Yesterday, I had a discussion with a couple of friends about photography and how light affects a photograph. I casually mentioned the inverse-square law as a part of the explanation and suddenly it was like I was speaking in foreign tongues. They instantly glazed over. I felt and heard their panic demands to change the subject. Anything else to talk about on a Saturday morning would have been better than talking about maths.

Maths escapes a lot of people. It is ignored, and even hated. If you are like most people, it is probably something like: numbers, equations, boredom, anxiety, and pain.

Then, there are the weird ones, like me, who enjoy maths and do not fear it, but try to understand it, and even embrace it to comprehend the world around us. In fact, some of my friends are math geeks (Simon, Erica, etc.) or use math everyday for work (Joe, Lee, Dave, etc.). Some even see themselves as mathematicians (Jim!, Ken!). When we casual admirers of maths think of maths, we think of shapes, colours, patterns, trends, creativity, freedom, insight, and the feeling of making deep connections.

Why is there such a huge gap between how the general population see and experience maths?

So many people simply give up on engaging with maths from a young age, deciding, “*I am just not a maths person, I’m not good at maths.*“

There is a deeply entrenched view in our society that the ability to learn and master mathematical understanding comes from an inborn innate gift rather than from hard work and perseverance.

When we hold this view from a young age, the maths classroom becomes a place where we are constantly being judged not good enough, a place of repeated failure and shame.

And with each failed test, with each time we are put on the spot to answer a question we are not prepared for, we think, “*See, I’m really not a maths person. Get me out of here, I don’t belong.*“

Then grows maths anxiety.

So, yesterday, I tried to describe the propagation of light in photography and used an explanation that called upon the physical properties of light govern by the inverse-square law.

Yikes, was that ever difficult to explain. I largely failed. I suspect that my friends saw me as a weird elitist trying to beat them to death with arcane knowledge that was of little value.

However, the exact opposite is the truth. Understanding our physical world empowers you to learn. Knowledge truly is power. I want my friends to be powerful. To be awesome. To be brilliant. If the process damages their brains, then that is the price we will all have to pay. [lol]

So, I decided to fix my own understandings, so I could do better the next time I used this term. [Oh god, can no one stop this horror?]

Last evening, I was using Google to research the inverse-square law and looked for video clips that helped to better explain it to the average person. My wife overheard one of the video clips and said, “*Oh, isn’t that interesting*“. While I have never considered her a maths expert, she does possess a PhD and therefore, she is able to learn quickly. She understood the narrative from the video instantly. I knew that I was on the right track with this video explanation. Now, I needed to formulate a better, simpler, and more concise explanation that I could use in the future to not put my friends to sleep in the coffee shop. Or, have them gang up and try to kill me if I attempt to explain the inverse-squared law again.

[*Note: The death of Julius Caesar in the photo below. Caesar was the last person to try to explain the inverse-square law last on 15 March 44 BC.*]

I know that my friends all enjoy connecting ideas and making sense of the world around them, just as I do. So, I felt that the failure was mine since I could not share this wonderful math equation properly. Sure, I had a wall to scale as high as the one that Trump wants to build with Mexico, but as we have all seen on TV, even that obstruction is easily climbed and overcome, even by children. At least by the Mexican kids anyway. Nothing can stop any of us when we are armed with a little bit of ingenuity and mankind’s endless drive and focused passion to overcome insurmountable obstacles.

Could I figure out how to explain the inverse-square law? I was hopeful. perhaps foolishly so…. it was a gamble.

The first point that I realized is that people easily understand anything that has a linear characteristic, but they struggle mightily to understand things that possess non-linear traits. So, I needed a visual idea to demonstrate non-linearity.

With the World Junior Hockey games on TV, I decided upon the trusty “hockey stick”. The hockey stick shape is something that every Canadian understands and it is commonly used to describe a non-linear curved model in maths.

Although, after Russia kick Canada’s butt yesterday at the IIHF game, it seemed like the hockey stick shape was a strange and ponderous object that our team had never seen before. But, I digress…

Next, I needed a safe and friendly description. Again, I planned to use an analogy of some sort to help explain the characterization of light propagation over a distance. I could use my background in radio frequency (RF) propagation, but RF is invisible. So, it is even harder to use the five human senses (see, hear, taste, touch, smell) to relate to it. People like ideas that can be imagined using our human senses – analogies that possess physical traits work best. While RF possessed the exact same physical properties as light, it is simply too abstract to describe. If I thought that light was challenging, RF would be far worse for my non maths friends.

*So, what is the inverse-square law anyway?*

*So, what is the inverse-square law anyway?*

The inverse-square law works as follows: If you double the distance between subject and light source, it illuminates a surface area four times greater than the one before.

In general, we therefore multiply the distance with itself in order to calculate the enlargement of that surface area. However, a larger surface area leads to a light intensity that is inversely proportional to the square of the distance, since the same amount of light must be distributed onto a larger surface area respectively.

Therefore, we see light fall-off, meaning a decrease of light intensity.

In technical terms the inverse-square law reads as follows: The energy (in our case: light intensity) at location A (subject area) decreases inversely proportional to the square of A’s distance to the energy source (for example, our flash head).

It only requires some basic math knowledge to write down the inverse-square law (its formula). However, the physics behind it is generally very complex. Because of this, we are only going to approach this law in an illustrative way and from the viewpoint of photography. For this reason, we are referring to the exposure of the image sensor or film and to the lighting of the subject. When using flash and spotlight, the inverse-square law is especially handy.

