Humor & Trigonometry

“Da Ya Think I Am √(1 + tan²c)?”
– Rod Stewart

A bit of lightness in the world of math goes a long way, especially when it comes to mnemonics. After all, a little humor and creative wordplay never hurt anyone.

Neuberg’s Theorem

A captivating result from geometry:

  1. Construct squares outwardly on the sides of triangle △ABC.
  2. Use the centers O1, O2, O3 of these squares to form a new triangle.
  3. Next, construct squares inwardly on the sides of this new triangle.

The surprising outcome? The centers of these inward squares will perfectly coincide with the midpoints I1, I2, and I3 of the sides of the original triangle △ABC.

For a deeper dive into the proof, check out this resource:
Neuberg’s Theorem – Detailed Explanation.

How penguins see the world

Sometimes, shifting our view is the key to seeing things more clearly.
This quincuncial Antarctic-centered projection offers a fresh take on the South Pole, allowing us to appreciate Antarctica from a different perspective, as well as the vastness of the world’s oceans.

Read more about quincuncial maps.

“The World on a Quincuncial Projection”, from Peirce (1879) – Available as prints.

The Cyanometer: A Tool to Measure the Sky’s Blueness

In the 18th century, Swiss scientist Horace-Bénédict de Saussure invented the “cyanometer,” a simple yet poetic device— a circular chart with 52 shades of blue, ranging from white to dark blue, to measure the sky’s blue hues. Inspired by his love for the Alps, especially Mont Blanc, Saussure climbed to great heights to study the deeper blue skies at higher altitudes.

Saussure believed the sky’s color was influenced by atmospheric particles. He tested the cyanometer at different elevations, noting that the summit of Mont Blanc matched the 39th shade. Later, explorer Alexander von Humboldt set a new record of 46 during his Andean expeditions.

Although Saussure’s theory linking sky color to atmospheric moisture didn’t pan out, his invention captured imaginations. Though it faded from science, the cyanometer lives on as an artistic and symbolic nod to our curiosity about nature. Modern versions even track air quality while celebrating the beauty of the ever-changing sky.

Next time you look up, think of Saussure and his ingenious little tool!

Read more.

World Map on a Dodecahedron

With the holiday season approaching, here’s a fun and educational activity to enjoy with your kids. Assemble a three-dimensional world map by cutting and folding a single-piece dodecahedron template featuring a gnomonic projection by Carlos A. Furuti.
Download the PDF template here.
A simple and creative way to explore geography while spending quality time together.