## Negative dimensions?

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### Negative dimensions?

Question: Cartesian product of a circle and what gives a point?
S1X?=S0

We know that Cartesian product of two circles give a Clifford torus/flat torus
Cartesian product of two discs give the duocylinder

Above question is conceived when I google for "negative dimensional space" and came across the following two websites:
http://alanmcculloch.blogspot.com.au/20 ... ative.html
http://forthelukeofmath.com/documents/W ... rkshop.pdf
Secret
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### Re: Negative dimensions?

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.
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### Re: Negative dimensions?

@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.

@Wendy:
And where did you come across -2 dimensions?

--- rk
Klitzing
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### Re: Negative dimensions?

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. ddds.PNG

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:@Wendy:
And where did you come across -2 dimensions?
--- rk ddds.PNG
Secret
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### Re: Negative dimensions?

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* ...

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

Accordingly:
1 - 14 + 24 - 12 + 1 = 0

Euler would rewrite the 3D case as:
F - E + V = 2

--- rk
Klitzing
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### Re: Negative dimensions?

A dimension of -2, turned up when i was discussing with Norman Johnson, that the Desarges Configuration could be read as a dyadic polytope, if one read the usually numbered elements at -1, and the "nulloid" at -2.

I really can't see how the down apex surtope (ie the one usually of -1), might be read as an empty set. It's nonsense, since every separate polytope has a seperate nulloid, whereas the empty set is singular across sets. Also, the set theory works with intersections, but not unions. The upwards incidence towards the contents, is also a series of intersections, so if the nulloid is the empty set, so is the contents.
The dream you dream alone is only a dream
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### Re: Negative dimensions?

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?
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions Prashantkrishnan
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### Re: Negative dimensions?

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.

--- rk
Klitzing
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### Re: Negative dimensions?

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?

Nothing. The nulloid is basically the element that contains nothing. It's really only useful in mathematical descriptions in order to provide an opposite of the element that contains the entire shape; it has no physical meaning.
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### Re: Negative dimensions?

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.

If a nulloid is of a dimension of -1, then can we take it as a (line segment)-1?
I am asking because the Cartesian product can be taken through the number series notation very easily.
Line segment ----- 1
Square ----- 11 (Cartesian product of two equal line segments, i.e. (line segment)2)
Cube ------ 111 (line segment)3
Even though I have not seen it anywhere, I have always assumed the number series notation of a point as 0. What would be the number series notation of a nulloid?
I got a reply from Klitzing as to the nature of the nulloid:
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.

If a nulloid has the value of zero, then I might consider it as an object of negative infinite dimensions, since we have lim(x-->0) log x = minus infinite.
So what has a finite negative dimension? How can we geometrically describe (line segment)-1? Is this a dimension of negative one? Is this correct logic?
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions Prashantkrishnan
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### Re: Negative dimensions?

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.
The dream you dream alone is only a dream
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### Re: Negative dimensions?

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.

If this is the case, then howw would the -2 dimensional objects be? What about dimensions less than that?
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions Prashantkrishnan
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### Re: Negative dimensions?

The number of dimensions less than zero arise from 'naming things'. That is, creating a cube from its environment also creates its nulloid.

When nulloids are involved in products, they assume the zeroth power, corresponding to a simplex of zero vertices. There is never any particular product that multiplies negative dimensions.

The thing of -2 dimensions happens in the desarge polytope, which is a pondered (dimension-reduced) simplex. Points have a handle with two numbers, lines with three. A nulloid has 1, and its bottom-incidence has 0, is of -2 dimensions.
The dream you dream alone is only a dream
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### Re: Negative dimensions?

The thing about negative dimensions is that they arise mainly by introducing abstract elements that are not part of the actual geometric construction (for example, these elements have no calculable location) in order to impart nicer closure properties to the mathematical structure underlying the object. I think it's a mistake to regard them on the same level as elements that can have "physical" realization (that is to say, they have shape, location, and extent in the ambient space), in the sense that we should not be misled into imagining that they somehow have an independent geometric existence apart from the abstract system that they are introduced under. That's not to say that these things are useless, of course -- closure properties are very nice to have in mathematics -- but one shouldn't be under the wrong impression that they somehow "exist" as independent entities.

