# Are all dimensions in an N-dimensional set homogeneous?

Hello and please excuse me if the title of the question makes little mathematical or English sense, my level in Maths is rather low and I am not a native English speaker, so I don't always choose the most appropriate words. I hope I'll be able to make the concept asked clear enough; maybe someone more knowledgeable than I can edit the title to a more appropriate one. Also, this question is related to physics too, but I am interested in the mathematical aspects of it, so I figure this might be a good place to ask it. I apologise if it is not.

In a three-dimensional set comprising three orthogonal spatial axes, I can transform a distance along one of the axes into a distance along an axis perpendicular to it by taking its vector product times a unit vector along the third axis, perpendicular to both. For example, if I have a distance along the X axis and multiply it times a unit vector along the Y axis, I get a distance with the same magnitude along the Z axis. I can do that regardless of the starting axis I choose, there is only one kind of transformation for all three, so I think of this 3-D set as being homogeneous, and please pardon me if that is not the best word to use in this case.

However, in a four-dimensional set comprising three spatial axes plus one reflecting how things change over time, and provided that there is an entity moving along some spatial axis at a rate greater than zero units of distance per unit of time, with respect to the origin of coordinates, I can transform a distance along the axis of motion into an interval along the temporal axis by dividing the magnitude of the displacement by the rate of motion. For example, if I have something covering 10 units of distance along the X axis with a velocity of 2 units per unit of time, I can divide one by the other and get an interval of 5 time units.

So in the 4-D set of space-time, there seem to be two different kinds of transformations applicable depending on whether we are transforming one spatial dimension to another (taking the vector product between two vectors) or transforming any spatial dimension to the temporal axis (dividing the magnitude of a vector by a rate of change). That suggests to me that the set is not homogeneous, in the sense that three of its constituent dimensions behave in one way and the other in a different way. And that may be how it is usually considered, but I am wondering, is there are more general kind of transformation that can be used between any two space-time dimensions, of which the two transformations described are only special cases? Can all dimensions in space-time be said to be homogeneous in that respect?

Thanks!

• In mathematics the word homogeneous is used with a variety of meanings. Your sense seems to be one of physical spacetime. If you are asking about physics and coordinate systems, this is the wrong place for such a question. Here we hope for questions that can be resolved with reasoned mathematical arguments. – hardmath Aug 21 '16 at 20:30
• Thank you, @hardmath. I thought this might be a good place to post my question, since I am not as interested -at this time- in the behaviour of an object changing its three-dimensional location over time but in the mathematical relationship between spatial and other dimensions. However, judging by the number of answers I've received from a mathematical perspective, you are probably right and this question would be more appropriate in a physics forum. I appreciate your clarification. Have a good day! – Carvo Loco Aug 23 '16 at 8:22
• Sorry to nag you with another question, @hardmath, but I am wondering about the tag "dimensional-analysis", described here as "the study of the relationships between physical quantities by identifying their units of measure and fundamental dimensions", that I chose for this question. Being a non-mathematician, non-physicist and non-native-English-speaker, I don't really know where to draw the line between physics and mathematics. Any hints? – Carvo Loco Aug 23 '16 at 8:28