In Praise of BMI: Not Perfect But an Exceedingly Useful and Convenient Parameter for Measuring Obesity

One year ago, I had published a review on Obesity  - a 'disease' that is now inflicting almost all societies.  Obesity is the cause of a multitude of ailments; it is not my intention to go over the details again - they are there to read in the review.  Suffice to say that the body mass index (BMI) has proved to be an excellent way to measure obesity and correlates well with observed consequences of obesity; BMI is well understood; easy to define and calculate, and is a great way for the GP to talk to her patients about lifestyle changes etc.- BMI gives a means to the patient to control and monitor her progress.   

I feel that I have to write this blog to critically examine the advice in the recent feature entitled 'It's time to change how we measure obesity'.  The article was also publicised through The Conversation with a title 'Stop measuring obesity with a ruler; we've discovered a far better predictor of health'.
Essentially the article is supposed to be summarizing a study in which a large number, at least 49 , of metabolites were selected for the study of the relationship between obesity and metabolic diseases.  I quote from the paper (my emphasis in red) :

'...However, our study found that metabolite levels did not provide predictive power for future weight changes.  Overall, the metabolome perturbations appear as a consequence of changes in weight as opposed to being a contributing factor.  While BMI correlates well, and to a large extent, with individual health outcomes, it does not have the sensitivity to identify outliers, some of which carry unique health consequences.'

So, the sudy says that BMI is not so good a predictor for outliers - may be less than 1% of the population.

Big deal- BMI has never been claimed to be an exact predictor - it is a useful index for measuring obesity.

I am perturbed by the apparent discrepancy between what is concluded in the research paper and what a senior author has tried to convey through the sensational headlines in the feature published in the World Economic Forum and the Conversation.

I have spent the past 12 years in talking to the community about science matters.  Generally, a non-specialist holds the experts in high regard; and listens and respects the message they are able to give - it is really difficult to translate a research work into language that a layman can understand.  It is our duty to be clear and not mislead the general population.  

BMI is a unique number that is easily understood by the general public.  Obesity is a real scrouge for the society and we need to find ways to control it. It is sad to read a feature article which comes dangerously close to undermining the usefulness of BMI index in the minds of the general public.  Reading the comments in the Conversation,  it seems that some of the readers do believe what the headlines of the features say.

There are better ways to promote one's research work.   

Over-population is 'Elephant in the Room'; Why Do We Not Address the Real Problem?

Over the past 20 years, I have seen innumerable reports about climate change, loss of bio-diversity, plastic pollution, atmospheric & river pollution, ocean acidification,  desertification, migration crisis, consequences of sea-level rise and many other issues. There has been no shortage of conferences, governmental level meetings, scenario-playing __ a bonanza for talking shops, kicking the can down the road and an excuse for not doing much until the next big meeting.

Until the end of the 20th century, North America and Europe were the biggest consumer of global resources and the impact was already visible.  Now a four times bigger population block from Asia has joined the consumption bonanza using up natural resources at a much faster rate.  
Not much else has changed since 2012 when I gave an extensive course on sustainability in Glasgow - the issues are still the same - just the time frame in which to act to 'Save the Earth' is getting vanishingly small. I am publishing my updated lecture on Population Dynamics in the following.  Other lectures and updates may be reached here ( Energy Resources, Food and Water Resources, Climate Change, How to Feed 10 Billion People, Future Water Crises).

The talk on population dynamics is presented below:

Sustainability - Population Dynamics –

What sort of world shall I inherit, daddy?

                                        (The talks were given in September 2012 in Glasgow) 

Population control is a sensitive topic and generates extreme emotional response in  people - partly due to the way most religions favour unfettered reproduction. This was fine when we were talking about a handful of humans on the Earth but now the problem of overpopulation and hence over-consumption must not be ignored.  

But this is what every body - well almost every body - does. 

When searching for solutions, it is usual to say that the Earth is under stress and somehow the humans will be able to find a way to save Mother Earth.

This is delusional thinking.  Humans are the problem - they are the 'cancer' that is eating away the fabric of the earth and a day will come when the earth will simply remove the irritant and carry on as it has for the last 4.3 billion years.

