The Tale of an Overweight Mathematician…

The Tale of an Overweight Mathematician…


I have decided to be rather honest in this article, and admit to the readers of emph Mathematics Today that in the spring of 2017, I acknowledged that I was overweight, certainly at medical appointments over the years I had been tutted at by the professional concerned. This had been despite a reasonably active lifestyle; I stand at work and typically walk around 20 miles per week. I had noticed that my waistline had increased by 4 inches (men’s trouser sizes increase in increments of 2 inches) in around a year, and decided enough was enough. A number of my colleagues and friends are aficionados of Slimming World™ groups, and I decided to join them.^1

The-Tale-of-an-Overweight-Mathematician-figure-1
Figure 1: Model of a human

At my first session the excellent consultant explained how the Slimming World™ approach of food optimising works, and we discussed a target weight. I did not really have a target weight, more a target waistline, and this seemed to throw the system. We therefore, eventually, agreed a target that would take me into a ‘healthy’ body mass index (BMI).^2 At the time of writing I have lost 2 stones in 8 weeks and have thoroughly enjoyed the experience. Surprisingly my meals have become more varied and interesting – and I have discovered that vast numbers of my friends are fellow travellers on the weight-loss journey, and are being thoroughly supportive. Despite this contentedness with my progress, I still have no idea whether I will once again be able to wear the clothes I could a few years ago — and it occurred to me that applied maths could help.

Like all good mathematical models, I made some assumptions:

  1. All lengths are measured in metres (m), volumes in metres cubed (m^3) and masses in kilogrammes (kg);
  2. The human body can be modelled with a cylinder and the waistline is considered to be the circumference of the cylinder;
  3. Human mass is equally distributed throughout the body with density 1000kg/m^3;
  4. Mass is lost and gained evenly in the x and y dimensions, but the z dimension (height) remains constant.

First, let us consider my specific situation. I am 1.85m tall, and when I began the weight-loss journey weighed 108.4kg.

Given my initial mass of 108.4kg, this implies the model has a volume of 0.1084m^3, and thus

(1)   \begin{equation*} 0.1084 = 1.85\pi r^2. \end{equation*}

Hence

(2)   \begin{equation*} r = \sqrt{\frac{0.1084}{1.85\pi}} = 0.137. \quad (3\ \text{sf}) \end{equation*}

Therefore, the model with my starting mass has waistline

(3)   \begin{equation*} 2\pi\sqrt{\frac{0.1084}{1.85\pi}} = 0.858. \quad (3\ \text{sf}) \end{equation*}

This is approximately 34inches^4 and was not my actual waistline (if only!); however I contend that, given the approximations made in the assumptions, increases or decreases in waistline can be considered as a proportion of this.

My target weight, which will take me into a healthy BMI (astute readers might be able to do some sums to calculate this!) is 85.7kg. Using the same model, this implies a volume of 0.0857m^3:

(4)   \begin{equation*} 0.0857 = 1.85\pi r^2. \end{equation*}

Hence

(5)   \begin{equation*} r = \sqrt{\frac{0.0857}{1.85\pi}} = 0.121. \quad (3\ \text{sf}) \end{equation*}

Therefore, at my target weight my model will have waistline:

(6)   \begin{equation*} 2\pi \sqrt{\frac{0.0857}{1.85\pi}} = 0.763. \quad (3\ \text{sf}) \end{equation*}

This is around 30 inches, and would represent an 11% (2 sf) reduction in waistline. If my own waistline were to be reduced by 11%, then it would reduce it by over 4 inches and return me to the waistline I had maintained most of my adult life. If this model is correct, then achieving my target mass would indeed return me to my target waistline also. I have the means to test this theory, I had some bespoke suits made in 2004 which at the start of this exercise no longer fitted me, if, when I achieve my target they do fit, I shall consider the assumptions valid.

It occurs to me that other potential slimmers would benefit from knowing what waistline their targets will bring them and, assuming the above assumptions are reasonable, I propose the following.

Assume the initial height is H, initial mass M_0 and target mass M_T, r the cylinder radius of the model at the start of the process, and R the radius of the model when the body has achieved target mass. The initial conditions are:

(7)   \begin{equation*} M_0 = 1000\pi r^2H. \end{equation*}

Hence

(8)   \begin{equation*} r = \sqrt{\frac{M_0}{1000\pi H}}. \end{equation*}

And, without much more ado, by the same reasoning at target:

(9)   \begin{equation*} R = \sqrt{\frac{M_T}{1000\pi H}}. \end{equation*}

Given I am interested in the percentage change in the circumference of the waistline, which is simply a constant (2\pi) multiplied by the radius, it simplifies things to consider the percentage change in the radius. The percentage change in the radius (and logically the waistline) is:

(10)   \begin{equation*} 100\frac{r-R}{r}\%. \end{equation*}

Substituting equations (8) and (9) into equation (10) gives us:

(11)   \begin{equation*} 100\frac{\sqrt{\frac{M_0}{1000\pi H}} - \sqrt{\frac{M_T}{1000\pi H}}}{\sqrt{\frac{M_0}{1000\pi H}}}\%. \end{equation*}

Delightfully, the \sqrt{1000\pi H} terms cancel (essentially a result of the property highlighted in note 3):

(12)   \begin{equation*} 100\frac{\sqrt{M_0} - \sqrt{M_T}}{\sqrt{M_0}}\%. \end{equation*}

Simplifying gives

(13)   \begin{equation*} 100\left(1 - \sqrt{\frac{M_T}{M_0}}\right)\%, \end{equation*}

which, as a sanity check, putting in my own measurements for initial and target masses, gives approximately 11% reduction in radius, and hence (as far as this model is concerned) waistline.

Rearranging, the target mass to give an N% waistline reduction is

(14)   \begin{equation*} M_T = M_0{\left(1 - \frac{N}{100}\right)}^2. \end{equation*}

When N/100 is small, then the maximum contribution of {(N/100)}^2 would be very small. Given the level of assumption used hitherto, it is perhaps justified to propose a simplified equation which will need caveats to be used only when N is less than, say, 20%^4:

(15)   \begin{equation*} M_T \approx M_0\left(1-2\frac{N}{100}\right). \end{equation*}

Thus, a target mass to give a reduction in waistline of N%:

(16)   \begin{equation*} M_0\left(1-\frac{N}{50}\right). \end{equation*}

I therefore offer to slimmers, and those who advise them, this simple model that could help them estimate expected percentage change in waistline for a given change in mass (equation (13)), and also the target mass required to give a required percentage change in waistline (equation (16)).^6

I do hope you see a lot less of me in the future!

Edward Rochead CMath CSci FIMA
Dstl

Acknowledgement

The author would like to thank Michelle Coombes, his Slimming World™ consultant, and those friends and colleagues who have encouraged him in his weight-loss journey, and Mike Lane and Linda Knutsen for taking the time to review this article.

The opinions expressed in this article are not necessarily those of Dstl, nor has it been endorsed by Slimming World™.

Notes

  1. Other slimming groups are available
  2. BMI healthy weight calculator, www.nhs.uk/Tools/Pages/Healthyweightcalculator.aspx
  3. Although a cylinder was used, any three-dimensional shape with these assumptions would scale similarly.
  4. Using a conversion factor of 1\,inch = 2.54\,cm.
  5. This would introduce a maximum error of 4\%.
  6. Any decision to lose weight should be taken with the advice of a medical professional.

Reproduced from Mathematics Today, April 2018

Download the article, The Tale of an Overweight Mathematician… (pdf)

Image credit: Fat business man use scale to measure his waistline © Tom Wang | Dreamstime.com
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