# Dimensional Features Summary (ConceptTopic, 3)

### From Hi.gher. Space

This page lists some general features of each dimension.

## 1D

- There are only two possible shapes: the point and the digon.
- The existence of any object blocks anything else from moving past it.
- Any form of energy can only travel in one direction, and does not disperse no matter how far it travels.
- A point wave source will cause a persistent wave to fill all space.

## 2D

- This is the only dimension where there are an infinite number of regular polytopes; in this case the regular polygons.
- Objects can now travel past one another, instead of colliding.
- Energy disperses proportionally to the distance it travels.
- A point wave source will propagate circularly, but there will be back-propagation of the wave front as well, causing the original signal to "echo" repeatedly at diminishing amplitudes.

- This is the first dimension in which there can be an angle between two objects.

## 3D

- There are only five regular polytopes in this dimension.
- Energy disperses proportionally to the square of the distance it travels.
- Due to this, this is the only dimension where stable orbits (as we know them) are possible.
- A point wave source will propagate spherically, and there will be no back-propagation of the wavefront; the signal will pass an observer exactly once. 3D is also the only dimension where the original signal does not distort when it reaches an observer.

- It is possible to have a solid connected object with an externally-visible hole in it (for example a torus). Due to this:
- Knots are possible. This is the only dimension where knots can be made with strings, because in lower dimensions, objects cannot get tangled, and in higher dimensions, string-based knots will simply fall apart due to the extra freedom.
- Chains (of objects which are topologically torii) are possible.
- Circuitry with wires crossing over are now possible.
- Bridges are possible for crossing rivers and for grade-separated road junctions.

- On a spherical planet,
*the wind cannot be blowing everywhere at once*. This occurs for all higher odd dimensions as well (i.e. 5D, 7D, etc.) - 3D is the first dimension where angles between two lines are visible.
- 3D is the only dimension where the cross product is a binary operator. Generalizations to other dimensions are either not a binary operator, or do not have vector-valued results.
- 3D is the first dimension where the alternated
*n*-cube is full-dimensioned. In 2D, the alternated square is a line segment, which is only 1D; in 1D, the alternated line segment is a point, which is 0D. The 0-cube cannot be alternated.

## 4D

- A lot of equations simplify due to the fact that √4 = 2 (as opposed to √2 and √3, which are irrational).
- This coincidence has many side-effects, such as the vertex radius of the tesseract being equal to its edge length, which results in the existence of the xylochoron.
- This also indirectly implies that 4D is the last dimension in which the n-cube pyramid is CRF (since the tesseract's circumradius being equal to its edge length means that the CRF tesseract pyramid will have height zero, and since for all subsequent dimensions the n-cube pyramid requires (n-1)-cube pyramids as facets, this means that from 5D onwards no n-cube pyramid can be CRF.)

- There are six regular polytopes in this dimension: analogs of the 3D regular polytopes, plus the xylochoron.
- Energy disperses proportionally to the cube of the distance it travels.
- This causes the only stable orbit to be a perfect circle, and even the slightest disturbances will cause the orbiting object to fall out of its orbit.
- A point wave source will propagate glomularly, but there will be back-propagation from the wavefront, causing the original signal to repeatedly echo in diminishing amplitudes.

- Structures, such as sponges, made out of sufficiently linear components attached to each other can appear to "pass through each other". This happens because in a wireframe lattice, there are gaps through which another lattice can pass.
- The closest 3D analog to this would be to imagine a series of parallel cylinders spaced apart with gaps wider than their diameters, with all the cylinders locked rigid in relation to each other by some "magical" force, and then have another such series of cylinders pass through the gaps in the first series. In 4D, this is exactly what happens, except that the rigidity is caused by physical connections in the object in the additional dimension, rather than magic.

- There are more complex forms of "holes" possible. Due to this:
- While turning a torus inside out results in a torus, turning a torisphere or spheritorus inside out results in the other one of the pair.
- This means that chains in 4D
*are*possible, but would probably have to consist of alternating torispheres and spheritorii, rather than only one type of component.

- This means that chains in 4D
- Knots in linear ropes are not possible, as they simply fall apart as mentioned above. It's possible to tie a sheet (2D rope) into a knot, but would be much more difficult to secure as there is a linear boundary rather than a two-point boundary.
- As seen here:
- It is possible to have "land-bridges" between two islands, while retaining water between the islands for boats to sail in.
- Bridges over rivers aren't necessary: one can simply walk "around" a river in the same way one walks around a pond.

