Point as a zero dimensional figure

Question

Is point just a miniature circle or a figure. We call it zero-dimensional because it doesn't have length, breadth, and height. But if you zoom it, it will appear like a figure.

Answer 1

Just as for truncation of regular polytopes you do have a combinatorical discontinuity for truncation depth zero. Or for Stott contraction the same: in the moment the edge type of application truely gets zero, you will have a combinatorical discontinuity.

And so too, you could start with any polytope you like. When scaling it down to zero, you wouldn't end with an infinitesimally small copy of your polytope, rather that combinatorical discontinuity just provides there a single point. Simply by identification of incident elements.

None the less, all these processes, when reversed, just happen to imagine according overlays of vertex points and according zero sized edges, etc. first, which just get expanded thereafter. This then would be just an application of the concept of abstract polytopes, with a coincidentally vanishing realization.

But you are fully right, THE zero dimensional polytope is the point. Sometimes even the empty set is being considered a polytope too, then having dimensionality -1.

This addition btw., together with the full dimensional bulk at the other dimensional end, makes it easy for example to describe the elemental counts of the general $n$-dimensional simplex by the numbers of the $n$-th line of the Pascal's triangle. - Or it makes the Euler equation much more symmetric, when including those 1's at either end: The alternating sum of elemental counts happens to be zero, no matter what the dimension is.

--- rk

Answer 2

Is point just a miniature circle or a figure.

Nope. All circles contain multiple different points, and have a diameter which is positive (which is to say, greater than $0$). The same goes for squares and other types of figures.

A point, on the other hand, does not contain multiple different points, and the diameter of a point is exactly $0$.

But if you zoom it, it will appear like a figure.

That's not true. If you "zoom in on" a point, it will still look like a point.

Answer 3

It is possible to map all sorts of things onto a geometric point - a circle, an abstract polytope, a topological discontinuity, but the point itself is independent of what you have mapped onto it and no matter how far you zoom in or try to tease it apart, it will always be just a point. For Euclid, that was in fact the definition of a point.

It is often the case that some object will appear as a point, but it is seldom correct to put it the other way round.