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The most famous Belgian comic hero is the brainchild of Herge, the pen name of Brussels author Georges P. Remi. The scene depicted is from The Calculus Affair.
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Be kind to cute creatures.
Professor Calculus knows that.
Little cute hedgehog from Paperdolly.
I bought Professor Calculus.
Gah, The calculus test was brutal. It'll be a freaking miracle if I get a D. Luckily, the class is early on in the day. The rest of my stuff is a joke. History? Already know it. Spanish? Fácil. Language Arts? A joke. And last but not least, Art. Basically, this might as well be recess. The teacher just tells us to draw and we do almost whatever we want. Not only that, I share this class with all my friends and Steph. Though she seems a bit late today. Nevertheless, the other three morons were at our table. These guys, Adam, Mike and Dom...oh boy, where to start? These guys are like a bad sitcom. Hilarious in it's own horrible way. When i walked up and grabbed a chair, they were all like "Heyyyy!" This is gonna be crazy...
Mike: "Been a bit, man, what's goin' on?
"Nothing. The gangwars held up something."
Dom: "Ah yeah. Freeze and his snowmen tearing shit up."
Adam: "About damn time Freeze lost. I was freezing my ass off with that damned arctic weather!"
Mike: "I was making spears from sticks and stones ready to hunt some caribou."
Dom: "You gotta be kidding."
Mike: "No shit I'm kidding. I'm not a caveman."
Adam: Really? You got hair like one."
Mike: "Dude, fuck you!"
Adam: "Lighten up, man. We know I'm not one to talk."
Dom: "Yeah, hence the hat. Seriously, how many detentions do you have because you wear it in class?"
Adam: "I lost count. Still, ain't the worst thing I've done this year."
Mike: "No shit. What was that you just blurted out in the middle of study hall? 'I have the weirdest boner'?"
Adam: "Yep!"
Dom: "Seriously, what the fuck? Why'd you say that?'
Adam: "Just to make everyone laugh."
Mike: "I've met 12-year olds more mature than you, y'know that?"
Adam: "Oh, I'm immature? Hey Tim! Take a look at fartknocker's sketchbook here!"
Oh god. This is gonna be stupid. Adam handed off the book to me and I opened it up, ready for virtually anything. What I got was....rather skillful drawings of boners. How charming. let's see, we've got giant monster boners, boners in knight armor, spaceboners...oh lord, is that a boner dressed like superman? I shouldn't be laughing at this. But yet, I am. I kept skimming what I'll now call the Encyclopedia Bonetannica, until a familiar voice made me shut the thing faster than I could blink.
"Hey guys!"
Adam/Mike/Dom: "Hey Steph..."
"What'cha got there, Tim?"
"Just Dom's book. Here ya go man.."
I quickly handed back the book back to Dom. Man, that was WAY too close. Thank god Steph didn't catch me looking at boners. While I gave a sigh of relief, I saw Adam lean in near Steph, and whispered just loud enough that I could hear what he was saying. His eyes met mine a few times as well.
Adam: "Hey babe, why don'tcha leave this nerd behind and go with a REAL man. I'll be behind the school. Eh?"
We all laughed. Me, him, Steph, everyone. Adam usually does this as a joke. He never means it. he's just here to make everyone laugh. Dom and Mike aren't that different. They all get along with Steph just great too, so it makes our lives much easier. Speaking of Dom and Mike...
Mike: "Boys and...uh...girl, i think we should consult the paper I found in my dad's trash bin last night?"
Adam: "You get your stuff outta your pop's trash? That explains alot..."
Mike: "Here's a rusty pen, go fuck yourself with it. Anyway...."
Mike took out a newspaper and flattened it on the table. It was the frontpage from a few days ago. On it "ROBIN AND SPOILER FOIL CITY BANK ROBBERY". Hehe, I remember that. Me and Steph stopping Roadkill's group from draining out the bank.
Mike: "Hell yeah, Robin doing what he does best!"
Adam: "Kicking ass and chewing bubblegum!"
Dom: "Don't know about the bubblegum, but kicking ass? Hell yeah."
Adam: "Dude, how much of a beast do you have to be, to be a kid, and fight with goddamn Batman?"
Mike: "A huge one. Remember when him and Batman took down the Jokerbot on live TV."
Dom:" Dude I saw that! Bad-fucking-ass!. And then there's his girlfriend, Spoiler."
Mike: "Hey, she can roll with Robin, she's gotta be a badass too, right?"
Adam: "She's kinda cute."
I looked up at Steph. She was looking back at me with a smug smile. Hell, I was smiling too. We kept listening to the three raving over our costumed personas. So, our best friends are also our biggest fans? What more could you ask for?
Usually I try to be a good student and pay attention and do my work, but it was Friday, and I just really, really really could not focus on anything that made me feel as stupid as calculus-based physics does.
Cuthbert wasn't participating in a We're Here outing today, but we were watching a comedian on YouTube talking about conspiracies. I hadn't heard of the "Benadryl Hat Man" conspiracy, but we thought we could create a photo depicting such a sighting.
The video: www.youtube.com/watch?v=q3JIRvP-7m8
Info on the conspiracy:
“You can't teach calculus to a chimpanzee. So just share your banana." -- John Rachel
Poppycock!
For every epsilon banana (ε > 0) there exists a delta banana (δ > 0) such that whenever Ιx - xₒΙ < δ then Ιf(x) - f(xₒ)Ι < ε.
Which is the Limit function, the cornerstone of Calculus. If only John Rachel had considered that two bananas are required, easy-peasy.
Our Daily Challenge - Over Your Head
Canon S.S.C. 50mm f/1.4, taken at f/1.4
Few months ago while I was doing my Calculus homework somehow this word “Evaluate” caught my attention. I knew that it inspired me to write something related to Christianity but I didn’t have a very clear idea what I should write about.
