Bodies with Minimal Constant Width and Tetrahedral Symmetry ___________________________________________________________ Fred Lunnon, Jan 2013 Ernst Meissner was an eminent applied mathematician in his day (1912). On first encountering this topic --- some 30 years ago I suppose, perhaps excusing numerous recent lapses of memory --- I wondered why a good engineer would be satisfied with something as ugly as the asymmetrical solutions now given his name. Failing to to appreciate the part played by minimal surface area and volume in his investigations --- discussed by Kawohl & Weber --- and preferring full tetrahedral symmetry, in I had waded; just as Patrick Roberts has, and no doubt a few more folk besides. Failing somehow to persuade my construction to hang together, I eventually was obliged to put it aside. This time round, Roberts has proved doughtier; sadly however, despite provision of a plethora of painstakingly prepared pictures, his report manages to follow such an circuitous logical path, that just having a another shot myself looked easier than attempting to check his. The process proves so gratifyingly straightforward that I can no longer imagine what my earlier difficulty could have been --- which must after all be preferable to the alternative! Stage (O): inflate the 4 faces of the unit-edge tetrahedron out to congruent caps of unit-radius spheres, meeting the neighbouring vertices, but with great circle boundaries to be specified later. The gaps over the 6 edges are to be filled by congruent lunes of cyclides: envelopes of pencils of spheres in two orthogonal ways, having circles of principal curvature, and 9 degrees of freedom. Stage (A): nail down the cyclide. Take a section through a plane of symmetry of the tetrahedron, and inflate the isosceles triangle out to constant width. Attach arcs of radius 1 over two sides, centred on triangle corners; and arcs of radius b over the base, c over the opposite corner, both with centre P at distance d along the altitude and outside the corner. [The disarmingly detailed diagram on page 2 of Roberts assumes without justification that d = 0.] Inspection of the lines joining P to the mid-point and corners of the base, together with Pythagoras and constant unit width, establishes b + c = 1 ; 2(c + d) + rt2 / 2 = 1 ; b^2 = 1/4 + (d + rt2 / 2)^2 ; which solve easily yielding c = d = (2 - rt2) / 8 ; b = 1 - d = (6 + rt2) / 8 . In particular, the completed small circle actually passes through the nearby triangle corner. It is a little harder to justify an assumption that the cyclide has pinch points at tetrahedron vertices, but we're going to assume that anyway. Now the cyclide is determined, and furthermore the tetrahedron edge lies on its surface --- a bonus feature suggesting that we're on the right track. Stage (B): take x-axis along the base, z-axis along the altitude, y-axis along the edge through the opposite corner. Using Pythagoras, the pencil sphere with centre (x, 0, z) and radius f(x) satisfies f(x) = z ; x^2 + (z + rt2 / 2 + d)^2 = (b - z)^2 ; whence f(x) = (1 - 4 x^2) d . Stage (C): consider a general diameter joining points on cyclides at opposite tetrahedral edges. This passes through the centres (x, 0, f(x)), (0, y, -f(y) - rt2 / 2) of pencil spheres tangent to the envelope at those points. By Pythagoras, the distance between their centres equals the square root of x^2 + y^2 + (f(x) + f(y) + rt2 / 2)^2 = (1 - f(x) - f(y))^2 , using f(x) definition, with (f(x) + f(y))^2 conveniently cancelling. Hence the width along that diameter --- the distance between cyclide surface points on the spheres --- equals constant 1 . Stage (D): inspect the boundary between cyclide and unit radius spherical cap over a tetrahedral face. This is a longtitudinal circle on the cyclide at x = +/- 1/2 with radius 1 - f(1/2) = 1 , coincident with a great circle of the sphere passing through the tetrahedral vertices. Furthermore cyclide and sphere meet tangentially, since both are perpendicular to radii from the vertex; so the composite surface remains differentiable, away from vertices. Bingo! Now what about the Minkowski mean of a pair of non-congruent Meissner bodies, mentioned by Kawohl & Weber? Elementary properties of Minkowski sums suggest --- now WFL has at last got his head around some, courtesy of strenuous combined efforts by DA and WDS --- that this also has minimal constant width, and composes spheres and cyclides as before. Any local alteration in curvature of such a surface would entail some counter-balancing alteration at its opposite side, in order to maintain the width at the expense of destroying the symmetry. Hence it looks a safe bet that Roberts = Minkowski holds, as conjectured by Dan Asimov, and earlier dismissed by myself with gratuitous contempt. A more detailed investigation along the lines above would not come amiss ... But now more generally, does minimal constant-width with tetrahedral symmetry uniquely determine the body modulo similarity --- why should no others exist? The singularities at the vertices may be avoided via expansion to a parallel surface at the cost of sacrificing minimal width, the larger surface composed of segments from 8 spheres and 6 cycloids. Though differentiable, this solution is less elegant than the analytical surface proposed by Fillmore. It would be nice to see analytical parametric forms or implicit equations for the latter, provided some heroic geek is prepared to decipher the spherical harmonic technicalities involved! [The Asimov discussion thread critique of the Fillmore n-space simplex corollary is inapplicable here, since we already are in possession of a continuous body for submission to Fillmore's finite analytic smoothing procedure.] References contributed by various math-fun sources: General page: en.wikipedia.org/wiki/Reuleaux_tetrahedron Discussion thread: http://mathoverflow.net/questions/54252/are-there-smooth-bodies-of-constant-... Newslist thread: [math-fun] rounded tetrahedron Demo video: http://www.youtube.com/watch?v=jYf3nOYM_mQ Patrick Roberts "Proof of Constant Width of Spheroform with Tetrahedral Symmetry" Corvallis Oregon, (August 2012) http://www.xtalgrafix.com/Reuleaux/Spheroform%20Tetrahedron.pdf Bernd Kawohl & Christof Weber "Meissner's Mysterious Bodies" Mathematical Intelligencer vol 33 (2011) 94--101 http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf Jay P. Fillmore "Symmetries of surfaces of constant width" J. Differential Geom. vol 3 (1969) 103--110. intlpress.com/JDG/archive/1969/3-1&2-103.pdf Thomas Lachand-Robert & Edouard Oudet "Bodies of constant width in arbitrary dimension" Mathematische Nachrichten vol. 7 (2005) 740--750 http://www.lama.univ-savoie.fr/~lachand/pdfs/spheroforms.pdf ___________________________________________________________