We woke at the thud, panicked.

My wife and I had been dozing in bed, squeezing the last seconds of sleep out of our pillows before the inevitable onslaught of little people. It’s something of a delicious defeat when you hear the door creak open and steps on the carpet. You know you’re trading something so precious to you (sleep) for someone so precious, who you just wish would be precious elsewhere. But then they’re all warm and soft and tousle-headed and snuggly and saying things like “daddy, can we have special cereal?” and poking their knees into your side and sitting on your face with a full diaper and sticking cold toes into your armpits.

This morning they didn’t get to do all those precious, horrible things to us because we woke up to the sound of something thwacking against our window. After shaking off the confusion (no, it’s not a child on the roof; they can’t reach the window), I recalled why I knew the sound. It was a bird. And from the sound, a pretty sizable one.

I said as much to my wife, as I tried to open my eyes but managed only a half-squint. She half-muttered sympathy, as if she cared deeply inside but couldn’t get past the early morning fog. I added that since it was a big bird, it probably got away a little dazed, no harm done. I tried to find her to give her a reassuring pat, but I was too drowsy and missed completely. I also couldn’t pry my eyelids open still; they ached a little, as if I hadn’t used them for a week instead of approximately six hours.

I’ve read that birds, reptiles, and some mammals have something called a nictitating membrane, a third eyelid that acts as a translucent shield when they are diving or flying. They can draw it horizontally across each eye to moisten and protect it. For instance, many birds use it when they are regurgitating food for their kids so they don’t get their eyes poked out by little beaks.

I think of things like this when I’m abruptly disturbed from sleep. Things like this, and demonic children’s songs that will not be exorcised for hours.

We turned over and buried ourselves in the blankets for three more minutes before the precious, horrible people came and got us.

Nadia was playing hopscotch out on the sidewalk later in the morning when she found the bird. Robins are always the first birds my children encounter and remember. It was easy for her to name the body she stumbled upon, one story below our window, and quite dead. We had a little flurry of don’t-touch-that’s and daddy-is-it-going-to-be-okay’s, and I managed to deposit it into a small terrarium for them to look at. Animals can die of diseases communicable to humans, but this one died upon impact, and it was still warm. So the kids and I examined it carefully.

It was larger than any bird I’d seen this close. The wings were tucked in beside its impossibly light body. Many of its bones were hollow, for ease of flight. The neck was at a weird angle, a wrong angle. The death in front of me was at a wrong angle. It always has been, but it just felt much more wrong suddenly, because it was also in front of my children. Blood speckled its beak, a dark decoration above the warm red tinges on its chest.

I realized that it was a young one, probably a teenager in robin years. Its feathers were perfect, straight and sleek, pinions just slightly larger and smaller on each side, forming beautiful strong wings. And the roses on the sides under the wings were just barely forming up into a speckled breast. It was about the size of an adult but without the experience, which is why it probably hit our window.

One eye was slightly open and looking like black jelly, like it would spill out of the socket as liquid if you tipped the bird over slightly. The other eye was closed. This teenage robin had eyelashes on the under-lid. It had whiskers. Its feet were curled and delicate.

It was too beautiful. It was too sad.

Nadia was watching me. She told me she was sorry that it died because she could see the sadness in my eyes; but I wasn’t really mourning after the bird. We touched its wings, feeling its perfect form beneath our fingers: the surreal softness, the preciousness of a young fleeting life snuffed out. Not a sparrow falls but that He sees it. God saw this.

Woodpeckers will close their nictitating membrane, then tighten it a millisecond before they hit the tree so that their eyeballs don’t fall out on impact. Perhaps we need similar shields.

Only a few days earlier we had discovered a young robin hopping around the pine tree down in the yard. We took pictures. The kids enjoyed watching it hop around. There was a hawk above, circling around, and I like to think we saved that little robin’s life. There was a little one then, alive and bouncing, and then a bigger one, dead and stiff.

God saw this.

What is it about seeing that is comforting? Wouldn’t catching the sparrow be more comforting? Wouldn’t solving the problem, ending the great horror – death – wouldn’t this be a more lasting comfort? You see me, God. You see my children. But if you were to let them fall…

Sharks close their third eyelid when they attack. This way they can see their victims without getting blood in their eyes.

Kai got out of the house when he was only one year old. We noticed when it was too quiet. He had slipped out the screen door moments after I arrived home from work, and neither Linnea nor I had seen him. I locked the door. It was five minutes before we noticed he was gone, and in those five minutes he was down the outer stairs of the apartment building and toddling along the street.

The terror in my heart: I was running down the sidewalk in my dress slacks, socks, and undershirt, desperately looking for my son, expecting to see him lying in the street, expecting to be one of those parents we pray for, expecting to live with the knowledge that we didn’t see him, we didn’t see him, and then it was too late to see him…

He liked stairs, so he had kept to the house side of the sidewalk, and a kind and observant couple had pulled over and were with him four houses down when I ran up. The man was on the phone with the police, and I heard him say, “Oh never mind, it’s okay. The father’s here.” I thanked them, breathless, and held my son, and held him walking back, and held him up the stairs, and held him for another hour until he didn’t want to be held anymore, promising him I would see him, always, I would see him.

God saw this. How can I say it differently, to ease the pain? God sees… me?

If we mourn this way, how must God mourn at all He sees, at all the flickering lights snuffed out, the bruised reeds broken, the sheep silent before the shearer? Death is nothing new to Him, and in fact, He knows it as an old friend, an old foe, better than us all.