However, a couple of days afterwards while I was waiting at the dentist office and reading Reader’s Digest, the staff of the magazine reported that Toronto is the second honest city in the world among all the cities they have tested. What they did is that in 2007 they travelled the world and put cell phones in public places in specific world famous cities and noted the percentage of the people who were honest enough to return the cell phones in each city. That immediately caught my attention because in 2007 while I was getting on the bus at the subway station (don’t you just love my subway station stories!) I found a cell phone on a seat beside the door. I picked it up and got off the bus trying to find its owner. I couldn’t find the owner because there were over 20 people who got off the bus and were heading in different direction in a hurry. So I gave it to the bus driver, since the bus services in Toronto offer a “lost and found” service. At the time I thought it was kind of odd that nobody found the cell phone while they were existing the bus since the seat was beside the door, or why would a person who is sitting right beside the door exist at the end!
So when I came home and was doing my Calculus homework again I remembered the word “Evaluate” and decided to photograph it. So I set my camera on “super micro” mode and got very close to the paper: the lens was about 0.5 cm away from the paper. And since the sun was setting its light was shining on the paper on a very steep angle so I didn’t have to worry about the lens overshadowing the paper.
After I transferred the image to my PC and viewed it I was amazed by the amount of details in the shot. It seems that under careful and very close observation the smooth surface of the paper looked more like the coarse surface of a rock! And I thought: Perfect…that’s exactly what the world “Evaluate” means to Christians!
You see, because I don’t have any close people to me, nobody knows what I am going through or what I am doing. That’s why I have to always go to my heavenly Father asking Him for guidance and correction. I had not always obeyed Him, but I had never disobeyed Him and not knew it before hand. Sometimes God’s will is not very clear and requires us waiting, but when it comes to sin He always makes His will known. Every time I sinned, sadly, I knew I was doing so. That’s why we feel guilt right after we sin—not because we became perfect right after sinning!—but because we realized that sin did not deliver what it promised as we believed Satan’s lie. In other words, we knew that sin promised something but God said that it was a lie, yet we believed Satan’s false promise—just as the Holy Spirit told us before we sinned.
When I became a Christian I started to realize that my old human standards governed by my conscious were obsolete because the new standards of God governed by the Holy Spirit have taken the definitions of sin and holiness to a whole new level. A level only God’s children living by the guidance and power of the Holy Spirit can live up to.
I didn’t take evaluating my life seriously until I came to Canada and that happened only after few incidents in my late teens.
When I came to Canada in late of 1999 I didn’t speak English, and being enrolled in high school I wanted to excel in studying as I did back home but English was a barrier so I decided to tackle it as best as I could as I went through high school. I started reading books, mainly novels. In the beginning I read those teen suspense novels, you know the ones that always have a weird character and then somehow the good looking guy or girl end up being the bad guy! You know, the ones that are always made up of 120 pages for some reason! My reading was very slow because I am a perfectionist by nature so I looked up every word and phrase, etc. But my ESL (English as a Second Language) teacher told me that when I come across a word that I should first guess its meaning from the contest of the sentence and not bother looked it up! That worked miracles, except that when my mom asks me now what does a certain word mean I have no clue how to explain it in Arabic! I think one semester I read about 38 such novels! Then as my English improved I moved up in my level in reading and started asking my mom what I should read and she told me to read books such as Great Expectations, Wuthering Heights, Jane Eyre, etc, and I loved them…especially Jane Eyre, and I thought that I wouldn’t read a finer novel afterwards! Then in 2002 my mom told me that I should read Gone with the Wind and I thought, “Yea, why not, another novel…” But it was not just another novel: it was THE NOVEL!
I mean, I cannot describe it to you if you haven’t read it! I’ve seen the movie and with all its awards it is nothing comparing with the novel! I remember when I lied in my bed starting to read it thinking to myself that this is different, this is on a whole new level by itself! I read so many wonderful pages in the beginning and they were simply setting up the settings for the first scene! I was so immersed into the story by then that when one evening I saw a Honda Accord passing by the window I kind of got disturb thinking, “What is this! There are no cars in 1860s!” The novel was so well written that it took me to its world rather than it came to mine!
I mean I couldn’t believe a human can write something like this! It was a masterpiece! I can’t describe how I felt when I was reading it. Sometimes I would choke trying to hold back tears. Sometimes I just wanted to scream at the characters especially Scarlet O’Hara saying, “What is wrong with you two! Why are you deliberately making a mess of your life! Why don’t you slow down and think about what you really want!” I was so frustrated at times that I didn’t want the novel to end because I knew it wouldn’t end the way all cheesy novels do! The characters were so real that there was no heroes, no perfect people--just plain ordinary people like me. Then I realized that I was not screaming inside at the characters but I was screaming at myself—it was me that I saw in those characters. I was completely stunned by everything: the reality of the civil war, the settings of the locations, the story, but most of all the characters and how they didn’t improve in a period of 10 years! That was the scariest thing that could’ve ever happen to me: that in 10 years or so I would look back at my life and find that I didn’t become more like Jesus and it was very much possible!
I took me few weeks to finish the novel and by then I was so worn out emotionally that I finished reading it at 5 AM (early in the morning but not sure exactly what time…it was still dark outside), and I broke down crying and crying and crying. I just cried out to God because I knew that I very much could be like those characters in 10 years. I can’t remember for how long I cried but it was a while because I can still remember where I was kneeling on the couch as I was crying. I stopped reading novels after Gone with the Wind.
Then, in 2003 after knowing a lady from work for few weeks she told me that she had been a Christian for 10 years, and she attends church regularly! I was absolutely stunned! I mean, I couldn’t have guessed she was a Christian in a million years! Not that I expect us, Christians, to walk with a halo above our heads, but when someone gets to know us they should not be shocked when they find out we are Christians. The least we could do is let our characters show neutrality toward righteous living and sin, not enjoying sin! Again I had to step back and take a look at my life.
