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I'm reading Basic Mathematics from Serge Lang and reached to the point with isometries in 3d space (that's in the exercise section). It got me thinking.

The question: If the intersection of two lines(in 2d space) is point and the intersection of two planes (in 3d space) is line, is there such thing as intersection of two 3d spaces?

Thanks in advance for everyone who reply! And please take in account that I'm reading Basic Mathematics(emphasis on basic). Answer like "google these things..." will be satisfying enough.

hardmath
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Rumata
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  • The insight you might be missing is how the "intersecting" 3D spaces are embedded in a higher dimensional space. – hardmath May 23 '17 at 01:21
  • So this mean that the 3D spaces preserve their properties. I still can't understand how they are embedded in higher dimensional space. – Rumata May 23 '17 at 01:57
  • You will get some tools to model this in studying linear algebra, but I'll post a brief sketch of how to do this. – hardmath May 23 '17 at 02:26
  • I think that I'm getting it (or my mind is just desperately trying to come with something so I'll stop bothering it). If we have four dimensions with directions ABCD and 3d space expanding in ABC and another 3d space expanding in dimensions BCD. They only common points will be on the plane expanding in directions BC. So they will intersect in plane? – Rumata May 23 '17 at 02:31
  • Right, you can think of the first three coordinates in four dimensional space versus the last three coordinates. With a bit of rigor the only points common to both "subspaces" will form a two dimensional subspace. If more than four dimensions are available to "move around in", the intersection can be smaller. In fact your "spaces" might be parallel (like parallel planes in 3D) and have empty intersection even in 4D. – hardmath May 23 '17 at 02:41
  • Thanks for your help! This is quite interesting. Can you tell me which branches of mathematics are focused in studying topics like this, at least at more basic level? – Rumata May 23 '17 at 03:02
  • It's also common to get stuck in the usual 3D space because that's the maximum dimension humans can physically imagine. But often mathematical analysis goes well beyond 3D space. As hardmath noted courses in linear algebra help to acquaint you to the concept of spaces of arbitrary dimension...even spaces where the usual notion of a vector in euclidian space is much different (like polynomial spaces). My favorite are the infinite dimensions of a Hilbert space used in solutions to quantum mechanical models. – Triatticus May 23 '17 at 04:51

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You started with the depiction of two lines in a plane intersecting in a point. It is also possible that two distinct lines in a plane have no intersection, i.e. the lines are parallel.

In the case of two distinct planes in three dimensional space we have the following possibilities for their intersection:

  • They intersect in a common line.
  • They do not intersect (parallel planes).

If we put two distinct planes (two dimensions) inside a four dimensional space, then in addition to the above we have the possibility:

  • They intersect in a common point.

Let's use the notation of Cartesian coordinates to explain this. Suppose we attach real coordinates $(x,y,z,t)$ to four dimensional space. One two dimensional subspace ("plane") consists of points $\{(x,y,0,0)\mid x,y\in \mathbb R\}$, and another (for instance) consists of points $\{(0,0,z,t)\mid z,t\in \mathbb R\}$. The only common point is the origin $(0,0,0,0)$.

In like fashion we can consider three dimensional subspaces of four (or higher) dimensional space. As the OP contemplated in a Comment above, taking $U = \{(x,y,z,0)\mid x,y,z\in \mathbb R\}$ and $V = \{(0,y,z,t)\mid y,z,t\in \mathbb R\}$ gives a two dimensional intersection:

$$ U \cap V = \{(0,y,z,0)\mid y,z\in \mathbb R\} $$

In addition to "Euclidean geometry" of various kinds (plane, solid, etc.), we use the term synthetic geometry to refer to studies where Cartesian coordinates are attached to points, allowing algebra to inform our solution of geometric problems. Such studies are customarily introduced in secondary school education and continue in college and beyond.

A related topic in undergraduate college courses is linear algebra. Courses in analysis may advance the studies from finite-dimensional spaces to their infinite-dimensional "cousins".

hardmath
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