Today, this day, my children are alive and real, perfect in form and surreally soft. They are the preciousness of young life, fleeting, and my God sees them. Lest you think me morbid, think on why this is precious, why as a parent, as a child, as a human I am determined to cherish this thing we call life, why I must defend it for those I love, why death is such an aberration and a destiny and a defeated foe. The thought of death is morbid, maybe, but so is the thought of life without the thought of death. For now, we need both. And God sees us, in both.

God sees us.

Peregrine falcons see too, because as they dive at over 200 miles per hour, they blink their nictitating membranes repeatedly to clear away the debris and clean each eye.

In the midst of holding onto my family’s break-neck life, I find that I am unable to bring myself to accept such a thing as death, and yet… I know that it’s what I must do, for we will all encounter it, sooner or later. This dark thing I wish to dismiss by forgetting, or fighting, or examining on the operating table of a small terrarium: Death will come for us all.  I know, in my spirit, it is good – a gift given so that our difficult, fallen lives will not last forever, but it will never lose its dread, even when it has lost its sting. And perhaps I will embrace it as a friend when more years have passed beneath my feet, or perhaps I will embrace my Savior, and He will carry me through it into the place where it cannot enter.

And my heart holds its breath for all the times I must try to prepare my children for that moment – the one I feel certain I will never be fully prepared for, the one that God sees. I know I am actually only trying to prepare myself. I know it is not up to me, and I wish I could take more comfort in that. Perhaps someday I will. Every day is another reminder that I am not in control, and death is the great Out-of-My-Control.

But I know this. Each day that I walk in the sight of God, He will train my eyes, and my sight can be strengthened by seeing life and knowing it itself is but a shadow of the life to come.

Nadia, my little avian maven, knows that birds need nests. So she settles herself in a patch of fragrant clover and picks flowers as a pillow for the bird. She is kneeling in her pink flowered dress, a lily ringed with gingham, examining the ground and plucking appropriate stems carefully, precisely, preparing for burial. Perhaps, preparing for flight.

And I blink back a tear.

Have you ever seen a rainbow? Until recently, I had looked at many rainbows, but had never truly seen a rainbow.

In his writings on faith and logic, religious scholar Bediuzzaman Said Nursi maintained that looking and seeing are two different things. In his book, The Words, he says: “The eye is a window through which the spirit looks at this world. If you use it on behalf of your carnal soul, without selling it to God Almighty, by gazing at transient, impermanent beauties and spectacles, it panders to lust and other carnal desires.”

Using Nursi’s reasoning, you might look at a magician who is right in front of you, for example, but never see how a trick is accomplished. The same thing happens when you look at a mathematical equation but don’t see what the x value is – unless you are good at mathematics.

After watching MIT physics professor Walter Lewin’s lecture “The Hidden Beauty of Rainbows,”[1] I am finally able to see the rainbow, instead of just looking at it. To share my fascination about this topic, and reinforce my own knowledge, here are some notes I took during Lewin’s lecture. Walter Lewin is the author of For the Love of Physics. This book is really a perfect book if you love Physics.

Figure 1.0

There are four fundamental facets we need to cover to understand a rainbow (Figure 1.0). The first is the “radius” of a rainbow. Since every rainbow has a radius, this means a rainbow is a circle. And every circle has a center point. A rainbow’s center point is below the horizon. Here, we can ask these follow-up questions:

1.1 How we can find the radius of a rainbow in degrees?

1.2 Do all rainbows have the same radius?

The second question is about the colors of a rainbow, specifically red. If you see a rainbow, there is always a red color. But:

2.1 Is the color red always on the inside or outside of the rainbow?

2.2 Does the position of the red color depend on the time of the day and year?

It’s obvious that light determines a rainbow. If there is a rainbow somewhere, there is an enormous difference in the brightness of the sky above and below a rainbow. We can now ask:

3.1 Where is it bright?

3.2 Where is it dark?

There are some additional questions we need to discuss regarding the second bow in Figure 2, specifically:

4 Have you ever noticed the second bow? If yes, then;

5 Where do you look for the second bow?

6.1 What is the color sequence of the second bow?

6.2 Is it the same as the primary bow, is it brighter, or is it reversed?

Figure 2.0

In order to understand rainbows better, we need to review some basic physics. Let’s start with reflection and refraction.

Mediums, reflection, and refraction

Definition: A medium[2] is the substance that carries a wave from one location to another.

When light goes from one medium to another, like from air to water, one of two things happens:

● Some of that light bounces back, which we call reflection. The light will reflect at the same angle at which it hit the surface.

● Part of the light doesn’t reflect, instead continuing into water at an angle. We call this refraction.

Each medium has an index of refraction, which tells us what the speed of light in that medium is. The index of refraction of air is 1.0003, and the index of refraction of a vacuum is 1.

That means the speed of light in air and in a vacuum is almost the same. That speed is 300,000 kilometers per second, and that’s why the index of refraction for air and a vacuum is 1. But in water, the index of refraction is approximately 1.33. The difference between 1 and 1.33 means that in water the speed of light is 33 percent slower than it is in air.

Snell’s law and an angle of incidence

There is also a connection between angle of reflection, angle of refraction, and the index of refraction. We call it Snell’s law.

Snell’s law illustrated:

Figure 3.0

To understand this formula, you need to know the definition of the sine of an angle.

Definition:The sine[3] is a trigonometric function of an angle. It is found by taking the length of the side opposite the angle and dividing by the length of the longest side of the triangle.

If α is 0°, then sin (α) = 0.

If α is 30°, then sin (α) = 0.5.