Also around that time I’ve done some things in my teen years that until today I regret—wish hadn’t done—and because I knew that I couldn’t change my past I wanted to control my present so I can change my future. And one day after listening to a sermon by Dr. Stanley I decided that from now on I have to evaluate my daily walk with God and listen to Him, and obey Him in whatever He brings to my mind. I don’t remember what the sermon was about but I remember Dr. Stanley saying that when he makes a decision one of the things he does is he asks himself when he will be 70 years old will he regret making that decision? I think he was talking in that sermon about how to discern God’s will in making decisions, or something like that. After that I started to regularly evaluate my steps in life and making sure they are aligned with His will for us Christians in general and for me personally.
It is very essential that we spend time with God alone and His word to evaluate ourselves, our motives, our actions—our lives. How many of us have seen on TV shows such as Cops a naked man running on the freeway across state lines and when stopped by the police and questioned, he would simply answer, “God told me to do it!” How many honest actors and actress think they are giving God the glory by thanking Him for “enabling” them to act in immortal movies!
(By the way I am listening to: SECRET GARDEN - SOLO VIOLIN www.youtube.com/watch?v=VOhjTpeQhXs&feature=related to help me write.)
It is important to evaluate our faith, that is: why do we do the things we do? What is driving us? What is motivation us? That’s why I often lie down quietly and ask God, “Lord, why do I want to do this?” or, “Why don’t I want to do this?” For example, a lot of relationships we build, the way we act in public, or even dress are motivated by a deep hurt from our past such as a rejection. And the saying “time heals all things” is non-sense to me. Only God heals; time just buries hurts so deep inside that we things are “healed” while they surface from time to time only to make us miserable and ruin our lives.
Let me give you an example:
Sometimes we go and tell people who have low self-image that they are beautiful and lovely and keep on doing so, and then their emotional security becomes wrapped up in us while all along only God’s love can heal them and make them emotionally whole. It is like giving a child chocolate and not expect him or her to become unhealthy! While all along they need healthy food…food only our heavenly Father can give. I am not saying that we should not admire people, but we must be sensitive to the work God wants to do first in other people’s lives. We too need to evaluate our motives.
Sometimes while we think we are doing God’s will, we are in reality simply hindering His work. And sometimes it is hard not to step in and do God’s work for Him, but like a heroin addict who is recovering he is starved from the drug and for a period of few days he may feel miserable without the drug, but once that period is over then he is free. But what we do is step in and give that addict a small quantities of the drug thinking we are helping them, while all along what we are doing is keeping him imprisoned to the addiction. That’s just a small example, and of course there are many.
Another area we need to evaluate ourselves is why do we obey God? Many times we, Christians, hear God’s voice and obey Him expecting something big to happen in our lives. But then when nothing spectacular happens we forget about God’s command and go back to whatever we used to do. Let me give you an example:
I think about 3 years ago after I ate my lunch I went to sit on some boxes in the warehouse I used to work at, to relax and read my Bible a bit. I could simply stretch on the boxes and rest my feet while had a quiet time with God. I opened the Bible and it was the book of James, so I tried to change the page! Then I thought to myself, “Why are you trying to change the page?” Well, the answer was simple: James tells us how to live for God, living for God requires change, change requires sacrifice, sacrifice makes us uncomfortable, and I didn’t want to feel uncomfortable! So I had to make a decision: read the book of James or ignore it? I knew that God often required me to do things that made me uncomfortable (who said carrying our crosses and follow Him is a comfortable thing to do?), so I decided to read the book of James. After some verses I came across this verse, James 4:8:
“Come near to God and he will come near to you. Wash your hands, you sinners, and purify your hearts, you double-minded.”
And I thought to myself, “Phew! Thank God, James can’t disturb my life here because I have never murdered or stolen so I don’t need to wash my hands!” But for some reason I couldn’t move past this verse! And I was thinking “James!!!” And I could clearly hear God asking me through my spirit, “Are your hands really clean? Have you really not stolen anything? Are you sure you don’t have any stolen things in your possession?” I was very confused at first because I had grown in a family that taught me not to steal, cheat, lie, always say “sorry” when I wronged a person, and even taught me not to throw garbage anywhere but in the garbage bin! So again I started to evaluate my possessions to see if I had something that belonged to someone else by may be a mistake!
After some thought I ended up thinking about what I have on my PC. My Windows was an illegal ((from now on I will just say stolen instead of “illegal”) copy, the music I listened to were stolen, and almost all the other software I used were stolen! I calculated it and it was about $5,000! If not counting the 3D software that each costs few thousands! I was absolutely shocked! I couldn’t believe I stole someone else’s hard work, intellectual property and rights, and live hood! So I knew what I had to do. Within a week or so I had removed every stolen piece of software and song from my hard drive and formatted it my hard drive, and broke all the stolen (copy) CDs I had.
You are probably thinking now, “WOW! What happened afterwards? Did you win $10,000 to buy all the software you want because you obeyed God voice?” No, nothing spectacular happened afterwards, even though in the beginning I thought that somehow God will give me the money to get all the software I want in the world! For about two years afterwards I didn’t do any photo editing because I couldn’t afford Photoshop, until one day I prayed asking God to please do something because I had obeyed Him and now I can’t even enjoy photo editing because I can’t afford those expensive software! Within few days I stumbled across Gimp and I absolutely love it :) (That’s why I always say “Thanks Gimp” when I write in regard to my photos.)
For a couple of years afterwards my computer was like a desert! It was so depressing to use it! But you know what? I didn’t get one single virus and I didn’t have any anti-virus or firewall! I learned that I could live without any of the software I had and I was just as happy! I learned that I didn’t need 95% of the software I had! I learned how to do things different ways because the legal software I had now were much older (the ones that came with my PC when I bought it). Now I am in university and so they give me licensed software.
