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 »

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
362
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
959
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