                (please click on a slide to see its full page-view - navigate with left/right arrow) 

It might appear that there is some hope of population stabilising at about 10 billion mark sometimes after 2050.  The difficulty arises with consumption and exploitation of Earth's resources.  For example, Standard of livings are improving and meat consumption is increasing - particularly in China and other emerging economies.  The number of livestock has increased in response for meat products.
Since 1960, in 50 years, the number of chicken has gone up by 300% to 12 billion in 2010, cattle numbers by 30% to 1.4 billion, pigs by 250% to 1 billion and sheep and goats by 50% to 1.8 billion.
These animals require land, water, food - exactly what humans need to live.  This is an additional burden on the Earth's resources and is necessary because humans love to eat animal meat.  Many studies have suggested that switching to a plant based diet will help the environment/ecosystem enormously but, due to commercial pressures,  no goverment is keen to openly come out in favour of such a transition.  

Technology can help in this regards and I have discussed this in my talks referred to earlier. However, it is moot point as to how long the earth carry the extra burden of us humans before a tipping point reaches and our civilization as we know collapses.

Just wish to mention the crazy notion of some eminent people that somehow, when the earth becemes too crowded and polluted, we can colonize othe planets and heavenly bodies in space. This is th e familiar Elephant in the Room scenario when a probelm is ignored and is allowed to get out of hand by irresponsible policy decisions.  I have discussed the matter of 'hype about space colonization' here.  

Reacreational Maths with Smartphone Calculator - 10 Games to Wow Your Friends

The problem with presenting an interesting party game is that your friends immediately ask for more.  Some will want to know how it was done.  For example, everybody gasps when the 4x4 magic square, after several arbitrary random choices, gives the sum as the correct birthday or marriage anniversary.  
Generally, you are asked - How do you get that?  Can you show some other games? etc.

I have decided to collect ten games that one can play on a four-function calculator that all smartphones have - and everybody carries one all the time!  After the initial groans when you ask them to take their smartphones out for some fun games, the interest builds up very quickly.  

Let us start:  Worked examples in red

Game 1:   On a piece of paper write 1089 and put the paper away. 
Now ask your audience to  

          1.  Choose a 3-digit number with different 1st and 3rd digits     357
          2.   Reverse the order of digits      753
          3.   Subtract the smaller from bigger  753 - 357 = 396  
          4.   Reverse the order of digits in the result.      693
          5.   Sum the numbers in steps 3 and 4              693 + 396 = 1089

The result is always 1089.  
Show the paper you had the prediction written on.

Note:  If the subtraction in step 3 is a 2-digit number then put a zero on the left side to make it a 3-digit number.

Game 2:   1.  Choose a 3-digit numer         425
                2.  Repeat the number to make it a 6-digit number.   425425
                3.  Divide the number by 7.    425425/7 = 60775
                4.  Add 715 to the result.         60775 + 715 = 61490
                5.  Divide by 11.               61490/11 = 5590
                6.  Subtract 65.                5590 - 65 = 5525
                7.  divide by 13.             5525/13 = 425

You have recovered your original number!!

Game 3:   1.  Choose a 4-digit number.            4798
                2.  Add digits.                4 + 7 + 9 + 8 = 28
                3.  Subtract sum of digits from the original no.  4798 - 28 = 4770
                4.  Cross one digit (except 0)  from the answer   4*70
                5.  Read out the remaining digits    4, 7, 0

In your head add the digits and subtract the sum from the next number that is divisible by 9.  
                      11; subtract 11 from 18 to get 7
                7.  The result is the deleted number.    

Game 4:   1.  Choose a 3 digit number (not all identical digits)     782
                2.  Form the biggest & smallest no. for these digits    872 and 278
                3.  Subtract smaller no. from the bigger number      872 - 278 = 596
                4.  Form the biggest and smallest no.   965  &  569
                5.  Subtract smaller no. from the bigger no.   965 - 569 =   396
                6.  Continue with steps 4 and 5 until you reach 495.
                7.  You has reached a limiting value that does not change anymore. 
You might need up to six steps to reach the limiting value.   

Game 5:   1.  Choose a 4 digit number (all digits not the same)     7821
                2.  Form the biggest & smallest no. for these digits    8721 and 1278
                3.  Subtract smaller no. from the bigger number  8721 - 1278 = 7443
                4.  Form the biggest and smallest no.   7443  &  3447
                5.  Subtract smaller no. from the bigger no.   7443 - 3447 =   3996
                6.  Continue with steps 4 and 5 until you reach 6174.
                7.  You has reached a limiting value that does not change anymore.
You might need up to 7 steps to reach the limiting value.