So what I want you to think about is this: let’s say that you obeyed God and that nothing spectacular happened afterwards, would you still obey Him? If not, why not? Ask yourself, ask God, get to know…whatever you do, don’t settle down for what you see and think “I guess this is the Christian life after all.” No, it is not! Open the Bible and read how God worked in the life of the Old Testament prophets, and how He worked in the life of the apostles and He can do the same in your life! Evaluate your steps, your thoughts, your words, your actions, yourself…even evaluate you faith, devotion and obedience to God.
Have God asked you to stop listening to a certain type of music that dishonours Him and when you did so you realized you were still the same person—you didn’t become a saint—and so you decided that may be it was not God’s voice you heard and went back to listen to the same ungodly music? My friend, if every time we expect God to do a miracle because we obeyed Him then we missed the whole point! Christianity is a relationship with God because we love Him because He loved us first! It is not a religion where you input things and you get some outputs! It is a relationship. Here is a passage of scripture for us to think about, Luke 17:7-10:
"Suppose one of you had a servant plowing or looking after the sheep. Would he say to the servant when he comes in from the field, 'Come along now and sit down to eat'? Would he not rather say, 'Prepare my supper, get yourself ready and wait on me while I eat and drink; after that you may eat and drink'? Would he thank the servant because he did what he was told to do? So you also, when you have done everything you were told to do, should say, 'We are unworthy servants; we have only done our duty.' "
That doesn’t mean we should not expect good things, such as blessings and rewards from God, He is our Father after all, but blessings and rewards should not be our motives behind obedience.
That’s it for this topic. I do not make promises, but God does. And I know that I know that I KNOW if you decide today that you will bring everything about you and in your life before God for evaluation, and you commit yourself to listen to His voice, read His word, get to know Him and obey Him that you will not be the same person next year, and the year after you won’t be the same person of next year…as time goes by you will become more like Jesus Christ. But don’t forget it is the work of the Holy Spirit in your life, and many times it is just a matter of letting Him take control of you; submit your tongue, your eyes, your mind, all of you to Him.
I will leave you with those verses:
"Each one of us will give an account of himself to God."
Romans 14:12
"Keep on the alert at all times, praying that you may have strength to escape all these things that are about to take place, and to stand before the Son of Man."
Luke 21:36
"If you are wise, you are wise for yourself, and if you scoff, you alone will bear it."
Proverbs 9:12
"I tell you that every careless word that people speak, they shall give an accounting for it in the day of judgment."
Matthew 12:36
“I have considered my ways
and have turned my steps to your statutes.”
Psalm 119:59
“Let us examine our ways and test them,
and let us return to the LORD.”
Lamentations 3:40
“This is what the LORD Almighty says: "Give careful thought to your ways.”
Haggai 1:7
“Examine yourselves to see whether you are in the faith; test yourselves. Do you not realize that Christ Jesus is in you—unless, of course, you fail the test?”
2 Corinthians 13:5
“If anyone thinks he is something when he is nothing, he deceives himself.”
Galatians 6:3
I just want to say one more thing: if we choose not to evaluate ourselves, we can pretty much live normally, you know, float in life from this to that and nothing will happen. We won’t have a big sign flashing in red telling us to stop and think, or do this or we will bear the consequences. After all we have God’s word and the Holy Spirit, if we choose to ignore their instructions we are pretty much on our own making a mess of our lives.
God will not come down from Heaven to stop you from drinking and driving while God the Holy Spirit lives in you and you are ignoring Him. Common sense still applies to us, Christians, just as much as physical laws such as gravity still apply to us! We break God’s laws, no matter how small they are, we bear the consequences. We steal, even if God forgives us, we are still found guilty under the law. We commit sexual immorality and God forgives us, we still get pregnant or have a sexual transmit disease, not to mention the damage that kind of sin imprint on our minds, and souls.
My point is this: choosing not to evaluate our lives is a very dangerous thing and we will suffer from the consequences and will regret our inactions now and for eternity. We will hurt people we love so much. We will battle against guilt. We will miss on God’s blessings. But most of all we will miss on knowing Him, experiencing His work in our lives, and fulfilling the privilege and honourable task of living for Him.
54.365
I have a huge Calculus test to study for so I just shot this real quick while studying at Taco Bell. Yes, I study for Calculus at Taco Bell.
STROBIST CALCULUS in a little pocket book ....
Hi folks,
have you ever wondered, how to calculate with HSS flash?
How to use gang light?
How to calculate with ND filters?
How to compare continuous light and flash light?
How to adjust an exposure with/without a light meter?
How to calculate with Guide Numbers?
How to benefit from the Inverse Square Law?
>>> Here you go, it is for free: :-)
I made this little pocketbook with RockyNook to say "thank you" to my trusty readers (and to say hi to all the hopefully will-be readers.:-):
fotopraxis.files.wordpress.com/2013/09/light-primer-engl-...
I hope, you enjoy and share this little pocketbook.
Cheers,
~gallo~
--
Tilo ~gallo~ Gockel
i hope the title conveys how stressed and frightened i am. i can't even tell if i'm prepared enough...
oh and sorry for the mess....
wearing:
h&m turtleneck
paisley dress
mum's belt
sweater tights
(pretend i have the brown flats on my feet, hehe)
your necklace
133/365
. . . seen from my house
__________________________
A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It takes the form of a multicoloured arc. Rainbows caused by sunlight always appear in the section of sky directly opposite the sun.
Rainbows can be full circles. However, the average observer sees only an arc formed by illuminated droplets above the ground, and centred on a line from the sun to the observer's eye.
