons of color experience, and potential tetrachromacy. Invited proceedings paper for the 2016 IS&T International Symposium on electronic Imaging (EI 2016). Technical session on Human Vision and Electronic Imaging.

 [5] West G, Brill MH, Conditions under which Schrödinger object colors are optimal, J. Opt. Soc. Am. 73, 1223-1225 (1983).

Michael H. Brill