Posted in Science & Nature

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.

Posted in Science & Nature

A Beautiful City

What makes a city or town aesthetically pleasing? Places such as Prague, Florence and Santorini are famous for their picturesque cityscape. Instead of specific famous buildings or tourist spots, postcards from these areas could just show any part of the city and they would still be beautiful. What sets these places apart? How is it that despite all our technological development, modern cities can’t compare to the beauty of cities that are hundreds or thousands of years old?

Korean architect Yoo Hyun-Joon proposes a theory regarding two factors: material and shape. Consider the following matrix using the two:

Out of these four, the combination that we find the most beautiful is when a city has simple materials but complex shape. For example, Santorini is made only of stone buildings painted white and blue. But because it is built on a volcano, the ground is uneven and the building shapes differ to accommodate for this. Florence is almost entirely made of bricks. Traditional Korean houses were made only of wood. This is because in the old days, due to labour costs and poor logistics, cities were usually built with materials abundant in the surrounding area. Instead of varying materials, architects would challenge the limit of materials with varied shapes.

Nowadays, thanks to trade and globalisation, it is much easier to obtain materials from all over the world such as glass, concrete and steel. Furthermore, we can use industrial vehicles to change the terrain to flatten the ground and we use tall rectangular buildings to maximise space. Thus, we end up with the ugly, chaotic combination of many materials and simple shape.

The solution to making a beautiful city is simple then – create a building restriction that unifies the building material to one. A good example is Newbury Street in Boston, USA. This shopping district is famous for its classy red brick appearance, thanks to a building restriction that ensures every new shop built on the street must have the side of the building facing the street built using red bricks.

Of course, just unifying the building material to any one thing does not solve the issue. For example, cities made of only concrete rarely are as appealing. What is important is to use local materials that best represent the context of the city and the land it was built on.

Posted in History & Literature

The Devil’s Interval

(See below NB for a simple guide to musical notes and tones)

In music, depending on what notes you use in a single chord, you can produce beautiful harmonies as the tones complement each other. The opposite of this is called dissonance and it results in a harsh, unpleasant sound. A famous example of this is a tritone – a chord made from two notes exactly three whole tones apart. In a standard C major diatonic scale (which doesn’t involve any flats or sharps), there is only one tritone per octave: F and B. But on the chromatic scale (all keys), any number of tritones are possible.


Historically, the tritone has been the black sheep of music theory due to its dissonance crashing any harmony of a song and being difficult to sing. It sticks out like a sore thumb among the sea of beautiful harmonies that other tones make. The tritone was hated so much so that it was named diabolus in musica (“the devil in music”) or the devil’s interval since the Middle Ages, even being banned in the production of music prior to the Renaissance. To this day, the tritone is suggested as an “evil”, “scary” sound.

Over time, composers worked around the tritone until they realised that thanks to the connotations, the tritone was a useful way to express “evil” in a musical way. The cultural association was exploited freely in works such as Franz Liszt’s Dante Sonata, where the tritone is used to depict Hell. The association is found in modern music as well to produce an unsettling feeling, such as the opening notes of Jimi Hendrix’s Purple Haze. The tritone is a common feature of heavy metal bands such as Black Sabbath.

Even though these songs use the devil’s interval, they are not at all inferior to “normal” major scale music. They are still beautiful in their own, interesting way. Perhaps the notion of good and evil have no place in judging whether something is beautiful or not.

NB: Musical tones are noted using the alphabet: C, D, E, F, G, A and B, with a flat(b) to denote a semitone lower, or a sharp(#) to denote a semitone higher. This is easy to visualise on a piano keyboard, where a single tone interval involves a white key, a black key in between and another white key. The interval between a white key and a black key is a semitone.


(Image: Portion of Hieronymus Bosch’s The Garden of Earthly Delights depicting musicians’ hell)

Posted in Science & Nature

Mathematical Beauty

What is the most “beautiful” mathematical equation? For millenia, many mathematical formulas and concepts have been described as beautiful (and some defining beauty, as the golden ratio does). In the mathematical world, the adjective “beautiful” is used in the sense that certain mathematical concepts, despite the fact they are rational and objective, are so pure, simple and elegant that they can only be described as art.

One such formula is Euler’s identity:


Renowned physicist Richard Feynman described it as “the most remarkable formula in mathematics”. What makes this array of symbols and numbers so beautiful? Firstly, it contains the three basic arithmetic operations exactly once each: addition, multiplication and exponentiation. It also connects five fundamental mathematical constants with nothing other than themselves and the arithmetic operations.

