Re: [math-fun] Closed curve puzzle (SPOILER)
I wrote: << In the plane: Let C be a C^oo simple closed curve. Let a "double-normal" be a line segment whose endpoints lie on C and which is normal to C at each of them. C must have a double-normal. (Proof: Consider the longest segment from C to C). Question: Let a "simple" double-normal be one that intersects C only at its endpoints. Must C have at least one simple double-normal? Prove or find a counterexample.
Consider an equilateral triangle T in the plane, and on each side build a 15-30-135 triangle, in the same sense, pointing into T. (Each side of T is the longest side of one 15-30-135 triangle.) Now remove the three original sides of T, leaving the 6 remaining sides of the obtuse triangles, which make a starlike hexagon H with 3-fold rotational symmetry. It's easy to check that if H is C^1-approximated by a smooth curve, then every double-normal segment (there are three) intersects the curve in its interior. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
Dan's closed-curve puzzle reminded me of a possible refinement to the idea of convexity. One property of convex plane figures is that a (straight) line cuts them into (at most) two pieces; while a (straight) line cuts a concave figure into various numbers of pieces, depending on the details of the figure & line position. Moreover, two straight lines cut a convex figure into at most four pieces, while concave figures can have more pieces. We could classify figures according to the spectrum of possible numbers of pieces from K cuts. Whether this is useful depends on what other properties we can connect to the classification. As a beginning, here's a conjecture: The intersection of two convex figures is convex. I think that the intersection of a concave figure (of cut-spectrum S) with a convex figure will have a cut-spectrum S' that's a "subset" of S. This is also related to the number and type of concavities for a figure. There may be theorems about the number of concavities for the intersection of two concave figures; maybe this number is limited to the sum of the concavity numbers of the two ingredients. Similar questions appear for the union of two figures; and for the pieces that result from cutting a concave figure. Rich ----------------- Quoting Dan Asimov <dasimov@earthlink.net>:
Let C be a C^oo simple closed curve. <snip>
another way to generalize convexity is to play with the quantifiers. S is convex if: for every x,y in S, the line segment L(x,y) between x and y is in S. what does it mean if: for every x,y,z in S, at least one of the line segments L(x,y), L (x,z), L(y,z) is in S? (some authors call such sets 3-convex, some call them Valentine convex) more generally, for any n points, if at least m of the (n choose 2) lines are in S? are there non-trivial examples of such sets? what are their properties? erich On Jan 10, 2010, at 9:20 PM, rcs@xmission.com wrote:
Dan's closed-curve puzzle reminded me of a possible refinement to the idea of convexity. One property of convex plane figures is that a (straight) line cuts them into (at most) two pieces; while a (straight) line cuts a concave figure into various numbers of pieces, depending on the details of the figure & line position. Moreover, two straight lines cut a convex figure into at most four pieces, while concave figures can have more pieces. We could classify figures according to the spectrum of possible numbers of pieces from K cuts. Whether this is useful depends on what other properties we can connect to the classification. As a beginning, here's a conjecture: The intersection of two convex figures is convex. I think that the intersection of a concave figure (of cut-spectrum S) with a convex figure will have a cut-spectrum S' that's a "subset" of S. This is also related to the number and type of concavities for a figure. There may be theorems about the number of concavities for the intersection of two concave figures; maybe this number is limited to the sum of the concavity numbers of the two ingredients. Similar questions appear for the union of two figures; and for the pieces that result from cutting a concave figure.
Rich
----------------- Quoting Dan Asimov <dasimov@earthlink.net>:
Let C be a C^oo simple closed curve. <snip>
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Say that the *concavity* of a polyhedron (2D or 3D) equals the number of points laying within the convex hull. Could (in principle) the measures you propose give different results? --> different 'types' or 'characters' of 'concavity'? Or, differently, the least count of points to be removed so as to *obtain* a convex figure. (more tricky) Wouter.
participants (4)
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Dan Asimov -
Erich Friedman -
rcs＠xmission.com -
wouter meeussen