Go figure: Nature's numbers are the keys to the cosmos

Posted by Jake Parks
on Monday, March 12, 2018

The Golden Ratio is an irrational number that often shows up in nature, appearing as an organized spiral pattern. The arms of the Whirlpool Galaxy (M51), as well as many other spiral galaxies, almost perfectly match the geometric shape produced by the golden ratio. //NASA/Hubble
Guest blog by Ben Palmer

Life is defined by numeric order. Our days, numbered by digits; our interactions, governed by probabilities; our universe, described by calculations. Am I making a Nostradamus-style prediction? Think again! Everywhere you look, the cosmos is alive with numbers. This very second, space is expanding, stars are dying, planets are orbiting, and you and I are breathing. All of these are behaviors that can be described by thousands of equations and a googol of numbers. Are they products of abstract manipulation? Mere symbols and numerals stitched together by centuries of human thought? Perhaps. But some special numbers make even mathletes quiver. Meet the not-so-hidden figures that call reality home.

1.618 — Good as gold

“The Golden Ratio” — 1.618 — possesses a Midas touch. Known as Phi in mathematics, this romanticized irrational number stakes a righteous claim as one of nature’s greatest figures. It stems from a simple premise: If the ratio of two quantities (A/B, where A is larger) is equal to the ratio of their sum over the larger quantity ([A+B]/A), then they are “in the golden ratio.” This is what we call Phi, equal to approximately 1.618.

But the lens of geometry reveals Phi’s true power. From early Euclidian doctrine to Penrose Tilling, centuries of geometric proofs unveiled strange patterns when Phi was charted, graphed, and visualized. But geometry isn’t just a human creation, it exists. And that’s where the magic happens. Armed with pictorial geometric concepts, people began looking at the real world through a different lens, and they found something shocking: Phi was everywhere. Flower petals, fruits, seashells, hurricanes, even DNA strands were discovered to obey Phi’s alluring golden proportions. A mathematical sensation, Phi became the Meryl Streep of numbers; it plays a starring role in almost everything — including astronomy.

Close to home, the Earth and the Moon share a golden ratio, with their dimensions forming a Phi-laced figure known as Kepler’s triangle. The planets of our solar system display characteristics of Phi in their orbital durations. RR Lyrae variables — a class of periodic variable stars — showcase Phi, expressed in the ratio between their primary and secondary brightness fluctuations. Even spinning (Kerr) black holes, which thrive in states of either heating up or cooling down, migrate between these states when the square of their spin times Phi equals the square of the black hole’s mass.

Backyard astronomers also routinely witness Phi’s power firsthand. Have you ever observed a spiral galaxy, or used a star chart with pentagram magnitude markers? Make sure to thank Phi, because the spiral arms and pentagram both showcase golden dimensions.

2.725 — Back(ground) to the future

To discover our next natural number, it took a singularity, cosmic inflation, and two guys with a (seemingly) broken radio telescope. In 1964, Arno Penzias and Robert Wilson thought Bell Lab’s Horn Antenna, which detected a constant and pervasive radio hiss, was going haywire. But as it turns out, two men’s static is really one of astronomy’s greatest revelations.

Today, we know 2.725 kelvins (-270 degrees Celsius, -454 degrees Fahrenheit) as the temperature of the Cosmic Microwave Background (CMB), the ubiquitous radiation remnant filling space itself.  The Big Bang’s living visage, the CMB throws back to the origins of the universe, when a point of infinite, fireballed density became our cherished cosmos.

By the 1960s, the theory war between Big Bang (an expanding universe that was born) and Steady State (an expanding universe that always was) reached a fever pitch. It boiled down to temperatures: A universe grown from fiery origins should exhibit a minute but measurable CMB — or a residual temperature — while a universe eternally expanding would present a 0 K temperature, no fiery birth, and no CMB. This number would be a figure in motion, constantly evolving and cooling under cosmic expansion.

Enter the Penzias and Wilson radio anomaly. Quickly determined to be cosmic in origin and uniform in distribution, the CMB’s measured temperature clocked in at just under 3 K, and has been refined over time to 2.725 K. The universe had been born. Courtesy of Penzias and Wilson, the road to the Cosmic Background Explorer COBE, ripples in space-time, and contemporary cosmology was set.

This map of the Cosmic Microwave Background, taken as part of NASA’s WMAP mission, shows the distribution of radiation leftover from the initial burst of the Big Bang. Over the past 13.8 billion years, the expansion of space-time has redshifted and cooled the radiation to just a few degrees above absolute zero. To calculate the angular dimension of unevenly cooled hot and cold patches, astronomers must rely on another natural number — pi. //NASA/WMAP Science Team
3.14 Easy as π

Everyone’s favorite irrational hardly needs an introduction. Pi, and its iconic leadoff digits (3.14159265…), pervade the heart of science, as well as pop culture. With its catchy name, cult following, and appointed holiday, the ratio of a circle’s circumference to its diameter has certainly made an astrophysical name for itself.

Where there’s a sphere, there’s Pi; and spherical entities are abound in astronomy. Stellar circumference, telescopic mirrors, and planetary disks all manifest that sacred 3.14. Searching for exoplanets requires hearty slices of Pi, whose formulaic presence helps determine densities, and even the atmospheres of alien worlds.

At the Large Hadron Collider, Pi leaps from the particle realm, contributing a numeric weight factor (1/16π2) to loops — a process where original particles shift to new particles, then return to their starting state. It even speaks the tongue of relativity’s equations, guiding how energy and momentum curve the fabric of space-time. And remember the CMB mentioned earlier? Pi thrives there too. In order to calculate the angular dimensions of hot/cold blotches in the microwave matrix, astronomers rely on a 4π ratio.

186,000 — Watch your speed

Up in the sky! It’s a bird, it’s a plane…but whatever “it” is, you can bet it’s not going faster than 186,000 miles per second. The speed of light in a vacuum, simply phrased, transforms the astrophysical and astrometric realms. Stargazers adore this cherished constant — and for good reason. Written simply as c, the speed of light helps us explore time dilation, length contraction, and mass-energy equivalency. At 299,792,458 meters per second, the speed of light establishes the heart of metric length, the meter. But whatever its denotation, light speed illuminates our cosmic condition more than any known value.

We usually view 186,000 from two angles: a yardstick and a speed bump. As a Rosetta stone of astrophysical measure, light speed contributions know no parallel. A key component of the light-year, c acts as both a surveyor and a time machine. From minute stellar distributions to the vast galactic web, the light-year allows all to envision, and marvel at the overreaching layout of the universe. The early cosmos, its age, appearance, and vivid evolution, would be impossible to comprehend without light’s finite speed.

Nothing reminds us to “know your limits” more than c. Under special relativity’s local laws, the physics police will ticket those who dare exceed the speed of light. Of course, under general relativity, distant galaxies race away in our expanding, dark-energy-laden cosmos, underscoring the speed-breaking stretch of space-time. In our universe, everything from particles to pulsars illustrates light speed’s astronomical proportions. On the information superhighway, even the march of data conforms to light’s limit. Just think, every supercomputer circuit must obey the speed of light. Every ground controller knows “real-time” spacecraft communiqués are illusionary — the information passed between probe and scientist is delayed by light’s omnipresent limit. And while contemporary experiments continue to test light’s limits, for now, 100-plus years of relativity stands.

At first glance, our numbers seem to be products of human agreement — 2 is 2 and 5 is 5 because you and I agree, beyond question, upon their assigned values. But their higher calling, an unspoken truth, offers something more. What we see as digits is really the language of the universe. And this cosmic dialect is so beautifully universal, it holds the power to unite us all.

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