The Cone of Perception: Empirical Evidence for Metaphysics and Magic

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I write this introduction with much work already completed in this 4th Edition of The Cone of Perception, primarily to frame the work and touch on what might be missing from it. Namely → though it is hardly lacking these, I’d like to add a few insights about the content of the work and its relevance to the subject and future technology. Why is this work still relevant? How can you use it in your wor/subject/area/field, and how am I supposed to read all of these equations?!!

This work is relevant, because is shows us that what we may have thought impossible is phenomenological and actualized. That there are, compressed within simple formulations of geometry and parameters of spatio-temporal experiencing, really a complex harmony of relations, coefficients, hexadecimal coefficients, equilibrium, dynamics, resonances, all of which are alternating with each other in a dance of higher dimensionality through many balancing solutions. All of this, and infinitely more is going on between the distance from me to you, and between each particle in between us.

You can use this in your life, work, or area of study, practically no matter what it is that you study, because these equations and insights apply to the fundament of the fabric of reality that we all share and live and the consciousness of it. Whether you are a physicists or a monk, these equations reveal answers and are the answers for the eternal questions of nature and reality.

Harmony with number, visual forms, etc. are a new language. Why have questions about God been so hard to express? Words don’t always do justice to the potents of philosophical debate. Much magic mystery and magic have been neglected because of this, but math, equations, visualizations, etc. can do more to reveal insights and converse on topics philosophers have been seeking for ages.

Also, this book reveals that what science is currently doing with particle accelerators, while offering its own set of insights, may actually be less productive, wasteful, unnecessary and dangerous that we think.

What The Cone of Perception offers is a mental micro/macroscope that will lead you on a journey from what is currently smaller than what physicists currently believe is the smallest mass/time unit all the way out to numbers and equations so large that they are not allowed by God for us to currently visualize their form. The Cone of Perception has revealed that there are particles of the photo electric effects with mass smaller than the quantum foam.

That is a revelation that ought turn the obtuse world of physics on its head. There are many other section in The Cone of Perception all with equally revolutionary material. This work is meant to revolutionize the current perception of the physical world. These equations are useful to those who study dynamics, seek understanding of balance, and those who seek visualizations, mystery and empirical proof of metaphysics and magic in this world.

 

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Chapel Hill Armed and Dangerous Person on Wednesday 12-2-15 Was False Alarm

From legitimate sources within the community, it was revealed that the police were unable to find any armed or dangerous people in the area. However, the police did show up in full force with M16/AR-15 pointed at the building. Police drew guns and pointed them at the ROTC building on UNC Campus. For emphasis, they pulled guns on a campus building on a false alarm with no evidence of an actual armed or dangerous gunman. NO SHOTS WERE FIRED.

This should be an embarassment for the entire Chapel Hill Police Department. For your information, it is not a crime to carry a firearm on a public sidewalk, and it is not probable cause to search individuals for the serial number on their firearm if they are simply openly carrying a firearm in North Carolina. Openly carrying a firearm in North Carolina is governed by the legislature, and it is perfectly legal. This is not to say that anyone actually was carrying a firearm that day.

According to McCracken, who is an alright fellow in my opinion, this whole thing of drawing live rifles on a UNC campus building based off of a 911 call with no visual confirmation from authorities is, “NORMAL PROCEDURE.” He said it’s normal procedure in an interview given just after the all clear was issued as you can see here:

http://abc11.com/news/lockdown-lifted-at-unc-after-report-of-armed-dangerous-person/1106443/

It sounds like a lot of assumptions were made today that weren’t necessarily correct. The response to draw multiple rifles in a civilian area was completely inappropriate. This action was taken by police without any actual visual confirmation that there was any threat.

It is unclear what evidence they had to justify any of these actions. Now we know what little excuses they will use to justify playing with their toys.

Also, I wish people would stop calling the police on people who decide to embrace their right to bear arms. Police claim they want safety, but they over-react and implement a double-standard on regular citizens. Society is also embracing the double standard of only people with badges are allowed to own and carry guns. This goes against the liberty that our fore-fathers fought and died to preserve.

I would like to see laws implemented that make it a felony for police to violate the 4th, 2nd and 5th and 1st amendments of the US Constitution and start using logic when obtaining probable cause.

