[math-fun] constant-width convex objects
1. The Minkowski sum of two constant-width convex objects, is another. Scaling and/or rotating one, yields another. 2. Consequently, once you find a few, then there should automatically be an infinite set of them. Also, one can perform a Minkowski sum of one object, plus itself rotated by an amount which is distributed via a "normal" distribution with very small "typical angular width" (infinite Minkowski sum) and in this way obtain a constant-width object which is Cinfinity-smooth, but is arbitrarily near in shape to any starting const-width object. 3. Video: http://www.youtube.com/watch?v=jYf3nOYM_mQ 4. One of the cites that has been mentioned: Bernd Kawohl, Christof Weber: Meissner’s Mysterious Bodies, The Mathematical Intelligencer 33,3 (September 2011) 94-101 http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf cites many other papers and mentions some remarkable theorems.
Item 1. below shows that starting with an arbitrary convex body W in 3-space of constant width and taking the Minkowsi sum S of the 24 bodies g(W), where g runs through the group G of symmetries of the tetrahedron (a subgroup of O(3)), guarantees that S will be of constant width and have the symmetry of the tetrahedron. If the original B were far enough from the sphere, it would be easy to show that S has no additional symmetries. Taking on faith that the Meissner tetrahedron is of constant width, that could be used as the B in the above procedure to definitely get a body of constant width with exactly the symmetry of the tetrahedron. Here's an article more technical than the one in the Intelligencer, but very readable (at least in the opening sections that I've read): < http://www.lama.univ-savoie.fr/~lachand/pdfs/spheroforms.pdf >. --Dan On 2013-01-11, at 9:34 PM, Warren Smith wrote:
1. The Minkowski sum of two constant-width convex objects, is another. Scaling and/or rotating one, yields another.
2. Consequently, once you find a few, then there should automatically be an infinite set of them. Also, one can perform a Minkowski sum of one object, plus itself rotated by an amount which is distributed via a "normal" distribution with very small "typical angular width" (infinite Minkowski sum) and in this way obtain a constant-width object which is Cinfinity-smooth, but is arbitrarily near in shape to any starting const-width object.
3. Video: http://www.youtube.com/watch?v=jYf3nOYM_mQ
4. One of the cites that has been mentioned: Bernd Kawohl, Christof Weber: Meissner’s Mysterious Bodies, The Mathematical Intelligencer 33,3 (September 2011) 94-101 http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf cites many other papers and mentions some remarkable theorems.
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On 1/12/13, Warren Smith <warren.wds@gmail.com> wrote:
1. The Minkowski sum of two constant-width convex objects, is another. Scaling and/or rotating one, yields another.
What appeared fiendishly involved in the special case M_V (+) M_F is immediately seen to be a one-liner when abstracted. Minkowski sum commutes with (ie. is invariant under) translation, so the diameter lines in both summands can be brought into coincidence; then the (1-space) mean width in that direction is the mean of the widths. QED & DUH! WFL
Below it should have been made explicit that we consider only a single parallel pair of summand diameters in each direction separately. Strictly speaking, both should furthermore be translated to meet the origin, in order for their sum / mean to become a formally 1-space affair. More generally, for convex summands, in any given direction only their extremal points are relevant to their sum / mean. Comments on the Low Rollers screed are invited. While it suffers from an absence of diagrams, that on page 2 of Roberts report is a partial substitute. And in that connection I have already spotted one misprint: Stage (A): for "d = 0" read "d = c" WFL On 1/13/13, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 1/12/13, Warren Smith <warren.wds@gmail.com> wrote:
1. The Minkowski sum of two constant-width convex objects, is another. Scaling and/or rotating one, yields another.
What appeared fiendishly involved in the special case M_V (+) M_F is immediately seen to be a one-liner when abstracted.
Minkowski sum commutes with (ie. is invariant under) translation, so the diameter lines in both summands can be brought into coincidence; then the (1-space) mean width in that direction is the mean of the widths.
QED & DUH! WFL
participants (3)
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Dan Asimov -
Fred lunnon -
Warren Smith