If α is 90°, then the sin (α) = 1.

Let’s do an example where we can apply Snell’s law (Figure 4.0).

Light comes from air to water at an angle of 60°. (This is possible, because the angle at which water comes in, the angle β, will be 40.6°.)

Definition: The angle of incidence[4] is the angle between the incoming light and the vertical on the surface.

Figure 4.0

Like professor Lewin said, “very simple, very straightforward.”

Figure 5.0 | Figure 5.0. Example of refraction and reflection. See also [5].[5]

Figure 6.0

Figure 6.0 displays a true wonder of water. Sunlight is coming from the left side, landing everywhere on half the surface of that raindrop. We can choose one very narrow beam of all that sunlight with a 60° angle of incidence. We call the angle of incidence “i” and the angle for the refraction “r.”

Of course, that narrow beam is the only a small fraction of the total.

At point A, two things happen (in accordance with Snell’s law);

● A little bit of light reflects.

● The remaining light refracts.

The light reaches point B inside the water drop. At point B, two things happen:

● Some of the light goes back into the air (signaling refraction).

● Some of the light gets reflected. (Therefore, at point B, the angles are the same, “r.”)

Then the reflected light goes on its way to point C. Here, a little bit of the light is reflected and goes back into the water. Most of it comes out of the water, which results in refraction.

At this point, if we use Snell’s law at point A, the reflection law at point B, and then Snell’s law at point C again, we will see that our angle for the refraction at point C will be, surprisingly, “i.”

Now, if we check the line that comes out from the water at point C, we will have a changing angle. We can call it φ (Phi). And for the angle φ, we have an algebraic formula, as follows:

φ= 4r — 2i

If you check the figure, you will see that we have 4 r and 2 i.

By the way, if we take the narrow beam which passes through the center of the raindrop, we can easily see that the angle of incidence is zero. In accordance to Snell’s law, this narrow beam goes straight through. Then it reflects back and finally it comes out of the water. This is true because:

if i = 0, then r = 0 and therefore φ = 0.

Let’s do something interesting and change the value of “i.” If the light hits the water drop higher up, the angle “i” increases. At some point, the angle φ also increases and reaches a maximum value.

Figure 7.0

If you recall, the index of refraction for water is 1.336. If we take any value of “i” on the table and apply Snell’s law, we can easily calculate values of r. And then, if we use the formula φ=4r — 2i, we can come up with the values of φ. This is illustrated in Figure 7.0.

If this doesn’t seem all that special, there is something else that you wouldn’t expect: φ reaches a maximum value, and when you go to higher values of i, then φ goes down again. This plays a key role in the formation of a rainbow.

The index of refraction depends on the color of the light.

Figure 8.0

As seen in the table, when the angle of incidence for red light is very close to 60°, the value of φ is at its maximum, which is 42.3°. In other words, the value of φ can be smaller than 42.3°, but it cannot be larger. For blue light, the angle of incidence is slightly lower, and the φ maximum is at 40.7°.

Water drop and a cone

A key question arises: Let us say we have one water drop, and the sunlight falling onto that water drop is coming from the left. What is that one water drop going to do with the white light from the sun that covers half of the rainbow?

Let us check first for red light and then for blue light.

Light beams will come out of this drop, and that cone will be filled with red light. But the angle for the light beams cannot be larger than 42°, meaning they can be smaller. Then we will see a cone of lights.

But why is it a cone? Because, if we go back to what was previously discussed, we will remember that one light beam comes in with an angle i, and then comes out with an angle i.

Figure 9.0

Figure 9.0 shows a representation of a water drop. The light that comes in at the angle of incidence is 60°. It refracts and reflects inside the droplet, then goes to point C and comes out of droplet at an angle of 42°. And, surprisingly, if the angle of incidence is 60°, when the light beam comes out, it has to be red light. On the other hand, there is not only one light beam coming in with an angle 60°. The entire surface of this water drop is illuminated by the sun, so there are an enormous number of beams for which the angle of incidence is 60°.

If we go back to Figure 9.0, we see the black line on this water drop. Any light beam that comes in anywhere on that black line will have an angle of incidence of 60°.

Figure 6.1

Now we will do the same for blue light. This time, we will have another cone which is completely filled with blue light, but when the blue light beams come out, the angle will be at a maximum of 40.7°.

Interesting fact: All the colors except for red and blue that are in the sunlight will be in between the red and blue.

Figure 6.2

A journey: refraction, reflection, and refraction

We know that when the light comes in, it goes through a journey. A light beam refracts, then reflects, and then refracts. That’s all. But, how about the light on the right side of the cone? Is there any light? When the sun shines in, does this water drop bring light there?

If the angle φ is larger than 42°, then, yes, it is possible. But we have already proven that the angle φ cannot be larger than 42°. So, it is impossible: There is no light outside the cone.

Another question… What will be the color of the light inside the blue cone? We should remember that the red light is everywhere in this cone, because as long as the angle φ is smaller than 42°, red light can be anywhere. The blue light is also almost everywhere because the blue light is allowed to be smaller than 40°. This means all the colors that we didn’t mention are coming out inside the blue cone and there will be just white light.

Figure 6.3

To summarize, if we have a water drop, and a bright light shining onto this water drop, we will see red, blue, green, white, and other colors inside the cone or circle. In other words, all the colors of a rainbow. And of course, there will be no light outside the cone. There are some experiments on YouTube that you can watch if you want.

Figure 6.4

You may think that at the location where blue comes out, if all the other colors can come out — which they can because the angle is smaller than 40° — the light should be also white, not blue. It is a nice approach, but wrong.

