I had a suspicion a couple of days ago, and the internet seems to agree with me, that a Circle is actually a 1 dimensional shape. So now I'm left wondering if a Sphere is a 3D or 1D shape. I'm guessing it's a 1D shape because it is made of many circles. What do you think?
I think your guess is wrong. How many coordinates do you need in order to specify a point within a sphere? And do you have a reference for wherever it was on the internet that suggests a circle is a 1D shape?
https://www.quora.com/How-many-dimensions-does-a-circle-have If Spheres really are 1D shapes then, that helps me understand Physics a lot more than I used to.
Ah I see what they are saying. That would make the surface of a sphere 2D, then. But a sphere itself is a 3D object.
Oops, I didn't know there was a more button, but anyway, my interpretation from this is that, a circle is 1D 'So, let me be very, very clear: a circle is a one-dimensional object.'
I think the language specifies that a circle is a 1D non-Euclidean surface, and the 2D shape is a disk (Euclidean). A circle encloses an area but does not include that area. Likewise there is a hollow sphere but a solid ball of dimensions 2 and 3 respectively. The generic language would speak of higher dimension being say a 4-sphere or 4-ball respectively, and so on up to any number of dimensions. So a 0-sphere is a pair of points, but a 0-ball is a line segment. See the wiki article on n-sphere.
I agree. Here's another way to look at it. Ask yourself: how many independent parameters are needed to uniquely specify a unique point on a circle? The answer is clearly one. For instance, the required parameter could be the distance around the circle from an arbitrary fixed point on the circle. If the cartesian equation for a circle is, say $x^2+y^2 =1$, then the fact that there is an $x$ and and a $y$ in the equation doesn't mean it is two dimensional. The equation has the effect of making the $y$ coordinates on the circle dependent on the $x$ coordinates (or vice versa). For instance, we can write $y=\pm\sqrt{1-x^2}$. So, again, we only need one independent parameter (say, the $x$ coordinate) to specify a given point on the circle. (You might think the $\pm$ there is a problem, but really that's a problem with the cartesian coordinates. If we used polar coordinates instead, that problem would go away.) Ask yourself: How many independent parameters are needed to specify a given point on a sphere uniquely? The answer is two, provided we're only dealing with a "hollow" sphere. For instance, specifying the lattitude and longitude of the point will do the trick. On the other hand, if you want to make it a solid sphere, then you'll need one more parameter (e.g. the radial distance of the point from the centre). So, spherical shells are two-dimensional, while solid spheres are three-dimensional. Oh, and a similar thing applies to circles. When I said a circle is one-dimensional above, I was referring to a "hollow" circle. If you want to include the "interior" of the circle as well as its circumference, then you'll be dealing with a two-dimensional object. The cartesian equation will need to change in that case, too, to something like this: $x^2+y^2 \le1$ and then it is no longer possible to uniquely specify $y$ in terms of $x$ using the equation. (There's now only a constraint, not a relation, between the two variables.)
Wrong. 0 = x^2 + y^2 + z^2 can only be satisfied if |x| = |y| = |z| = 0. Hence it's the parametric equation for a zero dimensional point.
Duh! Silly of me. You're right, of course. I saw the zero there and some quiet alarm bells went off in my head about that, but I foolishly ignored them.
See Q-reeus's post, above. Only a single point is described by that equation, so it has zero dimensions. If your equation had been $x^2 + y^2 + z^2 = a^2$, with $a$ being any non-zero real number, then you'd be dealing with a two-dimensional spherical surface.
What? Q-reeus told you that your equation with $R=0$ describes a zero-dimensional point (see post #12). I told you that if $R\ne 0$ then your equation describes a two-dimensional surface (see post #15). Is this clear for you, now?
No. Q makes a good point, but asserting something with three variables as a two dimensional object is illogical.
Merely having three spacial coordinates doesn't automatically specify the object's dimensionality. Only that it's located and oriented somewhere in a 3D space, whatever dimensionality it actually has. Which situation can be generalized to N dimensions, mathematically at least.
