Geometric Shape as a Metaphor for My Life
Geometry is all around us, and we are surrounded by myriads of geometric forms, shapes, and patterns. Every living organism and all nonliving things have an element of geometry within.
The above quote is from my article Geometry, All Around Us, which 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 rereading 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.
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).
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 halfarch 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 pendulumregulated 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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Komentari
Lada 🏡 Prkic
prije 1 godinu #22
I'm serious about that. My colleague is a yoga teacher (she is an IT engineer) and familiar with these techniques. Thank you for your suggestions. 🤗
Ken Boddie
prije 1 godinu #21
If you're serious about getting started with Tai Chi and Qigong, Lada, I suggest that you search for the following on YouTube:
☯️
Lada 🏡 Prkic
prije 1 godinu #20
Thanks again for a beautiful comment. My previous comment was ‘dark’, but I see the light somewhere in the darkness. It's my nature. Dealing with hardships affects my generally optimistic state of mind, but we can't be positive all the time. To me, it is simply not natural. I know my pendulum will slow down.
Maybe I need to start practising tai chi and qi gong. I have read about their health benefits after you mentioned these techniques in your previous comment. Thank you.
Ken Boddie
prije 1 godinu #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
prije 1 godinu #18
Ken, your comment made me think about the choices I've made lately and how they affected the amplitude of my life's pendulum. I decided to grapple with difficulties (my mom's serious health condition) instead of transferring them to someone else. My metaphorical pendulum bob started its swing at the highest position from equilibrium. On that one extreme, there is the hard stuff  sickness, suffering, sadness. On the other end of the pendulum, there's a big reward  my mom's smiling face when she looks at me by her bed and my feelings of fulfilment that I am doing everything I can to help her.
Everything will pass. This will pass too. My pendulum will rest for a while before the next swing to the "bad" side. Life is about a constant movement back and forth. There is Yin and Yang, positive and negative, dark and bright side, life and death.
Metaphorically speaking, since I wrote about the cycloidal pendulum, no matter how far we swing in one direction (success, wealth), the curve of life would lead us down in a specific period of time. The final destination is the same for every one of us.
Lada 🏡 Prkic
prije 1 godinu #17
Thank you for reading and commenting here and on LinkedIn, Franci. The swinging angle of my metaphorical pendulum is pretty large at the moment  swinging high between extremes. I hope the amplitude of swings declines soon (closer to equilibrium). 🙂
Zacharias 🐝 Voulgaris
prije 1 godinu #16
So true! In data science too, vectors play an important role. Matrices and tensors too. Math is truly liberating and liberated when it comes to applications…
Lada 🏡 Prkic
prije 1 godinu #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
prije 1 godinu #14
Yes, John. We can say that math and geometry describe nature. Like you, I feel comfortable within my expanding circle. 🤗
Preston 🐝 Vander Ven
prije 1 godinu #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
prije 1 godinu #12
Ken, I'll respond as soon as I can. Thank you as always for a great comment. 🤗
Zacharias 🐝 Voulgaris
prije 1 godinu #11
Awesome. I look forward to this!
Lada 🏡 Prkic
prije 1 godinu #10
Thank you, Neil. :) There's nothing more satisfying than to inspire readers. If my article made you think more about the metaphor for your life, my goal has been achieved. 🤗
I always try to find images or videos that best fit the topic. Readers, including myself, remember visual information longer and better than they remember written information.
Lada 🏡 Prkic
prije 1 godinu #9
Zacharias, I intended to mention spirals, helix, and vortex. Since this article is written for BIZCATALYST first, I decided to write only about some basic geometric shapes, including a cycloid curve which I thought would be interesting to readers.
When I have more time for writing, I intend to write a sequel on beBee where we have more freedom to include images of our choice (with a proper citation).
Franci 🐝Eugenia Hoffman
prije 1 godinu #8
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
prije 1 godinu #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
prije 1 godinu #6
That's a good point! Indeed, if anything, the pendulum can be a means to the teaching of equilibrium. The latter is a dynamic matter though, something we often tend to forget. Perhaps practicing Tai Chi and Qigong are better as a strategy for understanding and internalizing this teaching. Otherwise, we'd have to watch and ponder upon the pendulum for hours! Cheers
Ken Boddie
prije 1 godinu #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
prije 1 godinu #4
😅 I spend quite a bit of my time at work going round in circles so that is my obvious option.
Zacharias 🐝 Voulgaris
prije 1 godinu #3
If you find it and decide to write an article about it, perhaps we can start a new group, focused on Geometry :)
Neil Smith
prije 1 godinu #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
prije 1 godinu #1
For me, the most intriguing and even mindblowing 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 (3D spirals) to sunflowers to galaxy formations. It's really hard to overstate the importance of the spiral in nature. Cheers