In a primary rainbow, the arc shows red on the outer part and violet on the inner side. This rainbow is caused by light being refracted when entering a droplet of water, then reflected inside on the back of the droplet and refracted again when leaving it.
In a double rainbow, a second arc is seen outside the primary arc, and has the order of its colours reversed, with red on the inner side of the arc.
OVERVIEW
A rainbow is not located at a specific distance from the observer, but comes from an optical illusion caused by any water droplets viewed from a certain angle relative to a light source. Thus, a rainbow is not an object and cannot be physically approached. Indeed, it is impossible for an observer to see a rainbow from water droplets at any angle other than the customary one of 42 degrees from the direction opposite the light source. Even if an observer sees another observer who seems "under" or "at the end of" a rainbow, the second observer will see a different rainbow - farther off - at the same angle as seen by the first observer.
Rainbows span a continuous spectrum of colours. Any distinct bands perceived are an artefact of human colour vision, and no banding of any type is seen in a black-and-white photo of a rainbow, only a smooth gradation of intensity to a maximum, then fading towards the other side. For colours seen by the human eye, the most commonly cited and remembered sequence is Newton's sevenfold red, orange, yellow, green, blue, indigo and violet, remembered by the mnemonic, Richard Of York Gave Battle In Vain (ROYGBIV).
Rainbows can be caused by many forms of airborne water. These include not only rain, but also mist, spray, and airborne dew.
VISIBILITY
Rainbows can be observed whenever there are water drops in the air and sunlight shining from behind the observer at a low altitude angle. Because of this, rainbows are usually seen in the western sky during the morning and in the eastern sky during the early evening. The most spectacular rainbow displays happen when half the sky is still dark with raining clouds and the observer is at a spot with clear sky in the direction of the sun. The result is a luminous rainbow that contrasts with the darkened background. During such good visibility conditions, the larger but fainter secondary rainbow is often visible. It appears about 10° outside of the primary rainbow, with inverse order of colours.
The rainbow effect is also commonly seen near waterfalls or fountains. In addition, the effect can be artificially created by dispersing water droplets into the air during a sunny day. Rarely, a moonbow, lunar rainbow or nighttime rainbow, can be seen on strongly moonlit nights. As human visual perception for colour is poor in low light, moonbows are often perceived to be white.
It is difficult to photograph the complete semicircle of a rainbow in one frame, as this would require an angle of view of 84°. For a 35 mm camera, a wide-angle lens with a focal length of 19 mm or less would be required. Now that software for stitching several images into a panorama is available, images of the entire arc and even secondary arcs can be created fairly easily from a series of overlapping frames.
From above the earth such as in an aeroplane, it is sometimes possible to see a rainbow as a full circle. This phenomenon can be confused with the glory phenomenon, but a glory is usually much smaller, covering only 5–20°.
The sky inside a primary rainbow is brighter than the sky outside of the bow. This is because each raindrop is a sphere and it scatters light over an entire circular disc in the sky. The radius of the disc depends on the wavelength of light, with red light being scattered over a larger angle than blue light. Over most of the disc, scattered light at all wavelengths overlaps, resulting in white light which brightens the sky. At the edge, the wavelength dependence of the scattering gives rise to the rainbow.
Light of primary rainbow arc is 96% polarised tangential to the arch. Light of second arc is 90% polarised.
Number of colours in spectrum or rainbow
A spectrum obtained using a glass prism and a point source is a continuum of wavelengths without bands. The number of colours that the human eye is able to distinguish in a spectrum is in the order of 100. Accordingly, the Munsell colour system (a 20th-century system for numerically describing colours, based on equal steps for human visual perception) distinguishes 100 hues. The apparent discreteness of main colours is an artefact of human perception and the exact number of main colours is a somewhat arbitrary choice.
Newton, who admitted his eyes were not very critical in distinguishing colours, originally (1672) divided the spectrum into five main colours: red, yellow, green, blue and violet. Later he included orange and indigo, giving seven main colours by analogy to the number of notes in a musical scale. Newton chose to divide the visible spectrum into seven colours out of a belief derived from the beliefs of the ancient Greek sophists, who thought there was a connection between the colours, the musical notes, the known objects in the Solar System, and the days of the week.
According to Isaac Asimov, "It is customary to list indigo as a color lying between blue and violet, but it has never seemed to me that indigo is worth the dignity of being considered a separate color. To my eyes it seems merely deep blue."
The colour pattern of a rainbow is different from a spectrum, and the colours are less saturated. There is spectral smearing in a rainbow owing to the fact that for any particular wavelength, there is a distribution of exit angles, rather than a single unvarying angle. In addition, a rainbow is a blurred version of the bow obtained from a point source, because the disk diameter of the sun (0.5°) cannot be neglected compared to the width of a rainbow (2°). The number of colour bands of a rainbow may therefore be different from the number of bands in a spectrum, especially if the droplets are particularly large or small. Therefore, the number of colours of a rainbow is variable. If, however, the word rainbow is used inaccurately to mean spectrum, it is the number of main colours in the spectrum.
The question of whether everyone sees seven colours in a rainbow is related to the idea of Linguistic relativity. Suggestions have been made that there is universality in the way that a rainbow is perceived. However, more recent research suggests that the number of distinct colours observed and what these are called depend on the language that one uses with people whose language has fewer colour words seeing fewer discrete colour bands.
EXPLANATION
When sunlight encounters a raindrop, part of the light is reflected and the rest enters the raindrop. The light is refracted at the surface of the raindrop. When this light hits the back of the raindrop, some of it is reflected off the back. When the internally reflected light reaches the surface again, once more some is internally reflected and some is refracted as it exits the drop. (The light that reflects off the drop, exits from the back, or continues to bounce around inside the drop after the second encounter with the surface, is not relevant to the formation of the primary rainbow.) The overall effect is that part of the incoming light is reflected back over the range of 0° to 42°, with the most intense light at 42°. This angle is independent of the size of the drop, but does depend on its refractive index. Seawater has a higher refractive index than rain water, so the radius of a "rainbow" in sea spray is smaller than a true rainbow. This is visible to the naked eye by a misalignment of these bows.
