Posted in Science & Nature

Thoughts On Sunlight

We live in a world where everything is powered by something. Our technology is fueled by electricity. Our cars are fueled by fossil fuel (although hopefully not for much longer). Our generators are fueled by everything from coal to running water to the splitting of atoms. We are fueled by food, which we break down to release energy.

But at the core of it all, the world is fueled by one main energy source: sunlight.

Let us retrace the steps.

The device you are using to read this is charged by electricity provided by a power generator. Whatever the source of electricity is, humans are required to power the machines and we are fueled by food. The food we eat are either plants, or meat from animals that consume plants. Plants generate their energy through photosynthesis, where sunlight is used to store energy in carbohydrates. 

Ergo, sunlight fuels us all – we are all made of and held together by sunlight.

The Sun is positioned 152 million kilometres from Earth. This means that sunlight travels 152 million kilometres – a distance that takes even light eight minutes to traverse – to feed Earth, brighten our days and make us feel warm and fuzzy. 

Sunlight also heats the earth and seas to power various weather cycles and currents, provides heat to keep life possible and most importantly, lets us see because it floods our day with photons. 

Just something to think about the next time we enjoy a delightful nap in a warm, cozy sunbeam.

The sunniest Magic painting: Endless Sands - Imgur
Art credit Endless Sands by Noah Bradley

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Airplane Game

You are cordially invited to a game that lets you earn money very easily. The game works like this:

  1. You pay $1000 to be recruited as a passenger to a plane.
  2. There are 8 passengers, managed by 4 crew members, who have 2 co-pilots above them, co-ordinated by a captain at the top.
  3. Everytime the “plane” is filled with 8 passengers, the captain retires and is paid out $8000.
  4. When the captain retires, the plane is split into two planes and everyone else is promoted one step higher (co-pilots each become a captain, crew become co-pilots, passengers become co-pilots).
  5. When each plane fills with 8 new patients, the captain of each plane gets paid out $8000 and retires.

This seems like a very easy way to earn money. Where else could you invest money and guarantee a 700% return, only needing to recruit 7 new people into the game?

The problem with the airplane game is that it is a classic example of a pyramid scheme. At first glance, it seems that the payout of $8000 is guaranteed because it seems that the promotions will keep coming.

But if you look at the mathematics, 8 people need to participate before the first player wins. 16 people have to participate for the second player to win. 80 people have to participate for the tenth person to win. If you are the one-thousandth person to join the game, you need a total number of 8000 people to be playing the game before you are paid out. At the end of the game, 87.5% of people playing will have lost money because they will never be paid out.

This is how simple exponential growth can result in a very real fraud, resulting in thousands of people losing their hard-earned money.

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Exponential Growth

Imagine that you have won a strange lottery where they give you two options of payment: they can either pay you one million dollars up front, or they can pay you one cent on the first day, then double the amount you have every day for a month (i.e. 1 cent on day 1, 2 cents on day 2 etc.). Which would you choose?

It may seem obvious that the $1 million up front is far better than accumulating a few cents every day. But by the end of the month (day 31), you would actually have accumulated $5.37 million. How did this happen?

The secret to this extraordinary increase is the power of exponential growth. If you double a number constantly at a regular interval, it grows at a staggering rate. Let us look at the above example again.

On day 1, you have 1 cent. By day 10, you already have 2(10-1) = $5.12. Now we can see that instead of mere cents, we are gaining $5 in one day.
By day 15, you have $163.84. Now the doubling nets you another $163.
By day 20, you suddenly have $10,485.76.
We pass $1 million at day 28 where we have $1.34 million.
Day 29 you have $2.68 million and you can see how we end up with $5.37 million – over five times the amount we would have received compared to the first option.

This shows the sheer power of doubling. It is an important principle to grasp as we see exponential growth all around us in life. Nuclear chain reactions undergo exponential growth to power nuclear reactors. Positive feedback in speakers undergoes doubling amplification, resulting in the sharp screeching sounds. Compound interest follows exponential growth, allowing investments to give substantial returns over time (or result in crushing debt). Bacteria divide in two each time, resulting in a rapid population boom.

Understanding exponential growth also helps us make sense of scary situations such as pandemics. Viral infections are spread from one person to multiple people, represented by a basic reproduction number (R0). In the case of the COVID-19 (2019 coronavirus) pandemic, the R0 was between 2 and 3, meaning that left unchecked, the number of infected individuals would essentially double every few days.

Although this seems obvious, if you didn’t know about exponential growth, it would be terrifying to hear that one day you have 8 cases in a country, but in a fortnight, there are over 1000 cases, with each day presenting increasing numbers of newly infected patients. The media preys on this effect by providing anxiety-inducing headlines. But in reality, the headlines might as well read: “virus continues spreading in predictable exponential fashion“.