Game 6:   1.  Choose a 4-digit numer         7136
                2.  Repeat the number to make it an 8-digit number.   71367136
                3.  Add 6570 to this number.    71367136 + 6570 = 71373706
                4.  Divide this result by 73.         71373706/73 = 977722
                5.  Subtract 90 from this number.             977722 - 90 = 977630
                6.  divide by 137.       9777630/137 = 7136
You have recovered your original number!!

Game 7:  On a piece of paper write 18 and put the paper away. 
Now ask your audience to  

                1.  Choose a 2-digit number with different  digits   51
                2.   Reverse the order of digits      15
                3.   Subtract the smaller from bigger  51 - 15 = 36
                4.   Reverse the order of digits in the result.      63
                5.  Sum the numbers in steps 3 and 4              63 + 36 = 99
                6.  Sum the digits of the number obtained     9 + 9 = 18
The result is always 18.  
Show the paper you had the prediction written on.

Game 8:   1.  Choose a 2-digit numer         28
                2.  multiply the no. by 37       1036
                3.  Subtract 357       1036 - 357 = 679
                4.  Multiply result by 13   679 x 13 = 8827
                5.  Add 2587             8827 + 2587 = 11414
                6.  Multiply by 3                11414 x 3 = 34242
                7.  Add 6162             34242 + 6162 = 40404
                8.  Multiply by 7        40404 x 7 = 282828

You have recovered your original number repeated 3 times!!

Game 9:   1.  Choose a 2 digit number (both digits not the same)     38
                2.  Form the biggest & smallest no. for these digits    83 and 38
                3.  Subtract smaller no. from the bigger number  83 - 38 = 45
                4.  Form the biggest and smallest no.   45 and 54
                5.  Subtract smaller no. from the bigger no.   54 - 45 = 9  
                     If result is a single digit then put a zero on its left
                6.  Form the biggest and smallest no.   90 and 09
                7.  Subtract smaller no. from the bigger no.   90 - 09 = 81
                8.  Continue with steps 4 and 5.  You will find that you are 
                     repeating a string of numbers in the following order 
                     9, 81, 63, 27,45, 9, 81, 63, 27, 45, 9, .... 

Game 10:   The Amazing Number 1729 - Also known as the taxicab number.  
The story goes that Ramanujan, the mathematical prodigy, was in the hospital when his professor Hardy came to visit him.  When Ramanujan asked what the taxi number was - Hardy replied that it was an ordinary number 1729.
Ramanujan then said that 1729 is an amazing number because it has the following properties.

(a)   It is the smallest number that is equal to the sum of cubes of 
       two different sets of positive numbers  
             1729 = 1^3 + 12^3  = 9^3 + 10^3

(b)  If you add the digits of 1729, you obtain 19
      
      Now multiply 19 by its reverse i.e., 91 and you would obtain 19 x 91 = 1729
     Only three other numbers have this property.  
     They are   1, 81 and 1458

Try 1458 to see that it works.

Final Note:   I have algebraic proofs of the above games to show how they work.  If you are interested write to me at ektalks@yahoo.co.uk and I shall send you the proof you are interested in.

Enjoy!

Additive and Multiplicative 3X3 Magic Squares - Construction and Some Not So Well Known Properties

I had discussed a couple of party games with 4x4 magic squares that have proved really popular among friends and in gatherings. Magic squares have fascinated people for thousands of years - they have an aura of mysticism and intrigue that is irresistible.

Wiki's article on magic squares comes with a lot of historic background etc. (also see 1, 2).  In this blog, I wish to concentrate on the additive 3x3 magic squares - particularly on their construction and its less well known relative - the multiplicative magic square (MMS), and describe some other interesting variations. I shall end this blog with a discussion of the majestic 16x16 magic square constructed by Benjamin Franklin more than 200 years ago.

Let us look at magic squares with sequential numbers.

 A general way to find the number in the central cell is described in the next slide

 I shall now discuss two variations of the magic squares that are not well known, but have really interesting properties.

The Multiplicative Magic Square:  In the magic squares that we have considered so far, the numbers in each row, column and diagonal add up to the same value. In a multiplicative magic square, the product of the numbers in each row, column and diagonal has the same value.