Compared with other areas of the country, Chapel Hill has more logical police officers, but many police officers lack training in philosophy, logic, ethics and the meaning of natural rights, even though they’ve sworn an oath to uphold the constitution.

What we saw today was a rogue police decision to implement an armed occupation of the ROTC with improper probable cause.

WE NEED TO CALL FOR A FULL INVESTIGATION OF THIS ISSUE AND REPRIMAND THE OFFICERS WHO PARTICIPATED IN THIS INCIDENT, BECAUSE IT WAS A GROSS OVERREACTION TO ASSUMED FACTS NOT IN EVIDENCE.

 

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BITCOIN ART FROM MATH EQUATIONS

The maker of these mysterious equations was discovered by Parker Emmerson in the fabric of space-time. From seashell forms found within the fabric of spacetime as large as a light second to particles smaller than the quantum foam responsible for the mass of the photo electric effect to solutions to spatial variable in terms of a 10-dimensional angle from an equality of derivative and angular velocity to an infinite angle and beyond. How did the solutions to the velocity in the Lorentz coefficient manifest when the Lorentz coefficient should have canceled out with itself? Why is nobody talking about this completely new, valid solution to general relativistic velocity? Parker Emmerson does an interesting job of combining these different, originally discovered equations in an Escher like maze of perspective. Also, this system of a circle’s folding in to a cone yields numerous paradoxes and mimics the algebraic form and relations of the bitcoin network!!!

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I got stopped for Open Carry in NC

I was illegally detained in NC for open carry in Chapel Hill, NC.

They took my firearm away from me, and then they threatened to charge me with a crime if I didn’t give them my ID.

Eventually, they returned the firearm and let me go.

Police nationwide have been documented on video/audio trying to discourage citizens from open carry of legal guns. Often they use the excuse that citizens become frightened by the sight of anyone without a badge carrying and call police, who sometimes falsely arrest the carrier and confiscate the firearm. At what point does the mere open carry turn into ” terror ‘ for anti-gun proponents? It is not the fault of the open carrier that many people today are alarmed at the sight of a gun in a holster without obvious LEo credentials, and of course cops will use any excuse to limit citizens from exercising their rights when it comes to guns, wishing to be the only armed people in public, despite the laws to the contrary.

In fact, if someone is simply open carrying , the police do not have any legal reason to detain and demand ID from the carrier, seeing as how no reasonable suspicion exists from the lawful act of open carry..the cops often say that they ” have no way of knowing if the carrier is a felon ” as an excuse to detain and ID a carrier, but if they do not have any reawson to suspect that the carrier is in fact a felon the stop is just a fishing expedition w/o RS or Pc of any kind and is an illegal detention. People should not have to prove to a cop that his unfounded suspicions have any validity and the excuse that the public may become alarmed should not be an excuse to limit the right to open carry. It is not the responsibility of the lawful open carrier to give up his rights because some citizens may have apprehensions, but the cops are using this as a reason to harrass carriers all the time.

Without ” articulable ” and reasonable grounds to believe that a crime is being committed, open carriers should not be detained whatsoever, and lawsuits may be the only way to reign in agressive cops who resent not being the exclusive carriers of guns. Flipping off a cop does not give the officer the legal right to detain or charge a citizen and carrying a holstered gun does not give them any legal authority to initiate a detention and interrogation either.

People do not have to prove that they are entitled to open carry just because they are carrying, or because of calls by citizens who do not comprehend the rights to carry and are afraid that an otherwise innocuous citizen is open carrying a gun. If unfounded alarm is enough to deny our rights, we are in a sad state of affairs indeed.

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What Happens to a Dream Deferred

By Shweta Mishra

Dia de los Muertos — October 31, 2009

Fine nighttime rain shimmered amid headlights bobbling into the yard. Nine years after becoming undocumented, Esteban Ginocchio-Silva was throwing a party.

Bare-chested, wearing a pink mask emblazoned with a cross, the 5-foot 21-year-old cracked his knuckles, rolled his neck and broke into a mix of yodels and hoots, spirit fingers raised and body spasming. This was his stage entrance for El Dia de los Muertos. He was a luchador.

“Our traditions in Peru celebrate life,” he said. “Quinceañeras, cumpleaños, bailes, polladas — dances, home-made fried chicken and beer offered to the whole neighborhood. Community was how things rolled.”

