INTRODUCTION
Fermat's principle of least time states that the path taken by light between two points is the path traversed in the least possible amount of time - this includes a path that is traversed over multiple media. Simple put, light follows the path of least time.From Fermat's principle of least time, the Law of Reflection and Snell's Law are derived.
The law of reflection applies an instance where light traverses over 1 media (being reflected off a non-transparent object), thus having a constant velocity. In such an instance, the path of least time is the path of least distance. With a reflected light ray, there are two sections: the incident ray and the reflected ray. In this case, the angle of incidence is equal to the angle of refraction.
Snell's law (also known as the law of refraction) applies to an instance where light traverses over 2 or more different transparent dielectric media. This law is not as simple as the Law of Reflection. With Snell's law, there is a definite relationship (based on the velocities of light in each media) between the angle of incidence and the angle of refraction. In an instance where there are two media being traversed, (which is the instance we will deal with in this project) it relates the incident and angle of refraction to velocity of light in 1st media and velocity of light in 2nd media. This relationship, of course, can be used in finding the path of least time.
In this experiment, ants will, quite frankly, be compared to light. More specifically: Can ants, like light, follow the path of least time though two media. It is well known that ants choose the shortest of routes to a food source. We know, though, that in a case where the path traverses two media, the shortest path is not necessarily the path of least time.
In a situation where ants are made to traverse a path of two media (one of which they can walk faster on than the other) will ants take the faster route as compared to the most direct one? Will ants take the path that obeys Fermat's Principle of Least Time?
Deriving Question:
How effective is the Ant Colony Optimization Algorithm in finding the path of least time.
How can this be applied to Network Engineering.
MATERIALS
For ant enclosure:
-Ants
-Smooth Plexiglass (to use as surface media)
-Smooth Polyester Felt (to use as surface media)
-Rough Polyester Felt (to use as surface media)
-Any other materials that could be used as surface media
-Balsa wood (to be used in building nest and enclosure walls)
-Hot glue gun
-Ruler
-Camera (to be used in recording ants velocity)
-Foam Board ( to be used for base of enclosure)
Software:
VirtualDub (video capture/ processing unit to be used in finding ants velocity)
visit: http://www.virtualdub.org/ for info
Desmos Graphing Calcultor(Online): https://www.desmos.com/calculator
diagram drawn by publisher
The image above represent the refraction of light. The formula at the bottom states Snell's law. n1 is the velocity of light in the 1st media, and n2 is the velocity of light in the 2nd media. With refraction, the shortest path is not necessarily the path of least time. The path found using Snell's law is the path of least time.
image retrieved from: https://chemicalparadigms.wikispaces.com/Unit+2+Refraction+of+light
The image to the left represent reflection. Notice that the angle of incidence equals the angle of reflection. With reflection, the shortest path is the path of least time.
Angle of incidence = angle of refraction
image retrieved
PROCEDURE
Build ant enclosure.
The ant enclosure will include an arena(where ants will be released to forage for food), a nest, and a location for food. Dimensions follow
Record ants velocity on different media.
(Using VirtualDub Software)
Using the recorded ants velocity on each media, calculate shortest path using Fermat's Principle of Least Time.
diagram drawn by publisher
The image to the left is a simple diagram of the ant enclosure. The objective for the ants is to get from point A to point B with the path of least time. Remember, that in the case of traversing 2 media, the shortest path is not necessarily the path of least time. In such case, many would infer that the orange path of this picture would be the one of least time - the
justification for this being
that the distance of travel over the slower media is minimized and the distance traveled over the faster media is maximized. Yet, this is, in fact, not the path of least time. This paths having the longest net distance overrules its having the smallest distance in the first media. They key is to find a perfect balance between the black path and the orange path. Otherwise put, at which point on the interface between the media should the ants make their cross?
To put this example in a more mathematical context, imagine that we have two right triangles (Refer to image below):
diagram drawn by publisher
It may look quite convoluted with so many variables, but be assured that the variables will be reduced. For now, understand that d1 is the distance over the slower media(v1), and d2 is the distance over the faster media(v2).Also, in this instance:
z = b, and d1 = d2.