It is wrong, because of the light intensity. If you check the graph for the intensity of the light, you will see that different colors have peak points where φ reaches the maximum value. If all the colors are present and equal intensity, we see white light. That’s why the inside of the cone is white. However, when the color red spikes up, it dominates its location. We don’t even notice the other colors.

Figure 6.5

Looking at the sky

Let’s say you are standing at point A and the sunlight is coming from the left. Since the light comes from infinitely far away, all these lines are parallel to each other. Let’s say you look at the sky in direction B and pick a raindrop at any point. Any point you pick will cast the cone in the direction of the sun with the angle 60°.

When you look at the sky in the direction B, will you see light? You will not, because you are outside the 42°.

Figure 10.0

What if you look in a different direction? We will look at point C (it doesn’t matter where), and take a rain drop on line C. Light beams will do the same thing. Again, they will produce a 42° red cone. However, you will see white light. Yes, you read that right, white light, because you are looking straight in the middle of the cone of white light.

Let’s say we choose another angle and look at point D. If we choose a raindrop anywhere on line D, it will make no difference. It will cast into the sky the famous 42° angle because that’s the angle that was chosen there. If we look in that direction of the sky, what will we see? Yes, only red. No other color!

Figure 10.1

And now, we know where we will see red light, blue light, or white light. And we also know which direction has no light.

A rainbow is a bow in the sky

Let’s suppose the sun is on your left and you are standing at point A (Figure 11.0). Since there is light, you will have a shadow on the ground. As long as you look 42° away from any direction, you will only see red. That is because a cone is a spherical object. And that explains why the rainbow is a bowl in the sky. As long as it is 42° away from a line in any direction, you will always see red light. And if you decrease the angle, you will see the other colors in the sky.

And so now, we can make a picture of a rainbow.

When we look at the horizon, sometimes we see a rainbow. And we will have a center point somewhere below the horizon. The radius with 42° has red light. The blue light is closer to the 40° radius. This tells us that the red bowl is always on the outside and the blue light is always on the inside. That’s what we always see.

Figure 11.0

Double Reflection in the Rainbow

We can also include two reflections in the rainbow. If we do, we get refraction, reflection, another reflection, and then another refraction. So, the light beams reflect twice. And if we do this hocus-pocus again and again, we can come to a conclusion: There is no maximum value for the various colors, but there is a minimum value. In other words, various colors cannot be lower than a certain value, but they can be higher.

And if we do some calculation for the angles, we will see something very similar. We will find out that the second bow is pointing at the sky, and the angle is 10° above the first one (which is 52°). And, surprisingly, the colors of the second bowl reverse. In other words, red is the one on the inside, and blue is on the outside this time.

Figure 11.1

Interesting fact: There is no light between the two bowls. This is because of the φ maximum and minimum. We call that dark part Alexander’s dark band.[6]Additionally, since the second bowl is formed by φ minimum, sunlight can come out at angle φ or larger and there will be sunlight above a rainbow.

Figure 11.2

Some examples…

Image 1: René Descartes’ sketch of how primary and secondary rainbows are formed (probable engraver: Frans van Schooten the younger).

Image 2: This diagram was drawn by the famous physicist Newton who was one of the first to understand the rainbow, from his book Optics.

Image 3: Double rainbow. The primary red bow is on the outside, and blue is on the inside. Alexander’s dark band can be seen in between the two bows. It shows up really dark. For the secondary bow, red is on the inside, and blue is on the outside.

Image 4: Walter Lewis’ backyard and his rainbow. When you water your garden when the sun is high in the sky, you can get a rainbow all the way around you. 42° away from the line is always red. If you turn around and look back, it is red.

Image 5 | A white rainbow. It is very unique and rare. It is also called a fog bow, because fog has small water droplets that create this effect.

Image 8 | Red rainbow. Let us imagine that there is a rainbow at sunset. What would you expect to see? We would only see red light because there is only red light. What would happen to the white light inside the bow? It would be red too.

The lotus, a radiant and graceful aquatic plant with a magnificent ability to repel dirt, has been the inspiration behind a range of self-cleaning and dirt-repelling technologies. Now a registered trademark, the “Lotus Effect” promises a future filled with objects that rarely, if ever, need washing. Finally, this means owning bathroom mirrors that unfog itself and shirts that shrug off ketchup and coffee.

And the earth, We have spread it out like a couch; and how excellent We are in spreading it. (Adh-Dhariyah, 51:48)

The factory of the universe and the guesthouse of the earth are so pure and clean and so untainted and fresh that … if … the act of cleansing is not attributed to the Creator of the universe, then … [e]ither all the creatures would have a share in the universal act of cleansing… or there would have to be a consultative committee the size of the universe in order to decide and regulate all those acts together… This is impossible not just once, but hundreds and thousands of times over. (4)

Have you ever come across a “stinky forest”? How about a “messy desert”? These questions are primarily posed as rhetoric since various means of sanitation is ubiquitous in nature (Stinky forests and lakes actually exist only thanks to pollution—a modern problem introduced by human intervention via misconduct of technology) where Lotus Effect is only one of the mechanisms contributing to the steady “house-keeping” in nature.

In muddy waters, the lotus plant stands out with its ever-clean leaves making it a symbol of purity in Asian cultures. Moreover, lotus leaves keep dry under even the heaviest monsoon rain. Such seemingly ironical feats are accomplished by capitalizing on a principle named after the plant itself: The Lotus Effect.