0 is the additive identity, as adding it to another number results in the original number. 1 is the multiplicative identity for the same reason as 0. Pi(π) is one of the most important mathematical constants in the history of mathematics that is ubiquitous in Euclidean geometry and trigonometry. Euler’s number(e) is the base of natural logarithms and is used widely in mathematical and scientific analysis. i(√-1) is the imaginary unit of complex numbers, a field of imaginary numbers that are not “real”, allowing for the calculation of all roots of polynomials. Euler’s identity neatly sums up the relation between these five numbers that are so crucial in the field of mathematics. It is also interesting to note that these five numbers were discovered at different points in history spanning over 3000 years.

Some people describe mathematics as a distinct language in itself. Not only that, but mathematics is considered the universal language as it is both universal and ubiquitous. If that is the case, than Euler’s identity can be considered an extremely pithy literary masterpiece.

Posted in Psychology & Medicine

Stendhal Syndrome

There have been recorded cases of people gazing upon a beautiful panorama of Florence or an exquisite painting and suddenly collapsing. The condition is known as Stendhal syndrome, alternatively called Florence syndrome or hyperkulturaemia (excess culture in blood). It has been described as causing tachycardia (rapid heartbeat), dizziness, confusion and fainting after being exposed to a particularly beautiful piece of art or scenery. It is named after French author Stendhal (penname of Henri-Marie Beyle), who upon visiting Florence in 1817 experienced the very condition.

Stendhal syndrome is most likely related to a very common phenomenon known as vasovagal syncope, where extreme emotions overwhelm the brain, induce a massive parasympathetic nervous response, causing the person to faint. There are two major nervous systems: the sympathetic and parasympathetic. The sympathetic nervous system is responsible for the fight-or-flight response and essentially prepares the body for physical activity. The parasympathetic nervous system does the complete opposite and is activated when you are resting or digesting food. Thus, a burst of parasympathetic nervous activity causes a sudden fall in heart rate and blood pressure, causing the brain to lose the oxygen supply needed to maintain consciousness. When the person faints, they collapse and blood flow is restored to the brain. Vasovagal syncope can be caused by anything from standing up very quickly, extreme emotions (e.g. stress, seeing blood or needles) and fatigue. It is the most common cause of collapse and is (usually) completely harmless.

When a person looks at a breathtaking view or a stunning work of art, their brain is overwhelmed by intense emotions of excitement and joy. In the case of Stendhal syndrome, this effect is so great the person is literally blown away by the sight.

The people of Florence have noted that this phenomenon is rather common in tourists visiting the beautiful city.

Posted in Psychology & Medicine

Muslin Disease

The French Revolution that occurred in the late 18th century had a significant impact on not only politics, but French society as a whole. Even after the revolution, there was much hatred against the nobility and especially the luxurious and extravagant lifestyle they lead. Men and women wearing too much clothes and jewellery were punished heavily. There was even a law stating that the weight of your clothes and accessories combined must not exceed 3.5kg. In the early 19th century, the fashion trend changed from the fancy dresses of the past with many decorations to a much more simple, clean and frugal type of dress. A point of interest is that women wearing petticoats (an undergarment worn to puff out skirts) – a key point of the Rococo style – were executed on the guillotine, causing women to quickly throw away their petticoats and try to look as slim as possible. To look thin, women did not even wear underwear. There was a fabric that suited this new fashion trend very well and that was the extremely thin cloth, muslin.

But to follow the fashion of then, it is not enough to simply wear a muslin dress. In early 19th century France, the trend was to douse your dress in water. Why did the women drench themselves in water? The reason being, muslin is a very thin and light cloth that becomes half transparent when wet, while clinging to your body. Women dampened their muslin dress to prove that they were wearing nothing underneath. Also, the ideal, beautiful woman of the time was an intellectual woman who looked fatigued from reading books all night long. Drenching yourself in water adds to this gaunt image, accentuating your fatigue and by extension, your beauty. The woman probably also intended to make the clothing cling to their body to show off their figure (much like the modern day wet t-shirt contests).

The problem was that muslin is an extremely thin material that is unsuitable in the winter or in Northern regions. Considering that women were wearing such a thin dress and even pouring water on themselves, one can imagine how cold they must have been. In fact, France suffered a heavy epidemic of pneumonia in the early 19th century with as much as 60,000 patients turning up with pneumonia every day. A high proportion of these patients were women who liked to wear wet muslin dresses. Thus, the pneumonia was nicknamed muslin disease.

Posted in History & Literature

Thirty-Six Stratagems: Chapter 6 – Desperate Stratagems

(For all 36 stratagems, click here:

Desperate Stratagems are last resort tactics that can be used when you are placed in a disadvantageous state or risk losing the war.