(This instance being that of the black path)
It was mentioned earlier that the key to finding the path of least time is finding the point at which the ants should cross the interface between the media. This can otherwise be put as: at what length of z is the path of least time(which includes distances d1 and d2) met. So, we know that length z will decrease, and length b will increase. (Watch clip)
Note(on clip): The black cord represent the ants path(d1 and d2) from point A to point B. Notice: length z decreases at the same rate as length b increases(This relation will be important later)
To find path of least time,
Find formula for time of travel on path:
First let's find the time of travel for each component(d1 and d2):
since velocity = distance / time, time = distance / velocity,
and
Therefore,
(T represent Total time of travel)
Based on Pythagorean Theorem,
and
Therefore,
y and c are constants, so
From here, note b in terms of z. Remember that as length z decreases, length b increases at the same rate. x will be that increase of decrease of z.
When x is subtracted from z, x is added to (b=z)
So
One could also find the minimum using the first derivative test:
In order to graph the function, it is necessary that we have only one variable on the right side. Assuming that z=b at starting instance:
From here, plug in the values of the ants velocities(v1 and v2), and two variables are left: T(Time of travel time for overall path) and x(change in length z)
Assume v1= 1mm/s, and v2= 3mm/s
The equation is now in function form and ready to be plugged in to a graphing utility:
graph produced by Desmos Graphing Calculator
Minimum of function: T = 25.03 at x = 7.71
The path of least time:
Place food source, release ants from nest, and record using mounted video camera.
Organize results.
SCIENTIFIC PRINCIPLE
Fermat's Principle of Least Time- The principle that light traverses the path of least time when going through two different media
Snell's Law- The law that describes how to measure the path of least time given a specific situation.
Ant Colony Optimization Algorithm- It all boils down to this. Computers prefer to use algorithms as opposed to Calculus. The Ant Colony Optimization Algorithm is a method proven effective by the ants. This is done using pheromone communication. Refer to video on homepage for in depth explanation.
Investigation Questions:
The indices of refraction of the media, labeled n1 and n2 and so on, are used to represent the factor by which a light ray's speed decreases when traveling through a refractive medium.
1)What do you observe about the path of the beam when you turn on a laser pointer and shine the light? Shining the beam in air, is there any way to cause it to bend? Is it possible to redirect the light by reflection from a surface?
-It curves as it passes through, and the beam is more visible.No it won't bend when you light the beam in the air. Yes it is possible to redirect the light by reflection from a surface.
2)Imagine that a projectile had made a hole in the side of a container, and the water inside the container was flowing out. Make such a hole in a plastic bottle and fill the bottle with water. Allow the water to flow out of the container through the hole. Can you observe total internal reflection in this stream of flowing water?
-Yes I can see the total internal reflection in the stream of flowing water because when the water is in laminar flow the water stream acts just like a fiber-optic and carries the light.The laser gets reflected inside the stream.
3)How does the speed of light in water compare to the speed of light in air? How do both these speeds compare to the speed of light in a vacuum?
-Speed of light in air as taken generally is 3 X 10 power 8 meter per second and in water it become 2.25 X 10 power 8 meter per sec.The fastest thing in the whole universe is the speed of light in a vacuum (like outer space!), clocking in at a great 2.99 x 108 m/s.
4)How can you calculate the indices of refraction of water and of air?
-We can calculate the indices of refraction of water and of air using the Snell's law.
5)What happens to light when it passes from one medium into another medium with a lower index of refraction?
-It won't be bend, or refractive
6)What is the critical angle of incidence? How does light behave at the critical angle? What tools or materials would you use, and how would you use them, to demonstrate the existence of the critical angle and the behavior of light near this angle and at this angle?
-The critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees.The behavior of light at a critical angle it continues in a direction that is tangent to the boundary of the mediums.Fill the aquarium with water. Then add the milk a drop at a time, stirring after each drop, until you can see the light beam pass through the water. If you use powdered milk, add a pinch at a time.The farther the beam is from perpendicular when it hits the surface, the more strongly it is bent. If the light is moving from a material with a low speed of light into a material with a higher speed of light (for example, from water into air), the bending is toward the surface. At some angle, the bending will be so strong that the refracted beam will be directed right along the surface; that is, none of it will get out into the air.
SAFETY REGULATION
Take serious precautions if allergic to species of ant being used in experiment. Either way, make sure that nest is secure, for escaped ants indoors could be dangerous. Wear gloves and short sleeve shirt when handling ants. A short sleeve shirt is recommended so that ants are easier to repel if brought into contact with skin. If possible, make sure that gloves are tightly gripped around skin so that the ants being handled cannot make their way into the cavern that could be the result of an insecure glove grip.