Another strength of knowing about exponential growth in a pandemic is that it lets us predict what would happen without any intervention. The number of cases would explode in a matter of weeks, resulting in catastrophic numbers of unwell people taken off the workforce, accompanied by mass casualties. Hospitals would be completely overrun, crippling the nation’s healthcare system and resulting in even more deaths as the infection runs rampant.

Therefore, efforts to reduce the spread of the virus through social distancing and effective quarantining are vital to reduce the rate of exponential growth, flattening the curve and making the number of cases more manageable for the healthcare system to deal with.

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You can hear the Kaua’i ʻōʻō endling’s call here:

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Centre Of The Universe

We often meet people who act as if they are at the centre of the universe. These egocentric people behave as if they are the most important people in the world and that their words and actions are more meaningful than they actually are, while assuming that they play an important role in other people’s lives. This is a common belief in children who are still learning to differentiate the world and other people from their own minds, but in adults, it is almost pathological.

Speaking of which, where is the centre of the universe?

In ancient times, the concept of “universe” was very different. Many cultures imagined the universe as consisting of the Earth where we lived, plus the heavens and the underworld (often supposedly where the good and bad end up after death respectively). These worlds would be connected by a central axis mundi, or world axis. An example of this is the mighty Yggdrasil, the World Tree, found in Norse mythology. It is said to be a gigantic tree that connects the Nine Worlds and is the centre of all life.

As the science of astronomy developed, we realised that we are not at the centre of the universe. Geocentrism – the model where Earth is at the centre of the world with the Sun, Moon and planets orbiting it – eventually gave way to heliocentrism – the modern model where the Solar System orbits around the Sun.

It took brave scientists such as Nicolaus Copernicus, Johannes Kepler and Galileo Galilei challenging the Church and Aristotelian science establishments to show that our understanding of the universe was wrong, despite pressure and punishment. Through scientific observation and inquiry, it was shown that we are not at the centre of the world, but the Sun is.

But as we discovered more about the heavens, we realised that the universe is far vaster than the Solar System. With the advent of the Big Bang Theory, we realised that the universe is expanding, with every object moving away from each other in all directions. This is an extremely difficult concept to visualise, but because the universe is expanding infinitely in all directions, it technically has no centre.

On a final note, the concept of the universe being infinite may not be relevant to us because we cannot observe the infinite universe. Instead, we often talk about the Observable Universe, which is the portion of the universe that we can physically observe with our eyes, telescopes and other instruments. The centre of the observable universe, like anything observable, is the observer.

Therefore, in some sense of the phrase, you are technically at the centre of the universe.

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Tip Of The Iceberg

Icebergs are deceptive things. You may see a small bump above the ocean surface, but beneath the surface hides a massive block of ice. Using Archimedes’ principle of buoyancy, we can calculate exactly what proportion of an iceberg lies under the surface. Pure ice has a density of about 920 kg/m³ and sea water has a density of 1025 kg/m³. Ergo, we can calculate that about 10% of the volume of an iceberg is above water. Therefore, whatever you see above the surface, there is nine times the volume hiding beneath it.

Tip of the iceberg” is a useful metaphor in describing many things. Our base instinct is to believe what we see at first glance. We rely on first impressions, we judge books on their covers and we tend to believe headlines before reading the full text of an article.

Although this is a useful way to process massive amounts of information that we are exposed to every day, it is certainly a flawed method because not only can we miss a vast quantity of information, also easily misinterpret or misunderstand things.

Take mental health for example. Because we cannot read minds, we take clues from people’s expressions, body language and what they tell us to gauge what is happening in their minds and hearts. We are reasonably good at gauging this, so we often make assumptions based on surface information.

We might assume our friend is happy because they are smiling, or that a couple’s marriage is harmonious because of cute photos on their social media. Conversely, we might assume that a stranger is rude to us because they are terrible people.

But the smiling friend may be suffering crippling anxiety and depression. The happy-looking couple may be at the brink of divorce because of relationship problems. The rude stranger may have lost a loved one just the day before. Things are not always what they seem and it makes an incredible difference to have the insight to see past the surface.

Another lesson to learn from the tip of the iceberg is that when we encounter a problem – whether it be with another person or even within ourselves – we should ask the question of what lies beneath. The problem we notice may just be the tip, with 90% of the issues hidden from plain sight.

For example, if you feel tense and easily triggered often, perhaps it is worth looking under the hood and going on an introspective journey to discover what past experiences and traumas may have caused the insecurities. If you keep feeling victimised, attacked or sensitive, examine what story your subconscious is telling you and try to correct the narrative, being the agent of your own story.

Avoid the fate of the RMS Titanic: look beyond the visible tip of the iceberg and be aware of the entire problem. You will be surprised how it changes your perspective of the world, the people you interact with and how you feel about yourself.

