"3-way dimensions"

[citc]

This is a rather unusual "theoretical" question.

 

I was thinking of basic arithmetic..of a certain kind and was wondering if there is (probably) an area of theo maths or physics that covers it.

 

The way we view the world as far as counting is concerned is that there are 2 ways to go, up or down. For example from number 3 i get to 4 by adding 1 or i get to 2 by subtracting 1. This is the definition of addition in the simplest form. (subtraction can be argued is addition of negative numbers)

 

Is there a 'space' where there are 3 ways to go, so that from element 'a' we can go to 'b', 'c' or 'd'? Adding 1 wouldn't really make sense then, or it would have to be defined differently.

 

Sure i can go to from a to b and then back to a , but what would x + y translate to when each of x and y can be changed (incremented is probably wrong) in 3 different ways each [instead of the 2 in usual arithmetic]

 

Dimensions is perhaps not really the right word coz even in one dimension there are 2 ways to go, so what i am thinking would perhaps be a "dimension" with 3 ways to go. Of course it wouldnt be 'vizualised' as a line, although connections are there

 

Just asking if there is a field of pure maths that has dwelled on this, or if it is defined in essence but differently.

 

 

 

 

 

[imatfaal]

Have you ever looked at imaginary numbers and the complex plane?

 

http://en.wikipedia.org/wiki/Complex_plane

[citc]

Have you ever looked at imaginary numbers and the complex plane?

 

http://en.wikipedia....i/Complex_plane

 

 

 

that would be 4 way. From 3+i5 , you can get to 3(+/- 1) + i5(+/-1). but you are right.

 

 

What i was thinking probably alludes to dimensions after all. A 3way system could be "maped" to 2d or 3d space. For example in 2d space. Each element is a point, that is connected to 3 other elements. So in 2d there are several ways to do this via e.g. lines at 120 degrees from each other, like the mercedes sign. The particular case would have some symmetries too as far as "covering" the space (2d in this case) we use to "visualise" it. If each element were connected to its 3 elements but the angles were different, arbitrary, it would still work, but it wouldnt cover 2d space the same way.

 

The only restriction, as i expressed it, was that you get from an element to one of its 3 branches and you can get back to that original element. We could relax that too, say, an increment would be defined by going to one of the directions (corresponding to a pre-defined angle as we see it in 2d)) by moving by some measure say exp(a*t) or even just a*t where 'a' is a global constant and t is time that always counts forward. This would imply that going from element k to any of the other 3 elements, could not get you back to k. But that doesnt really make each element the same as the others, you would have to define a fixed origin and the collection of all the elements depend on the starting point. Still possible and natural in some universe, just difficult to see i guess. This last example would probably be a case where there could be really no subtraction as we see it, only addition. There is no guarantee that you can get back to the original point for arbitrary a,x and "angles" defining the connected elements, except by chance ofcourse. We are probably digressing though, what i was looking for was a counting (co-ordinate) system.

 

So sticking to the restriction that you can go back to the original point form each of the 3 branches and to give another example, in 3d space this time, it could be a mesh of tetrahedra (like a four-sided die).

[which is said to be the the best 3d shape to (randomly) fill an arbitrary volume of 3d space so that most of it is covered- think of packing a container with boxes and trying to find the box design that would maximise your utilisation of the space of the container, if you just threw them in randomly without having to order them (or it would be cubes and we are done with that)]

 

 

So I think that it all falls mostly in the realm of set theory utilising geometry and just about any field after that, maybe even fractals if we choose to make it 'naturaly' exciting so thats that, i guess. Thank you for taking the time to answer. I do agree that the question looked like a Bart Simpson moment on my part, probably still is..

[Bignose]

sounds like a vector space to me.

 

i.e if you start with [1,1,1], you can't just add "1" to it. That is, you can't add a scalar and a vector. You have to specify in which coordinate to add one: You can add [1,0,0] or [0,1,0] or [0,0,1] to get [2,1,1], [1,2,1] or [1,1,2], respectively.

[DrRocket]

This is a rather unusual "theoretical" question.

 

I was thinking of basic arithmetic..of a certain kind and was wondering if there is (probably) an area of theo maths or physics that covers it.

 

The way we view the world as far as counting is concerned is that there are 2 ways to go, up or down. For example from number 3 i get to 4 by adding 1 or i get to 2 by subtracting 1. This is the definition of addition in the simplest form. (subtraction can be argued is addition of negative numbers)

 

Is there a 'space' where there are 3 ways to go, so that from element 'a' we can go to 'b', 'c' or 'd'? Adding 1 wouldn't really make sense then, or it would have to be defined differently.

 

Sure i can go to from a to b and then back to a , but what would x + y translate to when each of x and y can be changed (incremented is probably wrong) in 3 different ways each [instead of the 2 in usual arithmetic]

 

Dimensions is perhaps not really the right word coz even in one dimension there are 2 ways to go, so what i am thinking would perhaps be a "dimension" with 3 ways to go. Of course it wouldnt be 'vizualised' as a line, although connections are there

 

Just asking if there is a field of pure maths that has dwelled on this, or if it is defined in essence but differently.

 

You can model something like what you are describing in any number of dimensions with the group [math]\mathbb Z^n[/math] which is just the set of integer lattice points in n-space.

 

There are lots of ways to study this, the theory of Fourier series in several variables being one.

 

 

 

 

 

[niai]

sounds like a vector space to me.

 

i.e if you start with [1,1,1], you can't just add "1" to it. That is, you can't add a scalar and a vector. You have to specify in which coordinate to add one: You can add [1,0,0] or [0,1,0] or [0,0,1] to get [2,1,1], [1,2,1] or [1,1,2], respectively.

In the terms of the person asking, with one axis giving us two directions, in the three-dimensional vector space there would be... I am not sure, either six or more likely an infinite number of directions, as each vector defines a direction in the space.

 

If you don't insist on infinity, a simple example is the 3-d Boolean cube - the (0, 0, 0), (0, 0, 1) and so on up to (1, 1, 1) vectors are "written" on the vertices of a cube in a specific order (well, partial order), so from one vertex you have three choices about the direction.

[the tree]

I suppose that on a poincaire half plane you can essentially only go left, right or up. Is that what we're talking about?