I have some suggestions for a more consistent toratope notation.
1) For any object A, the string "(A)" means the set of all points at a certain distance from A, and "[A]" means the set of all points
within a certain distance from A. In general, (A) is the boundary of the solid [A].
2) For any objects A and B, concatenation "AB" denotes the Cartesian product, assuming A to be completely perpendicular / independent of B.
3) The string "I" denotes a
point in one dimension.
I'll provide some examples. I think this is the right idea:
Marek14 wrote: In the same way, an empty string corresponds to a point (it can be added to anything to denote cartesian product by a single point, i.e. identity), while () is "surface of point", i.e. an empty set.
empty string = a point in 0D
() = nothing in 0D (the set of all points at a non-zero distance from the origin; doesn't include the origin)
[] = a point in 0D (the set of all points within some distance from the origin; includes the origin)
I = point in 1D
(I) = pair of points in 1D (the set of all points with x^2 = a^2)
[l] = line segment (the set of all points with x^2 <= a^2)
((I)) = four points in 1D
(((I))) = eight points in 1D
[(I)] = pair of line segments in 1D
([l]) = pair of points in 1D (the set of all points at a certain distance from a line segment)
II = point in 2D
(I)I = pair of points on x-axis
I(I) = pair of points on y-axis
[l]I = line segment on x-axis
I[l] = line segment on y-axis
(I)(I) = four vertices of a rectangle
[l][l] = rectangle (interior)
[l](I) = pair of parallel line segments
(II) = circle (boundary; x^2+y^2 = a^2)
[II] = disk (interior; x^2+y^2 <= a^2)
((II)) = concentric circles
[(II)] = annulus
([II]) = circle (the set of all points at a certain distance from a disk)
[[II]] = disk (the set of all points within a certain distance from a disk)
((I)I) = pair of circles on x-axis
(I(I)) = pair of circles on y-axis
([l][l]) = boundary of a rounded rectangle (the set of all points at a certain distance from a rectangle)
III = point in 3D
(III) = sphere (boundary)
[III] = ball (interior)
(I)(I)(I) = eight vertices of a box
[l][l][l] = box (interior)
[l][l](I) = pair of rectangles parallel to the xy-plane, displaced along the z-axis (non-coplanar)
[II](I) = pair of disks parallel to the xy-plane
[II][l] = solid cylinder
(II)[l] = lateral surface of cylinder
(II)(I) = pair of circles parallel to the xy-plane
(II)I = circle in the xy-plane
((II)I) = torus (boundary)
[(II)I] = solid torus
[II]I = disk in the xy-plane
([II]I) = hollow pancake shape (a pair of xy-disks z-displaced, and the outer part of a torus connecting them)
[[II]I] = solid pancake shape
IIII = point in 4D
(IIII) = glome
(II)(II) = Clifford torus
[II][II] = duocylinder
((II)(II)) = tiger (boundary of a neighbourhood of a Clifford torus)
[(II)(II)] = solid tiger (neighbourhood of a Clifford torus)
etc.
Some of these objects are ugly, like ([l][l]) and [[II]I]; in general, these can be avoided by applying [] only at the higher levels, not inside of another [] or ().
ICN5D wrote: Though, now I see the issues with arbitrary rules. If (I) is supposed to be two points as cut of hollow circle, then it won't hold up with (II)(I). The cut of duocylinder (II)(I) has three dimensions, so it should be a 3D cylinder. It becomes well defined when, and only when, we spherate with closing brackets to ((II)(I)). So, maybe to overcome this little obstacle, if necessary for clarity sake, would be to introduce new brackets [] for solid open prisms. But, maybe establishing more clear rules with what (I) means is the key, without new notations. If so, the ' I ' is a line , and (I) is spherating the line, making two points.
With the changes I suggested, (II)(II) is not a duocylinder but a Clifford torus. Its 3D cut (II)(I) is not a cylinder (solid nor surface) but a parallel pair of circles.
A spherated line cannot be a pair of points; the line is infinite and fills all of 1D, so there are no points at any distance from it. This is one inconsistency that would be solved by redefining "I". Likewise, a spherated square would not be a circle, but a rounded square, with straight edges and curved corners; this would also be solved by redefining "II".
PWrong wrote: So the main issue with the 1D sphere is that it's disconnected. This leads to other problems, like the fact that a square is not differentiable (note that the parametric equations for a square are generally piecewise), so its tangent space and normal space are discontinuous. This is why we don't usually allow square # circle but we do allow duocylinder # circle (are we still using # for the spheration product or is there something new?)
The boundary of a square cannot be obtained with my notation. The interior of a square is differentiable.
PWrong wrote: I would also prefer to stay away from multiple types of brackets if we can help it. The main reason for this is that if we want to do mathematical operations on these shapes, we'll have to use brackets. For example if we have some function f that acts on toratopes (like volume, tangent spaces, homology groups or whatever), then it would be less confusing to write f[A] than f(A). Otherwise, you get readers wondering "does f(22) mean f(duocylinder) or f(tiger)?" The curly brackets { and } are generally used for set theory.
Yes, that is a problem. It could be solved by simply not making [] solids, or by using more standard notation, like replacing "AB" with "AxB". There should be some named function for "the neighbourhood of A within distance b", so that "[A]" could be replaced with "N_b(A)". The standard notation for "boundary" is the "curly d" used for partial derivatives, so "(A)" could be replaced with "[curly d]N_b(A)". Of course, this would make the expressions longer and harder to read.
Perhaps you could just name the object "A", then apply the function "f" without parentheses: "fA".
wendy wrote: One of the points of the paper is that the products of sets does not have a unique euclidean representation. Where one is going to count the cylinder in the mix, why not count the cone, which is also straight-forward product? There are four different ways you can present a product AB geometrically. The prism (or 'cartesian') product is one, the torus is another, the tegum a third, and the pyramid the fourth.
The relevant sets here do not include content, and the product of A in R^m * B in R^n is producing AB in R^{m+n-1}, because the relevant sets here are not the solid space, but the surface: ie A bounded by R^{m-1} by B bounded by R^{n-1} gives AB bounded by R^{m+n-2}.
Cone: For each point in set A (a point), for each point in set B (a circle), there is a unique member (line) in the product AB (cone). This is the general process of the draught process.
Points, not lines, are elements of the cone. The Cartesian product of a point and a circle is a circle.
If A is in R^m, and B is in R^n, the Cartesian product treats the two spaces as completely independent, so the result is in R^(m+n) (though its intrinsic dimension may be lower; for example, a Clifford torus is a 2D subset of 4D space.) You may define other products that work differently, such as that which produces a standard torus (in 3D) from two circles. This is not a Cartesian product.
How does your pyramid product work? Here's my guess: Assume that A and B are in the same space, then take C to be the set of all points c = (1-t)a + tb, where a is in A, b is in B, and t is between 0 and 1. This is a
linear interpolation between A and B. A different possibility is the
convex hull of A and B.
I'm new here, so I have no idea what you mean by "tegum" or "crind" products. (The wiki doesn't explain them.)