# Evenly spaced vectors in N-dim

• Gerenuk
In summary, the conversation is discussing the most convenient way to obtain a set of unit vectors in N dimensions with a fixed angle between neighboring vectors. Possible methods include using a N-dimensional cube or a cross-polytope. It is suggested that this problem is similar to a sphere-packing problem, where the normalized vectors are considered to be on a higher-dimensional sphere. The desired solution is an approximate one that covers all the surface area.
Gerenuk
What is the most convenient way to get a set of unit vectors in N dimensions, such that the angle between two neighboring vectors is about a fixed value?

To be more precise I'd like to find a small number of vectors so that all possible other unit vectors are at most at a certain angle from a vector of this set.

It doesn't have to be precise, which probably would work for platonic solids like cubes only. Some rough method is to take all points on a N-dimensional cube and normalize the vectors. A whacky idea would be to do the same with a cross-polytope, but I don't know how to parametrize that.
But maybe one can do better?

Sounds like a sphere-packing problem, where you're packing (N-2)-spheres into an (N-1)-sphere. What I mean is, in 3 dimensions, your normalized vectors live on S^2, and their angular neighborhoods are 1-spheres (circles).

Yes, it is. However a full sphere packing problem is highly non-trivial, but I would be satisfied with a more approximate solution. The requirement is to find a reasonable solution which covers all the "surface area".

## 1. What are evenly spaced vectors in N-dim?

Evenly spaced vectors in N-dim are a set of vectors with equal spacing between each vector in an N-dimensional space. This means that the distance between any two adjacent vectors is the same.

## 2. How are evenly spaced vectors in N-dim different from evenly spaced vectors in 2-dim?

The main difference between evenly spaced vectors in N-dim and 2-dim is the number of dimensions. N-dim refers to an N-dimensional space, while 2-dim refers to a 2-dimensional space. In N-dim, there are N axes and N components for each vector, while in 2-dim, there are only 2 axes and 2 components for each vector.

## 3. What is the significance of evenly spaced vectors in N-dim?

Evenly spaced vectors in N-dim have many applications in mathematics, physics, and engineering. They can be used to represent physical quantities, such as position, velocity, and acceleration, in a multi-dimensional space. They also have important uses in data analysis and machine learning algorithms.

## 4. How are evenly spaced vectors in N-dim used in machine learning?

In machine learning, evenly spaced vectors in N-dim are often used to represent data points in a multi-dimensional feature space. This allows for more complex relationships between the data points to be captured and can improve the performance of machine learning models.

## 5. Can evenly spaced vectors in N-dim have negative components?

Yes, it is possible for evenly spaced vectors in N-dim to have negative components. In an N-dimensional space, each vector has N components, which can be positive, negative, or zero. The spacing between vectors is determined by the magnitude of the components, not their sign.

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