Except for the beer keg, this celebration was unusual for downtown Carrboro and Chapel Hill, North Carolina. In the affluent, well-meaning but neatly self-segregating college towns, Esteban had brought together Latino, black, Chinese, Australian, South Asian, British and Greek young adults with disparate class backgrounds. Only a quarter of the guests were white suburban Americans. But the conviviality was just human. Frida Kahlos, skeleton brides and Che Guevaras streamed in, not navel-baring nurses, but underneath they were just excited young women. In the kitchen, transsexual grannies with British accents cradled screwdrivers and chomped on traditional Mexican bread of the dead, flashing smiles from under their bonnets, lifting their floral dresses too high. Partiers on the veranda offered sugar skulls, tea-lights and flowers to Michael Jackson’s luminous death altar.

Put on a Happy Face

Amid the international throng, Esteban’s golden face, shrewd bright eyes and Cheshire Cat smile belied the skeletons in his closet. Researchers at the Department of Public Policy at the University of North Carolina at Chapel Hill found that immigrant Latino youth, particularly those without papers, suffered from clinical depression and anxiety at statistically higher rates than their white and black American counterparts. The Centers for Disease Control and Prevention corroborated these results by finding that Latino youth also planned and attempted more suicides per year. Why so distressed? The UNC-Chapel Hill study pointed to “high poverty levels prior to migration, stress during migration, and racial and ethnic discrimination upon settlement in the US.”

Esteban was not just an immigrant. He belonged to an even murkier subset of this high-risk category. Once the US Immigration and Naturalization Service sent vital paperwork to the wrong address right before the deadline, he and his family had become disabused of any hope they had entertained in the American Dream. No Mother of Exiles could be expected to wait for them by any golden door, with any mighty torch in her hand. The pleased face Esteban put forward testified to his unusual resilience and tenacity. When Halloween ended, he quietly resumed studying, not at a university of his choice but Durham Technical Community College, which he paid for himself by working full-time at Mediterranean Deli, a popular downtown restaurant. It was his third year doing this, and he was dog-tired. But he had dreams he had to fight for.

“I always wanted to be a business owner,” he said. “I wanted to be my own boss, you know, the whole romanticized aspects of the American dream. I thought a degree in economics would be the way to get there.”

Great Expectations — 1988 – 2000

At first no one had thought about moving to the United States. Three months after Esteban’s birth on Dec. 9, 1988, his family settled in Chosica, a promising city near Lima, Peru’s capital. Enrique Ginocchio Reeves got a job preaching at Bethel Baptist Church. He and his wife, Fernandina, enrolled their son in a private Christian school. Esteban flourished.

“But it sheltered me from getting a more real experience of the world,” he said.

While three written warnings meant a rap on the rear with a paleta, that never actually happened. It was only when he played hide-and-go-seek on the streets that he saw kids whose teachers did beat them, and others who spent schooldays begging for food.

“I saw homeless children basically spacing themselves on top of cars at stoplights to clean their windshields, and risking either getting run over or not getting paid,” Esteban said.

He saw brick houses next to cardboard shacks, people with electricity and others without drinking water.

“I could definitely tell the difference at a young age, and that was part of my upbringing,” Esteban said. It made him compassionate when his own fortune turned. When the specter of joblessness caught up with Esteban’s parents, Enrique turned to church friends who lived in North Carolina. In 2000, the family moved to Graham, NC, with a tourist visa. Enrique became a preacher at The Church of Nazarene. Esteban and his sisters learned English.

The family’s prospects withered when the US Immigration and Naturalization Service sent paperwork to the wrong address one day before the deadline.

Revision of the Future — 2010

But Esteban didn’t dwell in the past long.

He said he held himself with dignity and hope rather than cowering as an undeserving fugitive because his philosophical mind recognized a self-serving moral relativism in American law and history, and that individuals involved were just pawns, exonerated or deported by circumstantial details. He saw through the sanctimonious bureaucratic double-talk of immigration politics of the day into the messy heteroglossia of America’s real past. He and his sister Loida analyzed their undocumented presence in relation to that of John Smith’s settlers in Powhatan territory in 1609. Unlike the Jamestown settlers, the Silvas had not violently raided food stashes of tribes, nor planned to plant their own flags on a bleeding ground.

It was with this outlook that, in the summer of 2011, Loida changed her Facebook profile picture to Asheville Mountain Express cartoonist Randy Molton’s caricature of an angry skinhead next to an Indian chief with a crown of feathers.