To be more specific, the nulloid (-1D) arises from the face lattice of a polytope, in which the maximal element is the entire polytope, which one may regard as the set of all of its vertices. Each sub-element that forms some subset of this maximal element, until we reach its vertices, which are the singletons. This is the extent of the measurable elements in the polytope, but it is mathematically unsymmetric because it doesn't have the empty set. Therefore, we introduce the empty set in order to make this a lattice (lattices, by definition, must contain a bottom element), thereby imparting nice closure properties (you don't have to constantly treat the empty set in a special way, but it is subsumed with the rest of the elements and treated in the same way), and invertibility (the dual polytope then has the inverted face lattice). Since each lower level in the lattice corresponds with elements of one dimension lower, then by inference the empty set should correspond with an element of -1 dimensions. Thus, the nulloid is born. Nevertheless, the nulloid shouldn't be confused with something physical present in the polytope; it's simply an abstract element we introduce in order to impart nice closure properties to the face containment hierarchy (i.e., turn it into a lattice, which has nice properties).

Seen from another angle, we may say that the polytope is a physical realization of an underlying, abstract face lattice, in which the nulloid has no physical counterpart. Indeed, this is the approach taken in modern studies of polytopes -- the physical realizations are stripped away and one considers only the abstract face lattices. This then allows one to generalize the definition of polytopes beyond what is physically realizable, leading to the so-called "abstract polytopes", a large part of which cannot be physically realized in any space. Thus, the distinction between the physical realization (in which the nulloid doesn't have any corresponding physical counterpart) and the abstract structure (in which the nulloid is given by definition) is, indirectly, acknowledged, and one chooses to work with the abstract structure where its nicer closure / symmetric properties lends itself better to abstract analyses.
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### Re: Negative dimensions?

Negative dimensions are necessarily quantum in nature, and can not be detected from the real world. It is only when one says that this bit is X and this bit is not X, that the bits that are X call down to a 'wessian' (being) of 'X'. This existance happens at a dimension one less than that of points.

Endoanalysis

This was a little experiment done, to determine whether solids really have an existance separate from the surrounding space, or whether they were a device of our creation. The results are quite interesting, because the gulf between 'real densities' (which measure how many times space is occupied), and "integer densities" (which tells us that there are things points and lines etc there), is never breached. Polytopes are devices of our mind, and negative dimensions are what gives them a thing to 'belong' to: a handle.

In the simplest case, one might consider that space has a varying density. We can not directly detect this density, but instead see the divergence of it: that is, we see when it goes from 1 to 0. So we get in maths-speak Out-vector = divergence of density, volume = moment of out-vector.

It is possible to consider for example, nebulus figures, like the gaussian distribution exp{-x^2-y^2-z^2}. Solids have discrete boundaries, where the density jumps by 'integers' from, eg 1 to 0. Things like the pentagram have density jumps from 2 to 1 (the core to the points of the star).

For polytopes, all such jumps occur in a small set of planes. Since the plane contains itself a polytope, we can equate the density on the plane with the transverse out-vector. That is, if the density drops from 2 to 1, the outvector has a strength of 1, and thus the surface-density is also 1. One can then for example, find the density of the pentagon (core = 2, points = 1), the surface is completely unit density.

Because the surface lies in a plane, which is a space of lesser dimension, we can continue the calculations. The density of the edges is 1 all the way to the vertices, and 0 outside the vertices. There is no change of density on the line where the two edges cross at the core, so the "surface" of the line is where on the line, it changes from 1 to 0 (ie at the ends of the points).

Endo-analisys can not tell us that the out-vector is due to one or two or three elements. All it tells us is that eg, the density drops from "2.0" to "1.0". or "1.0" to "0.0".

In order to "create" a pentagon, we need to take a marker and draw points and lines and things. In other words, the decision that "2.0" is the result of one element, (as in the pentagram), or two elements, (eg in the hexagram), is a concious decision which we need to make. Even in the hexagram, we could consider the thing as a single element, with two surfaces. And herein lies the paradox.

A hexagram could be seen as two overlapping triangles or a single thing with a disjoint surface. But depending on what we call it, means that it descends down to two wessians (or nulloids), or one. Where it is two overlapping triangles, there are two separate incidence diagrams, two volumes, and two anti-volumes (or wessians). When it is read as one, there is a single incidence diagram, and a d2 volume and a d2 anti-volume (wessian).

The Empty Set

Norman Johnson posited that it might be an 'empty set'. One can see that there can be several different things, each having its own wessian. This is different to the way the empty set happen. The same empty set exists for all families of sets. Moreover, it is possible for wessians to have different densities, because the wessian acts as an anti-bulk, or dual of the bulk.
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