The Lotus Effect

At first sight, the cleansing of the leaves by the rainfall seems utterly trivial. However, the cleaning of the lotus plant (Nelumbo nucifera) by the downpour is not something to be taken for granted. Curiously, when raindrops encounter the lotus leaf, they adopt an almost perfectly spherical shape, resembling ball bearings, and start rolling off the surface carrying away all the dirt (Figure 1). Easier said than done, the lotus plant always keeps pristine, even at the microscopic level.

In the early 1970s, soon after the electron microscope (which can yield vivid images of the ultra small details that are at length scales on the order of a billionth of a meter) became commercially available, German botanist Wilhelm Barthlott (of University of Bonn, Germany) started imaging plants using the newly discovered technology. Sample preparations for electron microscopy normally demanded tedious cleaning procedures since even a speckle of dust could ruin the portrayed landscape at such minuscule scales. To Barthlott’s surprise, some plants apparently were “self-cleaning”: They required very little (or sometimes none whatsoever) cleaning for detailed inspection with the microscope and the lotus plant was a prince of these (1). Further intrigued by the fact, Barthlott looked at the lotus leaves through the electron microscope to find out what renders the lotus plant remarkably competent to repel even the tiniest dirt.

Water’s love-hate affair

Before delving into the secrets for lotus plant’s sanitation, let’s look at the interaction of water with other materials. At the molecular level, the electrical charges are unevenly distributed across a water molecule (i.e. water is highly “polar”) which becomes entangled in a love-hate type of relationship with other materials: Some materials “love” water tending to maximize their interaction with it, whereas others “hate” it, trying to avoid their encounter with water as much as possible.1 To put things in perspective, we can immediately tell from everyday experience that oil is “water-hating” since oil and water do not mix, while sugar is “water-loving” because sugar can dissolve in water without much effort.

Macroscopically—that is, one can immediately realize by touching a lotus leaf—the lotus leaf surface feels waxy, and should therefore be water-hating. However, the waxiness is not enough to equip the lotus plant with its unusual capabilities to remain clean since the lotus leaf is not alone among plants in its leaves’ surface waxiness. Indeed, there is more to the lotus leaf’s curious surface properties than that, a property which was first recognized by Barthlott under the electron microscope.

Super water-hating surfaces

When Barthlott looked at the lotus leaf, besides its spotlessness in the microscopic sense, the leaf surface was decorated with numerous bumps a few micron sizes each (a micron is one millionth of a meter). Such bumpiness served to enhance the water-hating aspect of the surface making it “super water-hating,” corresponding to a contact angle that is close to 170 degrees. As a result, water encountering the lotus leaf surface rapidly beads up forming a nearly spherical shape (akin to the scenario when one drips water on a hot cooking pan) and drops roll off the surface even with an ever-so-slight inclination of the leaf. Apparently, a rolling water droplet is much more effective in picking up the surface dirt than one that is merely sliding, and all the surface debris is thus wiped off.

Although discovered first in the lotus plant, the super water-hating surfaces are serving a passive yet effective means of cleaning for animals like butterflies, dragonflies and other insects that are not able to clean all their body parts actively. For plants, preventing the coverage of their leaves by water (or other contaminants) is important to maximize the exposure to sunlight which would otherwise cause reduced photosynthesis. Another great biological relevance of surfaces of such nature for all these creatures is that it provides protection against the growth of pathogens by keeping the surface dry at all times.

Inspirations from biology for technology

There are ever-growing biologically inspired technologies, so-called biomimetics, and the lotus effect provides a nifty example. Unfolding the mystery behind lotus’ exceptional competence in self-cleaning, Barthlott patented the idea of artificially manufacturing microscopically-raised, water-repelling surfaces to mimic the lotus leaf. “Lotus Effect” is now a registered trademark which underlies commercial products such as self-cleaning windows and fabrics, as well as a dirt-repelling paint. Other applications that are waiting around the corner are: swimsuits that stay dry for days allowing prolonged underwater excursions, coatings on metals to avoid the deposits of marine bioorganisms which would enable up to 40% reduction on fuel consumption by decreasing friction (2). Metal coatings will also find applications to prevent ice formation on plane engines alleviating their wear-and-tear (3).

Apparently, lotus does not collect dirt, but only patents.

Note

1 Contact angle is a metric for the water propensity of a surface. Water drop displays a higher contact angle (hence lower contact area) on a water-hating surface when compared to a lower contact angle (hence higher contact area) on a water-loving one.

References

1. Forbes, Peter. August, 2008. “Self Cleaning Materials,” Scientific American.

2. http://www.basf.com/group/

corporate/en/innovations/events-presentations/nanotechnology/basf

3. “Water-Repelling Metals,” Prachi Patel, MIT Technology Review, 2008, http://www.technologyreview.com/energy/21530/

4. “The Thirtieth Gleam,” Bediuzzaman Said Nursi, Risale-i Nur Collection.

The appeal of the Golden Ratio to the human eye and brain has been scientifically tested. When subjects are presented with a range of rectangles, people invariably pick out as most pleasing ones those whose sides are of the Golden Ratio. This golden number is the basic number at work in aesthetics; there are even claims that the Golden Ratio was used by Leonardo da Vinci when painting the Mona Lisa, and by the Greeks in building the Parthenon. But the surprising thing is that a number deemed aesthetically pleasing by human beings also crops up in nature and science. In a newly published article in Physical Review B, it is stated that the Golden Ratio appears in the structures of some metals. The Golden Ratio is seen in the arrangement of seeds on flower heads, in the spirals of sea shells and galaxies, even in black holes. This ratio can be found almost everywhere in the universe.

