Full Wolf Moon: New Year’s Supermoon Is the Biggest of the Year

 Credit: Shutterstock.com


New Year’s Day is a time for resolutions and hangovers, but this year, it also provides a chance to see the moon in all its glory.

The first day of 2018 brings a “Full Wolf Moon” — the biggest of two supermoons that will rise in January.

Skywatchers, take note! On Jan. 1, 2018, Earth will be closest to the moon at 4:54 p.m. EST (2154 GMT), according to EarthSky.org. The moon will be full at 9:24 p.m. EST (0224 GMT on Tuesday Jan. 2).

A supermoon occurs when the moon is at perigee — its closest point to Earth in its monthly orbit — around the same time as a full moon. The moon looks slightly larger and brighter than average at these times.

The moon that will be visible on New Year’s Day will appear bigger than usual, but most people will not notice the difference. However, thanks to a phenomenon called the “moon illusion,” the moon may appear bigger when it’s close to the horizon, so this New Year’s supermoon may be most impressive when it’s rising.

The Full Wolf Moon gets its name from the hungry wolves that would howl outside Native American villages during these January full moons, according to the Farmer’s Almanac. However, some people argue that the names for the full moons actually come from Anglo-Saxon culture, according to timeanddate.com.

In any case, wolves do not howl more during the supermoon, studies show. In fact, wolves do not howl at the moon at all, but rather at each other to communicate, according to “The Wolf Almanac, New and Revised: A Celebration of Wolves and Their World” (Lyons Press, 2007). [6 Wild Ways the Moon Affects Animals]

Many supermoon myths have tied the unusually bright celestial object to a variety of spooky outcomes. Some believe supermoons make people go crazy. Others claim supermoons trigger natural disasters. The vast majority of these supposed effects (such as increased emergency-room visits) have not been borne out by studies, Live Science previously reported.

Though the moon will be at its closest point to Earth on Jan. 1, the moon will gradually appear bigger and brighter over the coming days. And those who were too busy working out or still nursing a hangover on the evening of Jan. 1 shouldn’t despair: Another supermoon (this one a Blue Moon, or the second full moon in a calendar month), will occur Jan. 31, Space.com reported. That later supermoon will also be involved in a total lunar eclipse, in which the Earth’s shadow totally covers the moon, making it a Blood Moon as well.

Editor’s Note: This article was updated to remove a reference to the Full Wolf Moon being 11 percent larger than average. That is how much much bigger the Blue Moon at the end of January will be. 

Originally published on Live Science.

Metaphysical Alliance “AIDS Survivors and Thrivers”

Published on Dec 30, 2017

Metaphysical Alliance AIDS Survivors and Thrivers monthly AIDS Healing Service at the Unitarian Church in San Francisco on January 27, 1987 featuring long-term survivors Ron Carey, Bobby Reynolds, and Dan Turner. Also featuring Mike Zonta as MC, Irene Smith’s closing meditation and David Hummel of the AIDS Interfaith Network.

The videographer was Richard Locke. This video was converted from a 30-year-old VHS tape so the quality is not as up to par as it might be. But it is a historically important video exemplifying the late ’80s San Francisco gay and AIDS community at its best.

It’s a little slow in the beginning, but it gets better and concludes with a powerful closing meditation by Irene Smith. People were so moved they didn’t want to go home.

The Story Behind Dylan Thomas’s “Do Not Go Gentle Into That Good Night” and the Poet’s Own Stirring Reading of His Masterpiece

“Rage, rage against the dying of the light.”


The Story Behind Dylan Thomas’s “Do Not Go Gentle Into That Good Night” and the Poet’s Own Stirring Reading of His Masterpiece

“Poetry can break open locked chambers of possibility, restore numbed zones to feeling, recharge desire,” Adrienne Rich wrote in contemplating what poetry does“Insofar as poetry has a social function it is to awaken sleepers by other means than shock,” Denise Levertov asserted in her piercing statement on poetics. Few poems furnish such a wakeful breaking open of possibility more powerfully than “Do not go gentle into that good night” — a rapturous ode to the unassailable tenacity of the human spirit by the Welsh poet Dylan Thomas(October 27, 1914–November 9, 1953).

