Posted in Science & Nature

## 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?

Posted in History & Literature

## Designing Under Constraint

You would think that the more freedom the designer has, the more their creativity can flourish and they can produce more original, greater ideas. But it is a well-known fact in the design world that the the best designs are produced when designing under constraint.

Consider the beauty of the canal houses of Amsterdam. In the 17th century, plots of land by the canal were allocated in narrow (but deep) portions to maximise the number of houses. Architects worked around this restriction, resulting in the narrow, tall houses of various shapes and colours that we see today. Another architectural example is Florence and Santorini, where building materials were limited to red bricks or stone painted in white and blue respectively, meaning the buildings shared a consistent colour scheme, while varying in shape – the ideal combination for building a beautiful city.

We see the same in other fields. Photography is limited in the realm of time, as you can only take a snapshot. But by using long-exposure or composite images, time can be represented in unique, beautiful ways. The artistic restriction of painting led to Pablo Picasso pioneering cubism, which attempts to represent the many faces of a three-dimensional object on a two-dimensional medium. Great literature can be produced from limitation also, such as haikus or flash fiction, such as the infamous six-word story by Ernest Hemingway: “For sale: baby shoes, never worn”.

There are many reasons why designing under constraint results in greater works.

Firstly, choice and freedom can be paralysing. When we have absolutely no restricitons, rules or guidance, we have difficulty processing the sheer number of possibilities, because there are too many things to consider. We find it much easier to make a decision and proceed when there are a limited number of choices.

Secondly, constraint often comes in the form of consistency. One of the basic rules of graphic design is to limit your colour palette and font types to avoid clutter and messy design. A consistent theme is much more aesthetically pleasing. This is a core principle of minimalism.

Lastly, limitations encourage creativity as the designer has to come up with a way to overcome the restriction only with the available resources.

A fine example is Gothic churches. It was very difficult putting in large windows in church walls as they would cause structural instability. So architects devised flying buttresses to help bear the load. But even then, the technology for building large, transparent glass windows had not been developed. So instead, they pieced together small, coloured glass pieces to make stained glass windows, introducing light in to the church while telling stories from the Bible.

Ironically, limits and restrictions can be the catalyst for something better. Instead of rebelling and fighting against constraint, try adapting and coming up with a creative way to overcome it.

Posted in Science & Nature

## Clarke’s Three Laws Of Prediction

The following are three laws conjectured by acclaimed science fiction author, Arthur C. Clarke, regarding predicting the future.

1. When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.
2. The only way of discovering the limits of the possible is to venture a little way past them into the impossible.
3. Any sufficiently advanced technology is indistinguishable from magic.

Posted in Life & Happiness

## Possimpible

Have you ever had a moment when something so unbelievable, so improbable that you never would have imagined it would happen, happened? When something you could only dream of actually happened in real life? When something so impossible that you must have stepped into a parallel universe for that thing to happen? The feeling that such a moment brings is indescribable.

Success is not about money and power. Success is not a product of luck. To become successful, one must change their state of mind first. The most crucial thing to understand is that the only limit is that there are no limits. Only when you dare to go past what is possible will you attain anything worthwhile. “To the impossible?” you may ask. No, true success lies beyond the impossible. A place where the possible and the impossible meet to become: the possimpible. Only when you have become the master of the possimpible will you be able to confidently say that you have succeeded in life.

Nothing, and everything is possimpible.