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Are there some English words that describe the dimension of objects changed, by combining some lower dimensional objects to form a higher dimensional object. For example, from the lower dimensional object (like a point in 0 dimension) to a higher dimensional object (like a line/string in 1 dimension)?

What I can come up with is the "dimensional transmutation." However, are there shorten single word for this concept? Any adjective for this concept? This is required to be used in a scientific or math article.

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I disagree with the suggested answer of "transposition". Transposing rearranges the dimensions without affecting dimensionality.

In the abstract sense, I would say that a lower dimensional concept generalizes into higher dimensions (e.g. an n-sphere is the generalisation of a circle) but that doesn't necessarily work for the specific act of combining lower dimensional structures to form higher dimensional ones of a specific dimensionality. I would simply call this extension: two one-dimensional lines arranged perpendicularly is an extension into the two-dimensional plane. Three perpendicular lines extend to a three dimensional space, and so on.

In the specific case of geometry, we can also talk about projection into higher-dimensional spaces.

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  • I would not suggest the word projection as it refers to a kind of mapping from a set onto a subset of itself. In other words, OP seems to be looking for an antonym for projection, which does not seem to exist. I have seen the verb "lift" used in some technical settings, e.g. math.stackexchange.com/a/310826/37705 – Brendan W. Sullivan Mar 22 '18 at 2:08
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I don’t believe there’s a general term for this, since it depends on the particular operation. One option is extend, for example, “a 3-dimensional Euclidean space can be formed by extending the xy plane with an additional perpendicular z axis”, or “extend the line L by one unit in the positive z direction to form a plane”.

I would suggest you use the most specific term available for each instance. If you have some particular means of combining objects that you’re using, then refer to that specifically. You might take a curve and create a surface of revolution about a particular axis. You might take a point, line, plane, or cube and extend or extrude it into a line, plane, prism, or hypercube, respectively; or take two and connect or fuse them according to some rule. The most general term I can think of is simply generalise—for example, take a space-filling curve and generalise it to a higher dimension.

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  • thanks, “a 3-dimensional Euclidean space can be formed by extending the x–y plane with an additional perpendicular z axis” this is the correct example. – wonderich Jan 23 '18 at 23:14
  • This is the best answer, especially considering paragraph 2. There isn't a generally applicable answer, but paragraph 1 is likely what you want. Well done. – jimm101 Jan 24 '18 at 19:38
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My understanding of what you are expressing is that 'transposition' would be more suitable than 'transmutation'. If I understand correctly there is no fundamental mathematical change. It is a dimensional change but the (combined) entity is transposed into another dimension.

The meanings of 'transposition' given in OED list two that appear, to me, to be relevant :

3b. Math. The interchange of each row of a matrix with the corresponding column.

and :

  1. Algebra. Transference of a quantity from one side of an equation (or one member of a proportion) to the other.
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  • But the dimension is different by 1. I dont know does it capture it – wonderich Jan 23 '18 at 2:57
  • @wonderich I think you are better placed than I to comment on that aspect. – Nigel J Jan 23 '18 at 3:14

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