There are infinite possible shapes in an infinite world. There are not infinite possible shapes in a world that is not infinite.
In other words, a world with finite discreet parts would not have infinite shapes and neither would a world that is finite in space, unless the finite space contains infinite parts- which would, at least on the surface, appear impossible.
There are infinite possible shapes in an infinite world, such as the world of the mind rather than the world science can directly investigate unless there is no end to the possible divisions of parts of existence or unless there is no end to space.
There is no end to the complexity of possible shapes in an infinite domain like that of the human mind. Though, in empirical or observable scientific reality, there may be an end to the number of shapes.
The number of shapes in our universe, for instance, would be practically infinite though not actually infinite. I say this because the combinatorial possibilities of any finite system must be finite. Though numbers are infinite, there is not, if science is correct about this universe, an infinite number of discreet parts in it.
If there are infinite universes, then there would be infinite shapes across the multiverse but not in an assumingly finite universe, such as the one we occupy (should our beliefs about this universe be true, in that, it is finite and made of some smallest atomistic parts like those studied in quantum physics).
A point has no shape given that it does not have extension or dimension in space. If a point had shape, then it would not be a point but a circle or sphere or something rather than a marker for not necessarily anything.
The number zero represents the absence of anything though the marker or symbol "0" is something. Saying there is a point here does not mean there is something here but rather refers a place in space with no space itself.
A shape must have at least one dimension. A line is the simplest shape. A spiral line is also shape. Shapes can be open (as in an "S" shape) or closed (as in a square) and they can come in infinite dimensions.
Though, imagining shapes in more than three dimensions becomes representationally challenging or, under some interpretations, principally impossible, for the human mind that is designed to understand a three dimensional reality in terms of a static image without regards to the dimension of time.
"Polygon" is a word for a shape that exists on a single two-dimensional plane and is composed of a finite number of straight lines.
A line, by itself, is not a polygon. A circle and ellipse are not polygons because they contain no straight lines, though a polygon can be drawn with enough equal sides so as to continually, as the number of equal sides is increased, approach the dimensions of a circle.
Although each polygon must consist of a fininite number of straight lines, the instances of possible polygons is infinite, given that one could always add another line on to an existing shape and, therefore, could continue to name polygons forever.
Three points anywhere in space creates the simplest two-dimensional and single planed shape: the triangle.
The triangle exists in a single two-dimensional plane. Two dimensions specify length and width but no height. So, in terms of the empirical reality huamns currently understand, two-dimensional objects do not exist. Likewise, one-dimensional objects, or lines, exist nowhere in empirical reality either.
Things move in linear directions, along lines, and lines can be drawn between things, but the lines themselves are ideal or mental creations that have no other space than the single perfect dimension.
A pencil-drawn line, for instance, has three-dimensions because the graphite of the line is actually, upon close inspection, a jagged mountain of atoms left behind the tip of the pencil on the paper.
A line can be considered the most basic shape because to have shape means to have some spatial dimension or spatial extension. Having spatial dimension means that something has some length, width, or height, in at least one dimension.
Interesting questions include: Does a thought have a shape? Perhaps, under a scientific-realist understanding of reality, the thought is a first-person representation of a shaped neural or electrical pattern.
But, does the first-person, introspective thought itself have a shape? Perhaps some thoughts, such as that of a RED SQUARE have a shape. But, does the representational, private thought of a piece of music or of the emotion of love have any shape at all?
I am currently working on this page to include exotic shapes and study of multidimensional objects and shapes. So far, only the surface has been scratched on this page concerning shapes.