Written in 1947, Thomas’s masterpiece was published for the first time in the Italian literary journal Botteghe Oscure in 1951 and soon included in his 1952 poetry collection In Country Sleep, And Other Poems. In the fall of the following year, Thomas — a self-described “roistering, drunken and doomed poet” — drank himself into a coma while on a reading and lecture tour in America organized by the American poet and literary critic John Brinnin, who would later become his biographer of sorts. That spring, Brinnin had famously asked his assistant, Liz Reitell — who had had a three-week romance with Thomas — to lock the poet into a room in order to meet a deadline for the completion of his radio drama turned stage play Under Milk Wood.

Dylan Thomas, early 1940s

In early November of 1953, as New York suffered a burst of air pollution that exacerbated his chronic chest illness, Thomas succumbed to a round of particularly heavy drinking. When he fell ill, Reitell and her doctor attempted to manage his symptoms, but he deteriorated rapidly. At midnight on November 5, an ambulance took the comatose Thomas to St. Vincent’s Hospital in New York. His wife, Caitlin Macnamara, flew from England and spun into a drunken rage upon arriving at the hospital where the poet lay dying. After threatening to kill Brinnin, she was put into a straitjacket and committed to a private psychiatric rehab facility.

When Thomas died at noon on November 9, it fell on New Directions founder James Laughlin to identify the poet’s body at the morgue. Just a few weeks later, New Directions published The Collected Poems of Dylan Thomas (public library), containing the work Thomas himself had considered most representative of his voice as a poet and, now, of his legacy — a legacy that has continued to influence generations of writers, artists, and creative mavericks: Bob Dylan changed his last name from Zimmerman in an homage to the poet, The Beatles drew his likeness onto the cover of Sgt. Pepper’s Lonely Hearts Club Band, and Christopher Nolan made “Do not go gentle into that good night” a narrative centerpiece of his film Interstellar.

Upon receiving news of Thomas’s death, the poet Elizabeth Bishop wrote in an astonished letter to a friend:

It must be true, but I still can’t believe it — even if I felt during the brief time I knew him that he was headed that way… Thomas’s poetry is so narrow — just a straight conduit between birth & death, I suppose—with not much space for living along the way.

In another letter to her friend Marianne Moore, Bishop further crystallized Thomas’s singular genius:

I have been very saddened, as I suppose so many people have, by Dylan Thomas’s death… He had an amazing gift for a kind of naked communication that makes a lot of poetry look like translation.

The Pulitzer-winning Irish poet and New Yorker poetry editor Paul Muldoon writes in the 2010 edition of The Collected Poems of Dylan Thomas:

Dylan Thomas is that rare thing, a poet who has it in him to allow us, particularly those of us who are coming to poetry for the first time, to believe that poetry might not only be vital in itself but also of some value to us in our day-to-day lives. It’s no accident, surely, that Dylan Thomas’s “Do not go gentle into that good night” is a poem which is read at two out of every three funerals. We respond to the sense in that poem, as in so many others, that the verse engine is so turbocharged and the fuel of such high octane that there’s a distinct likelihood of the equivalent of vertical liftoff. Dylan Thomas’s poems allow us to believe that we may be transported, and that belief is itself transporting.

“Do not go gentle into that good night” remains, indeed, Thomas’s best known and most beloved poem, as well as his most redemptive — both in its universal message and in the particular circumstances of how it came to be in the context of Thomas’s life.

By the mid-1940s, having just survived World War II, Thomas, his wife, and their newborn daughter were living in barely survivable penury. In the hope of securing a steady income, Thomas agreed to write and record a series of broadcasts for the BBC. His sonorous voice enchanted the radio public. Between 1945 and 1948, he was commissioned to make more than one hundred such broadcasts, ranging from poetry readings to literary discussions and cultural critiques — work that precipitated a surge of opportunities for Thomas and adrenalized his career as a poet.

At the height of his radio celebrity, Thomas began working on “Do not go gentle into that good night.” Perhaps because his broadcasting experience had attuned his inner ear to his outer ear and instilled in him an even keener sense of the rhythmic sonority of the spoken word, he wrote a poem tenfold more powerful when channeled through the human voice than when read in the contemplative silence of the mind’s eye.

In this rare recording, Thomas himself brings his masterpiece to life:

For more beloved writers reading their own work, see Mary Oliver reading from Blue Horses, Adrienne Rich reading “What Kind of Times Are These,” J.R.R. Tolkien singing “Sam’s Rhyme of the Troll,” Frank O’Hara reading his “Metaphysical Poem,” Susan Sontag reading her short story “Debriefing,” Elizabeth Alexander reading “Praise Song for the Day,”, Dorothy Parker reading “Inscription for the Ceiling of a Bedroom,”and Chinua Achebe reading his little-known poetry.