This is from Wiki:

Sum of Products of Rows and Columns in a Magic Square:  This is a property of magic squares that is relatively unknown.  It was analysed by Professor Hahn in 1975.  Essentially, in an additive magic square, the sum the products of numbers in each row is equal to the sum of products of numbers in each column. 
Hahn shows, in a rather formal looking paper, that this property is always true for a 3x3 magic square but only holds for some (balanced) 4x4 and higher order magic squares. I refer you to the paper that is available online to read.

A second important point here is that the sum of products of numbers in rows or columns is not equal to the sum of products of numbers in the diagonals. 

I could algebraically prove these results for a general 3x3 magic square but the calculation is too long and tedious to present here.
I give some examples in the following:

The Majestic 16x16 Magic Square:  Professor Bill Richardson has described this magic square that was constructed by Franklin more than 200 years ago - without the advantage of computers!!  I refer you to the 1991 publication for all the details.  In the following is a brief summary:
The 16x16 magic square is shown in the slide:

https://en.wikipedia.org/wiki/Magic_square

https://ektalks.blogspot.com/2016/03/variations-on-magic-squares-interesting.html

Obesity in Children - Weight at Ages 3 to 6 is Critical in Determining Obesity in Later Life

A year ago, I had reviewed Obesity that has reached epidemic proportions globally and is projected to increase significantly in the future.  Obese adults find it almost impossible to reduce weight, and are at greater risk of many illnesses like high blood pressure, diabetes, gout, heart disease, strokes, some cancers, joint problems, osteoarthritis, breathing problems, problems with sleep, gallstones and more. The association between obesity and depression is also well established. 

It is widely appreciated that an overweight child is more likely than a child of normal weight to be obese as an adult. Childhood obesity has been increasing an an alarming rate with a 10 fold increase in the past 40 years and this does not bode well for the global health in the future.    

A study of over 51000 children  has looked at the dynamics of BMI changes in children and adolescents and come to some interesting conclusions that your weight between ages of 3 and 5 is an important determinant of your adult weight.  This study may pave the way to control the growing scrouge of obesity in the world. 

I present a slide to define BMI before discussing the results of the study. 

  
We shall assume that 95th percentile defines the boudary of obesity.  The chart shows that a 20 year old with BMI of 30 (95th percentile) is obese, while a child at age 10 with BMI of 20 is in the 95th percentile and may be classified as obese. Children in the 5th percentile are underweight. 

The body-fat content of a healthy full-term infant rises from 10-14% at birth to 25-30% at 6 months of age - with a consequent increase of 30% in BMI. This is to support normal physical body and brain development.  After peaking at 6 to 12 months of age, BMI declines to a low value at 5 to 6 years of age and then rebounds (adiposity rebound) - rising throughout late childhood and adolescence.

I refer you to the study for detailed analysis but present their results in the following.

1.  90% of children obese at 3 years of age were overweight or obese in adolescence.
2.  Among obese adolescents, greatest acceleration in annual BMI increments occurred between 2 and 6 years of age.
3.  53% of obese adolescents had been overweight or obese from 5 years of age onwards with BMI increasing further with age.
4.  Most of the adolescents with normal weight had always had a normal weight throughout childhood.

Points 1 and 4 are crucial results that indicate that your weight as a young child (age 3 to 6 years) predisposes you to a particular body weight as an adolescent and, it might be safe to state, also as an adult.

How to control childhood obesity:  From slide 1, one notes that obesity started its upward march from about 1970.  Lifestyle has changed a lot since then and one wonders if there is some serious relation between these two.  My previous blog has looked at some sensible steps that one can take to control obesity increase.  The present study highlights the need to act when the child is between 3 and 6 years of age.  Advice about nutrition is fine but it is only a small part of the overall picture.  Regular exercise is very important in maintaining good body weight.
Use of hand-held electronic devices must be rationed for children as they encourage sedentary habits.  Sleeping times that synchronize with circadian rhythm might be something to work on as this could help in weight control and fostering general health.

Parents are too busy these days to generally provide good wholesome care for the children. 
I would be very interested in your ideas about how to control weight gain in young children.   
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http://ektalks.blogspot.com/2017/10/the-disease-called-obesity-what-is-it.html

Perimeter and Area of Regular Polygons - From Triangles to Circles - The Iso-Perimetric Theorem

Index of Blogs    Who Am I?

