(4/3)Pi 1^3 1 ---------------- = --------------------- = one quadrillionth (4/3)Pi 100,000^3 1,000,000,000,000,000
x4 (integral y) = --- y =x3 y'=3x2 y''=6x y'''=6 y(4)=0 4 cube planes lines points null (4-D) (3-D) (2-D) (1-D) (Zero-D)A big problem occurs when we try to integrate something cubic ( x 3 ) into the fourth dimension...in this case ( integral y ), we have an exact mathematical representation of it ( x 4 /4 ) and if x = 1 we know this is equal to 1/4's worth of fourth dimensional volume (tesserarea?) but, what shape is it? Is it a snapshot in time? Is it an hypercube? Is it a mysterious visitor from the fourth dimension?
The cube isn't solid
...So, the basic building blocks of particle construction must be line segments or strings. We could actually have 2-D planes but, they would have to be made of strings and therefore be mostly space. Their arrangement makes all the difference... The way this actually works is by using axial directions as dimension. The basic unit of the x 3 variety is the X-Y-Z axes shaped particle, this is three strings joined at their centers. If in this example y = x 3 and x = 1, then the three axis will be 1/3 in length (X, Y and Z are 1/3 long) with a total length of one. All widths are infinitesimal. When we differentiate x 3 we get 3x 2 , this is 3 plus sign or XY axes shaped particles with a total length of 3, that makes 6 axis with 1/2 length... this is also the exact amount of length needed to add one quadratic XY particle to every axis of the original differentiated particle giving the XYZ an instantaneous change. ...Working this in reverse we see that as we integrate something into a higher power it changes the shape by adding an axis and it shrinks in size giving it a greater density. So the mysterious fourth Dimensional object may still look exactly like a supposed 3-dimensional cube but it will be of much greater density and its basic (atomic) structure will be of tetrahedral axes shaped particles, 1/4 total length f (1) = x 4/4 = 1/4 with 1/16 length axis (one fourth of 1/4 total ), it gets small very rapidly. More particles are needed to fill any volume because of the shorter lengths and tighter pack, ergo higher density. Now it is easy to see... in a field of 2-D ( XY ) particles we can only traverse horizontally and vertically. When we bump this up one dimension into 3-D we also have the toward and away axis, alas we still can't move on a diagonal, for that is reserved for the higher dimensions.
An Abbott Flatlander living on a two dimensional plane would actually be living on an infinity of dimensions if he can turn or move through every angle or vector direction on the supposed plane. Remember...there can't be an actual continuous plane. An actual working two dimensional model of space would be an infinite array of 2-D axes shaped particles arranged in a plane with the negative or expansive force vibrating through their continuum (matrix). In this 2-D model light is coerced into traveling in straight lines in only two directions (This model can also warp or flex, forcing the curvature of light). If you took enough 2-D particles and curved and connected them into a spherical surface shape, it would be misinterpreted as 3-D. Nothing is actually 3-D and/or solid. Now that we know the basic workings of particles at the quantum level and we know vibrations occur in every possible direction, a 3-dimensional particle will not sufficiently transmit vibrations along a diagonal, 3-D doesn't work. The particle capable of angular conveyance must be of higher dimension, have the most efficient shape to pack space with the least amount of material and have the smallest angle of symmetrical invariance. It turns out to be a particle with 10-dimensions or ten axes, any more axes and they won't connect or pack. This can be visualized as ten axes joined at their centers with every individual axis terminating on the vertices of a Dodecahedron or the set of twenty points......
( +-x/y, +-xy, 0) where y = (Ö5+1)/2 = 1.6180339887498948482... ( +-xy, 0, +-x/y) x = G /(20Ö3) ( 0, +-x/y, +-xy) G = 1 /(10Ö26-1)c ( +-x, +-x, +-x)This allows for vibrational conveyance in every possible direction, is stackable hyperbolically and will pack space "face centered cubic" this corresponds to a lattice (arrangement, not size) which the center of every sphere or particle is the set of all points (x,y,z) where x, y, and z are integers adding up to an even number. This axial concept allows for an actual visual of higher dimensions. Now it is easy to see how any length, for instance g = 1/(10Ö26-1)c, divided up correctly can be a direct representation of a particle(s).