The Binary Code of Body and Spirit: Computing Pioneer Alan Turing on Mortality

“The body provides something for the spirit to look after and use.”

(BrainPickings.org) The Binary Code of Body and Spirit: Computing Pioneer Alan Turing on Mortality

“The void horrifies: so we are all immortal,” Simone de Beauvoir scoffed at the religious escapism of immortality in explaining why she is an atheist, adding: “Faith allows an evasion of those difficulties which the atheist confronts honestly.” But there exists a certain orientation of spirit that is both unreligious and lucid in contemplating mortality. Einstein touched on it in his beautiful letter to the Queen of Belgium, in which he wrote: “There is, after all, something eternal that lies beyond the hand of fate and of all human delusions.”And yet he conceded that such an orientation toward mortality is reserved for those “who have been privileged to accomplish in full measure their task in life.”

To make sense of the untimely loss of a young and unrealized life is a wholly different matter, one which haunted computing pioneer Alan Turing (June 23, 1912–June 7, 1954).

Young Alan Turing

Turing’s decryption of Nazi communication code is estimated to have shortened WWII by two to four years, consequently saving anywhere between 14 and 21 million lives. But despite his wartime heroism, Turing was driven to suicide after being chemically castrated by the U.K. government for being homosexual. More than half a century after his disquieting death, Queen Elizabeth II issued royal pardon — a formal posthumous apology that somehow only amplifies the tragedy of Turing’s life and death.

Tragedy had been with Turing from a young age. At fifteen, while attending the Sherborne School, he fell deeply in love with a classmate named Christopher Morcom. For the awkward and ostracized young Alan, who was bullied so severely that a group of boys once trapped him under the floorboards of a dorm dayroom and kept him there until he nearly suffocated, Christopher was everything he was not — dashing, polished, well versed in both science and art, and aglow with winsome charisma. Alan’s love was profound and pure and unrequited in the dimensions he most longed for, but Christopher did take to him with great warmth and became his most beloved, in fact his only, friend. They spent long nights discussing science and philosophy, trading astronomical acumen, and speculating about the laws of physics.

Alan Turing (far left) with classmates at Waterloo Station on the way to the school carriage at Charing Cross Station, early 1926 (Turing Digital Archive)

When Christopher died of bovine tuberculosis in 1930 — a disease he had contracted from infected milk, for which there was no common vaccine until after WWII — Alan fell to pieces. He was able to collect himself only through work, by burrowing so deep into the underbelly of mathematics that he emerged almost on the other side, where science and metaphysics meet. Sorrow had taken him on a crusade to make sense of reality, of this senseless ruin, and he spared no modality of thought. Most of all, he wanted to understand how he could remain so attached to someone who no longer existed materially but who felt so overwhelmingly alive in his spirit.

All the while, young Turing remained in touch with Christopher’s mother, who had taken a sympathetic liking to her son’s awkward friend. After Christopher’s death, he visited the Morcoms at their country home, Clock House, and corresponded with Mrs. Morcom about the grief they shared, about the perplexity of how a nonentity — for Christopher had ceased to exist in physical terms — could color each of their worlds so completely.

Alan Turing and Christopher Morcom. Art by Keith Hegley from The Who, the What, and the When, an illustrated celebration of the little-known inspirations behind geniuses.

That sorrowful puzzlement is what Turing explored in a series of letters to Christopher’s mother, originally included in his first serious biography and brought to new life in astrophysicist Janna Levin’s exquisite novel A Mad Man Dreams of Turing Machines (public library) — a masterwork of fiction that swirls philosophical poetics around the facts of Turing’s life.

Levin paints Turing’s struggle to conciliate the materialism of his scientific devotion with his spiritual devotion to Christopher even after the material cessation of his existence:

The future, present, and past of every material object is subject to the laws of physics. The orbit of every celestial body, the fall of every drop of rain. His own body a collection of molecules. His desire a cauldron of hormones whose chemistry has just been scientifically documented. His brain a case of matter, blood, and bone.

But he feels direct experience of his own soul, his spirit. He cannot accept that as an aggregate of flesh, a clump of matter, that his future, past, and present are already determined by the laws of physics. He cannot crush out the intuition that he makes choices, influences the world with his mind and spirit.


Chris had shown him the reaction between solutions of iodates and sulfites. Holding the mixture in a clear beaker near his face, he watched Alan’s response as the solution turned a bold blue, tinting Christopher’s hair and deepening the hue of his eyes. To Alan it seemed the other way around, as though Chris’s beautiful eyes had stained the beaker blue.


