2015年2月3日 · The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point x of the circle is contained in at most two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle but with simple overlaps.

2011年5月6日 · A circle is a one-dimensional object, although one can embed it into a two-dimensional object. More precisely, it is a one-dimensional manifold. Manifolds admit an abstract description which is independent of a choice of embedding: for example, if you believe string theorists, there is a $10$- or $11$- or $26$-dimensional manifold that ...

2015年5月25日 · A line, a circle, an ellipse can all three be defined in 2D space, where they are determined by 2, 3 and 5 parameters respectively, but they are 1D entities as they can be mapped to the real line (you can graduate them with a single number). $\endgroup$ –

Yet, in school math we deal with a lot of geometric objects whose measures only have the first power of $\pi$ in their expression: length of a (1d) circle, area of a (2d) disk, area of a (2d) sphere and volume of a (3d) ball.

2016年3月12日 · A circle is just the curved line which forms the boundary of the disk. The circle is 1D while the disk is 2D. Another way to think of dimension is how many degrees of freedom you have in your movement if you were stuck in a circle or a disk. If I was confined to living on a circle, then at any point on the circle, I have only one degree of freedom.

2023年2月1日 · Writing a sum over a square lattice as sum over 1D chains in a circle. Consider the sum \begin{equation} S = \sum_{m=1}^\infty \sum_{n=1}^\infty V(m,n). \end{equation} Here we are summing over a square grid of points $\left\{(m,n)|m,n \in \mathbb{N}\right\}$. Alternatively, these points can be constructed from many 1D chains at different angles.

2022年2月6日 · Consider a thin cylinder. On this cylinder a circle can move. From a distance this looks as if a point-particle moves on a line. This is somewhat reminiscent to Kaluza-Klein, except that KK brings a true classical point particle in. We can describe this situation as one dimension of a 2D space curled up to a circle.

2023年5月21日 · To tackle this, I first calculated the radius of the large circle. 🙂. If we draw a square from the center of the 4 smaller circles, we can see the square's sides are of length $4$ and its diagonal is $4\sqrt2$.

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Circle 2 (inner circle $ \rightarrow x^2 + y^2 = 1 $ ): $ x^2 + y^2 = 1 \rightarrow r^2 = 1 $ Putting this ...