Posted in Science & Nature

## 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.

Posted in Science & Nature

## A Simple Task

A plague struck the ancient Greek island of Delos. As the disease ravaged the island, the people went to the oracle at Apollo’s temple for help. This is what the oracle said:

Double the volume of the cube-shaped altar in Apollo’s temple

People considered this a simple task and made a new altar where each side was double the original length. However, instead of disappearing, the plague worsened and people were confused.

Reason being, given that the length of one side of a cube is a, the volume is a³; if one side is 2a, the volume becomes 8a³, or eight times the original volume. Therefore, to double the volume of a cube, the number ³√2 is required. The problem is, whether ³√2 can be found using only compass and straightedge construction (where only the two tools are used to solve a geometric problem).

This problem, also known as the Doubling the cube problem, is one of three geometric problems known to be unsolvable by compass and straightedge construction. In other words, without the help of other mathematical methods, the answer cannot be found.
However, the solution to the above story is very simple.

Find a new god.

Posted in Psychology & Medicine

## Doppelganger

A doppelgänger is popularly known as a “double” of oneself, with the exact same appearance but usually with a completely different personality.
It comes from German folklore (the word is German for “double walker”), where it is believed that seeing a doppelgänger is an omen of bad luck, possibly death. It is also said that meeting one’s own doppelgänger will result in one of the two dying.
However, in modern times this concept of doppelgängers have been largely replaced by the first meaning, that of an alter ego who is only identical physically, and completely different persona-wise. This can sometimes overlap with the concept of an “evil twin”, as a doppelgänger is sometimes described to have an inverse personality, values and motives.

Although in most cases a doppelgänger is just a look-alike, it could be the result of the brain projecting an identical image of oneself in another space.
It has been discovered that electrical stimulation of an area of the brain called the left temporoparietal junction causes the sensation that someone is next to you. This “someone” has the same shape and body posture as the subject, and is sometimes described as having a completely different personality. It is reported that this sensation is extremely unpleasant.
The cause of this phenomenon is that the area of the brain is associated with self-image, such as body posture, location and proprioception (sense of what oneself appears as in space). Therefore, a disruption leads to the self-image being projected elsewhere, which the brain then interprets as a doppelgänger.

There are similar mental disorders such as the Capgras delusion (belief that someone close has been replaced by a doppelgänger), Fregoli delusion (belief that many people are the same person in disguise) and subjective doubles (belief that one’s own doppelgänger is acting on its own to interfere with one’s life).

Doppelgängers are common plot devices in many works, especially fantasy and science fiction where the aforementioned “evil twin” concept is used. However, the doppelgänger may also be described as benevolent or simply a clone.
They are notably used in How I Met Your Mother, where each character has a doppelgänger completely different to their persona (e.g. Stripper Lily, Fertility Expert Barney).