The quantity of any finite number of points in a space is simply the cardinality of the set of those points.

Let R

_{1}be the infinite quantity of points in the half-open interval [0,1).

In general, R

_{1}is to be assumed to obey all the algebraic laws normally valid for positive real numbers. Conclusions about the size(quantity of points) of more complex shapes are drawn from the above definition, and the principle that the size of any set measured in this way will be translation-invariant, rotation-invariant, and reflection-invariant within the relevant space. Shapes that cannot be directly constructed will be taken to be the limit of appropriate approximations to those shapes (appropriate in the sense that the troll proof of pi=4 by using rectangular approximations is an inappropriate approximation. The first-order derivative at the boundary must also converge in the limit)

R

_{1}

^{2}is then the half-open square of [0,1)x[0,1). The size of the closed interval [0,1] is R

_{1}+1, and the size of the closed unit square is R

_{1}

^{2}+2R

_{1}+1. The size of [0,n) is n*R

_{1}, the size of an n x m half-open rectangle is n*m*R

_{1}

^{2}, and so on for higher dimensional "rectangles".

Arbitrary polygons can be decomposed into triangles, and arbitrary triangles can be formed by cutting across the diagonal of a rectangle, the size of which is defined above. But care must be taken to ensure that the two halves have exactly the same number of points, including the points along the perimeter. An R

_{1}

^{2}term, when taken to be describing a triangle, is satisfied by a set containing the entire interior of the triangle, exactly half of the perimeter points, with 2 discrete endpoints on that section of perimeter. But any other construction that can be shown to be the same size (by translation/rotation/reflection of subsets) is valid as well. The most convenient such construction is a set containing the entire interior of the triangle *less* 1 interior point, and a half-open interval from the midpoint of each edge to the corner. One extra endpoint is given so that each edge has its own half-open interval, but this is accounted for by removing one point from the interior.

Given this construction, we can draw conclusions about arbitrary arrangements of polygons. Any two triangles joined at an edge will each contribute half the length of that edge, overlapping at the midpoint. This extra point can be shifted to fill the missing interior point of one of the triangles. With the shared edge completed, and one interior hole filled, the resulting quadrilateral is constructed in much the same way as the triangles were. There is one hole in the interior, and half of each edge, with no corners filled. The same logic extends to arbitrarily large connections of polygons. Since every exterior edge is missing precisely a half-interval from corner to midpoint, the entire boundary is completed by adding an R

_{1}

^{1}term, leaving only a single point to fill the interior hole and complete the bounded shape.

But something interesting happens when we consider polygons arranged to form the boundary of a polyhedron. Each face(F) has one missing interior point. Each edge(E), at which two polygons meet, will have an extra point where the half-open intervals overlap. And by construction each vertex(V) is left initially unfilled. So in order to complete the polyhedron, we must add a number of discrete points equal to F-E+V, which is exactly the classical definition of the Euler characteristic.

Looking at the shapes we've tried so far, the number of discrete points always matches the Euler characteristic. For a discrete set of points, it's the cardinality. For a closed interval, it's 1 (the missing point from the half-open interval of the R

_{1}term). For a polygonal perimeter, it's 0. For any filled polygon, it's 1. And taking simple smooth shapes like the circle, disc, and sphere to be the limits of their polygonal/polyhedral approximations, the correct values hold. After much difficult visualization of how to apply similar generic construction rules to 3d shapes, I determined that a pure R

_{1}

^{3}term interpreted as a polyhedral solid would have an extra point in its interior, which is in keeping with the extension of the classical Euler characteristic to higher dimensions. Proving it rigorously proves thus far out of my grasp, but it seems to be a reasonable conjecture and operational assumption that the finite term of the size of a given set will be equal to the Euler characteristic of that set.

It is also fairly obvious from the examples thus far explored that the coefficient of the R

_{1}

^{n}term of an n-dimensional object is equal to its measure in the traditional sense. So that just leaves the intermediate terms.

It's easy to show that the R

_{1}

^{n-1}coefficient is half of the traditional measure of the boundary of the shape (perimeter in 2d, surface area in 3d, etc). This can be seen from the exercise of splitting the half-open n-cube. The boundary between the two halves is part of the interior of the original shape, so the two halves must each get half of that boundary, which is consistent with the fact that the half-open n-cube already included half of its boundary.

But R

_{1}

^{m}where 0<m<n-1 is more difficult to figure out. The first time such a term even exists is in the case of finding the coefficient of R

_{1}

^{1}for a 3d solid. Although it is extremely difficult(maybe impossible) to deduce directly from slicing up the half-open cube, the result can be reasoned out by assuming that *some* construction rule must exist that will be consistent with the n-cube itself, and that has the property that pasting two such shapes together at a shared face will yield a larger solid that fits the same construction rule.

