Chasing the Pot of Gold

For those interested in a mathematical take on a natural occurrence.

This was originally a paper written for a math modeling class.

Everyone knows the pot of gold at the end of the rainbow is a myth, don’t they? Though this may be accepted as fact, it is quite impossible to prove. First, a rainbow does not occupy a point or region in space; instead it ‘floats’ along a path that is based on the position of the observer relative to the Sun. This means that, though two people may both see a rainbow at the same time in approximately the same place, there will be a slight difference in the observed position of the rainbow. Second, the ‘end’ of the rainbow is merely the point(s) at which it disappears behind the horizon. In reality, the refraction of light that is observed as a rainbow is a circle, the visible arc of which varies with the elevation of the Sun.

A rainbow is the manifestation of the refraction of sunlight by water droplets in the atmosphere. The angle of the projected prism is approximately 42 degrees to the angle of sunlight, with the angle decreasing toward the red end of the spectrum. This variation in the angle of refraction for the different parts of the spectrum is what splits the white light into its component colors and causes the ‘ROYGBIV’ arrangement of the rainbow.

Despite the tenuous nature of the rainbow, it still may be tempting to quantify or measure it in some way. One way would be to measure its height or width based on incline or azimuth from the observer’s point of view. On deeper reflection, this proves impossible. Not only is there no way to measure the distance to the plane of the rainbow in order to calculate from the other two measurements, it turns out that these angular measurements are purely a factor of the elevation of the Sun, as long as the elevation of the observer relative to the landscape does not change. In essence, the rainbow can be thought of as a cone projection in reverse; the sun strikes water vapor and is refracted at 42 degrees. If the observer is thought of as the point of this cone, the base is the circular rainbow projecting back the sunlight from behind the observer. The ‘bow’ shape is merely the part of the base of the cone that is above the horizon; the higher the sun in the sky, the less the arc length visible. As the distance from the observer to the plane of the rainbow increases, the diameter of the rainbow grows, but the angles remain the same, it is similar to taking slices perpendicular to the cone at different heights. If one were to look deeper and try to use the thickness of the rainbow to determine distance, the same property arises. Think now of this cone as two concentric cones, one of 40 degrees from the main axis to the sides, the other at 42. As planar slices are taken, the region between the surfaces of the cones is the span from red to violet; again this width or thickness increases with distance, but the angular measurement from the observer’s point of view is unchanged. Effectively, there is no means by which to use the observation of a rainbow to measure either its size or its distance. On the other hand, since the angles involved are known if either size or distance is given, the other can be calculated. If one, for the sake of superstition, argued that the pot of gold existed at the ‘apparent’ base of the rainbow, this point could be found from angular measurements, or from the known location of the Sun, and the distance to the horizon.

Say that the horizon is 2.5 miles away and the sun is at 20 degrees above the horizon behind the observer. Since the radius of the rainbow is the opposite side of a right triangle with angle 42 degrees and adjacent side 2.5 miles, determining the total size of the rainbow is simple trigonometry: Tan (42 degrees)*2.5miles=2.25 miles. Since only 42-20=22 degrees of elevation exist between horizon and top arc of the rainbow, we have a visible height that is 2.25 miles-tan (20)*2.5=1.34 miles, which relates to a ‘y-intercept’ of .91 miles. Now, with a circle of radius 2.25 and a y-intercept, the x coordinates are needed. Since a circle is y^2+x^2=r^2, we have .91^2+x^2=5.0625, x=2.058 miles. So our distance along the z-axis is 2.5 miles, in the x-axis 2.058 miles and our straight line distance to the pot of gold is 3.24 miles at a heading of +/-42 degrees. As long as the observer leaves his eyes behind, he can see himself reach the end of the rainbow.

CTM lit

I have published a partial list of my writings online-click on the title or copy and paste address. I will update as I progress on this endeavor. Eventually, I hope to post these all as pages of a single website and may possibly produce a serial (in my wildest dreams, right?). Please forgive the variety in formats...and please, enjoy:

Money Matters (published in The First Line)-http://docs.google.com/View?id=dvgtv7j_1hscnkmgv

The Human Condition-http://docs.google.com/View?id=dvgtv7j_2sb59jbfb

Seven-http://docs.google.com/View?id=dvgtv7j_3d7n54hhb
Snow is Best-http://docs.google.com/View?id=dvgtv7j_4hqb3r5c4
Junior's Creek-http://docs.google.com/View?id=dvgtv7j_5dt623rfc
Interaction-http://docs.google.com/View?id=dvgtv7j_6d3vtp82q

Insomnia-http://docs.google.com/View?id=dvgtv7j_7cqhz32gc

Mist-http://docs.google.com/View?id=dvgtv7j_8c2skr2hs

Anatomy of a Yard Sale-http://docs.google.com/View?id=dvgtv7j_9csjc3bgs

I should have a couple of regular essay posts shortly...

Intellectual debate part 1

Intellectual debate-part 1.

At a time when technology, a large population across a vast swath of land, specialization, consolidation of economic sectors and increasing international trade have made issues more complex and resolving them more critical, we seem less capable of doing so in a thoughtful manner. Our political leaders and partisans in the media have honed their sound bite skills to a razor’s edge and most of us seem content to quote them unthinkingly without questioning the logic or reason (or lack therein).

This brief essay addresses one of the weaknesses in statements regarding the current health care reform debate.

I recently heard someone say that they were against universal health care because it was too much like socialism. There are a couple of problems with this statement and both point to the lack of intellectual activity on the part of the average American.