He often tries to re-create the moment when Chris’s spirit seeped out of the portals of his eyes and infused the room, a stunning concentration of his soul trapped in the indigo liquid in the beaker. He knows the simple form of the chemicals and the rules of their combination, but he can’t shake the force of the impression that Chris makes on him. He can’t limit the experience to the confines of ordinary matter.

That unshakable sense of spirit beyond matter is what 20-year-old Turing articulates in a letter from April 20, 1933:

My dear Mrs. Morcom,

I was so pleased to be at the Clockhouse for Easter. I always like to think of it specially in connection with Chris. It reminds us that Chris is in some way alive now. One is perhaps too inclined to think only of him alive at some future time when we shall meet him again; but it is really so much more helpful to think of him as just separated from us for the present.

Turing visited Clock House again in July, for what would have been Christopher’s twenty-second birthday. Seeking to reconcile the irrepressible spiritual aliveness felt in grief with the undeniable definitiveness of physical death, as much for himself as for Christopher’s mother, he wrote in another letter to her under the heading “Nature of Spirit”:

It used to be supposed in Science that if everything was known about the Universe at any particular moment then we can predict what it will be through all the future. This idea was really due to the great success of astronomical prediction. More modern science however has come to the conclusion that when we are dealing with atoms and electrons we are quite unable to know the exact state of them; our instruments being made of atoms and electrons themselves. The conception then of being able to know the exact state of the universe then really must break down on the small scale. This means then that the theory which held that as eclipses etc. are pre-destined so were all our actions breaks down too. We have a will which is able to determine the action of the atoms probably in a small portion of the brain, or possibly all over it.


Then as regards the actual connection between spirit and body I consider that the body by reason of being a living body can “attract” and hold on to a “spirit” whilst the body is alive and awake and the two are firmly connected. When the body is asleep I cannot guess what happens but when the body dies the “mechanism” of the body, holding the spirit, is gone and the spirit finds a new body sooner or later perhaps immediately.

As regards the question of why we have bodies at all; why we do not or cannot live free as spirits and communicate as such, we probably could do so but there would be nothing whatever to do. The body provides something for the spirit to look after and use.

First page of “Nature of Spirit,” in Turing’s original handwriting (Turing Digital Archive)

How Turing’s ideas evolved over the course of his life as he tussled with this paradox is among the many profound and possibly unanswerable questions examined with enormous intellectual elegance in A Mad Man Dreams of Turing Machines, another thread of which explores how the mathematician Kurt Gödel shaped our ideas of truth. Complement this particular thread with Marcus Aurelius on mortality and the key to living fully, Mary Oliver on the measure of aliveness, and Oliver Sacks on death, destiny, and the redemptive radiance of a life well lived.

Bruce Lee’s Never Before Revealed Letters to Himself About Authenticity, Personal Development, and the Measure of Success

“Where some people have a self, most people have a void, because they are too busy in wasting their vital creative energy to project themselves as this or that, dedicating their lives to actualizing a concept of what they should be like rather than actualizing their potentiality as a human being.”


“This is the entire essence of life: Who are you? What are you?” So wrote young Leo Tolstoy in his diary of moral developmentBruce Lee (November 27, 1940–July 20, 1973) was around Tolstoy’s age when he turned to this central question of existence more than a century later and approached it with the same subtleness of insight and sincerity of spirit with which he approached all of life.

Bruce Lee (Photograph courtesy of the Bruce Lee Foundation archive)
Bruce Lee (Photograph courtesy of the Bruce Lee Foundation archive)

Revered by generations as the greatest martial artist in popular culture, Lee is increasingly being recognized as the unheralded philosopher that he was, from his famous metaphor for resilience to his recently revealed unpublished writings on willpower, imagination, and confidence. But his most intently philosophical work was the personal credo statement he wrote in the final year of his life, at the age of thirty-one, as a series of letters to himself under the heading “In My Own Process.” The piece underwent nine drafts, never finished and never published, which I’m delighted to share for the first time with special permission from Lee’s daughter, Shannon Lee, and the Bruce Lee Foundation.