Image credit:

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Rainbows have been associated with wonder and the heavens throughout the history of humanity. The Norse believed that the rainbow bridge, Bifröst, connects the realms of men and gods. The rainbow is mentioned in the Bible as a sign from God to signify to Noah that the flood had ended. Irish leprechauns are said to hide their pots of gold at the end of a rainbow. It is now adopted as a symbol for LGBT movements, symbolising diversity.

The massive scale and brilliant colours of a rainbow is awe-inspiring (famously captured in the Double Rainbow video). We now know that it is the result of sunlight interacting with water droplets: reflecting, refracting and dispersing.

Sunlight refracts (bends) as it enters the droplet. It then reflects off the inside wall of the droplet and refracts once more as it exits. Because each wavelength refracts slightly differently, light disperses and each colour can be seen separately, much like a prism breaking apart white light into colours.

Because of water’s refractive index being constant, the returning light is most intense at 42°, making the rainbow always form in a circle with an angular radius (angle of light compared to your eyes where a circle is seen as a specific diameter) of 42° surrounding the point opposite the sun. If you are standing exactly at this spot with the sun behind you, you will see a beautiful rainbow. Otherwise, the rainbow disappears.

Angular radius can sound like a complicated concept, but in this case, it results in something quite interesting. To capture a full rainbow with a camera, your camera lens must have a field of view (cone of light that the camera will photograph) of 84°. Most smartphone cameras have smaller fields of view than this (iPhone X has a 65° horizontal field of view for instance), meaning that it would be impossible to capture all of the rainbow in one photo.

Another impossible thing when it comes to rainbows is finding the mythical pot of gold at the end of a rainbow. Because rainbows are the result of optics, they are different to every observer and how they are positioned to the sun and water droplets. This means that no two people observe a rainbow in the same way and a rainbow is not static.

You can also never approach the rainbow as it will disappear given the angular radius mentioned above.

Furthermore, there is no end to a rainbow because it is actually a full circle that extends through the horizon. We cannot see it as there is ground between us and the rainbow, but you can sometimes see a ring rainbow from a plane.

However, because the rainbow is technically just light from the sun bouncing off water and into your eyes, we can imagine it not as a circle, but a double-ended cone that ends in your eyes. By this logic, your retinas that sense the rainbow (and by extension, you) are the pot of gold at the end of the rainbow.

The End of the Rainbow
(Image source:

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Blue Rose

One of the holy grails of horticulture is the blue rose. A variety of rose colours have been cultivated using various techniques such as hybridisation, ranging from the classic deep red to bright yellow, to even a mix of colours. However, there has never been a successful case of breeding blue roses.

This is why blue roses have become synonymous with the longing for attaining the impossible. It was a symbol of the Romanticism movement, representing the desire and striving for the infinite and unreachable; a dream that cannot be realised. The flower meaning for the blue rose is secret, unattainable love.

The reason why blue roses are impossible to produce naturally is that they do not have the gene for the protein that makes a blue hue. The biochemistry of flower colours is complex, but essentially, the blue colour seen in flowers such as pansies and butterfly peas is produced by the chemical delphinidin. Roses lack this pigment and only contain pigments that produce red and orange colours.

Because blue roses have always been deemed impossible, florists have had to resort to using blue dye on white roses to produce artificial blue roses. But this all changed with the introduction of genetic modification technology.

In 2005, scientists reported that they created the first true “blue rose”, by genetically engineering a white rose to produce delphinidin and using RNA interference to shut down all other colour production. However, the results were disappointing and the so-called “blue rose” turned out to be more of a mauve or lavender colour, due to the blue having a red tinge.

This is because rose petals are more acidic than true blue flowers such as pansies. Delphinidin is degraded by acid, meaning that you cannot produce the deep blue found in pansies in roses without finding a way to reduce the acidity. This chemical phenomenon can also be seen in hydrangeas, where the red and pink petals turn blue and violet when you acidify the soil that it is growing in.

Although we now harness powerful tools to modify nature in ways deemed impossible in the past, nature still proves to be tricky and elusive.

The Suntory Applause rose
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Periodical Cicada

In certain parts of eastern North America, it has been noted for centuries that some summers seem to bring a massive swarm of cicadas. Observant naturalists such as Pehr Kalm noted in the mid-1700’s that this mass emergence of adult cicadas happened every 17 years. Since then, a similar pattern has been observed with many different broods of cicadas, with precisely 17 or 13 years between emergences of mature cicadas.

What could possibly explain such a specific, long gap between these spikes?

This phenomenon has been well-researched and the species of cicadas (Magicicada) are known as periodical cicadas. They can be distinguished by their striking black bodies and red eyes. Like most cicadas, periodical cicadas start their lives as nymphs living underground, feeding on tree roots. They take 13 or 17 years (depending on the genus) until they emerge all at once in the summer as mature adults – far longer than the 1-9 years seen in other cicadas. After such a long period of growth, they emerge for a few glorious weeks in the sun to mate, before laying eggs and disappearing.

