Liberal Arts Blog — Geometry (I): Ode to the Triangle

Liberal Arts Blog — Monday is the Joy of Math, Statistics, and Numbers Day

Today’s Topic — Geometry (I): Ode to the Triangle (and triads in general)

Triangles are magic. Simplest shape. Huge power. Infinite implications. The Eiffel Tower is a network of triangles. The inclined plane is a triangle. All angles of any triangle add up to 180 degrees! How is that possible! And then the square of the hypotenuse of a right triangle is equal to the square of the other two sides! How crazy is that? Thales calculated the height of the pyramid at Giza using the principle of the similarity of triangles. Do you know how to balance a triangle? Experts — please chime in. Correct, elaborate, elucidate.

THE SIMILARITY OF TRIANGLES SOLVES A MYSTERY

1. How high is the pyramid at Giza? Thales harnessed the sun and his own shadow to do the calculation.

2, Waited until the time of day when the length of his shadow equaled his height.

3. Then, at the same time, he measured the length of the shadow of the pyramid.

NB: At that moment, the length of the pyramid’s shadow (plus half the base of the pyramid) was equal to its height. Wow!

HOW TO BALANCE A TRIANGLE — the centroid

1. Draw a line from the mid-point of each side to the opposite angle.

2.The point of intersection is the “centroid” — the point at which the triangle balances.

3. Don’t even need to draw three of the lines. Two will suffice.

NB: Try it with a piece of paper.

PYTHAGOREAN TRIPLES — (3,4,5) or (5,12,13) or (7,24,25) or (8,15,17)

1. Positive integers (a,b,c) for which it is true that “a squared” + “b squared’ = “c squared”

2. Works for any multiple of these — eg. (6, 8, 10).

3. Oldest example goes back to a Babylonian clay tablet from 1800 BC.

NB: To me the triangle is the perfect metaphor for the power of the number 3 in general. Think the “rule of three” for public speakers. Or the “rule of thirds” for artists and photographers. Or the Welsh or Irish triads of medieval manuscripts such as: “Three things not easily restrained, the flow of a torrent, the flight of an arrow, and the tongue of a fool.”

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

Pythagorean triple

https://en.wikipedia.org/wiki/Rule_of_three_(writing)

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

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

YOUR TURN

Any thoughts on triangles?

Please share the coolest thing you learned this week related to math, statistics, or numbers in general.

Or, even better, the coolest or most important thing you learned in your life related to math.

This is your chance to make someone else’s day. And to consolidate in your memory something you might otherwise forget. Or to think more deeply than otherwise about something dear to your heart. Continuity is key to depth of thought.