The reason the returning light is most intense at about 42° is that this is a turning point – light hitting the outermost ring of the drop gets returned at less than 42°, as does the light hitting the drop nearer to its centre. There is a circular band of light that all gets returned right around 42°. If the sun were a laser emitting parallel, monochromatic rays, then the luminance (brightness) of the bow would tend toward infinity at this angle (ignoring interference effects). (See Caustic (optics).) But since the sun's luminance is finite and its rays are not all parallel (it covers about half a degree of the sky) the luminance does not go to infinity. Furthermore, the amount by which light is refracted depends upon its wavelength, and hence its colour. This effect is called dispersion. Blue light (shorter wavelength) is refracted at a greater angle than red light, but due to the reflection of light rays from the back of the droplet, the blue light emerges from the droplet at a smaller angle to the original incident white light ray than the red light. Due to this angle, blue is seen on the inside of the arc of the primary rainbow, and red on the outside. The result of this is not only to give different colours to different parts of the rainbow, but also to diminish the brightness. (A "rainbow" formed by droplets of a liquid with no dispersion would be white, but brighter than a normal rainbow.)
The light at the back of the raindrop does not undergo total internal reflection, and some light does emerge from the back. However, light coming out the back of the raindrop does not create a rainbow between the observer and the sun because spectra emitted from the back of the raindrop do not have a maximum of intensity, as the other visible rainbows do, and thus the colours blend together rather than forming a rainbow.
A rainbow does not exist at one particular location. Many rainbows exist; however, only one can be seen depending on the particular observer's viewpoint as droplets of light illuminated by the sun. All raindrops refract and reflect the sunlight in the same way, but only the light from some raindrops reaches the observer's eye. This light is what constitutes the rainbow for that observer. The whole system composed by the sun's rays, the observer's head, and the (spherical) water drops has an axial symmetry around the axis through the observer's head and parallel to the sun's rays. The rainbow is curved because the set of all the raindrops that have the right angle between the observer, the drop, and the sun, lie on a cone pointing at the sun with the observer at the tip. The base of the cone forms a circle at an angle of 40–42° to the line between the observer's head and their shadow but 50% or more of the circle is below the horizon, unless the observer is sufficiently far above the earth's surface to see it all, for example in an aeroplane (see above). Alternatively, an observer with the right vantage point may see the full circle in a fountain or waterfall spray.
MATHEMATICAL DERIVATION
We can determine the perceived angle which the rainbow subtends as follows.
Given a spherical raindrop, and defining the perceived angle of the rainbow as 2φ, and the angle of the internal reflection as 2β, then the angle of incidence of the sun's rays with respect to the drop's surface normal is 2β − φ. Since the angle of refraction is β, Snell's law gives us
sin(2β − φ) = n sin β,
where n = 1.333 is the refractive index of water. Solving for φ, we get
φ = 2β − arcsin(n sin β).
The rainbow will occur where the angle φ is maximum with respect to the angle β. Therefore, from calculus, we can set dφ/dβ = 0, and solve for β.
Substituting back into the earlier equation for φ yields 2φmax ≈ 42° as the radius angle of the rainbow.
VARIATION
MULTIPLE RAINBOWS
Secondary rainbows are caused by a double reflection of sunlight inside the raindrops, and are centred on the sun itself. They are about 127° (violet) to 130° (red) wide. Since this is more than 90°, they are seen on the same side of the sky as the primary rainbow, about 10° above it at apparent angles of 50–53°. As a result of the "inside" of the secondary bow being "up" to the observer, the colours appear reversed compared to the primary bow. The secondary rainbow is fainter than the primary because more light escapes from two reflections compared to one and because the rainbow itself is spread over a greater area of the sky. Each rainbow reflects white light inside its coloured bands, but that is "down" for the primary and "up" for the secondary. The dark area of unlit sky lying between the primary and secondary bows is called Alexander's band, after Alexander of Aphrodisias who first described it.
TWINNED RAINBOW
Unlike a double rainbow that consists of two separate and concentric rainbow arcs, the very rare twinned rainbow appears as two rainbow arcs that split from a single base. The colours in the second bow, rather than reversing as in a secondary rainbow, appear in the same order as the primary rainbow. A "normal" secondary rainbow may be present as well. Twinned rainbows can look similar to, but should not be confused with supernumerary bands. The two phenomena may be told apart by their difference in colour profile: supernumerary bands consist of subdued pastel hues (mainly pink, purple and green), while the twinned rainbow shows the same spectrum as a regular rainbow. The cause of a twinned rainbow is the combination of different sizes of water drops falling from the sky. Due to air resistance, raindrops flatten as they fall, and flattening is more prominent in larger water drops. When two rain showers with different-sized raindrops combine, they each produce slightly different rainbows which may combine and form a twinned rainbow. A numerical ray tracing study showed that a twinned rainbow on a photo could be explained by a mixture of 0.40 and 0.45 mm
droplets. That small difference in droplet size resulted in a small difference in flattening of the droplet shape, and a large difference in flattening of the rainbow top.
Meanwhile, the even rarer case of a rainbow split into three branches was observed and photographed in nature.
FULL-CIRCLE RAINBOW
In theory, every rainbow is a circle, but from the ground, only its upper half can be seen. Since the rainbow's centre is diametrically opposed to the sun's position in the sky, more of the circle comes into view as the sun approaches the horizon, meaning that the largest section of the circle normally seen is about 50% during sunset or sunrise. Viewing the rainbow's lower half requires the presence of water droplets below the observer's horizon, as well as sunlight that is able to reach them. These requirements are not usually met when the viewer is at ground level, either because droplets are absent in the required position, or because the sunlight is obstructed by the landscape behind the observer. From a high viewpoint such as a high building or an aircraft, however, the requirements can be met and the full-circle rainbow can be seen. Like a partial rainbow, the circular rainbow can have a secondary bow or supernumerary bows as well. It is possible to produce the full circle when standing on the ground, for example by spraying a water mist from a garden hose while facing away from the sun.
