Brainy Quote of the Day

Showing posts with label Geometry. Show all posts
Showing posts with label Geometry. Show all posts

Tuesday, November 12, 2019

Eratosthenes to Starfish...

Sir Isaac Newton's impact on Optics. Link below.
Topics: Geometry, History, Science, Research

Every day, we conduct science experiments, posing an “if” with a “then” and seeing what shakes out. Maybe it’s just taking a slightly different route on our commute home or heating that burrito for a few seconds longer in the microwave. Or it could be trying one more variation of that gene, or wondering what kind of code would best fit a given problem. Ultimately, this striving, questioning spirit is at the root of our ability to discover anything at all. A willingness to experiment has helped us delve deeper into the nature of reality through the pursuit we call science.

A select batch of these science experiments has stood the test of time in showcasing our species at its inquiring, intelligent best. Whether elegant or crude, and often with a touch of serendipity, these singular efforts have delivered insights that changed our view of ourselves or the universe.

Here are nine such successful endeavors — plus a glorious failure — that could be hailed as the top science experiments of all time.

Eratosthenes Measures the World
Experimental result: The first recorded measurement of Earth’s circumference

When: end of the third century B.C.

Just how big is our world? Of the many answers from ancient cultures, a stunningly accurate value calculated by Eratosthenes has echoed down the ages. Born around 276 B.C. in Cyrene, a Greek settlement on the coast of modern-day Libya, Eratosthenes became a voracious scholar — a trait that brought him both critics and admirers. The haters nicknamed him Beta, after the second letter of the Greek alphabet. University of Puget Sound physics professor James Evans explains the Classical-style burn: “Eratosthenes moved so often from one field to another that his contemporaries thought of him as only second-best in each of them.” Those who instead celebrated the multi-talented Eratosthenes dubbed him Pentathlos, after the five-event athletic competition.

That mental dexterity landed the scholar a gig as chief librarian at the famous library in Alexandria, Egypt. It was there that he conducted his famous experiment. He had heard of a well in Syene, a Nile River city to the south (modern-day Aswan), where the noon sun shone straight down, casting no shadows, on the date of the Northern Hemisphere’s summer solstice. Intrigued, Eratosthenes measured the shadow cast by a vertical stick in Alexandria on this same day and time. He determined the angle of the sun’s light there to be 7.2 degrees, or 1/50th of a circle’s 360 degrees.

Knowing — as many educated Greeks did — Earth was spherical, Eratosthenes fathomed that if he knew the distance between the two cities, he could multiply that figure by 50 and gauge Earth’s curvature, and hence its total circumference. Supplied with that information, Eratosthenes deduced Earth’s circumference as 250,000 stades, a Hellenistic unit of length equaling roughly 600 feet. The span equates to about 28,500 miles, well within the ballpark of the correct figure of 24,900 miles.

Eratosthenes’ motive for getting Earth’s size right was his keenness for geography, a field whose name he coined. Fittingly, modernity has bestowed upon him one more nickname: father of geography. Not bad for a guy once dismissed as second-rate.

The Top 10 Science Experiments of All Time, Adam Hadhazy, Discover Magazine

Wednesday, August 22, 2018

Euclid and Parallax...

Topics: Astronomy, Astrophysics, Education, Geometry, Mathematics

This is something that I don't think is well known. It's a good example of the power of mathematics to students that may not see its value, or practical application. Feel free to use it in class.


NASA: Stellar Parallax
Wolfram Math World: Triangle

Tuesday, August 29, 2017

Diabolo...

Image Source: Link below
Topics: Geometry, Mathematics, Optical Physics

A ring-shaped optical cavity has degenerate resonant modes, because clockwise and anticlockwise waves resonate at the same frequencies. The degeneracy can be lifted, and the frequencies split, by a perturbation such as a physical rotation or the presence of a molecule or nanoparticle. Typically, the frequency splitting is proportional to the perturbation’s magnitude, as illustrated in the top panel of the figure for a hypothetical complex-valued perturbation ε (that is, one that can affect both the light’s frequency and its phase). Because the plot’s shape resembles a yo-yo-like toy called a diabolo, the degeneracy has been dubbed a diabolic point. The mode splitting around a diabolic point is the basis for optical gyroscopes, and it’s been explored for other sensing applications.

There’s another type of degeneracy, called an exceptional point, where not only do resonant frequencies coincide but their resonant modes do too. In the case of the ring resonator, inserting reflectors to scatter light from the anticlockwise mode into the clockwise mode (but not vice versa) creates an exceptional point with a single resonant mode, the clockwise-traveling wave. Perturbing the system splits that mode into two resonances, each with a small admixture of the anticlockwise wave, and the frequency splitting scales with the square root of the perturbation magnitude, as shown in the bottom panel.

