Posted in Science & Nature

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?

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Hurricane

We often hear on the news of cataclysmic storms with oddly common names such as Hurricane Sandy, Katrina and Harvey. It seems weird that we give such devastating forces of nature a basic name, let alone naming them human names at all.

A hurricane is the name given to tropical storms that occur in the Atlantic Ocean. For reference, a hurricane is essentially the same as a cyclone or typhoon. The history of naming hurricanes dates back over a hundred years, with residents of the Caribbean Islands naming hurricanes after the saint of the day from the Catholic calendar. Initially, American meteorologists named hurricanes by the geographic location that the storm originated in.

However, during World War II, military meteorologists in the Pacific started using women’s names for hurricanes. This made communication much easier as hurricanes could be identified by name and much easier to say. There are some apocryphal stories about the origin of women’s names for hurricanes, such as wishing that the hurricane will be calmer and of better temperament, or that they were named after the meteorologists’ wives and girlfriends. This practice soon spread to the rest of USA and became the default method of naming hurricanes. From 1979, it was decided that the gender of the names would be alternated.

In the present, there is a rolling six-year roster of 21 names each year in alphabetical order that is used to name hurricanes (see below for list). For example, the first hurricane of 2019 was called Andrea, the second Barry, the third Chantal and so on. In 2020, the first hurricane will be named Arthur, then Bertha, et cetera. The same names would be used in 2025 and 2026.

The one exception to this rule is that when a hurricane is particularly devastating and results in many deaths, the name is “retired” in honour of those who have lost their lives or livelihoods to the hurricane. For example, there will be no more hurricanes named Katrina or Harvey in the future.

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Grasshopper Mouse

Grasshopper mice are a species of New World mice found in deserts throughout North America. They are small-to-medium sized, growing up to 13cm in size and weighing around 40-50g.

Despite their cute appearances, they are carnivorous, ferocious hunters. They feast on various insects, but are also known to hunt other mice.

Grasshopper mice have interesting adaptations that make them seem more like a miniature wolf or mongoose rather than a mouse. For example, they stalk their prey like a cat before pouncing. They hunt highly venomous insects such as scorpions and centipedes because they have evolved to convert the deadly toxins of a scorpion sting into harmless chemicals.

An interesting feature of the grasshopper mouse is that they often howl like a wolf to ward off competitors and to communicate with each other. It has been nicknamed the werewolf mouse because they are known to howl in the night with their heads thrown back, communicating over the vast desert with a high-pitched howl.

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Compound Interest

When is the best time to invest? Is it when you have sufficient income and savings that you feel that you have a surplus to invest with?

The correct answer is much simpler: yesterday, with the second best time being today. Because of the magic of compound interest, investing early is the best strategy possible.

Thanks to a simple mathematic rule, compound interest rewards early, small investments more than late, large investments.

The way compound interest works is that after a given time interval (e.g. year), interest (as a percentage of the original investment) is paid out. The next year, interest is paid out again but as a percentage of the new amount. As an example:
1000 x 1.08 = 1080 (end of year 1)
1080 x 1.08 = 1166.40 (end of year 2)
1166.40 x 1.08 = 1259.71 (end of year 3)
…until end of year 10

If we use mathematical shortcuts and convert this into a formula, we can express it as:

(A = future value, P = present value, r = interest rate as decimal, n = number of periods/years)

For example, if we invest $1000 (PV) at an interest rate of 5% (r=0.05) for 10 years, then:

$1000 x 1.08^10 = $2158.92,

meaning we have earned $1158.92 over 10 years. Taking it further, in 30 years our investment would have grown to $10062.66 – ten times our original investment.

Because the formula uses exponents (or powers) for the time, your investments grow exponentially with time. This means that the earlier you invest, the greater your returns become disproportionately. This is why within 10 years, we have more than doubled our initial investment despite a reasonable interest rate and not doing anything else.

A rule of thumb for calculating how long it will take your investment to double is to divide 72 by the interest rate in % (e.g. 7). This is the number of years it will take for your investment to double (e.g. 72/7 = 10.3 years).

On top of this, if we invest small amounts every year, then we can benefit even more from the exponential growth of our investment. For example, just by adding in $100 every year, we end up with an additional $564.55 of investment earnings at the end of 10 years – a 50% increase in returns.

Unfortunately, mathematics works both ways and compound interest also applies to certain loans, such as credit cards. This means that your debt will grow exponentially unless you aggressively pay it back, making it seem impossible to pay off your credit card debt sometimes.

(This graph shows that investing early and consistently is the best strategy to maximise your eventual earnings. Compare the grey and purple line and you will see that despite investing a third of what Lyla invests total, Quincy ends up with a higher portfolio by retirement.)
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Posted in Science & Nature

The Closest Planet

Which planet is closest to Earth? If we look at a typical model of the Solar System with each planet neatly lined up, we can see that Venus approaches Earth closer than any other planet. However, this is only one interpretation of the question.

