- #1

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1) Does this actually work?

2) If it does work, what kind of geometrical shapes can you form a basis for?

- Thread starter crownedbishop
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- #1

- 20

- 1

1) Does this actually work?

2) If it does work, what kind of geometrical shapes can you form a basis for?

- #2

Fredrik

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It doesn't work. Every non-empty subset of ##\mathbb R^3## spans a subspace of ##\mathbb R^3## that includes points that aren't on the unit sphere.

Last edited:

- #3

Stephen Tashi

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Are you are asking if you can represent a sphere by using the usual method of treating vectors as 3-tuples of numbers? No, that wouldn't work. The sum of two vectors on the sphere would be off the sphere, so the set of vectors on the unit sphere wouldn't satisfy the axiom a vector space that says the sum of two vectors in the space must also be in the space.So I was thinking and I was wondering if we could have a set of vectors that spanned just the unit sphere, and nothing else beyond that.

However, the mathematical definition of a vector space is more general than the usual way of dealing 3-tuples of numbers. It would interesting to see what you can come up with.

A general approach to represent a surface as a set of vectors would be to find a 1-to-1 mapping F from the surface to the 2-D plane. Then for points P and Q on the surface, you define the operation P+Q to be: Use F to map P and Q to vectors in the 2-D plane. Do the addition on the vectors in the 2-D plane the usual way. Then map the answer back to the surface by using the inverse function of F.

This might not be the kind of thing you're looking for. However, it is mathematically legal to define the addition of vectors in a vector space in a complicated way, as long as the addition satisifies the mathematical axioms.

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Svein

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