by wendy » Thu Aug 07, 2014 8:44 am
The real issue in here is that people are seeing a repetition-product, and poking in a cartesian product. There is no cartesian product. Something like III is not resolvable in this sense.
A repetition product produces for each point in A and each point in B, a point in AB. That is, there is a full copy of A for each point in B, and vice verca. The difference here is that the cartesian product applies to all-space, including the content, whereas the comb product only produces a surface of AB that is the repeated-product of surfaces of A and B.
A product is an operation that reduces to an algebraic product P(A×B) = P(A)P(B), would make whatever is represented by × into a product. Here P() represents the Euler Characteristic of the surface (ie the surtope equation, like 6x^2 + 12x^1 + 8x^0 for the cube.). The difference is that the prism or cartesian product adds in a content term, 1x^3+6x^2+12x^1+8x^0 = (x^1+2x^0)^3, where the comb product does not.
Instead, the polygon p looks like px^1+px^0, and the torus formed by p×q is pqx^2+2pqx^1+pqx^0. This property differs from the cartesian product in that we do not add a term x^2 for the interior of the polygon: instead, we would restore the polyhedral nature by adding an x^3 to the finished product.
Such a polyhedron could be made from the cartesian product of the nets (a net folds up to be a surface). One takes a p×q rectangle, folds it up into a tube q round, and then connects the ends of the tube into a loop p round.
One can relax the condition of flat-surface, and still let the product work. A polygon thus includes a circle, and a polyhedron includes any three-dimensional solid, whose surface might become polygons of various kind. A cartesian product of polygons gives a polychoron or 4-solid, while a comb-product of polygons (bounded by 1-fabric), gives a polyhedron (bounded by 2-fabric).
The comb product looses a dimension on every application, and it is the loss of extra dimensions that makes the large variety seen in the higher dimensions. If you look at the cross-section of a torus, you will see two circles, separated by a distance. The planes of the two circles contain a shared axis, and it is this shared axis is "lost" to the product.
[b]A Wheel and a ring [/b]
The 'z' axis represents "up". The 'y' axis represents "forward". So the notional wheel in any dimension, has a rotation in the up and forward space. This is the large axis of the wheel, represented by a circle, say [zy]. There is a smaller section represented by the tyre. This has a radial component in the zy plane, and an "across" section in the x plane, so the section of the tyre is (ox). We can make [] a diameter of 26 inches, and () a diameter of 2 inches. The brackets are parametric, and meant to be identical in style, the different forms are simply to refer to different circles.
The wheel, is then the comb product of the rim circle [] and the section circle (). One could draw red lines parallel to [] and blue lines around each section, and each red circle would cross every blue circle, just once, and vice versa. The wheel is then (zy)(ox), say.
A ring is intended to go around a (linear) finger, so we might suppose that it is constraining the z-axis in the plane z=0. So we need a large circle to completely surround a point in z=0 (ie an n-1 sphere), and take the comb product of that with a circular section in the plane (oz). So we get a figure [yx..](oz).
In four dimensions, we add an extra axis 'w'. Of the wheel, we do not want the cabin of the vehicle to be rotating, so we want everything other than [zy] to be fixed. So the extra term is (oxw), which gives the wheel a spherical cross-section, and the large shape is round.
For the ring, the concern is now for the point escaping a closed surface in the yxw plane. So we need a sphere here [yxw] and (oz) remains the same. We have produced two topologically different figures, whose surface is identical in shape.
One can make a tri-torus, by supposing that the centre of the ring is actually to be contained in a torus in the zyx space, and the finger is pointing in the 'w' direction. This gives rise to a torus say [zy](ox), and we wrap the surface up in the space ow, to get a tritorus [zy](ox){ow}. Because this is the repetition-product of three circle-surfaces, it represents a cyclic portion of the cubic space, and is called thus a 'tri-torus'.
[b]The Dyad or 1D circle[/b]
This turns up more often than not in sections, so it is best to discuss it. A one-dimensional polytope is a line-section. A one-dimensional circle is the space between +r and -r, say (z).
When we multiply two one-dimensional figures, with an euler characteristic of (2x^0). we get eg (4x^0), or two line-segments. It is the repeated application of this that gives rise to ever-increasing powers of two.
