Illuminating a Convex Solid At Higher Dimensions

[gavinksong]

This is a follow-up to  do not know the answer to this problem.

 

How many lights are necessary to completely illuminate a convex solid in N dimensions?

 

Same conditions as before:

• We are working in an otherwise empty N-dimensional space.
• The lights are point sources.
• The solid is illuminated at every point on its surface where a straight line can be drawn uninterrupted between it and a light source.

Let me also add:

• Lights cannot be placed "at infinity".

[bonanova]

N+1. Add a dimension, add a light.

Point - 1 light

Line - 2 lights.

Circle - 3 lights

Sphere - 4 lights

Hyper sphere - 5 lights , but I can't draw the picture.

Etc...

[gavinksong]

N+1. Add a dimension, add a light.

Point - 1 light

Line - 2 lights.

Circle - 3 lights

Sphere - 4 lights

Hyper sphere - 5 lights , but I can't draw the picture.

Etc...

 

Can you prove it?

[harey]

 

Point - 1 light

 

Can you prove it?

 

I hesitate between 1 and 2...

[bonanova]

@gavinksong No, I can't.

 

For solids it just seems intuitive.

 

For points and lines, I am ready to change my answer.

Points and lines have no surfaces to illuminate.

In that sense they need 0 lights.

 

Shining 0 lights on a point or line suffices to leave no surface un-illuminated.

[harey]

Not so quickly...

 

If we need (in 3D) four lights for a sphere, a cylinder can be illuminated by three lights only. (I spent a long time to find a way to show that the edges can be smoothened and still illuminated until I consulted http://en.wikipedia.org/wiki/Convex_body and realized that a cylinder is a convex solid).

 

For a cone, two lights are enough.

 

What bothers me at most: Consider a unit cube in the xyz coordinates and a line (0;0;0) - (1;1;1). If we need 0 lights to illuminate a point or a line, the whole cube would be illuminated by just two lights placed on this line just slightly outside the cube.

Edited January 26, 2015 by harey

[bonanova]

@harey, all good points. I totally agree.

The answers given previous to your post apply only if we interpret the OP to ask

 

How many lights are necessary to completely illuminate any/a general convex solid in N dimensions?

 

Otherwise, we can enumerate special cases, as you have.

And a complete answer would be a complete enumeration.

 

These thoughts suggest what I think is an interesting question:

 

for each N, find an example of a convex solid that can be illuminated with the fewest lights.

So far we have,

 

N=0 point. zero. (a point may not qualify as a solid)

N=1 line segment. zero (a line or line segment may not qualify as a solid)

N=2 triangle. two. (a triangle may not qualify as a solid.)

N=3 two (truncated cone, cube lighted outside two opposite corners)

N=4 three?

N=5 four?

.

N=Q Q-1?

 

We could eliminate 0, 1 and 2 if we require the object to have non-zero volume.

Then we wouldn't have to debate whether dots, lines and circles are solids.

[harey]

These thoughts suggest what I think is an interesting question:

 

for each N, find an example of a convex solid that can be illuminated with the fewest lights.

 

That's the start to the original question

 

What bothers me with points and lines:

 

We do not illuminate an object as such, we illuminate it's surfaces. (Already a problem with a sphere and alike: in this case, we may not consider the sphere has one surface, but infinity.) If a surface is illuminated, does it imply that it's edges and vertexes are illuminated, too? One answer would be that this question does not have a sense, but it seems a little week as argument.

 

[bonanova]

The solid is illuminated at every point on its surface where a straight line can be drawn uninterrupted between it and a light source.

[harey]

This assumes that a point is (at least partially) illuminated if a straight line can be drawn uninterrupted between it and a light source.

[gavinksong]

A point, whether or not it is on a surface or on an edge or a vertex, is illuminated if a straight line can be drawn uninterrupted between it and a light source.

 

Thus, for example, in the three-dimensional case, a cube could be fully illuminated by two lights.

 

Like bonanova said, you may assume that the OP is asking for the fewest number of lights required to guarantee that any convex solid can be fully illuminated by some positioning of the light sources relative to the solid.