Stratagem 31: No weapon could beat seduction
This is the strategy where you use a beautiful woman to tempt the enemy to extract information or undermine their will to fight. It is a deadly strategy that never fails.

Stratagem 32: Empty fort strategy
Purposefully empty your fort to show the enemy that you are defenceless, causing them to think that it is a trap and retreat. It was effectively used by Zhuge Liang in The Three Kingdoms.

Stratagem 33: Countermine the enemy’s spy
Use the enemy’s spy to spread false information or use them as a double agent that can extract the enemy’s intelligence. The “double spies” and “dead spies” from Sun Tzu’s The Art of War fit under this stratagem.

Stratagem 34: Sacrifice yourself to comfort the enemy
Inflict injury on yourself to win the enemy’s trust. For victory, you may have to sacrifice even the dearest things such as your wife or most loyal servant.

Stratagem 35: Chain stratagems
This is an elaborate strategy where you first restrict the enemy’s movement then make use of a series of tactics one by one to decimate the enemy. The most famous example is from The Three Kingdoms in the Battle of Red Cliffs. All of the ships of Wei were linked by chains and could not move freely, making them vulnerable to a fire and causing Cao Cao to lose the battle in an instant.

Stratagem 36: If all else fails, retreat
It is often misquoted as “retreat is the best option”. The Thirty-Six Stratagems explain that retreating is not the most ideal option, but if there is no chance of winning and the cost will be too much, it is wiser to retreat then fight again after reinforcing your forces. Knowing to retreat when you are disadvantaged and the cost is too high is true wisdom.

By utilising these thirty-six stratagems wisely, you will be able to win no matter what situation strikes. Remember: life is a fierce battlefield and without effective strategies and tactics you cannot seize victory.

Posted in History & Literature

Hot Waitress Economic Index

What happens when an economy is going into a depression? Unemployment goes up, inflation goes up, housing markets tank… There are many (miserable) indicators of a waning economy, but none are as strange as the Hot Waitress Economic Index. Simply put, this index suggests that the worse the economy is doing, the more attractive the waitresses are on average.

Despite sounding incredibly shallow and sexist, there is sufficient data to support this theory. It can be explained by the fact that when the economy is doing fine, attractive women are more likely to be in higher paying jobs as they are favoured by employers (unfair, but statistically true). When the economy is doing poorly, unemployment rates rise and these attractive women are pushed down to low-paying jobs such as waitressing as actual skill becomes a higher priority when hiring. This causes an apparent increase in the overall attractiveness of waitresses in the country. Some studies suggest that the Hot Waitress Economic Index is even more accurate in predicting the state of the economy than unemployment as attractive people tend to be the first to earn jobs, acting as an immediate indicator for the economy. For example, when the economy dips out of the depression and starts to rise again, attractive people are the first to be re-hired into higher paying jobs, causing the Hot Waitress Economic Index to change before the unemployment rate does.

Interestingly, there is no data on how the economy affects the average attractiveness of waiters.

Posted in History & Literature

Eros And Psyche

Once upon a time, there lived a beautiful girl named Psyche. Psyche was so beautiful that she even caught Aphrodite’s eyes. Despite being the goddess of love, Aphrodite was known to be very jealous and felt threatened by Psyche’s beauty. She commanded her son Eros, the god of love, to put a spell on Psyche. Eros uses the Bow of Love and anyone shot by his golden arrows falls immediately and helplessly in love with the first thing they see.

Aphrodite came up with a devious plan to have Eros shoot Psyche and have frogs around her, making her fall in love with frogs and fall into a despairing relationship. But her jealous plan was overthrown by an unexpected event. When Eros first saw Psyche’s face, he became entranced and accidentally pricked himself with his own arrow. Thus, Eros became madly in love with Psyche. Aphrodite, enraged by this, cursed Psyche to never find a mate for the rest of her life. Eros became depressed from not being able to see Psyche and gave up shooting golden arrows. After he gave up his job, no animals or human fell in love and no new life was born. Aphrodite could not bear to see such a scene and begged Eros to start shooting arrows again, offering him one thing that he wanted. Eros said that he desired Psyche without hesitation and Aphrodite reluctantly allowed them to meet.


