Symmetry: How To Really “See” It?

Symmetry is one of those concepts that people tend to struggle with cognitively but appreciate instinctively. When I tell someone that a geometry is created by applying such-and-such symmetry, they tend to look perplexed. But when I show them the result, the light bulb turns on and they immediately respond:

“Oh! That’s what you meant. Why didn’t you say so in the first place?”

“Why does this happen?” you ask? Well that is a question (among others) we will be answering as part of this essay.

As a heads-up, it turns out that “symmetry” is a generic word that refers to a whole bunch of things. Certain kinds of symmetry appeal to human instincts whereas others don’t.

Before we get into that discussion, we will be starting with the basics using simple illustrations. Next, we will explore the manifestations of symmetry in nature and technology. Following this, we will dive into the role of symmetry in art. Finally, to celebrate your symmetrical knowledge, there is a fun bonus puzzle waiting for you.

Without any further ado, let us begin.

This essay is supported by Generatebg

What is Symmetry, Exactly?

“Symmetry” of geometric figures/objects refers to the property of “sameness”. When a geometric figure is said to be symmetrical, it means that its geometry remains unchanged after a “symmetry operation” is applied to it.

That’s enough theory; let us get practical. Imagine a neutral font for the English alphabet without italics, emboldening, etc. The letter ‘A’ looks the same when you place a mirror beside it. We refer to this property of sameness as “vertical symmetry”.

The letter ‘B’, on the other hand, does not possess this property. However, when you place the mirror below the letter ‘B’, its reflection looks the same. We refer to this property as ‘horizontal symmetry’, something that the letter ‘A’ does not exhibit.

More Alphabets and Richer Symmetry

So far, we have been looking at symmetry under “reflection”. There is another symmetry operation that we can apply to geometries: “rotation”. For instance, if you rotate the letter ‘S’ 180° in either direction, it looks the same. This property is also known as two-fold symmetry.

The more kinds of symmetry a geometry exhibits, the richer we consider its symmetry to be. The following letters possess all of the symmetry-types we have discussed so far:

H, I, O, and X.

Among these, ‘X’ is considered to be richer in symmetry than ‘H’ and ‘I’, because the two lines of ‘X’ meet each other at right angles. This means that if we rotate ‘X’ clockwise or counter-clockwise by 90°, it would still look the same. This property is known as four-fold symmetry.

You would get no prizes for guessing that ‘O’ has the richest symmetry of them all. Because of its perfect circular form, it looks the same regardless of any rotation or reflection.

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Human Association to Vertical Symmetry

I mentioned in the introduction certain kinds of symmetry appeal more to human instincts than others. Well, we have got the physics of our universe to thank. Because of the way gravity acts on Earth, we experience vertical symmetry far more than horizontal symmetry.

Think about it; be it the human body or a cute cat’s face, vertical symmetry is abundant. As a result, human inventions (such as cars and planes) feature vertical symmetry abundantly as well.

Most recently, I picked up on a social media trend, where viral short-form videos reflected with vertical symmetry tend to viral once again. It appears that human beings like the similarity and the novelty that such vertical symmetry offers.

The same, on the other hand, cannot be said for horizontal symmetry. It turns out that our instincts are terrible when it comes to horizontal symmetry.

Where is the Missing Slice?

When you and I are shown an inverted picture of a person and are asked to identify them, the first thing we tend to do is to rotate our heads to get an upright view. In other words, we are terrible at processing information upside down (or horizontal reflection).

Artists often use this to their advantage when they wish to gauge the colours of a scene. If they look at a scene normally, they might be biased with their gauging.

Instead, they elegantly stand with their legs spread wide, bend over, and look from underneath their legs to disassociate themselves from their biases. When they do this, the scene does not look like a scene anymore and they can just abstractly gauge the colours.

Earth’s moon has negligible gravity. So, vertical symmetry is perhaps not as abundant on the moon as on Earth. When astronomers take pictures of the moon, they consciously choose pictures where the sunlight illuminates the craters from above.

It is equally possible to take pictures where the sunlight illuminates the craters from below, but the astronomers just don’t choose to take them. “Why?” you ask? Well, the craters illuminated from below appear as if they were mesas rising above the surface.

To get a practical understanding of what is going on here, check out the illustration below. It appears as if a piece of the pie is missing. So, where is the missing slice? Well, turn the image upside down, and you will find the missing slice (keep looking until you can find it)!

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Symmetry and Art

Speaking of optical illusions, you must have seen images that show a man when viewed normally and show a woman when viewed upside down. These kinds of images take advantage of our disassociation with horizontal symmetry.

Historically, political cartoonists have used such “inversion” art to communicate hidden insults to public figures. But politics aside, in one of the most impressive feats of art I have ever seen, Gustave Verbeek ran a comic strip column titled “The Upside Downs Of Little Lady Lovekins And Old Man Muffaroo” in the early 1900s.

These comic strips read from left-to-right and top-to-bottom; nothing special about that. But once you are done reading them normally, you could invert the strips, and the story would continue! At his peak, Verbeek was producing one such comic strip per week. When I came to know of this, I was truly dumbfounded.

I have worked intensely with symmetrical geometries before, and I remember the kind of intense dreams and nightmares I used to have during this period. I have nothing but respect for Verbeek.

Compared to artwork that rotate 180°, those that rotate 90° are rarer. But they do exist; apparently the Germans were fond of them during the Renaissance period. For a practical illustration of such art, check out the duck-rabbit illustration below.

Bonus Puzzle

I promised a bonus puzzle to flex your practical “symmetry skills”. So, here we are. We have been discussing purely geometrical figures so far in this essay. But in a sudden plot twist, we turn to numbers now!

Consider a basket that contains more than a dozen fruits. Some of these are apples while the others are oranges. Let ‘x’ be the number of apples and ‘y’ be the number of oranges in the basket.

When you sum up ‘x’ and ‘y’, and rotate the resulting number by 180°, you get the product of ‘x’ and ‘y’. Your task is to figure out how many fruits are there in total in the basket?

This is a simple yet fun puzzle. If you have worked out the answer, post it in the comments section for others to compare. But please also remember to post a “spoiler alert” so that you don’t spoil the fun for other readers!

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Reference and credit: Martin Gardner.

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Further reading that might interest you: How To Actually Run A Growing Science Blog? and How To Really Choose Between Favourite Experiences And Trying New Ones?

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