Friday, September 6, 2019

Reflection on “Dark Spectrum Part II”

As a new experiment for ISCC News, my column in Issue 487 (reproduced here), together with that of Carl Jennings, comprise an interdisciplinary dialogue within a single issue.

In the course of writing “Dark Spectrum Part II” for the current ISCC issue, Carl Jennings asked me for comments. In response, I began to think about the optics of Newton’s vs. Goethe’s experiment. My thought process changed through the dialogue, especially as it related to Figure 3 of Carl’s essay. This Hue Angles summarizes the essentials of our email discussion, which seems to reveal some heretofore unremarked differences between the experiments of Newton and Goethe.

I started off with the idea that Newton’s prism experiment passes collimated (uni-directional) light from the Sun through a hole in a light-blocking shade (like a window shade), and through a prism. The prism disperses the sunlight into a spectrum according to the various refrangibilities of the wavelength components of the light. Then, in one version of the experiment, the dispersed spectrum hits a screen, and is reflected as a multicolored pattern to the observer. Collimation is necessary because light from two directions incident on the same point will provide different banding, and the bands from multiple directions will superimpose to wash out the pattern. I was convinced that collimation, being essential to Newton’s experiment, also figured in Goethe’s experiment. The only difference, I thought, was that Newton looked at a narrow beam through a hole or slit, and Goethe looked at a broad beam with narrow blocking elements that would cast shadows the prism would refract differently according to wavelength. Accordingly, I reacted as follows to Carl’s Figure 3 and its caption (see below for figure):

Mike: The caption of Figure 3 states: “A pair of scissors against a bright white winter sky in Munich, through two prisms simultaneously. (Source: author).” A bright white winter sky is about as non-collimated as you can get, and on the face of it this seems incompatible with the color bands in Figure 3.  The only way to assure collimation is to position the prisms on the light path that includes the scissors and the eye. In that case, if the distance between the prisms is long enough, only light going nearly parallel in one direction through the first prism will intercept the second prism and hence get to the camera.

Carl: You discuss the color bands in the scissor image (Fig.3) as being incompatible with non-collimated light - but that is exactly the point - it happens when it shouldn't! None of the banding should happen, collimated or non-collimated, but the fact is it is there and is easily observable. Both prisms used in the photo were between the camera and the scissors, so no light was collimated. I found that two prisms made the banding more distinct, though it is observable with one, if you use a good prism.

Mike: I now think the paradox of color-banding with light from a white winter sky is not a paradox after all. Newton needed collimated light because Newton’s prism images the spectrum directly on a screen. In Goethe’s geometry, there is another element that must be in the optical train: a lens. A lens provides a point-to-point transfer from an object to an image (in respective object and image planes), whether or not the light is diffuse.  The plane of Goethe’s shadowing components was the object plane, the lens was in his eye, and the image plane was his retina (or a tangent plane thereof).  In your scissors example, the lens was that of the camera.  Of course, the eye’s lens is implicit in all these demonstrations, but it is physically essential in Goethe’s experiment in which the eye looks directly at the diffuse light through the prism(s). Newton’s experiment does not have the eye looking directly at the light through the prism, and no lenses are needed between the slit and the screen, so collimation of the spectrum-separated light is essential.

In other words, a lens (be it eye or camera) is essential for the diffuse white sky light to show bands when it passes the scissors (which should be in the object plane of the camera lens).  That role of the lens is essential to Goethe’s experiment. A lens is also part of Newton’s experiment because Newton used his eye to see the card-reflected spectrum, but the lens plays a different role here. It is a subtle point but should be understood.

Incidentally, Figure 3 suggests to me that, although a diffuse white sky exists in front of the camera, there must be very little light from behind the camera or there would be a white desaturating reflection from the front surface of the scissors.

The discreteness you have noted of the band colors---as opposed to their presence at all---is still a perceptual effect, as you have said before. I have no further thoughts on this matter now.

Carl: That is very interesting - I have never come across a description of boundary colors (even colorimetric ones, as in Koenderink or Bouma) that discuss the role of the lens. This is certainly a key feature to Goethe's phenomenological approach, but as far as I can tell does not exist in the literature.
One more question. Would sunlight passing through a hole in a window shade be already collimated? I ask because in Newton's own diagram of his experiment you can see that he has placed a lens in front of the prism, presumably to collimate the light.

Mike: Good question. The Sun is very far away (93 million miles), but it has a diameter of 0.864 million miles, which causes the Sun to subtend about half a degree of visual angle. The Sun’s rays depart from collimation by as much as ¼ degree.  Collimation is almost—but not quite—completed without the lens, and Newton obviously sought to do better.

Michael H. Brill

Figure 3. A pair of scissors against a bright white winter sky in Munich, photographed through two prisms simultaneously. (Reproduced with permission from Carl Jennings)

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