Sciences In the Mural Of Life

In the spring of 2016, I attended a STEM conference. Expanding the acronym as science, technology, engineering and math doesn’t shed too much light on the intentions and philosophy of STEM. The premise is that math, science and technology subjects should not be taught in isolation; there should be more integration and emphasis on applications. All of this is largely inspired by the job market’s need for a larger number of better-trained people in these specialised fields. It all seems reasonable as long as the approach is not taken to an extreme.

A society, regardless of its bent, functions best when a wide range of talents are cultivated, even if they seem to serve no practical purpose. Similarly, we have a healthier situation in schools and colleges when educators don’t sail on the same ship. There was at least one organiser at the conference who shared my views because a particular lecture went against the grain of STEM. 95% of the auditorium featuring the lecture was empty and attended mostly by the speaker’s university students, a couple of bird-lovers and a little cluster of Canada Wide Science Fair attendees. Rudely, the latter group even walked out before it ended. But if you stick to the premise that attendance at public events is very often inversely proportional to its quality, you don’t worry about numbers.

Parentese or “baby talk“ is far from being just indulgence on the part of parents. In humans it helps draw attention from babies and promotes the learning of speech. Regardless of language, there are universal characteristics of parentese. Voice pitch is modulated; speech is slower, more repetitive and more attention-grabbing than adult-talk.

Babies-Zebra-Finch — Image of baby zebra finch from http://www.singing-wings-aviary.com/zebrafinches.htm

A similar situation arises in a small Australian bird known as the zebra finch. For those of you unfamiliar with the small bird, one of its distinguishing characteristics is its song, which is reminiscent of the squeaky sound of a rubber duckie. With their form of baby talk, adult zebra finches change their vocalisations when singing to young birds. They slow them down and use more repetition. The juvenile finches in return pay more attention to such songs than to those used between adults.In their young lives, they the simpler versions. With time, in the physical presence of adults, the chick’s song converges with that of their tutor. When zebra finches were isolated and exposed to mere recordings, they developed different songs.

The social interaction between tutor and chick stimulates communication between the midbrain’s ventra legmental area (VLA) and regions of the cerebrum. The VLA is partly a reward centre and uses dopamine. When humans acquire language, neural bridges of that type are also made. In case of the finch, the evidence for such a pathway comes from the fact that in the absence of tutor’s physical presence, a marker for gene expression of catecholamines (which include dopamine) remained inactive.

Another revelation which made my morning was that the zebra finch researcher had originally majored in economics, reinforcing my notion that to get to an island you don’t have to board any specific boat.

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There’s something truly beautiful about the internal consistencies of mathematics. Thanks to its nature we get to hike on such a variety of paths to reach the same destination. Math becomes a mental traveler’s paradise.

Consider the trigonometric addition formulas:

sin(x + y) = sin(y) cos(x) + sin(x) cos(y) and

cos (x + y) = cos (x) cos(y) – sin(x) sin(y) .

These lead to highly useful double angle formulas, which are needed to evaluate integrals. They are also used in Fourier analysis and in communication systems. How do we derive them? I’ll first explore a slightly more difficult route. Secondly we will briefly refer to a conventional geometrical approach and finally we’ll examine my own offbeat path (but I can’t guarantee it’s original) .

(1) A circuitous route using slightly higher mathematics

Knowing the rate of change of sine, cosine and natural base functions allows one to express sin(w), cos(w) and e^w as power series. Replace the w with xi, where i is the square root of -1, the so-called imaginary number, and e^xi turns out to be the sum of the cosine series and i times the sine series, namely Euler’s relationship:

e^xi = cos (x) + i sin(x).

If we subsequently use the above to express e^i(x+y) we obtain

e^i(x+y) = cos (x + y) + i sin(x + y). equation(1)

But using exponent laws, e^i(x+y) = e^xi (e^yi ) , and then applying Euler’s formula again:

e^i(x+y) = e^xi [e^yi ] = [cos(x) + i sin(x) ] [cos(y) + i sin(y) ] and expanding the right hand side:

e^i(x+y) = cos(x) cos(y) + i sin(y) cos(x) + i sin(x) cos(y) – sin(x) sin(y).

If we regroup the above by pairing and separating the real components from the imaginary ones :

e^i(x+y) = cos(x) cos(y) – sin(x) sin(y) + i [sin(y) cos (x) + sin(x) cos (y)]. equation(2)

we can compare equations (1) and (2) and have to conclude that:

cos(x + y) = cos(x) cos(y) – sin(x) sin(y).

sin(x + y) = sin(y) cos(x) + sin(x) cos(y).

(2) A Geometric Approach

Elementary math can be used if we begin with the following diagram. The details of this particular derivation are found at the site indicated in the caption.

TrigAnglesWeisstein_1000 — http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html

(3) A simpler off beat path from working backwards

After deriving the Sine Law from simple trig ratios, I subsequently thought of using the same diagram to make substitutions into the sine addition formula. Surely enough, after some manipulations, I obtained the Sine Law, implying that the latter can be used to derive the other.

A derivation of the sine addition formula

By using the same approach with the cosine addition formula, I obtained another unexpected starting point. Apologies for not typing it out. The convenience of simply photographing my handwritten versions proved to be too tempting!

Sciences In the Mural Of Life

Zebra Finches At a STEM Conference

One More Path To Trigonometric Addition Formulas