Invisible objects

There are a lot of unscience fiction stories about invisibility cloaks and so forth, bu is it possible to build an object that is for all intents and purposes invisible.

Mathematicians say it is so and we give here a very simple example of an invisible object. Our definition of invisible is different from transparent. We would say that light passes through a transparent object. We will say that light is diverted around an invisible object.

In this case we will consider light coming in from infinity so that all the light rays are parallel.




Now observe that light from infinity is diverted around the object if we have constructed the angles properly. This construction involves isosceles triangles with base angles of 30 degrees such that the bases of the two facing triangles are a distance of 1 apart and the length of each base (the vertical line) is sqrt(3)/2.

Under these circumstances, can you show that the path of the line at the beginning and end are in the same direction (or line).

See Mathematics Resolution on invisible objects for more information.

A Simple Railroad Car Problem

Consider the simple railroad car problem

Two trains headed by engines wish to pass each other so that the trains are in the same order
after reassembly. There is one siding which will hold one car at a time. The engines may disconnect and go into the siding or they may push a car into the siding but only from the right side of the diagram.

This is similar to a sliding block puzzle except that that certain blocks or cars can only move if attached to the engines.

Give a solution to the problem. Is your solution the best (shortest number of moves) possible where a move is a position where a car has been put into the siding or removed from the siding.

Can you complete this task?

A supertask is a countably infinite sequence of tasks to be performed in order, say

{T₁,T₂,T₃,...}

such that we first perform task T₁ and then task T₂ and so forth.

As an example one might consider the case of Zeno's Paradox where an arrow is shot at some target. The arrows first task is to complete the journey half way to the target. The second task is to complete the remaining journey half way to the target and so on.

Now consider the case of Jon Perez Laraudogoitia's beautiful supertask: There are a sequence of unit point mass balls stationary at x=1/2,x=1/4,x=1/8 and so on. A unit point mass at x=1 and time t=0 is moving with a velocity of 1 unit per second to the left (i.e. toward x=1/2).

We assume that there is an elastic collision at x=1/2 and so the moving ball is left stationary at x=1/2 and the ball at x=1/2 moves with velocity 1 unit per second in the direction of the ball at x=1/4. It is clear that at t=1 all the balls have been hit by the preceding ball.

Assuming that all the collisions are elastic and behave in exactly the same manner as the first collision, is there a ball that passes through x=0?