The question is about spatial dimensions.
It seems to me that every dimension adds something fundamentally new to space:
If you have one dimension, you have different positions and the ability to move an object (like a point). By adding the second dimension, the objects have the possibility to go around each other - they mustn't go through each other to change positions. The third dimension allows to tie a knot. Is there anything new like this if you add the fourth, fifth, ... dimension?
In four dimensions, you can not knot a line.
On the other hand, it is possible to build a city with roads, railways and footpaths, on the level with none of these crossing each other.
Seasons are still possible in four dimensions, but the day is 12 hours everywhere, and the world would be perfectly round. What does happen though, is that one gets 'season zones' like 'time zones'. One can think of 3d world as a line where the S hemisphere is one season, and the N hemisphere is six months later. In four dimensions, there is always somewhere that is in early spring, or late autumn, or mid-winter, or whatever.
You can weave 2d sheets in 4d, in general, to cut and weave one needs N-2 dimensional sheets and blades.
In four dimensions, it is possible for topologically different figures to have the same topological surface.
In five dimensions, it is possible to knot, but not weave, 2d sheets.
With four dimensions, you can have an orthogonal transformations with determinant 1 which can not be expressed as a rotation around a given axis by a given angle. It is up to you whether or not you want to call such a thing a rotation nevertheless.
The number of regular polytopes in $n$ dimensions (generalised from regular polyhedra in $3$ dimensions by the rule that all of their vertex figures (or equivalently faces) must be regular $n-1$-dimensional polytopes) is given by $\mathrm{A060296}$$(n)$, in $n\geq3$ dimensions there always exist the simplex, orthoplex and hypercube, and in $3$ there also exist the icosahedron and dodecahedron (which can be easily proven to complete the full set of regular tilings of the spherical $2$-space).
Expanding on MvG's answer, for each pair of points on the boundary of a unit $n$-dimensional sphere, there exists a unique rotation (that maps one to the other while keeping all perpendicular points invariant), so given a point arbitrarily defined as the identity, each point uniquely corresponds with an orientation, if and only if $n\in\left\{1,2,4\right\}$ (for which there exist the group algebras $\left\{\pm1\right\}$, the unit complex numbers and the unit quaternions (versors), respectively).
$1$, $2$ and $3$ dimensions are the only ones for which the algebra of rotations is isomorphic to that on such a sphere, $1$ and $2$ are their own but $3$ is described by the versors, where the action of a quaternion $\text{q}$ upon a vector $\text{v}$ is $\text{q}*\text{v}*\text{q}^{-1}$ (not an identity due to their non-commutativity), where $\text{v}$ is interpreted as an imaginary-only quaternion (and so too is the output). ($4$-dimensional rotations are describable in terms of a different versor acting on either side, and in general, to rotate the quaternions by angle $a$ from some versor $\text{v}$ to a perpendicular versor $\text{w}$ (while leaving the perpendicular plane invariant (instead of line)), you need the map $x'=\left(\cos\left(\frac{a}{2}\right)+\sin\left(\frac{a}{2}\right)*\text{w}*\text{v}^{-1}\right)*x*\left(\cos\left(\frac{a}{2}\right)+\sin\left(\frac{a}{2}\right)*\text{v}^{-1}*\text{w}\right)$. Rotations can be uniquely decomposed into left- and right-acting versors (called left- and right-isoclinic rotations), and have ${4\choose2}=6$ degrees of freedom but aren't described by a sphere in any dimension, because the period of motion across a sphere's surface is $2*\pi$ divided by the angular velocity, whereas for these it is the $\mathrm{lcm}$ of the two isoclinic rotations' periods, if one exists at all, so orientations are described by points on the surfaces of a pair of spheres.)
This has the consequence that if a "default orientation" is assigned for a $3$-dimensional space, and a regular polyhedron is oriented such that pointing in this orientation from the origin will be looking directly towards one of its vertices, the set of rotations under which the polyhedron is invariant (ie. the $5$ orientations per vertex in the icosahedron) are equivalent to a set of versors that behave as a group, and (if viewed as points in non-quaternionic $4$-space) are themselves regular polytopes. In this sense, the $120$-cell is the icosahedron's and dodecahedron's rotation group (called the icosians), its dual is the $600$-cell, and the $24$-cell is a subset of it that also comprises a group.
I mention this because in $n\geq4$ dimensions, there exists no such group algebra on the ${n\choose2}+1$ sphere equivalent to $n$-dimensional rotations, so no such constructions of $n+1$-dimensional regular polytopes form the $n$-dimensional ones, and for $n\geq5$, it turns out that there is no other way and $\mathrm{A060296}$$(n)=3$ (from the simplex, orthoplex and hypercube).
Also on the note of quaternions and things that cease to be possible in higher dimensions, the dot and cross product were first derived from the fact that for two quaternions $\text{v},\text{w}$,$$ \text{v}*\text{w}=\left(\mathrm{Re}(\text{v})*\mathrm{Re}(\text{w})-\mathrm{Im}(\text{v})\cdot\mathrm{Im}(\text{w})\right)+\left(\mathrm{Re}(\text{v})*\mathrm{Im}(\text{w})+\mathrm{Im}(\text{v})*\mathrm{Re}(\text{w})+\mathrm{Im}(\text{v})\times\mathrm{Im}(\text{w})\right), $$so where two vectors $\text{v},\text{w}$ are interpreted as imaginary-only quaternions, $\text{v}*\text{w}=-\text{v}\cdot\text{w}+\text{v}\times\text{w}$, and the Cayley-Dickson construction (that allows the derivation of $2^n$-dimensional multiplication methods from its recursive application), which means a $2^n-1$-dimensional cross product exists for all nonequivalent orderings of the $2^n-1$ dimensions under action of cyclic shift (which for $3$ dimensions are only $(x,y,z)$ and $(z,y,x)$, and correspond with whether it follows a left- or right-hand rule, since for each pair, there is a unique other dimension).
Note that there also exist tilings of $2$-, $4$-, $8$-, and $24$-dimensional space (the triangular, $4$D orthoplex, $E_8$ and Leech lattices, respectively) such that when each vertex is replaced by a sphere such that all connected by an edge are tangent to each other, and all are removed except for one sphere and all of its tangential neighbours, there is no room for these neighbours to be 'wobbled' about, making each of these optimal (as maximum kissing neighbour counts, or kissing numbers), though no other dimensions $n>3$ have known kissing numbers (only upper and lower bounds).