A circular rainbow should not be confused with the glory, which is much smaller in diameter and is created by different optical processes. In the right circumstances, a glory and a (circular) rainbow or fog bow can occur together. Another atmospheric phenomenon that may be mistaken for a "circular rainbow" is the 22° halo, which is caused by ice crystals rather than liquid water droplets, and is located around the sun (or moon), not opposite it.
SUPERNUMERARY RAINBOWS
In certain circumstances, one or several narrow, faintly coloured bands can be seen bordering the violet edge of a rainbow; i.e., inside the primary bow or, much more rarely, outside the secondary. These extra bands are called supernumerary rainbows or supernumerary bands; together with the rainbow itself the phenomenon is also known as a stacker rainbow. The supernumerary bows are slightly detached from the main bow, become successively fainter along with their distance from it, and have pastel colours (consisting mainly of pink, purple and green hues) rather than the usual spectrum pattern. The effect becomes apparent when water droplets are involved that have a diameter of about 1mm or less; the smaller the droplets are, the broader the supernumerary bands become, and the less saturated their colours. Due to their origin in small droplets, supernumerary bands tend to be particularly prominent in fogbows.
Supernumerary rainbows cannot be explained using classical geometric optics. The alternating faint bands are caused by interference between rays of light following slightly different paths with slightly varying lengths within the raindrops. Some rays are in phase, reinforcing each other through constructive interference, creating a bright band; others are out of phase by up to half a wavelength, cancelling each other out through destructive interference, and creating a gap. Given the different angles of refraction for rays of different colours, the patterns of interference are slightly different for rays of different colours, so each bright band is differentiated in colour, creating a miniature rainbow. Supernumerary rainbows are clearest when raindrops are small and of uniform size. The very existence of supernumerary rainbows was historically a first indication of the wave nature of light, and the first explanation was provided by Thomas Young in 1804.
SCIENTIFIC HISTORY
The classical Greek scholar Aristotle (384–322 BC) was first to devote serious attention to the rainbow. According to Raymond L. Lee and Alistair B. Fraser, "Despite its many flaws and its appeal to Pythagorean numerology, Aristotle's qualitative explanation showed an inventiveness and relative consistency that was unmatched for centuries. After Aristotle's death, much rainbow theory consisted of reaction to his work, although not all of this was uncritical."
In Book I of Naturales Quaestiones (c. 65 AD), the Roman philosopher Seneca the Younger discusses various theories of the formation of rainbows extensively, including those of Aristotle. He notices that rainbows appear always opposite to the sun, that they appear in water sprayed by a rower, in the water spat by a fuller on clothes stretched on pegs or by water sprayed through a small hole in a burst pipe. He even speaks of rainbows produced by small rods (virgulae) of glass, anticipating Newton's experiences with prisms. He takes into account two theories: one, that the rainbow is produced by the sun reflecting in each water drop, the other, that it is produced by the sun reflected in a cloud shaped like a concave mirror; he favours the latter. He also discusses other phenomena related to rainbows: the mysterious "virgae" (rods), halos and parhelia.
According to Hüseyin Gazi Topdemir, the Persian physicist and polymath Ibn al-Haytham (Alhazen; 965–1039), attempted to provide a scientific explanation for the rainbow phenomenon. In his Maqala fi al-Hala wa Qaws Quzah (On the Rainbow and Halo), al-Haytham "explained the formation of rainbow as an image, which forms at a concave mirror. If the rays of light coming from a farther light source reflect to any point on axis of the concave mirror, they form concentric circles in that point. When it is supposed that the sun as a farther light source, the eye of viewer as a point on the axis of mirror and a cloud as a reflecting surface, then it can be observed the concentric circles are forming on the axis." He was not able to verify this because his theory that "light from the sun is reflected by a cloud before reaching the eye" did not allow for a possible experimental verification. This explanation was later repeated by Averroes, and, though incorrect, provided the groundwork for the correct explanations later given by Kamāl al-Dīn al-Fārisī (1267–1319) and Theodoric of Freiberg (c.1250–1310).
Ibn al-Haytham's contemporary, the Persian philosopher and polymath Ibn Sīnā (Avicenna; 980–1037), provided an alternative explanation, writing "that the bow is not formed in the dark cloud but rather in the very thin mist lying between the cloud and the sun or observer. The cloud, he thought, serves simply as the background of this thin substance, much as a quicksilver lining is placed upon the rear surface of the glass in a mirror. Ibn Sīnā would change the place not only of the bow, but also of the colour formation, holding the iridescence to be merely a subjective sensation in the eye." This explanation, however, was also incorrect.[65] Ibn Sīnā's account accepts many of Aristotle's arguments on the rainbow.
In Song Dynasty China (960–1279), a polymath scholar-official named Shen Kuo (1031–1095) hypothesised—as a certain Sun Sikong (1015–1076) did before him—that rainbows were formed by a phenomenon of sunlight encountering droplets of rain in the air. Paul Dong writes that Shen's explanation of the rainbow as a phenomenon of atmospheric refraction "is basically in accord with modern scientific principles."
