wendy wrote:Dimensions of -1 and -2 have been recorded, but they do not participate in cartesian products.
A dimension of -1 is the 'nulloid', which functions as an identity element in the products of draught. For example, the product of two points in a pyramid product is the line that connects them. In order to get the terminals on the line, one has to include a nulloid in both sides: ie a nulloid stretched towards an X gives X itself, while a point stretched to an X gives an X-pyramid.
Klitzing wrote:@all:
Also in the alternating brutto sum formula of elements of a D-dimensional polytope (extension of euler's formula) you'd benefit from the nulloid (-1 dimensional element):
sum(i = -1, ..., D) [(-1)^i * count(i-dimensional elements)] = 0
being then valid uniformely for any dimension (at least for convex shapes - and even several ones beyond, i.e. having no tunnels etc.). Here both the body (D-dimensional element) and the nulloid (-1 dimensional element) contribute by count=1. I.e. for even dimensions, those extremals cancel out each other, while for odd dimensions those combine for that else somehow to be encounterd summand of 2.
Klitzing wrote:@Wendy:
And where did you come across -2 dimensions?
--- rk
Secret wrote:...Klitzing wrote:@all:
Also in the alternating brutto sum formula of elements of a D-dimensional polytope (extension of euler's formula) you'd benefit from the nulloid (-1 dimensional element):
sum(i = -1, ..., D) [(-1)^i * count(i-dimensional elements)] = 0
being then valid uniformely for any dimension (at least for convex shapes - and even several ones beyond, i.e. having no tunnels etc.). Here both the body (D-dimensional element) and the nulloid (-1 dimensional element) contribute by count=1. I.e. for even dimensions, those extremals cancel out each other, while for odd dimensions those combine for that else somehow to be encounterd summand of 2.
*Is confused* ...
Klitzing wrote:Cuboctahedron has
1 3D element: its body
14 2D elements: 8 triangles and 6 squares
24 1D elements: its edges
12 0D elements: its vertices
1 -1D element: the nulloid
Prashantkrishnan wrote:Klitzing wrote:Cuboctahedron has
1 3D element: its body
14 2D elements: 8 triangles and 6 squares
24 1D elements: its edges
12 0D elements: its vertices
1 -1D element: the nulloid
What exactly is a nulloid? What does it look like?
wendy wrote:Dimensions of -1 and -2 have been recorded, but they do not participate in cartesian products.
A dimension of -1 is the 'nulloid', which functions as an identity element in the products of draught. For example, the product of two points in a pyramid product is the line that connects them. In order to get the terminals on the line, one has to include a nulloid in both sides: ie a nulloid stretched towards an X gives X itself, while a point stretched to an X gives an X-pyramid.
Consider the point as being the "one of multiplication" within the cartesian product.
The nulloid then is kind of the zero.
Or consider the pyramid product more in the sense of an addition of something atop.
Then the point atop would still produce the pyramid, it still adds the height dimension.
Whereas the nulloid, i.e. the empty set, just adds nothing atop.
So it serves as neutral element of that addition.
Thus again kind a zero.
wendy wrote:There is a thing, whose dimension is -1 dimensions, whose nature is that it is incident under anything that belongs to it. We shall need to create things and separate them from the background, for them to exist, and thus a nulloid. It is known to be of -1 dimension, it is because it is part of the various draught product.
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