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Years ago in math class I thought I heard that n dimensional objects live in n + 1 dimensional space.

So a point must at least exist on a line. A line must at least exist on a plane. A plane must at least exist in a three dimensional space.

Is this true? Is there a place I can learn more?

ash
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    "Dimension" is one of those terms that mean different things in different branches of mathematics. In my experience of the different meanings of the term, an "object" (whatever it means in the given context) of dimension $n$ can exist totally independently without an "object" of dimension $n + 1$, but can always be extended to an "object" of dimension $n + 1$ if needed. – Theo Bendit Sep 19 '18 at 01:29
  • Hi, welcome. What is your background in math? If you're asking for a place to learn more about this topic, it would help to explain where you are now. – Matthew Leingang Sep 19 '18 at 01:35

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If by living you mean it could be embedded in an $n+1$ dimension, your answer is yes, we can extend the dimension and have our old object live in higher dimension.

But a line passing through the origin by itself is a one dimensional space and it does not have to be viewed as a subspace of a plane to be studied.

Same with a plane passing through the origin which is a two dimensional vector space and it is studied without the need to be seen as a subspace of a three dimensional vector space.

Mohammad Riazi-Kermani
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