Exceptional points make for exceptional sensors
At just the right locations in parameter space, resonant frequencies are ultrasensitive to tiny changes.
Johanna L. Miller, Physics Today

Thursday, April 13, 2017

Equiangular Geometry...

Igor Balla for Quanta Magazine
Topics: Geometry, Mathematics, STEM, Research

Imagine a set of many lines as in a dream. The lines intersect at a point and radiate outward. There’s something perfect about the way they’re spaced that you can’t quite put your finger on. You start counting them, but before you can finish you wake up with a question hanging on the fringe of your mind: Just how many were there?

For at least 70 years, mathematicians have been trying to answer a question like that one. The sets of lines they’re interested in share a basic feature: Any two lines from the set intersect to form the same angle. Such sets of lines are called “equiangular.” Mathematicians want to know just how big those sets can get as you move past the 3-D space of our everyday experience and into higher dimensions.

Equiangular lines are much more than a curiosity — they’re an almost elemental way to think about geometry. Maximal constructions of equiangular lines often align perfectly with the vertices of highly symmetric shapes, which make them a way to discover the existence of those shapes in the first place. In addition, radiating equiangular lines would pass through the surface of a surrounding sphere at equidistant points. This property makes the lines important for so-called spherical codes, which have important applications in applied mathematics and computer science.

Last spring a team of mathematicians found the maximum number of equiangular lines possible in any dimension, given certain conditions. They proved that that number is much smaller than previous best estimates. Benny Sudakov, a professor of mathematics at the Swiss Federal Institute of Technology Zurich and one of the lead authors, credits the breakthrough to the wide range of mathematical techniques he and his coauthors were able to apply to the problem.

“It’s like when you’re cooking something, we suddenly found we had the right ingredients,” said Sudakov.

Quanta Magazine: A New Path to Equal-Angle Lines, Kevin Hartnett

Wednesday, July 6, 2016

Beauty and Symmetry...

Figure 13.8, the center vertical position for the unusual mirrored structure in the parliament building in Berlin
Jim Zuckerman on Composition: Symmetry
Topics: Geometry, Mathematical Physics, Quantum Mechanics, Theoretical Physics


DATE: Saturday, June 4, 2016
TIME: 2:00 PM-3:30 PM
VENUE: NYU Skirball Center for the Performing Arts
MODERATOR: John Hockenberry
PARTICIPANTS: Robbert Dijkgraaf, David Gross, Alan Lightman, Maria Spiropulu
From a bee’s hexagonal honeycomb to the elliptical paths of planets, symmetry has long been recognized as a vital quality of nature. Einstein saw symmetry hidden in the fabric of space and time. The brilliant Emmy Noether proved that symmetry is the mathematical flower of deeply rooted physical law. And today’s theorists are pursuing an even more exotic symmetry that, mathematically speaking, could be nature’s final fundamental symmetry: supersymmetry. Join some of the world’s preeminent scientists to explore the core role symmetry plays in our unraveling of nature’s deepest secrets—and catch a glimpse of profoundly important symmetries that may be awaiting us just over the horizon.

The Big Ideas Series is supported in part by the John Templeton Foundation.

Image Credit: Miles Verkade

Saturday, March 14, 2015

Pi Day and Einstein...

We're wearing them!
Topics: Blerd, Circle, Circumference, Geek, Geometry, Math, Nerd, Pi Day

...and yes as you can see, we have our official T-shirts!

Pi Day is celebrated on March 14th (3/14) around the world. Pi (Greek letter “π”) is the symbol used in mathematics to represent a constant — the ratio of the circumference of a circle to its diameter — which is approximately 3.14159.

Pi has been calculated to over one trillion digits beyond its decimal point. As an irrational and transcendental number, it will continue infinitely without repetition or pattern. While only a handful of digits are needed for typical calculations, Pi’s infinite nature makes it a fun challenge to memorize, and to computationally calculate more and more digits.

Ultimate nerd out: It's Albert Einstein's birthday! It's also the 100th anniversary of the General Theory of Relativity, that led to the discovery of Black Holes. Trivia: Black Holes was a subject - like quantum mechanics he couldn't bring himself to believe in, though his work contributed to both.

Info from Celebration Site: PiDay.org
Biography.com: Fascinating Facts About Pi Day & Birthday Boy Albert Einstein
NBC News Weird Science: Pi Day Hits a Milestone, Alan Boyle