Although the Greek mathematician Euclid first defined the Golden Ratio in around 300 BC, the followers of Pythagoras probably knew of it two centuries earlier. Euclid defined it as a line that can be divided into two unequal parts (Figure 1), where the ratio of the smaller part of the line to the longer part is the same as the ratio of the longer part to the whole. This ratio is 1.6180339887…, the Golden Number.

What makes the Golden Ratio special is the number of mathematical properties it possesses. The Golden Ratio is the only number whose square can be produced simply by adding 1 and the reciprocal of which can be arrived at by subtracting 1. If you take a Golden Rectangle – that is a rectangle where the length-to-breadth ratio is equal to the Golden Ratio, and take out a square, what remains is another, smaller Golden Rectangle. Also, think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. If you start with 7 and 11, then what you have is:

7, 11, 18, 29, 47, 76… When you have written down approximately 20 numbers, calculate the ratio of the last to the penultimate: the answer should approximate the Golden Number.

In mathematical terminology it is (an/an-1) equals to the Golden Ratio.

Another example of the Golden Ratio is the Fibonacci Numbers, that is a number series where each number is simply the sum of the previous two numbers. These numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… The ratio between any two successive Fibonacci numbers approaches the Golden Ratio as the numbers get larger. We can find the Fibonacci series particularly in the spirals of sea shells and in the arrangement of seeds on sunflower heads.

It was the elusive nature of the Golden Ratio that led the Italian friar and mathematician Luca Pacioli to equate it with the incomprehensibility of God. In the 15th century, he wrote a three-volume treatise, Divina Proportione (Divine Proportion), that was crucial in the dissemination of the Golden Ratio beyond the world of mathematics. After him, many artists, architects, and musicians used the Golden Ratio in their works; for example, musicians such as Debussy and Bartok and the architect Le Corbusier.

Let’s examine the arrangement of leaves on the stem of the plant Phyllotaxis. As each new leaf grows, it does so at an angle offset from the leaf below. The most common angle between successive leaves is 137.5 degrees – the Golden Angle; 137.5=3d 360-360/G, where G is the Golden Ratio. Why does the Golden Ratio play a role in the arrangement of leaves? It is all down to the irrationality of the number. A new leaf must collect sunlight without throwing too much of a shadow on the leaves below. A plant must arrange its leaves in such a way that the greatest number can spiral around the stem before a new leaf can sprout immediately above a lower one – that is at 360 degrees. If the leaves were arranged at an angle of 120 degrees, then the leaves would grow as 3 separate columns with large gaps between them. This would effectively block out the sunlight to the lower leaves. If the angle were 50 degrees, then there would be spaces between the leaves. But, with an angle of 137.5, the maximum amount of leaves can be arranged with a minimum of space being left between the leaves.

The Golden Ratio also crops up in hard sciences. Let’s take a look at the growth of “quasi-crystals.” These maintain a five-fold symmetry, which means that they make a pattern that looks the same when rotated by multiples of one-fifth of 360 degrees. Since the time when these crystals were discovered in 1984, many physicists have been researching their properties. In Brookhaven National Lab in New York State, Tanhong Cai imaged the microscopic terrain of the surface of such crystals made from alloys of aluminum-copper-iron and aluminum-palladium-manganese. It is found that flat terraces are punctuated by abrupt vertical steps. The steps come in two predominant sizes, with the ratio of the heights of these two steps being the Golden Ratio. This fact was discovered in 2002.

The most surprising place where the Golden Ratio appears is in black holes, a discovery made by Paul Davies of the University of Adelaide in 1989. Black holes and other self-gravitating bodies, such as the sun, have a negative specific heat. This means that they get hotter as they lose heat. In a spinning black hole there is an outward centrifugal force acting to prevent any shrinkage of the hole. The force depends on how fast the black hole is spinning. It turns out that at a critical value of the spin (when the ratio between the square root of the mass value and the square root of the spinning parameter is equal to the golden ratio), a black hole flips from having negative to positive specific heat. In other words, the Golden Ratio determines the character of the black hole.

Figure 2 shows the finger bones of a hand. The ratio of each bone at the top of the hand to the bones at the bottom of the fingers is the Golden Ratio. Pine-cones show the Golden Ratio spiral clearly (Figure 3). If one looks carefully at an ordinary cauliflower, one can see a center point where the florets are smallest, and the florets are organized in spirals around this center in both directions (Figure 4). The flower, Echinacea Purpura, has the same spirals (Figure 5).

With the help of developing science, new examples of the Golden Ratio are waiting to be discovered in the universe. The latest discoveries demonstrate that by using this ratio in technology, new products which make our lives easier will soon be available to all. This mystery that is spread throughout the universe may be an opportunity to renew and change our points of view on life.

Reference

• Chown, Marcus, “Why Should Nature Have a Favorite Number,” New Scientist, 21-28 December 2002, 55-56.

In this article, we will delve a little deeper into this magnificent miracle of God: the mathematics of nature.

FLOWERS:

For example, look at the pictures given below. For 1-petalled flowers, we offer white calla lilies; for 2-petalled flowers, we offer the very rare euphorbia; and for 3-petalled flowers trilliums, lilies, and irises.

<cellpadding=”15″ cellspacing=”15″>White Calla Lilly

Trilliums

Did you ever wonder why 4-petalled flowers are so rare, and why everyone gets excited when they find a 4-leaf clover? The reason for this is because such flowers are very rare, for 4 is not a Fibonacci number. Some violets and bluets also have 4 petals.

