## Tuesday, July 20, 2010

### Notes from the NPA: New platonic solids, new visions of Doppler shift

Michael H. Brill
[Here are some notes from a most unconventional convention. Perhaps they will make the ISCC blog light up… …MHB]

The Natural Philosophy Alliance is even more diverse than the ISCC. Its members--- artists, lawyers, physicians, chemists, physicists and mathematicians---all give voice via their own expertise to challenge current scientific theories. They agree with each other less often than ISCC members. (Yes, that’s possible.) I attended their meeting (NPA17) [1] in Long Beach, CA the week after ISCC met in Princeton. Two NPA nuggets might intrigue ISCC members: a geometrical shape with interesting optical properties, and a thought experiment to clarify the Doppler effect.

Artist Michael R. Evans [2] dubs the “atom” out of which he builds his creations the Trion-Re’. Realizable in paper or clear acrylic plastic, the Trion-Re’ looks like a shortened weaving-shuttle. You can make one by cutting out the 60°-arc-limited parts in the figure below. Assemble the parts so A is preserved, the three B vertices coincide, edge 1 meets 1’, 2 meets 2’, and 3 meets 3’. [Exercise for the reader: Is this construction mathematically possible or must it be forced? See “Trion Re’dux” below.]

On a mathematical note, the Trion-Re’ has the minimum number of faces (F), vertices (V) and edges (E) that satisfies Euler’s formula V + F - E = 2 for polyhedra. Although the Trion-Re’s faces are not flat when assembled, given the above construction they start out flat and are never stretched (i.e., have zero Gaussian curvature).

Another NPA nugget was Physicist Francisco J. Műller’s paper [3] asking, if we think we understand Doppler red-shifts of light, then how do shifts of non-light happen (e.g, for a Fraunhofer absorption line incurred by an interstellar cloud)? Let a stationary Earth E look at a star S (possibly receding) through a cloud C (possibly receding). Say the recession red-shifts 656 nm to 670 nm. (I will speak of shift rather than scaling because the wavelengths here are not very different from each other.) There are three cases.

Case I: If S and C recede together from E, then the S’s spectrum and C’s absorption line are red-shifted together (the latter to 670 nm). C receives 656 nm light from S with no Doppler shift (because C and S are not in relative motion). The cloud absorber stops that radiation, and the rest of the light is passed Doppler-free. The lengthening path from C to E then shifts the entire spectrum including the absorption line.

Case II: If S recedes while C and E are stationary, then S’s spectrum is red-shifted but not C’s absorption line. Now C is in the same frame as E. S’s light is already red-shifted upon reaching C, and in particular 642 nm light is shifted to 656 nm and is stopped by the absorber. The absorption is at 656nm, but the rest of the spectrum is shifted 14 nm higher.

Case III: If S is stationary relative to E and C recedes, then C’s absorption line is red-shifted but S’s spectrum is not. Light received through C outside C’s absorption band has no interaction with C. For these wavelengths, the cloud does not exist, so S and E are static and have a vacant path between them—incurring no Doppler shift. But light emitted at 670 nm is blocked by C (at 656 nm), and that line is Doppler-shifted back to 670 nm due to the recession C from E.

Műller and I seem to agree on the results of these three cases, but whereas I model Doppler shifts based on how the whole optical path stretches in time [4], Műller has a different view. I thought this problem would be interesting for ISCC thinkers. In a sense the assumptions are simpler than the ones we take for granted in color science. At first I saw the cloud as a filter, but Műller correctly noted that even a transparent filter interacts with the light at all wavelengths, unlike in parts of the Doppler example.

Oh, by the way, much is said in NPA about Einstein’s relativity theories, and I added to that this year [5]. Back to normal stuff next issue!

1. http://conf17.worldnpa.org/ and click Abstracts to read any paper.
2. M. R. Evans, The geometry of light, Proc Nat. Philos.Alliance, 7, 149-153 (2010).
3. F. J. Műller, The Doppler effect of absorption spectral lines in moving astronomic bodies (How can it happen?) Proc Nat. Philos.Alliance, 7, 336-342 (2010).
4. M. H. Brill, Doppler effect: surprises from the time domain, J. Nanophotonics 4, 041520 (4 Feb 2010).
5. M. H. Brill, Cochetkov’s speeding bola---yet another entanglement for special relativity, Proc Nat. Philos.Alliance, 7, 62-63 (2010).

P. S. Trion Re’dux

My Trion Re’ puzzle (above) has several levels of answer---so I won’t wait for the next issue to clue you in. Can the three flat leaves be rolled up into a 3D convex figure with three-fold symmetry about the axis AB? Yes, but….

Examined along the AB axis, any cross-section of the Trion Re’ is an equilateral triangle, and the triangles are all centered on the axis and have the same orientation. Any progression of triangle sizes (as a function of position on the AB axis) is enforced by the shape of the unrolled leaf (whether or not limited by 60° arcs). Since the corresponding triangle sides are straight and parallel, each face rolls and unrolls between 3D and flat. So the answer to my puzzle is “Yes.”

A this point, it bothered me that the lining-up of the triangles implies that the edges of the Trion-Re’ are plane curves in 3D. They are also plane curves when unrolled flat. So how do you roll a planar 60° arc out of its plane so it becomes again a plane curve? I was able to show this is not possible if you use a circular cylinder as a “curling iron”. But I still don’t know the cross-sectional shape of the cylinder that makes it work. Maybe some ISCC geometer can find it.

Then I heard from artist Michael Evans (through physicist Greg Volk): The faces of the acrylic Trion Re’ are not Gaussian-flat, but are closer to being parts of spheres! So we have two distinct constructions, folks, the paper-folding one (with its neat math problem) and the acryllic one (with its neat optical property). They may be “artistically equivalent,” a matter to be decided by Interest Group 3!