Technically, Venus is the planet that comes closest to Earth. However, as they do not orbit in synchrony, this approximation happens about once a year. At other times, Venus will orbit away from Earth and can go on the other side of the Sun, making the distance between Earth and Venus vast. In those times, Mars may seem like the next obvious choice to be closest to Earth.

But then again, Mars has the same issue where it and Earth are often on opposite sides of the Sun. Because of the nature of circular orbits, the distances between the planets swing and fluctuate, meaning that the real question should be:

Which planet is closest to the Earth most of the time on average?

The answer to this question happens to be Mercury. If we look at a “top-down” model of the Solar System, we can see that Mercury – being closest to the Sun – orbits rapidly around the Sun and often lies between Earth and the two other planets, Venus and Mars. If we plot the distance between each of these three planets and Earth, we can see that on average, Mercury is closer to Earth because the distance fluctuates less.

Interestingly, if we take this question further, we find that Mercury is also Mars and Venus’ closest neighbour on average. This is a property of the Solar System being formed of concentric circles, meaning that Mercury’s smallest orbit makes it average a closer distance to all of these planets.

Fascinatingly, if we go even further than that, we find that the same pattern holds for every other planet in the Solar System, despite the vast distance between Mars and Jupiter due to the Asteroid Belt. Even Pluto (not formally a planet anymore) with its massive elliptical orbit has Mercury as its closest neighbour on average compared to the other planets, due to the unique property of concentric circles.

No matter the distance, if you are orbiting the Sun, Mercury is the closest planet to you.

This video from CGP Grey explains it in a concise and informative way, complete with clean diagrams and animations!
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Shooting Star

When an object from outer space enters the Earth’s atmosphere, it starts to burn up and creates a brilliant streak in the sky, which we call a meteor or shooting star. Contrary to popular belief, this is not due to friction with the air in the atmosphere.

An object entering the atmosphere is typically travelling at extraordinary speeds. Most meteors are travelling around 20km/s (or 72000km/h) when they hit the atmosphere. At these speeds, air molecules do not have a chance to move out of the way. The meteor will instead collide into the air molecules, pushing them closer and closer to each other, compressing the air in front of it.

As we know from physics class, compression increases temperature in gases as per the ideal gas law (PV=nRT). The impressive entry speed of these meteors result in so much air compression that their surface can heat up to 1650 degrees Celsius.

The heat boils and breaks apart the contents of the meteor, turning it into superheated plasma that gives off a glow. This is the streak of light that we see in the night sky when we wish upon a shooting star.

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Oobleck

If you mix 1 part water to 1.5-2 part corn starch, you create a strange mixture called “oobleck“, named after a Dr. Seuss story. It is so simple to make, yet it exhibits some very strange properties that makes it a popular science experiment.

Oobleck is what is known as a non-Newtonian fluid, where the viscosity (or “thickness”) changes with how much stress it is under. If you press your finger gently into it, it will feel like water, but if you strike it with a hammer, it will behave as a solid. It will stiffen when you stir it, but run when you swirl it.

Related image

You can even run over a tub of oobleck as long as you change steps quickly enough to apply enough pressure to keep the fluid under your feet solid. This is because oobleck becomes very viscous under high stress, making it behave more solidly (shear thickening).

We can learn from oobleck not only some interesting physics principles, but also how to interact with people.

Much like a non-Newtonian fluid, people will tend to react stiffly and with more resistance if you apply stress or force. But if you apply gentle pressure and be assertive, you will find people generally react more softly and fluidly.

This simple change in your approach will lead to much better conflict resolution and constructive outcomes when dealing with other people.

Image result for oobleck run gif

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Pringles

Pringles are a beloved snack well-known for its addictiveness (“Once you pop, you can’t stop“). There are a few other interesting factors that set Pringles aside from other potato chips.

Firstly, Pringles have been called many things, because it is not strictly a potato chip. When it first debuted, other snack companies complained that it was not technically a potato chip as they were made from dried potatoes, so they were labelled “potato crisps“. Ironically, the company successfully argued in 2008 that Pringles were not “potato crisps”, using the logic that they were not of natural shapes and only contained 42% potato as they are made from potato-based dough. This was so that they could avoid the British tax on potato crisps.

Secondly, Pringles chips have a characteristic saddle-shape, known in mathematics as a hyperbolic paraboloid. This creates a uniform shape, meaning they can be stacked neatly in a tubular container for efficient and reliable packaging, as opposed to most potato chips that are packaged in bags. Furthermore, the shape is structurally sound, preventing the chips from breaking under the weight of the stack.

Finally, the inventor of the cylindrical container was a chemist named Fredric Baur, who started the process of making Pringles. His dying wish was to have his ashes buried in a Pringles can and this wish was respected by his children.

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