I am looking for a theorem that states approximately the following:(adsbygoogle = window.adsbygoogle || []).push({});

An n-dimensional object, while appearing perfectly regular within the n-dimensional space to which it belongs, can actually be bent or distorted in (n+1) dimensions.

Please forgive my ignorance of the proper terms. I'm a newbie and want to learn.

Here is the my take on the problem on a simple model of Flatland suspended in 3 dimensions, exemplified with a sheet of paper in our familiar world (everything is examined in the framework of a hypothetical 3D Euclidean space) :

In the simplified case of Flatland, the 3rd dimension emerges orthogonally from its 2D sheet. If I sample 2 points on Flatland's 2D plane, by drawing a line through each pointinto the 3rd dimension, orthogonal to the plane, and then compare the angle between the resulting 2 lines, then, if this angle is 0, it means that they are parallel to each other and the area between these 2 sampling points is flat. If not, this means that the 2D plane is curved in 3D between these 2 points.

That part was simple.

Now, in 4D, a 3D object has 3 bounding planes, orthogonal to each other (XY, XZ and YZ).

Looking at this 3D objectfrom within the same 3D(i.e. playing with a cube in our familiar 3D world), with nothing distorted:

XY⊥XZ, XY⊥YZ , XZ⊥YZ

and if I draw a linea ⊥XY, lineb ⊥XZ and linec ⊥YZ,

these 3 lines,a bandc, arenotparallel to each other. (they are in fact⊥to each other).

Now looking at 3D objectfrom the 4th dimension, no distortions:

again I draw 3 lines orthogonal to XY, XZ and YZ, but this time I draw theminto the 4th dimension:

a ⊥XY,b ⊥XY,c ⊥YZ

in the 4th dimension, these 3 lines,a bandc,are parallelto each other.

In other words, from the point of view of the 4th dimension, the 3 planes XY, XZ and YZ belong to thesame 4-plane-?

.. and if it is not flat (determined by similar test to Flatland above) then the 3-dimensional object in question is distorted in 4 dimensions -?

Intuitively this appears evident to me, but.. instead of me re-inventing this bicycle, I'd like to avail myself of an already existing, properly formulated topological theorem. It gotta be out there. What's it called and where can I find it?

Thank you for your feedback :)

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# Which theorem? Determining distortion of a 3d-object in 4D

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