I Intersection of a 4D line and a 3D polyhedron in 4D

AI Thread Summary
The intersection of a 4D line segment and a 3D polyhedron in 4D is indeed a point if they intersect, assuming they are not confined to the same 3D space. This conclusion aligns with the concept that a line in a higher dimension can intersect a lower-dimensional object at a single point. To prove this, one can analyze the situation using a coordinate system that aligns the 3D polyhedron with the axes. The reasoning parallels that of a line intersecting a plane in 3D, extended to an additional dimension. Thus, the intersection remains a singular point in 4D space.
LCDF
Messages
15
Reaction score
0
Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.
 
Mathematics news on Phys.org
If they are not in the same 3D space then the intersection will be a point. You can even use the full 3 D volume and a full line and it will still be a single point.

You can look at the line segment in a coordinate system where the 3D object is aligned with three axes, for example.
 
Thanks. Is there a mathematical proof for your observation? A hint would also work.
 
My second paragraph is a description how to get to a proof.

It's basically the same as with a plane and a line in 3D, just with one more coordinate.
 
  • Like
Likes LCDF

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
View full post »

Thread 'Six Pencil Puzzle'

Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
View full post »

Similar threads

Equator of a Planet with Four Spatial Dimensions is a 4D Torus
Replies
9
Views
4K
I A way to comprehend the geometry of a hypercube
Replies
2
Views
2K
I How Is Intrinsic Curvature Measured in Higher Dimensions Like 4D Spacetime?
Replies
19
Views
357
Wrapping a 3D space wihtin a 4D space
Replies
4
Views
3K
I Round 3-sphere symmetries as subspace of 4D Euclidean space
Replies
7
Views
954
B What Is the Fourth Dimension and How Does It Work?
Replies
3
Views
2K
Perspective sketches from 4D to 3D?
Replies
2
Views
2K
B 4th spatial dimension thought experiment
Replies
8
Views
2K
Is the 4th Dimension More Than Just Time?
Replies
9
Views
3K
Map of a 4D planet
Replies
8
Views
1K

Hot Threads

Recent Insights

Back
Top