June 15, 2018 Friday
Bedtime Story
On 3-4-5 Triangles
Last night we were talking about the
Egyptian professional rope-stretchers who acted like the surveyors whose main
work in today’s world is to estimate the terrestrial three-dimensional
positions of points and the angles and distances formed between them.
These rope-stretchers had to ensure that the
rope was stretched taught so that it did not sag, and thereby leading to errors
in calculations.
Many of us may not be aware but modern
surveying has a strong basis on both mathematics and sciences relying upon
elements of geometry, trigonometry, regression analysis (statistical process
for estimating the relationships between the variables), physics, engineering,
metrology (not, it is not the science of meteors but rather the science of
measurements), programming languages and the law.
Very much like their modern counterparts,
they made use of the 3-4-5- triangles and the plummet for their measurements.
A 3-4-5 triangle is a special ‘side-based’
right-angle triangle in which the sides of such triangles form ratios of
specific whole numbers of 3 : 4 : 5.
These set of three numbers also go by the
name of Pythagorean triple.
The 3-4-5 triangle is unique in that it is
the only form of right-angle triangle with sides in arithmetic progression.
Moreover, the triangles that are based on
Pythagorean triples are Heronian, meaning triangles whose side lengths and
areas are both integers.
This peculiar name of such triangles come
from the Alexandrian mathematician by the name of Hero who was born in 10 AD and
is most well known in mathematics for his formula for calculating the area of a
triangle.
The beauty of this formula that goes by the
name of Hero’s formula or more often Heron’s formula is that this area
calculation does not depend on choosing any arbitrary side to be base or height
which is commonly known to all of us as the product of half times the base times
the height.
For a triangle with the sides a, b and c,
Heron’s formula for the area of the triangle is:
A = √s(s-a)(s-b)(s-c)
Where s is the semiperimeter of the
triangle and calculated as
s
= (a + b + c)/2
I shall not go into the proof of it,
suffice it is to know the reason why certain kind of triangles have been named
after him.
In a way, these surveyors of the past were
too using the principle of parsimony while using taught knotted ropes and
plummets.
Yet, the person who most clearly enunciated
this law was, or at least to who the Western world gives credit to, is the
French mathematician Pierre Louis Maupertuis.
We shall look into the life and the work of
this French mathematician Maupertuis in the nights to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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