The rule that works is that a proportion of each edge is included, and that proportion is equal to the proportion of the circle subtended by the dihedral angle around that edge. So, for the cube, the dihedral angle is tau/4 (tau=2pi), so 1/4 of each edge is included. These edges can be cut-and-pasted to have 3 full edges, 1 along each axis, as there would be for the direct construction of the cube based on the product of half-open intervals. And, when constructing larger polyhedra using tetrahedra, much as one could use triangles to construct polygons, the only restriction on the number of tetrahedra sharing a given edge in the interior of the resulting solid is that their dihedral angles must add up to a complete revolution, or else the edge would still be exposed to the exterior of the solid. When this condition is met, the fractions of the edge contributed by each tetrahedron will by definition sum to 1, meaning the entire edge is now included, since it's just another part of the interior.

The only edges that aren't filled are those on the exterior. And then, in addition to the proportion contributed by the R

_{1}

^{3}term, exactly 1/2 of the total edge length will be contributed by the R

_{1}

^{2}term (the R

_{1}

^{2}term will be half the surface area, and any R

_{1}

^{2}term of a polyhedral surface area will include all the edge length. This particular identity I've shown to myself will hold in a general construction, where it doesn't matter "which half" of the surface area comes from R

_{1}

^{3}or R

_{1}

^{2}, but I won't go into it here). When taking polyhedral approximations to a smooth surface, the dihedral angle at the surface approaches tau/2, so the R

_{1}

^{3}term will contribute half of that edge, and the R

_{1}

^{2}term will contribute the other half, meaning the entire edge is included. So taking the limit of approximations to the sphere ends up needing no additional term for edge length, which is of course expected since a sphere has no edges. So far the system is consistent with expected results.

This approach to working out what proportion of the exterior edges are included in the R

_{1}

^{3}term can be extended to higher dimensions. The notion of the dihedral angle can be extended to be the proportion of the k-sphere subtended by the interior of the solid, where k is whatever dimensionality is necessary for the angle. In general, the proportion of the m-dimensional boundary of an n-dimensional set that is included by the R

_{1}

^{n}term is the proportion of the (n-m)-sphere subtended by the (n-m)-dimensional "dihedral angle" around that boundary, or the integral of said angle for all points on that boundary, if it is not constant. This even extends to the R

_{1}

^{n-1}case. Only the R

_{1}

^{0}case doesn't fit easily, but that's fine because it doesn't make sense to take a proportion of a single point anyway, and we already have the Euler characteristic as a general way of finding that term.

Now that there's a way to determine what proportion of each type of boundary is included by the higher dimensional terms, they can simply be subtracted off from the total value to find the coefficient of the relevant power of R

_{1}. Meaning there is now a general method of calculating the size of all manner of sets.

It also seems to work reasonably for fractals. When the value of a coefficient diverges in the limit, define the coefficient itself to be an infinite quantity obtained by taking the approximation where edge lengths are R

_{1}

^{-1}, or areas are R

_{1}

^{-2}, or whatever's appropriate for the construction. Based on the principle that n*R

_{1}is supposed to be the number of points in the half open interval of length n, 1/R

_{1}is the "length" that results in a single discrete point, making a perfect approximation to the desired shape. Likewise for reducing areas to discrete points, and so on. This hand-wavy technique is apparently coherent enough to give the correct fractal dimension of the Koch curve (the only example I worked out), so it seems to work.

So, is this a well-known thing that I simply didn't know what term to search for, or am I onto something new here, or is it completely broken in some way that I haven't seen?

EDIT:

Actually, I'm not sure what I was thinking when I said that there ends up being no length term on the sphere, and that that is an expected result because the sphere has no edges. The sphere has no vertices either, but it still has a finite term. And upon actually trying to construct various smooth solids as limits of their approximations, I do see length terms appearing that don't correspond to actual edges of the resulting solid. But they do seem to be consistent regardless of the specifics of the sequence of approximations, and they seem to be meaningful 1-dimensional characteristics of the shape in question. The height of a cylinder, the length along the surface from the vertex to the base of a cone, half the circumference of a sphere, etc. Also, the general method of finding such terms by subtracting out the amount included by the higher dimensional terms does in fact work for the 0-dimensional case. This seems to define a generalization of the Euler characteristic, where the regular Euler characteristic is the 0-dimensional term, and can be computed by the same generalized method as the higher-dimensional terms.

tl;dr

I define a number R

_{1}to be the quantity of points in a half-open unit interval, assume R

_{1}+1 != R

_{1}and so on for higher powers of R

_{1}, and use various shenanigans to build a method of computing the exact quantity of points in arbitrary shapes in Euclidean space given this notion of infinite quantities of points.