- You can easily build a road or canal through the middle of a town without it dividing the town in two.
- There is no need for grade-separated junctions: lanes can switch places without actually colliding all while at the same height level.
- Stable vehicles would probably use a cross
*between*road and rail: a planar rail would hold vehicles on the road, while one could still steer within that plane just as steering is done in 3D. - Complex junctions can be far more compact: all "crossovers" can take place at the same height level, removing the need for long gentle slopes, and all bends can be banked perpendicular to the plane of rotation, removing the need for wide curves. The only limit to bend radii would be the tolerability to G-forces by drivers, passengers and the vehicle itself. This would still depend on speed, but likely not to the extent that 3D designs do.
- Caves can contain planar passages as well as linear ones.

- While turning a torus inside out results in a torus, turning a torisphere or spheritorus inside out results in the other one of the pair.
*Clifford rotations*are possible.- On a glomic planet, the wind
*can*be blowing everywhere at once. This occurs for all higher even dimensions as well (i.e. 6D, 8D, etc.) - A glomic planet can now be considered in two different ways:
- having two-point poles and a spherical equator, with a spherical sweep between them
- having a circular pole and a circular equator, with a toric sweep between them
- This interpretation is thought more likely, and leads to all sorts of interesting effects. While in 3D we have time zones going east to west, and a northern and southern hemisphere, in 4D we would have time zones going east to west
*plus*"climate zones" going marp to garp, instead of the two discrete hemispheres.

- This interpretation is thought more likely, and leads to all sorts of interesting effects. While in 3D we have time zones going east to west, and a northern and southern hemisphere, in 4D we would have time zones going east to west

- The Schroedinger equation for the hydrogen atom has no local minima in 4D, which means atoms as we know them would not exist. Matter would have to be made of something more exotic.
- Here's one attempt at a particle system, though it isn't very formalized, more just collecting a bunch of ideas.
- Stable hydrogen
*is*possible, if you can put up with changing some equations. It's way over Keiji's head, though.- To achieve this requires modifying the laws of electromagnetism such that a charged particle still has a 1/r potential (equivalently, electromagnetic attraction/repulsion still obeys an inverse square law instead of an inverse cube law)
- This has the side-effect of making electromagnetism non-local at the microscopic scale (microscopic causality may be violated, even though macroscopic causality continues to hold).

## 5D

- There are only three regular polytopes in 5D and above: the simplex, the cross polytope, and the measure polytope, which are analogs of the 3D regular polytopes, minus the rhodomorphs.
- No regular star polytopes are possible in 5D and higher.
- The n-cube pyramid is non-CRF in 5D and higher.
- 5D is the first dimension where the demihypercube (alternated hypercube) does not coincide with a regular polytope: in 2D, the alternated square is the line segment, which is regular; in 3D, the alternated cube is the regular tetrahedron; in 4D, the alternated tesseract is the aerochoron.
- Energy propagates according to an inverse quartic law.
- A point wave source will propagate in concentric 4-spheres. There is no back-propagation from the wave front, so the signal will only pass an observer once; however, the signal the observer receives is the first derivative of the original signal, rather than the original signal itself.

- The unit ball in 5D has the largest volume of all (integer) dimensions. (Proof: [1], [2].)

## 6D

- 6D is the first dimension where the Gosset
*k*_{21}polytope diverges into a distinct family of uniform polytopes. In 5D, it coincides with the demihypercube; in 4D it coincides with the rectified pentachoron; and in 3D it coincides with the triangular prism. - Energy propagates according to an inverse fifth power law.
- A point wave source will propagate in concentric 5-spheres, and there will be back-propagation from the wavefront, causing the original signal to repeatedly echo in diminishing amplitudes.

## 7D

- The surface area of the unit hypersphere is the highest in 7D compared to other (integer) dimensions. The actual maximum is actually for dimension d=7.25695... (proof: [3]).

## 8D

- 8D is the last dimension in which the Gosset
*k*_{2 1}polytopes exist. The 9D 5_{2 1}polytope is a tesselation of 8D space. - The E
_{8}lattice is the optimal sphere packing in this dimension (ref: [4]).

## 9D

- Starting from 9D, the only family of axially-demihypercubic monostratic CRFs that remains is the one whose height is 1/sqrt(2). This is nothing but the (n+1)-demihypercube itself, and its (partial) Stott expansions thereof. (proof: [5]).

## 10D

- From 10D onwards, only 8 families of axially-simplectic monostratic CRFs exist (proof: [6]). Their heights are limited to: h=1, h=sqrt[(d+1)/(2d)], and h=2/sqrt(2d), where d is the dimension.

## 19D

- 19D is the last dimension in which the n-cube || 2-tomo-n-tope (x4ooo...||o4xoo...) segmentotope is CRF. [proof]
- The surface of the 19D sphere is less than 1 solid radian. The prism-radian is larger than the surface of the sphere in all higher dimensions.

## 24D

- The Leech lattice is the optimal sphere packing in this dimension (ref: [7]).