The Iso-Perimetric Theorem (IP Theorem; iso - same perimeter) concerns with the relation between the perimeter and the area of two dimensional (2-D) surfaces - plane figures.  For a historic account click here. A rigorous proof of IP Theorem is complex (see also) - I shall present here proofs that only need school level mathematics. 

IP Theorem may be stated as follows:

a. Among all 2-D shapes with the same perimeter, a circle has the largest area; or equivalently - 
Among all 2-D shapes with the same area, a circle has the shortest perimeter.

A short 2 minute video on youtube has a demonstration of the IP theorem.

For polygons (2-D shapes with n sides; n ≥ 3), IP Theorem may be stated as:

b.  For an n-sided polygon (fixed n; n-gon) with the same perimeter, the regular polygon (all sides of the same length) has the largest area.
c.  Among all regular polygons (any n) with the same perimeter, the one with the largest number of sides (largest n) has the largest area.
A circle may be considered a regular polygon approaching infinite number of sides and has the largest area for a given perimeter (points b and c). Statement c is thus equivalent to statement a of the theorem. 

A corollary of the IP Theorem may be stated as

d.  For a regular polygon for arbitrary n, the ratio of its area to its perimeter is equal to half the radius of the inner circle (half its apothem).

About Polygons:  The following two slides list the properties of regular polygons.  They relate the area and perimeter of n-sided polygons to their basic parameters like side length, circumradius and the apothem.  Centre of a regular poygon is the point that is equidistant from all vertices of the polygon - it is the centre of the circumcircle.

(Click on a slide to see full page view, Escape to return to text)
Slide 1

Slide 2

Eq.7 gives us the ratio of the area A to the perimeter of a regular polygon.  For a given parameter P, the maximum value the ratio (A/P) may have, is r/2 and that is for a circle for which n tends to infinity and cos (𝜋/n) = 1. For all other values of n, area A of the polygon is less than Pr/2. I show this in the next slide


Slide 3

We can make an interesting observation, from the graph on this slide:  The ratio of the area and the perimeter of a regular polygon is equal to half the length of the inner circle radius (the apothem). This result is independent of the number of sides in the polygon. Length of the apothem y does increase with n according to eqns. 1 and 2; namely y = r cos(𝜋/n).


Slide 4

We also notice that the circle has the largest area for a given perimeter (A/P approaches 0.5 r as n tends to infinity).
In the above, we have proved statements a, c and d of the isoperimetric theorem. Next let us look at statement b.

Regular n-gons have the largest area for a given perimeter:  To understand this, we use a heuristic approach and look at triangles formed on a chord in a circle with the vertices of the triangles on the circumference. See the following two slides:


Slide 5

Slide 6

 The Equilateral Triangle:  We shall use algebra to show that 
for a triangle (n = 3), for a given perimeter an equilateral triangle has the largest area.  
I shall use the well known Heron's formula for area of a triangle.

Slide 7

Slide 8

The Square:  First we look at the general four-sided polygon (a quadrilateral) and show that the rectangle (all internal angles equal to 90 degrees) has the largest area for a given perimeter.  Then we use the 
 AM-GM theorem (the arithmetic mean of a set of numbers is always greater than or equal to their geometric mean) to show that of all rectangles, a square (a regular 4-gon; all sides are equal and angles = 90 degrees) has the largest area.
Here, I have followed the analysis by Martin Joseffson in Geometricorum 13, 2013, 17-21

Slide 9

Slide 10

An interesting historical remark is that the formula K = (a+c)/2 x (b+d)/2 was used by the ancient Egyptians to calculate the area of a quadrilateral, but it’s only a good approximation if the angles of the quadrilateral are close to being right angles. In all quadrilaterals but rectangles the formula gives an overestimate of the area, which the tax collectors probably didn’t mind!

AM-GM Theorem: I now show that the largest area for a rectangle is for a square when all four sides are equal. 
Slide 11

Notes:  
1.  I have considered convex polygons in the discussion.  For completeness, I present a slide to explain the difference between a convex and a concave polygon.

Slide 12

2.  I find the case when y = 2 very interesting and have prepared a slide to show the numbers for some polygons. Please refer to slides 1 and 2 for notations.
Slide 13

In a circle, the inner and circumradius are always the same.  

3. I have drawn all the slides in this blog.  You can use them but acknowledge the source of this blog as 
http://ektalks.blogspot.com/2018/09/perimeter-and-area-of-regular-polygons.html