history, photography, STEM

Beyond Measuring the Earth

Geometry began with practical measurements over moderate distances.  Boundaries of Egyptian farmers’ fields had to be restored after the Nile’s annual floods.  A taut rope between two posts marked where an edge of the base of a pyramid would be laid.  And so on.  This prosaic technology inspired ancient Greeks to create something weird and wonderful.
 
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People like Pythagoras and Euclid reimagined the pyramid builders’ rope as perfectly straight (not sagging a little), so thin that it had no thickness at all, and extending forever beyond the posts.  Crazy.  They called it a “line” and found that they could reason about such things, proving new statements by deductions from what they already knew.

Those ancient geometers discovered much that was true and good and beautiful in the imagined world of points and lines, and a few of them took the first tentative steps toward using their discoveries to help answer questions about the experienced world of posts and ropes and much else.  Eratosthenes kept the promise made by “geo”+”metry” when he measured the circumference of planet Earth, even tho it was impractical to try to wrap a tape measure around it.

Modern STEM is rooted in ancient geometry (among other things), and a long hard slog has progressed from measuring the Earth to understanding it.  Our understanding is not perfect and never will be, but maybe it is good enough to help us save the Earth.  From us.  I hope we can rise to that challenge, and that I have risen to this one:

Geometry ~ Pic and a Word Challenge #269

Pythagoras_BlueMarble_840x842

Image Sources

  • The colorful frame around the image is upsized from my much smaller diagram for Bhaskara’s elegant proof of Pythagoras’ Theorem.  The resulting fuzziness of the points and line segments is a reminder that we cannot experience the ideal perfection of geometric shapes.  But we can refer to the shapes when we tell each other stories about what we experience!  (Tho often hard to read w/o wrangling equations, scientific theories are among the best stories we can tell.)  The colors of the line segments tie the image to the theorem’s bottom line w/o using letters that would clutter the diagram:
      a² +  b²  c²
  • The Blue Marble image overlaid on the diagram was downloaded from NASA Visible Earth: The Blue Marble.   Making NASA’s image cost a lot more than making mine.  That’s OK.  It was money well spent.

haiku, history, humor, math, politics

Bhaskara for President!

Fooey.  He has hardly any name recognition, was not born a US citizen, and has been dead for centuries.  Being more reality-oriented than those who handed Donald Trump the job, I cannot seriously promote Bhaskara.  What a pity.  His elegant old proof helps me stay sane in the Age of Trumpery.
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Who’s Bhaskara?  We will get to that question shortly.  First, consider whatever gadget U are using to read this post.  It depends on many things, discovered over many years by many people who (unlike many pols) preferred building up to tearing down.  With many steps omitted (and “depends on” abbreviated to ), a few of those dependencies go like this:

Your Gadget quantum physics coordinate systems Pythagoras’ Theorem

Back in high school, Pythagoras’ Theorem may have seemed like a little fact about right triangles that may have been mildly interesting but did not deserve the effort of slogging thru the book’s tedious proof.  I could read the proof line by line, observe that it was valid, and be glad that I never needed to retrieve it for a test.  Hardly anybody could remember it for more than a few minutes.

Pythagoras’ Theorem turned out to be essential to blogging (and much else), so it would be nice to have a proof that mere mortals could remember, appreciate, and be inspired by.  Enter Bhaskara, 1114-1185.

Bhaskara replaced the usual picture (of 3 squares glued to the sides of 1 triangle) with a picture of 4 copies of the same triangle, arranged to form a big square with a little square inside it:

Pythagoras
(a+b
=
4 · ( ½ · a · b) + c²

The proof is sometimes displayed more tersely, with just the figure.  I prefer to write out a little algebra (while not belaboring why the angles do add up the way the figure suggests).  Tho he did not have modern notation, Bhaskara did have an elegant way to provide more detail for the mathematically fastidious.  He displayed another figure that also puts the 4 copies of the triangle inside a big square with sides a+b.  In the other figure, the area not covered by copies of the triangle amounts to a²+ b² because it consists of 2 small squares.  But the not-covered area amounts to c² in the figure displayed above, so we can conclude that

  a²+ b² = c²

w/o bothering with algebra and how to compute areas of right triangles.  We just need to bother with drawing both figures.  Wanna try your hand at drawing the other figure?  U can find the answer by following the link provided by Sieglinglungenlied in the comment section.

Googling reveals some variation in what is attributed to Bhaskara. The 1-figure proof I displayed appears in several places (sometimes attributed to Bhaskara and sometimes w/o attribution).  A similar 1-figure proof is commonly attributed to Bhaskara, with a big square of length c.  The 2-figure version that avoids algebra is attributed to Bhaskara in Math in 100 Key Breakthroughs, a nicely illustrated book by Richard Elwes.  Historical accuracy is not crucial at the moment, so I went with the best story w/o worrying about who got it right.

OK, I admit that having written a proof of mind-blowing elegance does not really qualify Bhaskara to be POTUS.  Too bad that many people think mind-blowing arrogance can hack it.

Clicking on the “politics” category or tag in this post will display all my uses of acidic humor to cope with the current state of US politics.  But acids are corrosive.  Sometimes, I forgo acid and contemplate some of the enduring (so far) glories of modern Western civilization, one of which is that it is not exclusively Western.  In particular, we got some elegant math from India and some elegant poetry forms from Japan.

One Way to Stay Sane in the Age of Trumpery
|Cherish all that is
|true and good and beautiful
|(like Bhaskara’s proof).