A 'curvy' shape with rational diameter:perimeter

[Twelfthroot]

I was wondering whether we can find a 'curvy' shape in the plane such that the ratio of the/a diameter to to the perimeter is rational. I figure we could construct one if we could find a simple curve γ : [a,b] -> R^2 fulfilling these requirements:

|γ(b)-γ(a)| is rational,
\int_\gamma\,\mathrm{d}s is rational, and
the curvature of γ is nowhere 0.

Or if we like, without loss of generality we can assume we're just looking for a nonzero-curvature rational-length curve connecting (0, 0) to (1, 0), but either way I'm not sure how to investigate this. My intuition is that there is no such curve, but how might we go about showing that (or is there one?)

[Giallo]

What do you mean by "curvy shape"? A subset of the plane bounded by a continuous loop? By a C^\infty loop? Or a C^\infty function with compact support whose ratio integral/length is rational?

If the first or the second I think any circle with rational radius should work...

[mike-l]

The ratio of a chord to the arc length is 2 sin theta/2 : theta, the ratio is continuous and non constant, so takes rational values. Pick any theta that gives such a rational value, draw a circle with radius 1/(2 sin theta/2) passing through (0,0) and (1,1).

Edit: Grammar

[Twelfthroot]

Ah, yes, I see that I failed to formalize what I was getting at, so I think I'll just throw out what I was looking at and see if someone else can help specify what I mean.

First, clearly any circle with rational radius has irrational perimeter. I suspect it is also the case that an ellipse with rational axis lengths has irrational perimeter -- is this true? I would think that without a factor of 1/π in the lengths, we can't 'cancel' the irrational π in the arc length, but stranger things have happened. I then thought about a shape made of smoothly joined identical parabolas with rotational symmetry about the origin, and (though I only considered the case of 4 parabolas, with 3 being the least possible) at a glance it seems that if the distance from a parabola vertex to either the focus or the origin is rational, the figure's perimeter must be irrational.

So the idea I'm working with is that if a shape is 'built from / specified by' rational lengths then the perimeter is irrational (unless of course the curve is made of lines). That is, in going from the rectangular to the curved we "lose rationality". But as was pointed out (more or less), between any two points we can find a (continuous) curve of any length, so I'm not sure what I'm getting at. Thoughts?

[mike-l]

In general, you can take any two shapes and continuously deform one into the other. Infinitely many 'in between' shapes will have rational perimeter. Depending on your definition of 'radius' you may be able to fix two shapes with constant radius and deform them with that radius staying fixed. (So you could, for example, go between a diamond and a circle)

[Qaanol]

I expect there are curves of constant width that have perimeter a rational multiple of that constant width. Whether or not we can explicitly define one with an algorithm is another matter, but I think we probably can.

[PM 2Ring]

On a related note, there are shapes you can construct using circles of rational diameter which have a non-transcendental or even rational area.
http://mathworld.wolfram.com/Lune.html

[gmalivuk]

I expect there are curves of constant width that have perimeter a rational multiple of that constant width.

That very article mentions the theorem that this is not the case, because the perimeter is always pi times the width.

[chenille]

So the idea I'm working with is that if a shape is 'built from / specified by' rational lengths then the perimeter is irrational (unless of course the curve is made of lines). That is, in going from the rectangular to the curved we "lose rationality". But as was pointed out (more or less), between any two points we can find a (continuous) curve of any length, so I'm not sure what I'm getting at. Thoughts?

I guess the main question is what it should mean to be specified by rational lengths. Something that fits my intuitive sense of the idea would be an algebraic curve - that is, one specified by a polynomial equation - where all the coefficients are rational numbers. In that case, you do get the odd thing like cardioids where the arc length is also a rational number. I know that example isn't smooth, but is it the sort of thing you were thinking of?
Edit: Or an astroid might be a better example, since there is an obvious diameter to work with.

[MartianInvader]

Ok, so your interested in some sort of "diameter" value and an arclength of the perimeter. Just to be clear, you're not looking for a rational ratio between the two, but rather a case where both are rational? Is that correct?

If you just want the ratio to be rational, then there do exist, say, ellipses whose arclength is a rational muliple of their major axis. That's because if you continuously increase the major axis while holding the minor axis fixed, the ratio of major axis to perimeter will continuously change (I guess it approaches 1 as the axis gets long), which means it hits a bunch of rational values along the way (archimedian principle + mean value theorem = yay!)

[mike-l]

Ok, so your interested in some sort of "diameter" value and an arclength of the perimeter. Just to be clear, you're not looking for a rational ratio between the two, but rather a case where both are rational? Is that correct?

If the ratio is rational, then normalizing makes both rational.

[MartianInvader]

Heh, good point. So there are ellipses where the perimeter and at least one axis is rational. And if you've got any other sort of shape where the perimeter varies continuously with any single "diameter" value, then there are examples of that shape where both are rational.

[Timefly]

Consider a 1 dimensional elastic band in the plane with a rational arc length. By deforming the elastic band continuously the diameter will also change continously. Therefore there must be a 'curvy' shape with rational perimeter and perimeter, depending on your definition of curvy.

[gmalivuk]

Perimeter is easy to define, and "curvy" isn't much harder. The difficulty is in defining "diameter" for non-symmetrical shapes.

[chenille]

The standard definition, used for both metric spaces and graphs, is the least upper bound of distances between points. Since the opening post said a diameter, I imagine it also allowed a width along an arbitrary axis. As pointed out, though, either of those makes it easy enough to find a curve with rational perimeter. I think the case where the whole shape has to be somehow specified by rational numbers is less clear but more interesting to explore.

[Twelfthroot]

Ah, yes, the astroid is definitely the sort of thing I was thinking about -- both in terms of its geometric properties and its defining equation. Very interesting; thanks to all for the info.

Perhaps by "specified by rationals" I mean an algebraic curve with rational coefficients. Now that I've seen the astroid perimeter I'm satisfied that my vague conjecture was indeed false, but for curiosity's sake, I wonder: is there a smooth (or at least everywhere once differentiable) closed simple curve with horizontal and vertical symmetry, which is the locus of solutions to one or more rational polynomial equations, with has rational arc length?

[Qaanol]

I expect there are curves of constant width that have perimeter a rational multiple of that constant width.

That very article mentions the theorem that this is not the case, because the perimeter is always pi times the width.

I roll corrected.

[eta oin shrdlu]

Perhaps by "specified by rationals" I mean an algebraic curve with rational coefficients. Now that I've seen the astroid perimeter I'm satisfied that my vague conjecture was indeed false, but for curiosity's sake, I wonder: is there a smooth (or at least everywhere once differentiable) closed simple curve with horizontal and vertical symmetry, which is the locus of solutions to one or more rational polynomial equations, with has rational arc length?

The best I've been able to do so far is by stitching together pieces of the cardioid [or astroid]. The parts of the cardioid r=1+cos(theta) lying between its minimum and maximum y extents have rational arc lengths, so you can cut off the part with a cusp and replace it with another copy of the smooth part, giving a symmetric, smooth oval shape. This gives an equation with an |x|, though, so it's not quite a polynomial; and squaring to eliminate the absolute values just introduces the parts of the cardioids I was trying to get rid of.