The astute reader would notice that both 13 and 17 are prime numbers (a number divisible only by itself or 1). Is this a sheer coincidence or a beautiful example of mathematics in nature?

This curious, specifically long period of maturation has been a great point of interest for scientists. The phenomenon of mass, synchronised maturation is a well-documented survival strategy known as predator satiation. Essentially, if the entire population emerges at the same time, predators feast on the large numbers, get full and stop hunting as much. The surviving proportion (still a great number), carry on to reproduce and the species survives.

One theory holds that the prime numbers are so that predators cannot synchronise their population booms with the cicadas. If the cicadas all emerged every 4 years, a predator who matures every 4 or 2 years could exploit this by having a reliable source of food in a cyclical pattern. 13 and 17 are large enough prime numbers that it would be very difficult for a predator to synchronise its maturation cycles with.

Another possible theory is that it is a remnant of a survival strategy from the Ice Age. Mathematical models have shown that staying as a nymph for a longer period increased the chances of adults emerging during a warm summer, rather than when it is too cold for reproduction. This resulted in broods of varying, lengthy cycles, but this created another problem: hybridisation. When broods of different cycle lengths intermingled, hybridisation could occur and disrupt the precise timing of maturation cycles, decreasing the brood’s survival rate. Prime number cycles such as 13 or 17 years have a much less chance of hybridisation, increasing the survival rate.

As Galileo Galilei said, mathematics is the language in which the universe is written. It is fascinating to see examples of how maths can influence natural phenomena, even the life cycles of insects.

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Grandi’s Series

In 1703, Italian mathematician and monk Guido Grandi posed a deceptively simple-sounding question:

What is the sum of the following infinite series?
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1…

With simple arithmetic, we can easily divide the series using parentheses (brackets):

(1 – 1) + (1 – 1) + (1 – 1) + (1 – 1)… = 0 + 0 + 0 + 0 +… = 0

But what if we changed the way we used the parentheses?

1 + (-1 + 1) + (-1 + 1) + (-1 + 1)… = 1 + 0 + 0 + 0 +… = 1

Because of the way negative numbers work, this solution is equally feasible. Ergo, both 0 and 1 are acceptable answers.

How can one series possibly have two different answers? Grandi used the fact that both 0 and 1 are possible from his series as proof that God exists, as something (1) can be made from nothing (0).

Grandi’s series becomes even stranger when a more advanced technique is applied.

Let us say that Grandi’s series is denoted by S (S = 1 – 1 + 1 – 1…).
We can then break down the series as 1 – (1 + 1 -1 + 1…), because the plus and minus signs can be inverted together.
Ergo, S = 1 – S → 2S = 1 → S = ½

Now we have three answers to Grandi’s question: 0, 1 and ½.
For over 150 years, mathematicians fiercely debated the answer to Grandi’s question. By the 19th century, mathematics had evolved and mathematicians had figured out better ways to solve infinite series.

The classic example is the solution to the series: 1 + ½ + ¼ + ⅛…
To solve this, you can add the partial sums, where you add each number to the sum of the previous numbers to see what number you are approaching (the limit).

1 → 1.5 → 1.75 → 1.875 → 1.9375… until we infinitely approach 2 (or 1.9999999…)

If we apply this method to Grandi’s series, we do not approach a single number because we keep swinging between 0 and 1. (1 → 0 → 1 → 0 → 1…)

So we can apply another method, where we average the partial sums as we go instead of adding.

e.g. 1 → ½(1 + 1.5) = 1.25 → ⅓(1 + 1.5 + 1.75) = 1.416 → ¼(1 + 1.5 + 1.75 + 1.875) = 1.531… until we approach 2.

Using this method on Grandi’s series:

1 → ½(1 + 0) = ½ → ⅓(1 + 0 + 1) = ⅔ → ¼(1 + 0 + 1 + 0) = ½…

Eventually, the series appears to converge on ½, showing that the answer to Grandi’s series seems to be ½.

The problem with this method is that Grandi’s series does not actually have a limit, but we are applying a solution as if it has a limit. This is similar to using a divide by 0 trick to prove that 1 + 1 = 3. In mathematics, when rules are bent, we end up with weird, paradoxical results.

To show this empirically, consider the thought experiment of Thomson’s Lamp:

Imagine a lamp that is turned on after 1 minute, turned off after ½ minute, turned on again after ¼ minute ad infinitum.
This incorporates both infinite series discussed above.
Ergo, we know that the sum of time is 2 minutes.
So, at the end of 2 minutes, is the lamp on or off?
If Grandi’s series solves to 0, the light is off; if it is 1, the light is on.
Then what does it mean if Grandi’s series solves to ½?
Is the light on or off?