Flowers with 5 petals are rather common. Among them are buttercups, wild roses, larkspurs, and columbines.

Examples of 8-petalled flowers are bloodroots and delphiniums. Examples of 13-petalled flowers are ragworts, corn marigolds, and cinerarias; those with 21 petals are daisies, asters, and chicories; and those with 34 petals are oxeye daisies, sunflowers, plantains, and pyrethrums.

Some families of daisies, such as the michaelmas daisies from the asteraceae family, have 55 and 89 petals.

Seed and flower heads

The echinacea purpura is a member of the daisy family native to the Illinois prairie. You can see in Figure 1 that the orange œpetals seem to form spirals curving both to the left and to the right. At the edge of the picture, if you count those spiraling to the right as you go outwards, you will notice that there are 55 spirals (a Fibonacci number). A little further toward the center, you can count 34 spirals (another Fibonacci number). If you count the spirals going the other way, you will see that the pair of numbers (counting the spirals curving toward the left and toward the right) are neighbors in the Fibonacci number series.

The same happens in many seeds and flower heads, among them sunflower seeds, daisies, pineapples, and pine cones. The reason for this is that such an arrangement packs the optimal number of seeds so that no matter how large the seed head is, the seeds are always packed uniformly at any stage. As they are the same size in any given area, there is no crowding in the center and no scarcity at the edges.

The spirals form a pattern: The œcurvier ones

appear near the center, while the flatter ones, which are more numerous, appear the further out you go. Thus the number of spirals we see in either direction differs according to the size of the flower’s head. On a large flower head, we see more spirals further out than we do near the center. The numbers of spirals in each direction are (almost always) neighboring Fibonacci numbers!

Let’s observe the spirals in these beautiful arts of the Infinite Artist. Consider the daisy. In the close-packed arrangement of tiny florets in the daisy blossom’s core, you can see the phenomenon in an almost two-dimensional form. As shown in Figure 2 there are 21 (a Fibonacci number) counterclockwise spirals and 34 (another Fibonacci number) logarithmic or equiangular spirals. In any daisy, the combination of counterclockwise and clockwise spirals generally consists of successive terms in the Fibonacci sequence.

The Math Behind It

Botanists have shown that plants grow from a single tiny group of cells right at the tip of any branch or twig belonging to a growing plant or tree. This tip is called the meristem. They grow in size after their formation, but new cells are formed only at such growing points. Cells further down the stem expand, and thus cause the growing point to rise. This means that a growing plant produces seeds at the flower’s center, and that those seeds then push the other seeds outward. Each seed settles into a location that turns out to have a specific constant angle of rotation relative to the previous seed. This constant rotation forms the spirals.

Consider the following case. There are n seeds in the arrangement. The nearest seed is seed 1, the next one is seed 2, and so on. If each seed has an area of 1, we have a total area of n and a circle with a radius of from the area formula (Area= ). Given this, the distance from the center to each seed is proportional to the square root of its seed number. If we call this angle alpha, the angle of seed k will be alpha*k. So we can easily describe the location of any seed in polar coordinates with and theta=k*alpha.

If we chose an angle of 0.15 (54A) the result will be far better. But we will be done again after 20 seeds, for 20*0.15=3, so after 3 complete turns we will be back on the same line. If we choose 0.48, which is a little bit better than 0.15, we will come back to the original line in 25 rotations.

Therefore, speaking logically, if we choose an irrational number we will not return to the original line. Of course we will get close to it, since every irrational number has some kind of rational approximation. In nature, we mostly observe the so-called golden ratio, an irrational (1.618 = ), which is the root of the equation , which is the limit of the ratio of two consecutive Fibonacci numbers.

With this angle of rotation, each seed is rotated approximately 1.618 revolutions from the previous seed (i.e., 0.618 revolutions or 0.618*360=222.5A). Notice how well distributed the seeds appear; for there is no clumping and very little wasted space. Although the pattern grows quite large, the distances between neighboring seeds appear to stay constant. In nature, you can see that plants grow their seeds simply where there is the most room.

It is really amazing that a single fixed angle can produce the optimal design no matter how large the plant becomes. For example, once a leaf’s angle is fixed, that leaf will œdo its best not to obscure the leaves below and œdo its best not to be obscured by any leaves that will grow above it. Similarly, once a seed is positioned on a seed head, the new seeds continue to push the older ones out in a straight line. However, it retains the seed head’s original angle. The seeds will always be packed uniformly on the seed head regardless of the head’s size. The principle that a single angle produces uniform packing no matter how much growth appears thereafter was proved mathematically only in 1993 by the French mathematicians Douady and Couder.

We frequently observe the golden ration in nature. In addition, we can try flowers and flowers, at least mathematically and as models.

Conclusion

We look at nature and see God’s creation. Most people just look at the general design, but others also study the seeds and their designs. As the Qur’an’s first verse tells us to œRead, in the sense of reading the signs in nature (the Qur’an was revealed to a largely illiterate people who had no significant body of written literature), we should understand this as an indication to analyze nature.

As a mathematician, I see that God has arranged everything in nature according to a mathematical order. He puts the maximum number of seeds in a minimum area. Bees use hexagons to store the maximum amount of honey in a minimal space. This fact is even mentioned in Qur’an 16:68: Your Lord revealed to the bee. Those who believe in evolution say that the picture is very clear. But there is a great art in front of us, one which is very well balanced and in perfect accord with mathematical harmony. This shows that there is no luck or chance in the world, and that everything is based on the rules that God has laid down for His creation.