Bruce Lee (Photograph courtesy of the Bruce Lee Foundation archive)

The timing of “In My Own Process” is also significant, for Lee began writing it at a pivotal point in his life. After years of being sidelined by the Hollywood studio system, which continued to cast Caucasian actors to play Asian lead characters, Lee finally got his big break and was cast as the lead in Enter the Dragon, the script for which he helped write. But when Warner Brothers pushed to cut out all the philosophy and turn the film into a mindless action movie, Lee refused to show up on set in protest — he firmly believed that the kung fu was merely the vehicle for the deeper philosophical message, rather than the philosophy being a distraction from the kung fu, as Warner Brothers implied.

Well aware that his principles could cost him the fulfillment of his lifelong dream, he stood his ground. After a two-week standstill, the studio relented and let Lee keep the philosophical elements, so production began.

Bruce Lee on set (Photograph courtesy of the Bruce Lee Foundation archive)

In the midst of this busiest and most tumultuous period of his career, Lee made deliberate time for self-reflection in drafting his credo. It was in these letters to himself, written in his third language over the course of several months on a colorful variety of stationery, that he arrived at the concept of being an “artist of life.” In them, he examines with great simplicity and wisdom some of the most elemental questions of existence. Decades before the Harvard psychologist Dan Gilbert made his memorable assertion that “human beings are works in progress that mistakenly think they’re finished,” Lee considers with acute self-awareness the mutability of what we experience as the “self.” Echoing the poet Laura Riding’s conviction that “nothing is really important but being oneself,” he maintains through the various revisions that all knowledge is self-knowledge — the seedbed of his oft-cited assertion that “the greatest help is self-help” — and that personal authenticity is the object of life and the only real measure of success.

“In My Own Process,” Draft 1 (Courtesy of the Bruce Lee Foundation archive)

In the first draft, he writes:

Any attempt to write a somewhat meaningful article — or else why write it at all — on how I, Bruce Lee by name, emotionally feel or how my instinctive honest reaction toward circumstances is no easy task. Why? Because I am a changing as well as an ever-growing man. Thus what I held true a couple of months ago might not [be] the same now.

“In My Own Process,” Draft 2 (Courtesy of the Bruce Lee Foundation archive)

In the second draft, after relaying the difficulty of conducting this self-examination in the midst of his grueling work schedule, he insists on the importance of personal authenticity above all else and considers the vital difference between what Hannah Arendt called being vs. appearing and Kahlil Gibran contrasted as the seeming self vs. the authentic self. Lee writes:

Of course, this writing can be made less demanding should I allow myself to indulge in the standard manipulating game of role playing, but my responsibility to myself disallows that… I do want to be honest, that is the least a human being can do… I have always been a martial artist by choice, an actor by profession, but above all, am actualizing myself someday to be an artist of life. Yes, there is a difference between self-actualization and self-image actualization.

“In My Own Process,” Draft 3 (Courtesy of the Bruce Lee Foundation archive)

In the third draft, he considers our chronic fear of the unfamiliar in a sentiment of particular poignancy at this political moment:

Among people, a great majority don’t feel comfortable at all with the unknown — that is anything foreign that threatens their protected daily mould — so for the sake of their security, they construct chosen patterns to justify.

“In My Own Process,” Draft 4 (Courtesy of the Bruce Lee Foundation archive)

In the fourth draft, Lee turns to the perpetual evolution of personhood, which renders the idea of static self-definition unnecessary and unhelpful:

I have come to accept life as a process, and am satisfied that in my ever-going process, I am constantly discovering, expanding, finding the cause of my ignorance, in martial art and especially in life. In short, to be real…

“In My Own Process,” Draft 5 (Courtesy of the Bruce Lee Foundation archive)

In the fifth draft, the revisits the inherent paradox of the quest to define himself and his process:

I don’t believe in the manipulation game of creating a self image robot.

“In My Own Process,” Draft 6 (Courtesy of the Bruce Lee Foundation archive)
“In My Own Process,” Draft 7 (Courtesy of the Bruce Lee Foundation archive)

In the seventh draft, he echoes Walt Whitman’s incantation to “re-examine all you have been told at school or church or in any book,” and writes in a passage of especial relevance to our present epidemic of unquestioned “alternative facts”:

Surely we all admit that we are intelligent beings, though in reality we are being crammed with ready-made facts handed down to us ever since [childhood]. Some of us even went through college but something is the matter because … some of these facts are examined in the form of self-inquiry, but in most cases we accept most of these facts unexamined.


We possess a pair of eyes to help us to observe as well as to discover, yet most of us simply do not see in the true sense of the word. However, when it comes to observing faults in others, most of us are are quick to react with condemnation. But what about looking inwardly for a change? To personally examine who we really are and what we are, our merits as well as our faults — in short, to see oneself as [one] is for once and to take responsibility [for] oneself.

