Investigating the Possible Existence of a Fourth Spatial Dimension

[Duongvo]

A spatial system has three dimensions, x, y, and z. If we project this three-dimensional spatial system onto a two-dimensional plane and then introduce a time axis perpendicular to that projection, which axis would be perpendicular to both the spatial projection and the time axis?


If such an additional axis could be identified, would it allow us to detect or demonstrate the existence of extra dimensions or a multidimensional space?

 

[WWGD]

A spatial system has three dimensions, x, y, and z. If we project this three-dimensional spatial system onto a two-dimensional plane and then introduce a time axis perpendicular to that projection, which axis would be perpendicular to both the spatial projection and the time axis?


If such an additional axis could be identified, would it allow us to detect or demonstrate the existence of extra dimensions or a multidimensional space?

Yes, as pairwise perpendicular axes are linearly independent, you would have a quartet (x, y,t,w) that are pairwise perpendicular, hence linearly independent. You need 4 linearly independent axes for that, meaning you're in at least 4- dimensional space. Conversely, consider 4 dimensional space, then apply Gram - Schmidt , and you'll have 4 vectors that are pair wise perpendicular. Hope I understood your question.

 

[WWGD]

It's a nice exercise to show, e.g., by induction , that a set of n pair wise perpendicular vectors are , form, a set of linearly independent set. For n=2, consider 2 perpendicular nonzero vectors v1, v2. Assume c1v1+c2v2=0 for Real c1, c2 nonzero, then dot the expression by v1 or by v2; say v1, then v1<c1v1+c2v2>=c1<v1,v1>+0=0, thus c1||v1^2||=0, so c1=0 ( 0 vector). Now complete the induction on the number of pairwise linearly independent vectors.

 

[Hornbein]

A spatial system has three dimensions, x, y, and z. If we project this three-dimensional spatial system onto a two-dimensional plane and then introduce a time axis perpendicular to that projection, which axis would be perpendicular to both the spatial projection and the time axis?


If such an additional axis could be identified, would it allow us to detect or demonstrate the existence of extra dimensions or a multidimensional space?

You can define all the perpendicular axis that you like to get as many identical perpendicular spatial dimensions as you please. Usually this space is considered flat so these dimensions are related in the familiar way known way back to Euclid. The relation of the time dimension to the spatial dimensions is hyperbolic, which is why it seems so strange to us.

In physics there is no evidence of anything other than the usual 3+1 dimensions. As for the additional 7 dimensions of string theory as far as I know nobody really knows what they are. But I don't know much about string theory.

I've written a hopefully entertaining book about what life on Earth would be like if our Universe were 4+1. Free! https://www.researchgate.net/publication/398234590_Elsewhere_Everyday_Life_On_A_Hypergeometric_Earth
But no such thing exists. It's just a fun intellectual exercise.