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        riddles >> medium >> What is the shape?
        
(Message started by: Christine on Oct 4^th, 2012, 9:16am)

Title: What is the shape?
Post by Christine on Oct 4^th, 2012, 9:16am

Question #1
A shape has property that every line through a given point P is a line of symmetry. Prove that shape must be a circle (with P its center).

Question #2
A shape has the property that for every direction there is a line of symmetry parallel to that direction. Must the shape be a circle?

Title: Re: What is the shape?
Post by rmsgrey on Oct 5^th, 2012, 5:46am

on 10/04/12 at 09:16:00, Christine wrote:

Question #1
A shape has property that every line through a given point P is a line of symmetry. Prove that shape must be a circle (with P its center).

Is an annulus a shape?

[hideb]It's fairly straightforward to prove that the shape must have radial symmetry with center P (given a point at distance r, every image of that point is also at distance r, and every point at distance r is an image of that point) but that's all that the "every line through P is a line of (reflective) symmetry" condition tells us.

You need other conditions to rule out a collection of concentric circles[/hideb]

Quote:

Question #2
A shape has the property that for every direction there is a line of symmetry parallel to that direction. Must the shape be a circle?

[hideb]If you take a single point, then any line of reflection not passing through that point produces a different image of that point from any other such line (the lines run in different directions, so the lines connecting point to image must too). Unless all the lines of reflection intersect at that point (in which case you have the setup from Question #1) the point has at least one distinct image point.

With two points, the only lines that don't create at least a third point are the one joining them and the perpendicular bisector of the line segment joining them - no other line passes through both nor reflects each onto the other. For each additional image point, at most two lines can map one or other of the two source points onto that image point, so there must be an uncountable number of image points.

At most countably many of those points can be isolated points (having some finite minimum distance to the closest other point); the remainder form a collection of densely filled regions of the plane (lines and areas). Assuming one such region has a non-fractal boundary (taking the boundary of a curve to be the curve itself), for most points on that boundary the curvature at that point must be well defined. For any such point where at most countably many lines of reflection intersect, there are uncountably many image points, each on the boundary of a densely filled region, and with the same curvature at that image point.

Again, those image points must include a densely filled region, entirely comprised of boundary points, so a curve of some description, with the same curvature at every point, and a curve with the same curvature at every point is an arc of a circle. An arc of a circle, the points of which are all images of a single point, must be produced by multiple lines of reflection (intersecting at the center of that circle) so cannot have endpoints (the image of an endpoint must the other endpoint, but each endpoint has more than one image) so must be a full circle.

By the same arguments as for a single point, either the images of a given set of points are those same points, or there are uncountably many distinct images of that set of points. Considering the images of a single isolated dense region (one for which there is a minimum distance, closer than which no points not part of the region are to any point of the region), either they are all the same region, or there are uncountably many isolated dense regions (which is impossible). So, for any given isolated dense region, either it's the entire plane, or each of its boundaries is a circle.

Any circular boundaries have to be concentric (each of them is centered at the mutual intersection of the lines of reflection) so again you have radial symmetry.[/hideb]

So, there are two possibilities:

1) [hide]The "shape" is the entire plane[/hide]
2) [hide]The "shape" has radial symmetry[/hide]

Note: I assume, but have not proved, that any collection of uncountably many objects in the plane cannot have each of them separated from all the others - that there must be (uncountably many) points in the collection for each of which, for any finite distance, you can find another point in the collection which lies closer.

Title: Re: What is the shape?
Post by peoplepower on Oct 6^th, 2012, 5:33am

Quote:

Note: I assume, but have not proved, that any collection of uncountably many objects in the plane cannot have each of them separated from all the others - that there must be (uncountably many) points in the collection for each of which, for any finite distance, you can find another point in the collection which lies closer.

Hint: [hide]How large is a collection of pairwise disjoint open intervals in the set of real numbers?[/hide]