While this happened, Aphrodite’s curse made no man come to propose to Psyche. Her parents became worried and asked the advice of the oracle at Apollo’s temple. The oracle stated that as she is destined to marry a monster, she must be placed atop a mountain in bridal attire. Accepting her fate, Psyche stood on the mountain but eventually jumped off a cliff in despair. But Zephyrus the West Wind caught her and brought her safely to Eros’ place, just as Eros planned. Psyche enjoyed a comfortable life in the beautiful castle with many maids at her service. However, her husband only came in the deep dark of the night. Whether it was because he feared Aphrodite’s wrath or the difference between a god and a mortal, he asked her to never try find out who he was and that if she truly loved him, she should trust him. But Psyche eventually fell victim to her jealous sisters’ scheme and her curiosity, leading to her accidentally dropping candle wax on Eros’ face as she took a peek at his face. Eros was awakened and became enraged. He chased her away and forbade her from coming back. Psyche fell in despair and threw herself into the river, but the river carried her to the riverside where the shepherd god Pan rescued and consoled her.


At first, Psyche tried to find Eros while avoiding Aphrodite, but eventually she decides to plea directly to her. Despite her bravery, Aphrodite threw challenging tasks one after another at her like a mean mother-in-law, ultimately commanding her to retrieve some beauty from Persephone, the queen of the underworld (since travelling to the underworld signifies death, Aphrodite must have truly hated Psyche). 

However, Psyche was determined to see Eros even at the cost of her life. Admiring her commitment, a tall tower before the underworld gave her a hint. It told her to place two coins on her tongue and bread in each hand when going to the underworld. The coins would be to pay Charon the ferryman while the bread would distract Cerberus the three-headed dog guardian. She succeeded in seeing Persephone, who gave her a box of beauty and told her to never open it. But wanting to look beautiful in front of her lover (Eros), Psyche opened the box. The box did not contain beauty, but instead contained a death-like sleep from the underworld, putting Psyche in a deep sleep. Although he chased her away, Eros came back to see Psyche and found her in this sleeping state. He took the corpse-like Psyche and kissed her softly on her lips, awakening her from the deep sleep.


Eros eventually sought help from the king of gods, Zeus, to persuade Aphrodite. After Zeus’ persuasion and seeing the love the two have for each other, Aphrodite accepted the relationship and Zeus gave Psyche the immortal drink ambrosia to make her into a goddess. The now immortal Psyche and Eros were married and had a daughter named Hedone (like in Hedonism), the goddess of sensual pleasures.

Eros symbolises physical and sensual love while Psyche is a Greek symbol for butterflies, the soul and emotional love. Ergo, Eros and Psyche represent the union of physical and emotional love into perfect love. Love and the soul are inseparable things. The most basic instinct of any organism is to reproduce and human beings have evolved that into the sacred concept that is love. To speak bluntly and without philosophy, from a purely biological perspective there is no greater purpose to life than to find a suitable mate and leave descendants. There is nothing more fundamental than pure love.

Posted in Science & Nature

Golden Ratio

The golden ratio is a magical number that divides a line into the most beautiful ratio. It bestows a mystical power in an object and allows for the creation of excellent architecture and art.
This magical ratio is (1 + √5)/2, or 1.618033988. If there is a line divided by the golden ratio called a + b, then b:a and a:(a + b) are both the same ratio.

We can find the golden ratio in countless values seen in animals and plants. A snail shell’s golden spiral allows for the snail to grow without changing shape, while the distribution of branches on a tree also follows the ratio. The golden ratio controls everything from the spiral pattern of galaxies to the pattern of our brain waves. The golden ratio is the law of the universe.

Using this magical ratio, we can find the most beautiful composition of a human being. The Venus of Milo, considered as one of the most beautiful figures in history, has a ratio of 1:1.618 between her upper and lower body (divided at the belly button) – the golden ratio. The same can be said for the ratio between the head and neck compared to the rest of the upper body, and the length from the belly button to the knee compared to the length below the knee. The exact same composition was used to construct the statue of Doryphoros, one of the most famous examples of ancient Greek sculptures. The diagram that illustrates these ratios is the Vitruvian Man by Leonardo da Vinci (Vitruvius was a Roman architect who utilised the ancient Greek knowledge of applying the proportions of a human being, i.e. the golden ratio, in constructing temples). 

The Great Pyramids of Giza, Solomon’s Temple and the Parthenon are all partially constructed according to the golden ratio. It is said that buildings constructed outside of the golden ratio will collapse over time. The same is seen in Eastern constructions, such as buildings and inventions from the Goryeo Dynasty of Korea. 

Interestingly, the golden ratio applies to intangible objects as well. For example, Chopin’s Nocturne pieces tend to climax at the point of the golden ratio (roughly two-thirds in). The ratio is still used in modern day design, with the standard credit card size being the best example.

The golden ratio is an eternal beauty that does not go out of fashion with time.