The first problem is with the definition of socialism. Socialism is government control over the means of production. This label may or may not apply to universal health care, depending on how it is done. Subsidizing of health premiums does not meet the definition. A single payer plan where the government collects taxes, fees or premiums and pays providers directly also fails to meet the criteria (though completely controlling the market for health insurance might be socializing insurance, it is not socialized medicine). In contrast, if the government ran the hospitals, hired the doctors, bought all the equipement, etc, that would be socialized medicine. The recent government bailouts of banks and automakers (and previous takeover of railroads) IS socialism.

The second part of the statement implies that the listener automatically assumes socialism is an undesirable condition. Though many may believe that a pure socialist state is a bad thing, even those do not really object to the socialism of certain elements of our economy. How many Americans would like our military to be a capitalist enterprise? Would we want to lose a war because the lowest bidders did not spend enough money on recruiting or training soldiers? How about legislation? We already have enough problems with PAC money influencing our elected officials; how bad would it be if we opened seats in congress to bidding?

The examples given may seem extreme, but they illustrate at least that there are some tasks best served by non-capitalist means and perhaps we should admit to this and argue for or against partial or full government control of services and economic sectors based on logical premises rather than hot button words and catch phrases that stir emotions, but don’t contribute to the debate. It might even be helpful if some of us would agree to discuss these issues without the use of inflammatory terms or exaggerations. If socialism is bad and the things that make socialism bad also apply to universal health care, then we should say, ‘Universal health care is bad because I object to paying for someone else’s care’ or ‘Government control of health care costs will stifle innovation’ and then be prepared to discuss the strengths and weaknesses of the arguments.

More to come on this…

To dance or not...

Why are Baptists so anti-sex? Because they are afraid it could lead to dancing.

J Vernon McGee, a man whose commentaries I have gained much insight from, once used the passage where David danced before the Lord as a basis for teaching that Christians should not dance. Similarly, I once sat in on a Sunday school lesson where the passage regarding not drinking to excess was used as a basis for teaching that Christians should not drink at all.

Without debating the merits of dancing and drinking, in both cases the Bible is used to enforce a rigid set of rules and behaviors that do not fit with the message of what Christianity is and is not.

All of this is a prelude to my point.

We often here the refrain of ‘love the sinner, but hate the sin.’ I have always believed that this is a PC way of labeling someone as less righteous than ourselves, while technically ‘not judging.’ I don’t believe that this fits with the teachings and life of Christ; if anything, it is the opposite.

It is a human need to feel good or worthy, even if it requires looking down on someone else, but Christians are instructed to be humble. Our role model in Jesus humbled himself to the most humiliating treatment and death offered in first century Israel. This punishment, the penalty of the sins of the world, was taken on by the one man who did not deserve it…willingly. If this is not an example of ‘love the sinner and forgive the sin’, then take a look at His interactions with others during His ministry. The only words of condemnation were for the religious leaders who considered themselves righteous and looked down on those who did not meet their (not God’s) standards. When dealing with sinners, Jesus was sympathetic and loving.

This need to define sins and consequently sinners is as prevalent as ever today. How many Christians would judge (even a little bit) someone who had an abortion harsher than someone who had a doctor create multiple embryos that will never see the light of day for fertility treatment? How many would judge someone who engaged in a homosexual act harsher than a heterosexual person who was promiscuous? Would we judge a drug addict as worse than someone who was proud?

So what is the right tack in dealing with sin? I am reminded of Romans 6, ‘do we go on sinning that grace may increase?’ Of course, the answer is no: a life of sin is not what God intended and God’s intent is the highest and best use for our lives. But this is personal, and the point here is interpersonal. When dealing with others in the world, we are to show God’s love, reach out and help, both spiritually and physically, in any way we can, without judgment. We are all sinners and if there are some sins that are worse than others, they are the pet sins of many Sunday Christians:

Matthew 25 tells us that Jesus will reward us for reaching out to those in need and will deny us if we don’t. Are we doing this to the best of our abilities? I know I am not. I even wonder if His instruction to the rich young ruler related to this in some way; are we really reaching the world around us if we have lots of ‘things’?

How about Revelation 3? We go to church on Sunday, throw something into the offering plate, perhaps read the Bible and pray and maybe even stick a fish on the back of our car. Does that qualify us as being on fire for God? In Revelation 3, God says that he would prefer us to be either cold or hot and that those who are lukewarm he would puke up (paraphrased).

These harsh judgments are, like those Jesus spoke during his time on earth, harsh and aimed at the believer. The sinner is a dead man walking, blinded and bound in his sin. Our job is to let Jesus work through us to reach him, restore his sight, release his bonds and breathe life into him. At some later point we may help him move away from his sin, but only once the relationship is there that we speak redeemed sinner to redeemed sinner. Even then, it is best if we merely seek God together and let Him highlight our sins. In other words, ‘love the sinner and let God worry about the sin.’

There are exceptions to this, such as those in a position to lead others astray, but once someone is cleaned up to that point, we don’t tend to look down on them too much, anyway.

I guess my point is that if we want to call ourselves Christians, we should live by the book as it is written, not as we have made carefully selected verses fit our philosophy. We need to ask ourselves if we have spent more time reaching out to the fringes of our society than worrying about gay marriage. We should ask if we have removed the log out of our own eye before worrying over the splinter in someone else’s. Most of all, we should recognize that living the life Christ redeemed us for means being freed from the burdens of sin; though we don’t ignore it altogether, if we spend everyday focused on what we should not do, we will never ponder what we SHOULD do. The Christian life is more about doing than refraining; Jesus said ‘go into all the world and make disciples’ and ‘follow me’ and ‘wash each others’ feet’ and ‘pray’ and ‘forgive’.