Do we live in a 3D Hypersurface?
How would we know about it? We can see the three dimensions. The surface aspect would come into play if those three dimensions were embedded into some extra dimension.
If there was something outside that Hypersurface, we would see it in the same way, a Soccer ball left under the hot Sun, would feel its heat.
That heat would be diffuse and since no direction within a 3D spatial manifold could be used to point to the actual source.
Luckily, we don't see that happening, which means we are not in a collision path onto another part of this Universe (4D spatial manifold).
So, it would seem that a fourth dimension would be unknowable..:) Einstein proposed that time is relative (time flow). That sound like a great sound bite...:) and clearly incorrect since we all describe the beginning of times to be around some 14 Billion years ago. So there is something timing the Universe expansion and that somehow always flow in the same way no matter where you are in the Universe.
The timing of the Universe expansion is mapped into the Cosmological Time in my theory. The fourth dimensional radius of the Universe is related to it by the observed speed of light. The fourth dimensional radius of the Universe is C (observed speed of light) times Omega (Cosmological Time).
In my theory, the actual speed of light is SQRT(2) C...:) since in my theory our reference frame is a lightspeed traveling reference framework.
Other Hypergeometrical Universes
In my theory, numbers and Number Theory plays a strong role, albeit accidental. I am not a Numerologist..;) Primes felt on my lap when I started my assignment of Hyperons.
Primes also appears in my reference to a Number Universe where the volume of each Number is always a product of vectors along Prime directions. This view of a Number Universe has a reflection on the Hypergeometrical Universe Cosmogenesis.
Remember that the Cosmogenesis started in Zero Dimensional Fluctuations (0 = 1-1= 2-2=7-7 etc).
Fluctuations of Zero Dimensions could be followed by One Dimension Fluctuation through a Dimensional Transition.
The next step is Two-Dimension embedded into 3D, Three-Dimension into 4D...
Remember that the Origin of the Universe is the decay of the Fundamental Fluctuation into a myriad of small fluctuations. Enthropy creates the barrier to the recombination into Oblivion. The question is how entropy barrier depends upon dimensionality. I am not going to dwell on that issue right now, but entropy increases with dimensionality (higher number of possible states), thus it is necessary a minimum level of Dimensionality Migration to create a long lasting Universe.
At this point we can define an Anthropic argument that our 3D Hypersurface (lightspeed expanding due to the partial recombination of the initial fluctuation) would would live long enough for us to exist... no mentioning of the precise value of the Gravitational constant, etc... just an Antropic argument based upon dimensionality dependent entropic volume implied a sizable residual Universe after initial recombination.
The next great question is if Primes controls the dimensionality of Universes... :) I would be my house (I live in an apartment..:).
It is an appealing concept. I will make it MP Conjecture..such that my name is attach to it..:)