Lada 🏡 Prkic

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Geometric Shape as a Metaphor for My Life

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Geometry is all around us, and we are surrounded by myriads of geometric forms, shapes, and patterns. Every living organism and all non-living things have an element of geometry within.

The above quote is from my article Geometry, All Around Uswhich I wrote almost five years ago. I have read it again after the Editor of Reflections, the journal of the Mathematics Association of NSW in Australia, contacted me recently with the request for republishing my article in the mentioned journal. I was thrilled and honoured.

I use mathematics and geometry my whole life. For me, geometry is such a fascinating subject. It has been one of my favourite subjects during my school years. Geometry has shaped the world around us, from human DNA to crystals. While re-reading the article, one thought came to my mind

What geometric shape could be a metaphor for my life?

I wasn’t thinking of complex geometric objects but basic geometric shapes to depict a complex biological system such as life.

I first thought of a straight line, which is a set of points between two milestones of our life – birth and death. It makes sense actually if we see life as a continuous journey. The path of life is anything but a straight line with its ups and downs, potholes and gorges, but putting into a larger timescale, it is straight. As a side note, when put into a broader perspective, human life is an infinitesimal dot.

Still, a circular shape best represents life. A circle might be a perfect metaphor for life. The ancient Greeks considered the circle superior to all geometric shapes and a symbol of balance in nature. It is a perfect shape where every point on its rib stands at an equal distance from the centre. As someone stated, a circle is the most ‘democratic’ geometrical shape.

Life is a continuous process of change. The symbolic metaphor of that process is a circle that grows by the power of life itself.

I can visualise myself inside my metaphorical circle pushing the boundaries of the circle outward.

As we learn new things, embrace new knowledge and ideas, stay curious about the world around us, our metaphorical circles expand, their radius and area increase. Once people stop learning, they begin to stagnate within the existing boundaries, and eventually, their circles even start to shrink.

To quote Gandhi, “We should learn as if we were to live forever.” Knowledge is considered the greatest treasure one can possess. I see it as the power that makes the boundaries of my circle expand.

We all are learning and growing at our own pace. Some people never stop expanding their circles. Some choose to exist within their closed limits. No matter the size of our metaphorical circles, they are all moving through time. When I wrote this, I instantly envisioned a circle rolling along the straight line and a cycloid curve.

A cycloid is such an elegant curve, traced by a point on the rim of a circle as the circle rolls along a straight line. I have been amazed at its beautiful geometric and mechanical properties since I first heard of the curve. It inspired generations of great mathematicians. Galileo gave the curve name ‘cycloid,’ from a Greek word meaning shaped like a circle.

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The curve is periodic, which means it repeats the identical pattern for each cycle. The length of the straight line for each cycle is equal to 2rπ (circle circumference).

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The above image shows the inverted cycloid curve with its repeated patterns. The inverted cycloid is brachistochrone, meaning the curve of the shortest time (the word brachistochrone originates from the Greek for ‘shortest time’).

The Brachistochrone problem, or the problem of the fastest descent curve, is one of the oldest problems in the history of calculating variations. As seen in the short video below, the shortest path and shortest time are not always the same thing. An object subjected only to gravity (without friction) will slide between two points in the least possible time along a curve, not a straight line. The curve that represents the path of fastest descent is the half-arch of an inverted cycloid.

Another amazing property of a cycloid is the tautochrone (isochronous) property, meaning the same time or equal time (in Greek). The time for an object (subjected only to the force of gravity and without friction) to slide down the inverted cycloid does not depend on its initial position. It will reach the lowest point of the curve in the same amount of time regardless of the point from which it starts (clearly explained in the video below). The cycloid is the only curve that possesses both brachistochrone and isochronous properties.

There is more about this amazing curve!

If a pendulum is suspended at the point between the two adjacent arcs of the inverted cycloid, and the length of a pendulum is equal to half the arc length of the cycloid, the pendulum bob will swing in the shape of the same cycloid as its shifted copy.

The video below shows five pendula with the same length and different amplitude swinging in a cycloidal trajectory under the influence of gravity. Due to the isochronous property of cycloid, all five pendula cycle through one complete oscillation (back and forth motion of a bob) in the same amount of time whatever the amplitude – meaning no matter how far from the central equilibrium position each bob is to start. All five pendulum bobs reach the lowest point of the curve (equilibrium) at the same time!

 

 

The pendulum-regulated clock was an application of that isochronous motion.

I could go further and talk about spirals, shapes most often found in nature and the universe, but I would like to stop here. Talking further about oscillations and isochronism will lead us even to the concept of quantum entanglement. Let just say that the universe is an interconnected oscillating space, and the cycloidal pendulum connects curve, time, and oscillation.

Those of you who still read this article (and did not get tired of curves and Greek names) may be wondering what a cycloid has to do with a metaphor for life.

Well, in my opinion, a cycloid and cycloidal pendulum, with its amazing geometric and mechanical properties, could be the perfect metaphor for my life.

A cycloid is a reminder that life is cyclical.

A cycloid is a reminder that the shortest path is not always the fastest.

A pendulum is a metaphor for life swinging back and forth between extremes as it goes through equilibrium. It is also a reminder that swinging is an act of balancing our life. Those five isochronous cycloidal pendula symbolize nature and life as a whole and agreement between the parts of a whole united by a mutual relation of motion.

***

I use geometry to shape and build engineering structures. Geometry as a metaphor shapes the way I understand the world around me. To quote the renowned American physicist Richard P. Feynman,

If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

And the language nature speaks is geometry.