Footnote

1. Fibonacci (Leonardo da Pisa, c1175-1250): The son of a Pisan merchant who also served as a customs officer in North Africa. He travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence. In 1200 he returned to Pisa and used the knowledge that he had gained on his travels to write Liber abaci, in which he introduced the Latin-speaking world to the decimal number system. Fibonacci numbers begin with 1,1 and the next term is the sum of the two previous terms. The first ones in the series are 1,1,2,3,5,8,13,21,34,55,89,

References

Mathematics Magazine 75:3 (June 2002): 163-73.

http://ccins.camosun.bc.ca/~jbritton/fibslide/jbfibslide.htm.

G. J. Mitchison, œPhyllotaxis and the Fibonacci Series, Science 1965, 270-75.

P. Stevens, Patterns in Nature (New York: Little Brown and Co.,1974).

When a honeybee discovers a new source of honey within about ten metres of the hive, she returns to the hive and regurgitates drops of honey which are eagerly drunk by other bees. Then, in order to indicate the source of the drops they have just drunk, she performs an interesting sequence of movements we call the ‘round dance’. She dances round in a circle, first in one direction, then reversing and going round in the opposite direction, again reversing and so on. During the performance, other bees follow her about, holding their antenna against her abdomen. After the dance, she flies back to the food source, normally a particular flower or group of flowers. The other bees who followed the dance do not fly after her but fly out in all directions. However, fairly soon, a great number of them find the new source. They do so because they have smelt the scent of the flowers that clung to the dancer’s body and so to recognize the right kind of flower. Observing this procedure of the bees’ communication with admiration, Frisch reported that the ‘round dance’ tells the other bees to go out and search in the near neighbourhood of the hive; and the scent on the dancer’s body tells them which flowers to look for. In one experiment, bees informed by a dancer at once found the right flowers in a section of Munich Botanical Gardens where 700 different plant species were blooming.

But honeybees search for food at a greater distance than 10 metres. They have been observed to fly more than 13 km in search of honey. Remembering the average that a honeybee is only about 13 mm long, that 13 km for a bee is the equivalent of about 1,600 km for a human being. It is clear that, even for distances considerably less than 13 km, the round dance would not be much use. If the discoverer of a new food source informed the other bees to go out and search in all directions for over several kilometres, they would never find the flowers. Thus, when a bee finds food at a considerable distance, say over 100 metres from the hive, she returns to the hive, offers the honey she has found, and then performs a different kind of special dance. She dances along in a straight line for a certain distance, wagging her tail vigorously and buzzing away by means of slight vibrations of the muscles that flap her wings in flight. At the end of this wagging run, she stops buzzing and wagging her tail, circles round to one side back to where she started, does another wagging run, circles round to the opposite side, does a third wagging run and so on. This sequence of movements is called ‘tail-wagging dance’. Showing special interest in wagging runs, a number of other bees follow her round. It is known that bees cannot hear sounds in the air, however they can feel the buzzing vibrations through the surface on which the dance is performed. Like the round dance the ‘tail-wagging’ dance tells the followers of the dance that there is food available, and what flowers to look for, from the smell. But it tells them much more than this; it tells them precisely how far away the flowers are, and in exactly what direction. And so, the bees who study the dance can fly with precision to the spot indicated, even at a distance of kilometers, and find the honey-bearing flowers.

The distance to the food is conveyed by the tempo of the dance. A quick-step tempo means a relatively nearby food source, a slow tempo means a more distant one. More precisely as the distance increases, so does the duration of each wagging run. The bees who follow the dance study several wagging runs and then appear to calculate the average duration which they then translate into distance by a mathematical rule.

The ‘tail-wagging dance’ is sometimes done on a horizontal surface just outside the hive. When this happens, the dancer indicates the direction of food by aiming her run in exactly the direction of the food. She can do this if she can see the sun (or the polarized light of the blue sky, which indicates the position of the sun to bees, though not to us). So, actually she is taking up a position in which she sees the sun as the same angle as during her flight to the honey source. The wagging run makes the same angle with the sun on her outward flight did. But the dance is normally performed in the dark inside the hive, on the vertical surface of the honeycomb. Here the angle between the flight path and the sun is translated into the angle between the wagging run and the vertical direction straight upwards.

Another remarkable fact about the bees communication is that if the bee reached her goal (and returned from it) by an L-shaped detour, she uses the angles and lengths of the two sections of flight to calculate the true direction of the goal by straight-line flight, and this is the direction she conveys in her dance, even when she did not this direct route herself.

As for the bees who are following the dance, they are working literally in the dark and can only use touch to find out the angle of the dancer’s wagging run against the vertical. They translate this back into a visual angle with the sun, fly off in this direction for the distance signaled by the wagging run tempo, and look for the flowers of the scent they smelt on the dancer.

So far we have described the language of the Carniolan race of honeybees according to the findings of Frisch. Other races have different ‘dialects’. German, North African, Caucasian, Italian honeybees all indicate distance and direction by much shorter dances than the Carniolans. How the honeybees have learned to communicate with each other is certainly remarkable! However, it is not a true language, because the system used by bees, unlike human languages, cannot generate new combination of structures and symbols to describe novel events. One example of their limitation is that bees appear to have no way of communicating height or depth beyond a few meters. Thus, if you take a bee to a source of food in a place, say 10 metres above ground level, she will return to her hive and try to indicate the food source by means of her dance. Then, the other bees will all fly out in the direction she indicated, but they will never find the source. No matter how near the source is, even if it is just above the hive, the result will be the same.