“In My Own Process,” Draft 8 (Courtesy of the Bruce Lee Foundation archive)

In the penultimate draft, he turns from the intellectual dimension of self-knowledge to its emotional rewards:

I am happy because I am daily growing and honestly not knowing where the limit will yet lie. To be certain, every day can be a revelation or a new discovery. However, the most satisfaction is yet to come to hear another human being say, “Hey, here is something real.”

He touches on the deeper significance martial art held for him as a spiritual practice and not the merely the decorative performance Hollywood made it out to be:

By martial art I mean, like any art, an unrestricted expression of our individual soul… The human soul is what interests me. I live to express myself freely in creation.

Bruce Lee (Photograph courtesy of the Bruce Lee Foundation archive)

Lee’s reflection on what it means to be a great actor applies equally to every art, as well as to the art of life itself:

An actor, a good actor that is, not the shallow stereotyped artist, is an ever-growing process of learning, expansion and constant discoveries… To be of quality in acting means … lots of painful hard work and lots of undivided dedication to practicing what one believes.

“In My Own Process,” Draft 9 (Courtesy of the Bruce Lee Foundation archive)

In the ninth and last draft — which is still a draft, for his untimely death intercepted the completion of the piece — Lee reassembles the mosaic of the intellectual, spiritual, and emotional dimensions of selfhood, and returns to his central ethos of personal authenticity:

Where some people have a self, most people have a void, because they are too busy in wasting their vital creative energy to project themselves as this or that, dedicating their lives to actualizing a concept of what they should be like rather than actualizing their potentiality as a human being, a sort of “being” vs. having — that is, we do not “have” mind, we are simply mind. We are what we are.

Complement with Lee on self-actualization and the crucial difference between pride and self-esteem and the philosophy and origin of his famous water metaphor, then hear Shannon Lee discuss her father’s work on “In My Own Process” with cohost Sharon Lee in this episode of the excellent Bruce Lee Podcast:

The Mandelbrot Set and Complex Numbers

This, which is from Wikipedia, goes into great depth about what the Mandelbrot Fractal is and how it functions. I have worked on this Fractal for many years now, and to this day I still find unique ways to further explore it.  Although it always has a certain size, the complicated edge that you see, called its “frontier,” is always infinitely complex, meaning that you can zoom into the edge at any part of the frontier and you will see new patterns and forms that are both self-similar to a degree, and unique in a noticeable way.  What is referred to as complex numbers have the y-axis a function of the square root of negative one, an imaginary number, referred to with the letter i. The x-axis is a function of real numbers. ~ Ben Gilberti

The Mandelbrot set is the set of complex numbers {\displaystyle c}c for which the function {\displaystyle f_{c}(z)=z^{2}+c}{\displaystyle f_{c}(z)=z^{2}+c} does not diverge when iterated from {\displaystyle z=0}z=0, i.e., for which the sequence {\displaystyle f_{c}(0)}{\displaystyle f_{c}(0)}{\displaystyle f_{c}(f_{c}(0))}{\displaystyle f_{c}(f_{c}(0))}, etc., remains bounded in absolute value.

A zoom sequence illustrating the set of complex numbers termed the Mandelbrot set.

Its definition and name are due to Adrien Douady, in tribute to the mathematician Benoit Mandelbrot.[1] The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.

Mandelbrot set images may be created by sampling the complex numbers and determining, for each sample point {\displaystyle c}c, whether the result of iterating the above function goes to infinity. Treating the real and imaginary parts of {\displaystyle c}c as image coordinates {\displaystyle (x+yi)}{\displaystyle (x+yi)} on the complex plane, pixels may then be colored according to how rapidly the sequence {\displaystyle z_{n}^{2}+c}{\displaystyle z_{n}^{2}+c} diverges, with the color 0 (black) usually used for points where the sequence does not diverge. If {\displaystyle c}c is held constant and the initial value of {\displaystyle z}z—denoted by {\displaystyle z_{0}}z_{0}—is variable instead, one obtains the corresponding Julia set for each point {\displaystyle c}c in the parameter space of the simple function.

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The “style” of this repeating detail depends on the region of the set being examined. The set’s boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization.


