Math as the Language of the Universe

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  • Member since
    October, 2013
Math as the Language of the Universe
Posted by Gujba.Phile on Thursday, October 31, 2013 9:56 PM

I appreciate the intent of revealing the all-encompasing nature of Math as the language for describing what we find wherever we look.  It is too true that with a limited comfort level for math, we are very easily intimidated by cosmic scales and relationships.

I was especially impressed that you interpreted the factors, and demonstrated in such detail the reciprocal relationships inherent in the Lorentz Transformation.  But then things fell apart in the detail on eccentricity of an orbit.

Just as movie producers employ a staff of "Continuity" observers to assure things like the placement of objects, time-indications on clocks, and so on are 'consistent' from one 'take' to the next, it is helpful to a reader if symbols representing specific values or properties of an illustrated relationship are kept consistent.

The discussion of Kepler's First Law lacks a similar level of instructional value.  The boxed illustration on page 28 presents a formula, but no labels within the diagram to impart meaning to the factors in the equation.  The  specific mention within the body of the text offers no direct elaboration of the terms, either.

The wording of the explanation on page 27 introduces terms that may themselves be unfamiliar, or non-intuitive, like 'semimajor axis'. 

So, if Mercury has an eccentricity of 0.20, the major axis of its orbit would have a length 5 times the spacing of its foci, but is that it's 'diameter'?

Then, help me with the illustration of Kepler's Second Law.  In the first illustration within the box, the arc segments 'A' and 'B' appear to be equal, as they should be if the orbit is circular.  But in the third illustration, they also appear to be equal!    Which results in a HUGE understatement of the area swept by the 'string' discussed in the text during the planet's passage along the arc 'B'.

Given the point of the illustration is to counter the misunderstanding illustrated as the second case, it might have been better to err on the side of exageration, and show an arc 'B' that was clearly twice as long as arc 'A'.  Yes, the picture reflects an increase in the angle subtended, but hardly an equality of swept area.

Don't be afraid to offer more of these.  I believe establishing the 'meaning' of math in young minds is a critical step in promoting access to careers in astronomy and the other sciences.  Making it plain that math is cool is of imense value to the entire human race.

 

  • Member since
    October, 2012
Posted by AstroBernd on Sunday, January 05, 2014 7:35 PM

SmileI want to reinforce Gujba's comments on the intent and general direction of the article. Well done!

But I also want to build on his comments regarding the definition of the terms used. That is a must when using mathematical equations to describe physical phenomena.

My main concern is caused by the box on p. 26 explaining some specific effects of the theory of special relativity. Although being called "equations", neither time dilation nor length contraction really are. They both miss the "equal" sign and the left-hand term. What is presented in the black boxes are only expressions from one side of each equation.

The equation for Time dilation should read:

t=t0/sgrt(1-v2/c2).    t0 being the time elapsed in the Observer's system, t being the (LONGER) time elapsed for the astronaut in the rocket, as v<c for all practical cases and hence the sqrt-term <1.

Length contraction should read:

d=d0*sgrt(1-v2/c2).   d being the distance/ size of the moving object, measured from the observers point of view. d0 being the size at rest (v=0).

Be aware that both equations are symmetrical with respect to who is the "moving" and who is the "resting" partner. The astronaut in the rocket would see the SAME time dilation and length contraction of the "Observer" system. That is what makes this theory counter-intuitive from a day-to-day perspective, but is a direct consequence of the physical assumption that the speed of light c is the SAME in all constantly moving systems, irrespective of their relative speed v.

There is certainly more to add, but I hope this clarifies a little bit.

Sincerely

Bernd

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