A simple 1D torus is then [z](o). It is represented by two line segments in z, separated at their centres by [], and the lengths are (). A quad-line is then [z](o){o}, and so forth. It consists variously of a [z](o)'s surface (end-points) used to centre line segments {o}, or a pair of (o){o} 1-torus separated by a length [z].
When we want to convert this into 2D or 3D, we add extra letters y, x, which do not replace the o, but just as extra segments. The effect of adding extra letters means that the linear parameters are seen in the additional axies. For example, a torus [zy](ox), one can see the full size of the wheel and the tyre in the z and in the y axis, but only the thickness of the tyre in the x axis.
In the [zy](o) space, the bi-segment [z](o) is rotated around the y axis, so the imprint on the y axis is [y](o). This is an analus, representing here the full extent of the rim and the linear thickness of the tube.
In the zx space, we see [z](ox). This produces two circles separated by [] in the z-axis, the tube radius is seen in the z and x sections. But x does not see any of the size of the rim, so [](ox) is only () thick.
[b]Linking[/b]
Linkages of chains in general, follow the solid comb product of two spheres. If A and B are spheres, and the comb product A×B is solid in a given dimension, then a chain might be made of alternating links A×B and B×A. Because the sum of surfaces adds up to N-1. one sees that the dimension of the bodies of the links adds up to N+1, eg circle + circle = 2+2 = 4 in 3d, but for 4d chains, you need 2+3 = 5 to make this link.
[b]Of Hoses and socks[/b]
In the form [zy](ox){ow}... one can suppose that the torus product can be made in two different ways. The way to add a term to the front is to suppose the circle [zy] is actually a cross-section of a long tube, which one joins the end as one might a garden-hose. This happens by supposing a space az, the loop is an even larger circle <az>[oy]... The z is transferred and in the [] section, the z there becomes a radial section from the directions in the az-space.
The sock section adds a tail segment <ov>, smaller than {ow}. The surface of what proceeds is covered by the new circle <ov> which is rolled down as one might take off a sock. An elastic band on the sock, is covered by the sock, and likewise, the surface of ...{ow} is covered by <ov>.
The two process produce identical results. Starting off with {zw}, and repeatedly applying hose-sections, gives (zx){ow} and then [zy](ox){ow}. The sock process applied to [zy] gives [zy](ox) and then [zy](ox){ow}.
[b]Of Tigers[/b]
{One needs to explain the pun involving japanish here. I don't know it outside the existence.}
One can use a polytope in the torus product as the reduction-axis for more than one polygon. That is [zy](xw){oo} makes some sort of sense, since one starts off eg with a circle {yx}, and then hose-fits the z axis to get [zy]{ox}, a torus. But now a second hose function might be made to the height or thickness of a tube, where the letter {x} is transferred to (xw){o} to give eg [zy](xw){oo}.
This is a tri-parametric figure too, but there is no intersection between zy and wx, these can be set to anything as long as they are larger than {}. One can suppose a circle centred at [] and () in the yx space, of radius {}. One rotates this circle in the zy space, to give a torus in the zy plane, and then rotates the result in the xw space.
The notation can get quite complex quite quickly, because one might have difficulty following
The fix is to turn the o's into brackets, and substitute what the o's are radial to. that is eg [zy](xw){oo} gives {[zy](wx}}. There is enormous profit in this, since one can see that the parameters represented by {}, [] and () are visible into an axis, only if the letter is inside that set of braces. Likewise, an inner set of brackets represents a length that is at least large enough to contain all of the brackets that contain it. So we see that [] must be bigger than {}, but not necessarily ().
One looses the simple dimensionality inherent in {oo} = circle, because you have, eg {[....](.....)}. A bracket set represents an additional dimension in the cross-section.
[b]Torotopic Notation[/b]
When one makes these notes, it is possible to reduce all letters to I, and all brackets to (). So the tiger is ((II)(II)). It has three opening brackets, so is tri-parametric. It has four verticals, so is four-dimensional.
[list]
[*]Wheel: [zy](oxw) = ([zy]xw) = ((ii)ii)
[*]Ring: [yxw] (oz) = ([yxw]z) = ((iii)i)
[*]Tritorus: [zy](ox){ow} = {([zy]x)w} = (((ii)i)i)
[*]Tiger: [zx](yw){oo} = {[zx](yw)} = ((ii)(ii))
[/list]