The first published picture of the Mandelbrot set, by Robert W. Brooks and Peter Matelski in 1978

The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. This fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups.[2] On 1 March 1980, at IBM‘s Thomas J. Watson Research Center in Yorktown Heights, New YorkBenoit Mandelbrot first saw a visualization of the set.[3]

Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980.[4] The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard,[1] who established many of its fundamental properties and named the set in honor of Mandelbrot.

The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books,[5] and an internationally touring exhibit of the German Goethe-Institut.[6][7]

The cover article of the August 1985 Scientific American introduced a wide audience to the algorithm for computing the Mandelbrot set. The cover featured an image created by Peitgen, et al.[8][9] The Mandelbrot set became prominent in the mid-1980s as a computer graphics demo, when personal computers became powerful enough to plot and display the set in high resolution.[10]

The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all the mathematicians who have contributed to the understanding of this set since then is beyond the scope of this article, but such a list would notably include Mikhail Lyubich,[11][12] Curt McMullenJohn MilnorMitsuhiro Shishikura, and Jean-Christophe Yoccoz.

Formal definition

The Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map

{\displaystyle z_{n+1}=z_{n}^{2}+c}{\displaystyle z_{n+1}=z_{n}^{2}+c}

remains bounded.[13] That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets. This can also be represented as[14]

{\displaystyle z_{n+1}=z_{n}^{2}+c,}{\displaystyle z_{n+1}=z_{n}^{2}+c,}
{\displaystyle c\in M\iff \limsup _{n\to \infty }|z_{n+1}|\leq 2.}{\displaystyle c\in M\iff \limsup _{n\to \infty }|z_{n+1}|\leq 2.}

For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26, …, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = −1 gives the sequence 0, −1, 0, −1, 0, …, which is bounded, and so −1 belongs to the Mandelbrot set.

The Mandelbrot set {\displaystyle M}M is defined by a family of complex quadratic polynomials

{\displaystyle P_{c}:\mathbb {C} \to \mathbb {C} }P_{c}:\mathbb {C} \to \mathbb {C}

given by

{\displaystyle P_{c}:z\mapsto z^{2}+c,}P_{c}:z\mapsto z^{2}+c,

where {\displaystyle c}c is a complex parameter. For each {\displaystyle c}c, one considers the behavior of the sequence

{\displaystyle (0,P_{c}(0),P_{c}(P_{c}(0)),P_{c}(P_{c}(P_{c}(0))),\ldots )}(0,P_{c}(0),P_{c}(P_{c}(0)),P_{c}(P_{c}(P_{c}(0))),\ldots )

obtained by iterating {\displaystyle P_{c}(z)}P_{c}(z) starting at critical point {\displaystyle z=0}z=0, which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot set is defined as the set of all points {\displaystyle c}c such that the above sequence does not escape to infinity.

A mathematician’s depiction of the Mandelbrot set M. A point c is colored black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c, respectively.

More formally, if {\displaystyle P_{c}^{n}(z)}P_{c}^{n}(z) denotes the nth iterate of {\displaystyle P_{c}(z)}P_{c}(z) (i.e. {\displaystyle P_{c}(z)}P_{c}(z) composed with itself n times), the Mandelbrot set is the subset of the complex planegiven by

{\displaystyle M=\left\{c\in \mathbb {C} :\exists s\in \mathbb {R} ,\forall n\in \mathbb {N} ,|P_{c}^{n}(0)|\leq s\right\}.}M=\left\{c\in \mathbb {C} :\exists s\in \mathbb {R} ,\forall n\in \mathbb {N} ,|P_{c}^{n}(0)|\leq s\right\}.

As explained below, it is in fact possible to simplify this definition by taking {\displaystyle s=2}s=2.

Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number c either belongs to M or it does not. A picture of the Mandelbrot set can be made by coloring all the points {\displaystyle c}c that belong to M black, and all other points white. The more colorful pictures usually seen are generated by coloring points not in the set according to which term in the sequence {\displaystyle |P_{c}^{n}(0)|}|P_{c}^{n}(0)| is the first term with an absolute value greater than a certain cutoff value, usually 2. See the section on computer drawings below for more details.

The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials {\displaystyle P_{c}(z)}P_{c}(z). That is, it is the subset of the complex plane consisting of those parameters {\displaystyle c}c for which the Julia set of {\displaystyle P_{c}}P_{c} is connected.

{\displaystyle P_{c}^{n}(0)}P_{c}^{n}(0) is a polynomial in c and its leading terms settle down as n grows large enough. These terms are given by the Catalan numbers. The polynomials {\displaystyle P_{c}^{n}(0)}P_{c}^{n}(0) are bounded by the generating function for the Catalan numbers and tend to it as n goes to infinity.

