Science

The Mathematics of Rainbows

Here’s a good resource with some helpful videos:
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=44.0

Rainbows are puzzling. I heard that rainbows are supposed to be sections of a circle. But I tried to fit the following picture of a rainbow sent to me by family in Duluth, and found that a parabola fit the curve much better than a circle.

I know that parabolas and circles are both conic sections. Would a better fit be an ellipse? I tried it out, and it seems that yes, an ellipse may fit even better than the parabola.

ellipse rainbow fit.jpg A colleague (thanks Matt!) that told me that if you look at a rainbow from above, it’s a complete circle:

Looking at a circle from an angle, do you see an ellipse? So are rainbows often following precise mathematical elliptical curves when not seen from directly above? What determines the center of a rainbow? Is the rainbow continuing out of sight in a complete circle from another perspective?

Here are the conic sections, from a Google Image search:

conic sections.jpg

Another colleague (thanks Carol!) reframed these questions in a neat way:

“What determines the center of a rainbow? Is the rainbow continuing out of sight in a complete circle from another perspective? I’m now fascinated by these! What does determine the center of a rainbow- and the plane that it’s on? Hurrah for ellipses!”

Another colleague (thanks Natalia!) taught me about an equation for the radius of a rainbow.

“Great problem, Erik! I knew the fact about rainbows being full circles with a center hiding below the horizon, but it never occurred to me to make a math problem out of it! Now I wish I was teaching precalculus just so I could use this problem!

Mathematically, the radius of the rainbow is determined by r = d*tan(42˚), where d is how far the water droplets responsible for forming the rainbow are from your eye. The 40-42˚ angles in the formula come from the refraction index for water. Raindrops reflect sunlight at all angles from 180° (straight back to the sun) down to about 138° (red) to 140° (violet).”

She also provided this picture, not her creation (which I believe has an error within it):

Screen Shot 2021-06-20 at 6.02.43 PM.png

What determines the plane the rainbow is on? Even if it has an error in the picture, it’s still a helpful visual. It’s so cool to see and think about an equation for the radius of the rainbow. Thinking about it now… if the picture is accurate, wouldn’t you have to take half the angle pictured? Wouldn’t it have to be r = d*tan(21˚)? But I think maybe the mistake is in the image itself. Maybe the angle shown in the image should be 84˚?

Here’s a website I just found with some good info, which seems to have the answer to the question about the center of the rainbow:

https://inlightofnature.com/how-rainbows-form/ (Links to an external site.)

Apparently, the center of the rainbow is the shadow of your head. That still doesn’t make total sense to me, but it’s making more sense than it did before!

Here’s a Geogebra thing I made because it didn’t make any sense to me how light separated out from a cloud of droplets. If you Zoom out, you can see that sure enough, the different wavelengths of light do separate out into bands of color. https://www.geogebra.org/m/pnraht55 (Links to an external site.)

Zoomed out view of how light separates into distinct color bands with enough distance.

Regarding the plane the rainbow is on, it seems from the article that the rainbow is the same no matter where the droplets are. That you can only see a rainbow if the sun is below 42 degrees in the sky and if you are looking at water at the right angle. But if you do see a rainbow, and it doesn’t matter how far away the droplets are from you, then shouldn’t all rainbows appear circular?

Update: it’s all starting to make a little more sense to me now. Rainbows appear circular or as ellipses. That’s because you are seeing the base of a cone. You can see the cone from an angle, and then it’s an ellipse. Now, would it be possible to view another conic section if you were looking just right, or are only elliptical (and circular) curves possible?

InSketchUp:

This shows a circle viewed from the side.

Handwashing Song

Because this song is AWESOME and INFORMATIVE, here it is, again. You might find yourself listening to it several times a day, or having it stuck in your head for as long as we’re dealing with the current situation. As they sing in the song,

“Together we shall beat the coronavirus.”

View this post on Instagram

#ghencovychallenge #handwashingmove #coronahanddance #VuDieuRuaTay 🌏 Because more international friends are coming to this post so I will change this to English for everyone: COVID-2019 disease is spreading, affecting people and social activities. Regular handwashing is considered a simple and effective method to protect the community from diseases (according to the World Health Organization). According to research by the Massachusetts Institute of Technology (MIT), 78% of people say they wash their hands often but only 25% actually wash their hands after going to the toilet, 20% wash their hands before cooking. To spread the habit of washing your hands to prevent this disease, I invite you to take part in the #ghencovychallenge challenge with me. Game rules: You perform the dance of the song Ghen Co Vy with 6 hand washing movements as recommended by the World Health Organization and the Ministry of Health, based on the music song COVID-19 prevention – Jealousy, cooperation between Institute of Occupational and Environmental Health, musician Khac Hung, singer Min and singer Erik. Take this challenge or share the following epidemic prevention habits: 1. Wash your hands often with soap or an antiseptic solution. 2. Do not put hands on eyes, nose and mouth. 3. Regularly clean personal hygiene, hygiene of utensils, houses and surroundings. 4. Wear a mask to go to public places, on vehicles or when you are sick. 5. Self-awareness to improve health for themselves, the family and the community. 6. People with symptoms of COVID-19 have high fever, cough, shortness of breath, etc. or close contact with infected person / person suspected of COVID-19 and limit contact with other people and contact local health facilities. After completing the challenge, SHARE + TAG immediately 2 friends want to join this challenge. ✌ 🌐 for news reporters and press who want to use my video, please feel free to do so. 🌐 for people want to dance my choreography, please feel free to do so, it’s all yours 🌐 join hands to spread this extremely useful message! 😉 #handwashdance #handwashingdance

A post shared by Quang Đăng (@im.quangdang) on Feb 19, 2020 at 8:12am PST

The Real Johns Hopkins Map of Coronavirus COVID-19 and the Viral Handwashing Song from Vietnam

I recently learned that various websites using this map have been phishing people, so I Googled Johns Hopkins Map to find the original (safe, non-scammy) source. Don’t get scammed! Google it yourself, or use the link below. If you want to make sure the link below is valid, you can hover over it with your mouse cursor, then look to the bottom left of your screen. It should say the same website.

https://coronavirus.jhu.edu/map.html

This map shows recoveries, not just deaths, which I appreciate. Also, this song is amazing:

Lyrics in English:

Let’s wash your hands hands hands

Keep them away from eyes-noses-mouths

And restrict going into the public

Together we shall beat the Coronavirus

Constantly improve your health

And clean your personal space

Let’s improve our social awareness

Together we shall beat the Coronavirus