According to Nader El-Bizri, the Persian astronomer, Qutb al-Din al-Shirazi (1236–1311), gave a fairly accurate explanation for the rainbow phenomenon. This was elaborated on by his student, Kamāl al-Dīn al-Fārisī (1267–1319), who gave a more mathematically satisfactory explanation of the rainbow. He "proposed a model where the ray of light from the sun was refracted twice by a water droplet, one or more reflections occurring between the two refractions." An experiment with a water-filled glass sphere was conducted and al-Farisi showed the additional refractions due to the glass could be ignored in his model. As he noted in his Kitab Tanqih al-Manazir (The Revision of the Optics), al-Farisi used a large clear vessel of glass in the shape of a sphere, which was filled with water, in order to have an experimental large-scale model of a rain drop. He then placed this model within a camera obscura that has a controlled aperture for the introduction of light. He projected light unto the sphere and ultimately deduced through several trials and detailed observations of reflections and refractions of light that the colours of the rainbow are phenomena of the decomposition of light.
In Europe, Ibn al-Haytham's Book of Optics was translated into Latin and studied by Robert Grosseteste. His work on light was continued by Roger Bacon, who wrote in his Opus Majus of 1268 about experiments with light shining through crystals and water droplets showing the colours of the rainbow. In addition, Bacon was the first to calculate the angular size of the rainbow. He stated that the rainbow summit can not appear higher than 42° above the horizon. Theodoric of Freiberg is known to have given an accurate theoretical explanation of both the primary and secondary rainbows in 1307. He explained the primary rainbow, noting that "when sunlight falls on individual drops of moisture, the rays undergo two refractions (upon ingress and egress) and one reflection (at the back of the drop) before transmission into the eye of the observer." He explained the secondary rainbow through a similar analysis involving two refractions and two reflections.
Descartes' 1637 treatise, Discourse on Method, further advanced this explanation. Knowing that the size of raindrops did not appear to affect the observed rainbow, he experimented with passing rays of light through a large glass sphere filled with water. By measuring the angles that the rays emerged, he concluded that the primary bow was caused by a single internal reflection inside the raindrop and that a secondary bow could be caused by two internal reflections. He supported this conclusion with a derivation of the law of refraction (subsequently to, but independently of, Snell) and correctly calculated the angles for both bows. His explanation of the colours, however, was based on a mechanical version of the traditional theory that colours were produced by a modification of white light.
Isaac Newton demonstrated that white light was composed of the light of all the colours of the rainbow, which a glass prism could separate into the full spectrum of colours, rejecting the theory that the colours were produced by a modification of white light. He also showed that red light is refracted less than blue light, which led to the first scientific explanation of the major features of the rainbow. Newton's corpuscular theory of light was unable to explain supernumerary rainbows, and a satisfactory explanation was not found until Thomas Young realised that light behaves as a wave under certain conditions, and can interfere with itself.
Young's work was refined in the 1820s by George Biddell Airy, who explained the dependence of the strength of the colours of the rainbow on the size of the water droplets. Modern physical descriptions of the rainbow are based on Mie scattering, work published by Gustav Mie in 1908. Advances in computational methods and optical theory continue to lead to a fuller understanding of rainbows. For example, Nussenzveig provides a modern overview.
EXPERIMENTS
Experiments on the rainbow phenomenon using artificial raindrops, i.e. water-filled spherical flasks, go back at least to Theodoric of Freiberg in the 14th century. Later, also Descartes studied the phenomenon using a Florence flask. A flask experiment known as Florence's rainbow is still today often-used as an imposing and intuitively accessible demonstration experiment of the rainbow phenomenon. It consists in illuminating (with parallel white light) a water-filled spherical flask through a hole in a screen. A rainbow will then appear thrown back / projected on the screen, provided the screen is large enough. Due to the finite wall thickness and the macroscopic character of the artificial raindrop, several subtle differences exist as compared to the natural phenomenon, including slightly changed rainbow angles and a splitting of the rainbow orders.
A very similar experiment consists in using a cylindrical glass vessel filled with water or a solid transparent cylinder and illuminated either parallel to the circular base (i.e. light rays remaining at a fixed hight while they transit the cylinder) or under an angle to the base. Under these latter conditions the rainbow angles change relative to the natural phenomenon since the effective index of refraction of water changes (Bravais' index of refraction for inclined rays applies).
Other experiments use small liquid drops, see text above.
CULTURE
Rainbows occur frequently in mythology, and have been used in the arts. One of the earliest literary occurrences of a rainbow is in Genesis 9, as part of the flood story of Noah, where it is a sign of God's covenant to never destroy all life on earth with a global flood again. In Norse mythology, the rainbow bridge Bifröst connects the world of men (Midgard) and the realm of the gods (Asgard). Cuchavira was the god of the rainbow for the Muisca people in present-day Colombia and when the regular rains on the Bogotá savanna were over, the people thanked him offering gold, snails and small emeralds. The Irish leprechaun's secret hiding place for his pot of gold is usually said to be at the end of the rainbow. This place is appropriately impossible to reach, because the rainbow is an optical effect which cannot be approached.
Rainbows sometimes appear in heraldry too, even if its characteristic of multiple colours doesn't really fit in to the usual heraldic style.
Rainbow flags have been used for centuries. It was a symbol of the Cooperative movement in the German Peasants' War in the 16th century, of peace in Italy, and of gay pride and LGBT social movements since the 1970s. In 1994, Archbishop Desmond Tutu and President Nelson Mandela described newly democratic post-apartheid South Africa as the rainbow nation. The rainbow has even been used in technology product logos including the Apple computer logo. Many political alliances have called themselves Rainbow Coalition.
WIKIPEDIA
professor calculus is always a bit confused, but tintin manages it all. sometimes he has to hurry up...
MIT calculus class with Prof. John Bush, 2009; photo by Christopher Harting, 2009; cps chcalculus; _DSC4682
3D prints of some graphs which are used as teaching material for calculus classes. The models were printed on the Replicator 2.
I haven't been stitching a lot this fall, and here's the culprit: calculus! It's been good for the grades, but somehow not as fun as working on an embroidery...