Basic properties

The Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 around the origin. More specifically, a point {\displaystyle c}c belongs to the Mandelbrot set if and only if

{\displaystyle |P_{c}^{n}(0)|\leq 2}|P_{c}^{n}(0)|\leq 2 for all {\displaystyle n\geq 0.}{\displaystyle n\geq 0.}

In other words, if the absolute value of {\displaystyle P_{c}^{n}(0)}P_{c}^{n}(0) ever becomes larger than 2, the sequence will escape to infinity.

Correspondence between the Mandelbrot set and the bifurcation diagram of the logistic map

The intersection of {\displaystyle M}M with the real axis is precisely the interval [−2, 1/4]. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,

{\displaystyle z\mapsto \lambda z(1-z),\quad \lambda \in [1,4].}{\displaystyle z\mapsto \lambda z(1-z),\quad \lambda \in [1,4].}

The correspondence is given by

{\displaystyle c={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right).}c={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right).

In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.

As of October 2012, the area of the Mandelbrot is estimated to be 1.5065918849 ± 0.0000000028.[15]

Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, they constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of {\displaystyle M}M. Upon further experiments, he revised his conjecture, deciding that {\displaystyle M}M should be connected.

External rays of wakes near the period 1 continent in the Mandelbrot set

The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard’s proof of the connectedness of {\displaystyle M}M, gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle.[16]

The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters {\displaystyle c}c for which the dynamics changes abruptly under small changes of {\displaystyle c.}c. It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting p0 = zpn+1 = pn2 + z, and then interpreting the set of points |pn(z)| = 2 in the complex plane as a curve in the real Cartesian plane of degree 2n+1 in x and y. These algebraic curves appear in images of the Mandelbrot set computed using the “escape time algorithm” mentioned below.

Other properties

Main cardioid and period bulbs

Periods of hyperbolic components

Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This main cardioid is the region of parameters {\displaystyle c}cfor which {\displaystyle P_{c}}P_{c} has an attracting fixed point. It consists of all parameters of the form

{\displaystyle c={\frac {\mu }{2}}\left(1-{\frac {\mu }{2}}\right)}c={\frac {\mu }{2}}\left(1-{\frac {\mu }{2}}\right)

for some {\displaystyle \mu }\mu  in the open unit disk.

To the left of the main cardioid, attached to it at the point {\displaystyle c=-3/4}c=-3/4, a circular-shaped bulb is visible. This bulb consists of those parameters {\displaystyle c}c for which {\displaystyle P_{c}}P_{c} has an attracting cycle of period 2. This set of parameters is an actual circle, namely that of radius 1/4 around −1.

Continue reading The Mandelbrot Set and Complex Numbers

Dawna Markova on the unlived life

“I will not die an unlived life.
I will not live in fear
of falling or catching fire.
I choose to inhabit my days,
to allow my living to open me,
to make me less afraid,
more accessible;
to loosen my heart
until it becomes a wing,
a torch, a promise.
I choose to risk my significance,
to live so that which came to me as seed
goes to the next as blossom,
and that which came to me as blossom,
goes on as fruit.”

― Dawna MarkovaI Will Not Die an Unlived Life: Reclaiming Purpose and Passion

Horace Mann on a victory for humanity

Horace Mann

“Be ashamed to die until you have won some victory for humanity.”

Horace Mann (May 4, 1796 – August 2, 1859) was an American educational reformer and Whig politician dedicated to promoting public education. He served in the Massachusetts State legislature (1827–1837). In 1848, after public service as Secretary of the Massachusetts State Board of Education, Mann was elected to the United States House of Representatives (1848–1853). About Mann’s intellectual progressivism, the historian Ellwood P. Cubberley said:

No one did more than he to establish in the minds of the American people the conception that education should be universal, non-sectarian, free, and that its aims should be social efficiency, civic virtue, and character, rather than mere learning or the advancement of education ends.[1]

Arguing that universal public education was the best way to turn unruly American children into disciplined, judicious republican citizens, Mann won widespread approval from modernizers, especially in the Whig Party, for building public schools. Most states adopted a version of the system Mann established in Massachusetts, especially the program for normal schools to train professional teachers.[2] Educational historians credit Horace Mann as father of the Common School Movement.[3]

More at:  https://en.wikipedia.org/wiki/Horace_Mann

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