The light intensity, for example, quadruples (4) upon halving (1/2) the distance to the light source and subject. Respectively, the light intensity decreases to a quarter if we double the distance. According to this, these exemplary pairs of digits are valid (distance: 3-fold; intensity: 1/9) and (4; 1/16) if we multiply the distance respectively.

In general, the inverse-square law explains the disproportionate light fall-off with increasing distance of the subject to the light source. This knowledge helps us to better understand how to correlate light and lighting with the distance to the subject and its brightness.

Can you imagine the “hockey stick” shape of the curve?

*Practical Application*

*Practical Application*

Because of the inverse-square relationship of the described law, the light intensity drops rather heavily when the subject is first moved further away from the light source. After that, it continuously decreases on a weaker level. For example: If we increase the distance between light source and subject from 1 meter to 2 meters, 75 percent of light intensity is lost on the subject. But when we increase the distance from 4 to 10 meters, we only lose 5 percent.

So, the light intensity close to the light source has especially high values. But in the distance, this intensity only reaches a tiny value. Here is how we create the appropriate lighting: At constant shutter speed the f-value increases, the closer the subject gets to the light source – the smaller the aperture, the less light enters the camera.

Vice versa too: the f-value decreases as the distance of the subject to the light source increases. In both cases the respective shots almost look the same: Simply because the same amount of light enters through the lens.

Simple, right?

*My Inverse-Square Law Analogy*

*My Inverse-Square Law Analogy*

So, for my analogy, I want to use “money”. Everyone understands money as we use it every day. So, hopefully it will form a good foundation to help explain this idea?

We start out with $100.00. We plan to travel 100 kilometres per day. After the first day of travel, we have all our money remaining, so we still have $100.00 as we have not paid for any of the travel costs. But we must pay for the first night. The farther we move from home, the lower it costs each night. So, the first night is expensive and it costs $75.00. We are left with just $25.00 the next morning. Ouch!

After the second day of travel of another 100 kilometres, we spend much less, and it costs us just $13.89, so we have $11.11 remaining on day three.

As we travel the fourth 100-kilometre day, we spend just $4.86, so we have $6.25 remaining in our pockets. After the fifth day’s travel, we spend $2.25 and we have $4.00 remaining. On the sixth day, we complete our travel and spend $1.22 and have $2.78 left. And so, it continues decreasing by smaller amounts each day as we journey further away from home.

Light decays over distance in the exact same way as our wallet is emptied.

Does this analogy make sense? Does it explain the Inverse-square law in simple language that is understandable by the average friend? I will test it this morning at coffee and let you know. Assuming I survive, of course?

————————–MJM ————————–

## References:

Dauner, J. (2016). *Understanding the Inverse-Square Law of Light*. PentaPixels. Retrieved on December 29, 2019 from, https://petapixel.com/2016/06/02/primer-inverse-square-law-light/

Unknown. (2017). *Why people hate maths and how to fix it*. Phys.org. Retrieved on December 29, 2019 from, https://phys.org/news/2017-10-people-maths.html

————————–MJM ————————–

**About the Author:**

*Michael Martin has more than 35 years of experience in systems design for broadband networks, optical fibre, wireless, and digital communications technologies.*

*He is a business and technology consultant. He is employed by Wirepas Oy from Tampere, Finland as the Director of Business Development. Over the past 15 years with IBM, he has worked in the GBS Global Center of Competency for Energy and Utilities and the GTS Global Center of Excellence for Energy and Utilities. He is a founding partner and President of MICAN Communications and before that was President of Comlink Systems Limited and Ensat Broadcast Services, Inc., both divisions of Cygnal Technologies Corporation (CYN: TSX).*

*Martin currently serves on the Board of Directors for TeraGo Inc (TGO: TSX) and previously served on the Board of Directors for Avante Logixx Inc. (XX: TSX.V). *

*He has served as a Member, SCC ISO-IEC JTC 1/SC-41 – Internet of Things and related technologies, ISO – International Organization for Standardization, and as a member of the NIST SP 500-325 Fog Computing Conceptual Model, National Institute of Standards and Technology.*

*He served on the Board of Governors of the University of Ontario Institute of Technology (UOIT) [now OntarioTech University] and on the Board of Advisers of five different Colleges in Ontario. For 16 years he served on the Board of the Society of Motion Picture and Television Engineers (SMPTE), Toronto Section. *

*He holds three master’s degrees, in business (MBA), communication (MA), and education (MEd). As well, he has three undergraduate diplomas and five certifications in business, computer programming, internetworking, project management, media, photography, and communication technology. He has earned 15 badges in next generation MOOC continuous education in IoT, Cloud, AI and Cognitive systems, Blockchain, Agile, Big Data, Design Thinking, Security, and more*.

Regularly, the topic of the inverse square law would come up in a university course that I taught for many years. I found that a fruitful, non-mathematical way to explain it was to imagine something like a pinhole in the side of a space ship that was slowly & steadily leaking a million molecules of air per second into space. If you had a 1 cm**2 vacuum hose available to catch them, the further away it was from the pinhole, the fewer you would catch per second. From there, it wasn’t much of a stretch to get them to realize that the fraction u could catch in that 1 square cm receiver was just the ratio of 1 cm**2 to the area of an imaginary spherical surface centered on the pinhole, ie, the rate per square cm would drop with the square of the distance. And that was always a good time to remind them of Gauss’ Law & the Divergence Theorem that most of them should have learned as sophomores, LOL. It’s also no stretch to replace the molecules of air with photons & then they are talking about optics.

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