The space of a 2 dimensional area is flat, a plane. In three dimensions it is a cube. That much is pretty straight forward.
Add a fourth dimension (or more) and you have a hypercube which would look like this if the observer and it were standing still and the dimensions were 4 of space:
If the observer (or it) moved around so that the hypercube could be seen from different perspectives, it would look like this:
Add the perspective of time (space/time instead of just space) and it would look like this:
Getting back to communications for a second, Shannon's model looks at the message space as a sphere - the sum of which would look like a gumball machine before a selection is made by the receiver. That part is very well established in information theory and also as information theory is applied to molecular biology.
Shannon sphere: A sphere in a high dimensional space which represents either a single message of a communications system (after sphere) or the volume that contains all possible messages (before sphere) could be called a Shannon sphere, in honor of Claude Shannon who recognized its importance in information_theory. The radius of the smaller after spheres is determined by the ambient thermal noise, while that of the larger before sphere is determined by both the thermal noise and the signal power (signal-to-noise ratio), measured at the receiver. The logarithm of the number of small spheres that can fit into the larger sphere determines the channel capacity (See: Shannon1949). The high-dimensional packing of the spheres is the coding of the system.
There are two ways to understand how the spheres come to be. Consider a digital message consisting of independent voltage pulses. The independent voltage values specify a point in a high dimensional space since independence is represented by coordinate axes set at right angles to each other. Thus three voltage pulses correspond to a point in a 3 dimensional space and 100 pulses correspond to a point in a 100 dimensional space. The first `non-Cartesian' way to understand the spheres is to note that thermal noise interferes with the initial message during transmission of the information such that the received point is dislocated from the initial point. Since noisy distortion can be in any direction, the set of all possible dislocations is a sphere. The second `Cartesian' method is to note that the sum of many small dislocations to each pulse, caused by thermal noise, gives a Gaussian distribution at the receiver….
For a molecular machine containing n atoms there can be as many as 3n-6 independent components (degrees of freedom) so there can be 3n-6 dimensions. The velocity of these components corresponds to the voltage in a communication system and they are disturbed by thermal noise. Thus the state of a molecular machine can also be described by a sphere in a high dimensional velocity space.
A simpler way to consider how that might work would be to meditate playing Tic-tac-toe on a Hypercube and how a "solution" there might be akin to the message encoding structure of the DNA (not the DNA itself which is double helix).