What about you? Do you understand this language? What geometric shape could be the perfect metaphor for your life?

 

Author's Note: This post is previously published on BIZCATALYST 360°.

Title Image: A giant pendulum in Tate's Turbine Hall. Photograph: AFP/Getty

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Comments
Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #22

Ken Boddie

Ken Boddie

3 months ago #21

Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #20

Ken Boddie

Ken Boddie

3 months ago #19

#18 

I’m not so sure thar the pendulum swing is constant for all of us, Lada, except when we ignore opportunities and let life pass us by. I believe that the

pendulum can slow down, or even temporarily be brought to a halt, by the choices we make, by either pushing or pulling on it. Holding onto the bob to maintain it in one position for too long can be damaging. Perhaps the solution to an equitable lifestyle is to maintain balance as best we can. I don’t think of the pendulum as what life delivers us, but rather as how we react. Sometimes we feel overwhelmed by problems, but they become much lighter when we think of them as opportunities to make choices. There is always light somewhere in the darkness and calm in the eye of the storm, but we may not see the yang if we hold on too tightly to the yin. 

Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #18

Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #17

Zacharias 🐝 Voulgaris

Zacharias 🐝 Voulgaris

3 months ago #16

Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #15

Physics is intertwined with math and geometry on many levels. Vectors are used in both physics and geometry. It is another great metaphor for life, having magnitude and direction. 

Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #14

Preston 🐝 Vander Ven

Preston 🐝 Vander Ven

3 months ago #13

I never thought of this topic in a manner of geometry. I usually thought of it as a manner of physics. That is why I would choose a Vector to describe life. We have our direction, and our magnitude or passion driving that direction. 

Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #12

Zacharias 🐝 Voulgaris

Zacharias 🐝 Voulgaris

3 months ago #11

Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #10

Lada 🏡 Prkic

Lada 🏡 Prkic

3 months ago #9

Franci 🐝Eugenia Hoffman, beBee Brand Ambassador

Most interesting post, Lada, and interesting how geometric shapes can be a metaphor for our life. I feel I've been in and out of the pendulum a few times. The spiral is the most fascinating because it's everywhere and especially noticeable in nature.  

John Rylance

John Rylance

3 months ago #7

Geometry is the knowledge that appears to be produced by human beings, yet whose meaning is totally independent of them. Rudolf Steiner.

Nature is very much the root of most of shapes which are artistically beautiful.

I dont mind which geometric shape I am, as long as I'm comfortable with it.

Zacharias 🐝 Voulgaris

Zacharias 🐝 Voulgaris

3 months ago #6

Ken Boddie

Ken Boddie

3 months ago #5

Interesting, Lada, how you take geometrical shapes beyond an association of problem solving solutions, and extend them into metaphors for life. The circle for me represents the ideal balance of yin and yang, with neither predominance of dark or light, hard or soft, negative or positive, cold or warm, wet or dry, feminine or masculine. Extending the circle into three dimensions represents the moon, which I regularly embrace in my daily Tai Chi and Qigong practices. Hence, the circle or sphere is the ultimate representation of life's balance. 

The most meaningful metaphor for my interactive work life experiences, however, is the pendulum, which represents how choices can influence the swing of the pendulum bob (or our metaphorical problem) backwards and forwards. The extent of each swing to the opposite side depends upon the starting point and angle above the vertical neutral plane.  If the pendulum bob is uninhibited by restrictive forces then it will swing backwards and forwards to equal distances beyond the neutral plane.  As the pendulum bob encounters resistance, however, it will come to a resting place on either side of neutral, depending upon the net degree of restraint, whether positive or negative, that is applied to resist the influence of gravity.  I am frequently amazed at how far out of balance life's pendulum bobs can be sustained by an initial solution that applies a total containment, and for how long this imbalance can be maintained, until change eventually occurs by either popular demand or rebellion.  Even stranger is the observation that the pendulum bob is often repositioned by an alternative solution (or restraint) that places it almost on the directly opposing side of the neutral axis. Such pendulum bob solutions can take many swings before an equitable solution can be found and the pendulum bob comes to a long term resting place much closer to the neutral plane. But such an equitable solution will only occur if both positive and negative forces are applied and balanced out. When such balance of yin and yang is absent, then the pendulum bob is often doomed to come to rest at an equally extreme position on the opposite side of the neutral plane, where it can be more easily dislodged into subsequent wild swings.

Neil Smith

Neil Smith

3 months ago #4

Zacharias 🐝 Voulgaris

Zacharias 🐝 Voulgaris

3 months ago #3

Neil Smith

Neil Smith

3 months ago #2

You have a habit of finding the most wonderful images for your articles. The pendulum photo is super. On the article itself I am now going to spend half the rest of the week pondering which is the most apt geometric shape for me.

Zacharias 🐝 Voulgaris

Zacharias 🐝 Voulgaris

3 months ago #1

For me, the most intriguing and even mind-blowing curve is the spiral (and yes, this is an oversimplification since there are countless different spirals out there!). I think it was already known in antiquity that one of the ways to trisect an angle (which we don't know with any precision as it's any given angle), is through a spiral, namely the Archimedian spiral (https://www.cs.princeton.edu/~chazelle/courses/BIB/trisect.html). Also, spirals are found anywhere, from DNA molecules (3-D spirals) to sunflowers to galaxy formations. It's really hard to overstate the importance of the spiral in nature. Cheers

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