No perfect circles in space...

[Jehanne]

Or, at least very few.  From 21st Century Astronomy, 4th edition:



My point in posting this is to demonstrate that Nature does not conform to our Aristotelian presuppositions.  Can anyone here comprehend a circle with its diameter times pi which does not equal its circumference?

[Anomalocaris]

Or, at least very few.  From 21st Century Astronomy, 4th edition:



My point in posting this is to demonstrate that Nature does not conform to our Aristotelian presuppositions.  Can anyone here comprehend a circle with its diameter times pi which does not equal its circumference?

Yes. A circle inscribed on a non-flat surface?

[BrianSoddingBoru4]

Since a circle is defined as that set of unique points equidistant from a centre point, would it be possible to draw such a shape on a non-flat surface?  If you could, would it still qualify as a 'circle'?

Boru

[Anomalocaris]

Since a circle is defined as that set of unique points equidistant from a centre point, would it be possible to draw such a shape on a non-flat surface?  If you could, would it still qualify as a 'circle'?

Boru

I think all points on the circle can still be equidistant from a center point, regardless whether the distance is calculated as a trace along the actual surface the circle is on or straight line directly between the points on the circle and center. So it should still qualify as a circle. But it seems easy to vary the contour of the surface in a ways to almost arbitrarily increase the circumference of this circle.

[BrianSoddingBoru4]

Well, you can increase the circumference of any circle, but how do you do that while keeping the radius the same?

Boru

[Jehanne]

Or, at least very few.  From 21st Century Astronomy, 4th edition:



My point in posting this is to demonstrate that Nature does not conform to our Aristotelian presuppositions.  Can anyone here comprehend a circle with its diameter times pi which does not equal its circumference?

Yes.  A circle inscribed on a non-flat surface?

Ha, ha; on a flat surface?  Or, for the above example, in our planet's orbit about the Sun?  Can you visualize that??

[Anomalocaris]

Well, you can increase the circumference of any circle, but how do you do that while keeping the radius the same?

Boru

As an example, take a flat surface, make a dimple on it.   Trace an enclosed line, exactly equidistant from some other point on the surface as measured along the surface, and passing through the dimple.   This line forms a perfect circle from the perspective of the surface.

But notice if you project this circle onto another truly flat surface, it won’t be circular anymore because where it passes though the dimple, it has to pinch in towards the center to accommodate for the fact the line of radius has longer to go to follow the dimple down to the circumference of the circle.  At the same time, the segment of the circle crossing the dimple is also longer than its project onto the surface, because that section of the circle has longer to go follow the dimple down and then back up.   So I just made a circle whose circumference is longer than its radius times twice pi.

Now to show I can arbitarily increase the circumference of this circle further from the perspective of the surface, all I have to do is to keep the same radius, against as measured by tracing along the surface, but add more dimples where the circumference goes.  I can also change the geometry of the dimple from a simple dimple to dimples with convoluted surfaces, or dimples that balloon outwards near their tips.

---

Yes.  A circle inscribed on a non-flat surface?

Ha, ha; on a flat surface?  Or, for the above example, in our planet's orbit about the Sun?  Can you visualize that??

Who said a planet’s orbit about the sun has to lie on a flat plane?   Gravitational purterbation from other bodies can make some pretty non-planar orbits.  Lots of asteroids have radically non-planar orbits Due to purterbations of the planets. There is at least one earth crossing asteroid that is locked into a strange orbit bobbing up and down many times per revolution by the the gravity of the earth right now.

The sun itself bobs up and down by hundreds of light years as it orbits Milky Way.

[Jehanne]

That's very true, but the point of my OP is that one cannot visualize the warping of space/time.

[polymath257]

That's very true, but the point of my OP is that one cannot visualize the warping of space/time.

I think this is more a matter of practice than inability.

Most people don't spend any time really attempting to visualize four dimensional objects, let alone curved four dimensional manifolds. For that matter, even curved *three* dimensional manifolds are beyond what most people spend any time thinking about. That doesn't make them impossible to imagine. It just means you have to work a bit harder to do so.

But to answer the OP: yes, I find it quite easy to imagine 'circles' (meaning the set of points equidistant from some center point) to not have the 'circumference' (meaning the arc length of the boundary curve' equal to pi times the 'diameter' (twice the defining distance).

Among other possibilities is changing the definition of 'distance'. The standard Euclidean distance (based on the Pythagorean equation) is not the only possible 'metric', even in a 'flat' plane. The taxicab metric is another, perfectly usable measure of 'distance' and the 'circles' for the taxicab metric are actually 'squares' in two dimensions, 'octagons' in three dimensions, etc.

Other types of 'metric' can easily be used, and are used, when discussing general relativity. Again, no fundamental issue is found, but you have to keep track of the definitions a bit more closely.

[robvalue]

Most of this is probably over my head, but I've always reckoned that perfect circles may well not exist because this would rely on the building blocks of matter being continuous rather than discrete. My understanding is that they are more like discrete ("stable wavelengths" and such).

So an abstract circle is perfect, but it is (maybe) only a rough model of any actual theoretical circle comprised of elements of reality.