“Protests should’ve been made about immigration years ago!!” said the skinhead in his speech bubble.
“Oh, at least a few centuries ago!” said the chief.

That is, Esteban thought, before the Battle of Fallen Timbers, Indian Removal Act, Trail of Tears. Before his ancestors, or at least genetic equivalents, were displaced from their traditional lands.

Of course, Esteban’s historian logic was absent in discussions among congressmen, whose composite face as digitally calculated by artist Rebecca Lieberman and developer Matthew Skomarovsky was that of a masculine, middle-aged white man with a receding hairline, the stereotypical son of colonial America. For this white face of congress, it seemed irrelevant that after the American Revolution and before the 1800s, the United States’ first immigrants — perhaps the congressmen’s ancestors — were themselves undocumented. That in the 1600s and 1700s, all settlers were foreign immigrants and fugitives who crossed myriad borders of myriad indigenous nations.

Esteban said that it was this amnesia of privilege that had allowed congress to glibly quibble over immigration reform for 11 years.

The DREAM Act, or Development, Relief and Education for Alien Minors Act, was a bill that would grant undocumented minors a pathway to citizenship contingent on higher education or military enlistment. If it had been passed in 2006, when Esteban graduated from high school, it would have allowed him to apply to universities and get private scholarships.

The Yellow Brick Road with an Impasse

Esteban’s hard work at Durham Tech paid off when the University of North Carolina at Chapel Hill admitted him as an undergraduate transfer in fall 2010. But it was a Pyrrhic victory as the Public Ivy classified him as an International Student, which meant he had to shell out nonresident tuition and fees of $12,640 per semester instead of the $3,333 for other state-dwellers. His status also meant ineligibility for university scholarships and the Free Application for Federal Student Aid.

Learning of this, Esteban’s boss thought, why not just marry an American citizen? Esteban could get financial aid, a driver’s license, a legal work permit.

“When I came here, I married a friend in three months exactly,” Jamil Kadoura said. “It used to be easy, but I guess then terrorism happened.”

Regardless, Esteban’s preferred solution was the DREAM Act. On May 18, 2010, he and four other local undocumented youth formed the North Carolina DREAM Team. One of the group leaders was Viridiana Martinez, a Mexican-born undocumented activist profiled in the Al Jazeera documentary “Activate – The Dreamer.”

“We met at Noodles & Company and made a plan to fight,” she said. “Esteban was there and Loida and Rosario and me and Manuel. We knew that we, the youth, had to sort of take more of a lead role in what was already happening, this whole talk of immigration reform. The Dream Act in particular because that was something that existed. It was a bill that was there since 10 years ago and we knew we needed to be at the forefront.”

Unlike the rest of the group, Esteban remained “in the shadows,” unwilling to publicly disclose his legal status, because he felt he had too much at stake.

“I knew what he was up against,” Viridiana said. “Having to work to pay all of this money for school, and activism on the side. But he was there. He wanted to do it, so that’s how NC dream team got started.”

Later that summer, Viridiana, Loida and Rosario led a strike in Raleigh.

“My own sister Loida decided to starve herself in the summer heat,” Esteban said. “Right in front of Senator Hagan’s office as long as she said no to the bill.”

The strikers stopped after 13 days. Esteban’s sister had been hospitalized for a heat stroke.

The Kindness of Strangers

Senator Hagan voted no anyway.

It seemed that, ultimately, no amount of activism made a difference in Esteban’s daily life. After the hoopla and heroism, he had to turn to his boss again. Influenced by a childhood in the Middle East, where his and his family’s survival as Palestinians had depended on the benevolence of the Red Cross, Jamil had a charitable spirit that was renowned in Chapel Hill and Carrboro. He thought Esteban was a nice kid, sharp, so he gave him an interest-free loan for half the amount Esteban needed to take two classes. Esteban paid the difference with personal savings and eliminated his debt within the semester. He also worked to pay his rent and bills. There was little time in his day to eat and none to study, and after the first term, he gave up. “I didn’t really have the wherewithal to push myself and needed a break for my own sanity,” he said. Anyway, there was no more money.

Go West, Young Man, and Grow Up with the Country — March 20, 2011

On the third floor of a colonial house in Echo Park, Los Angeles, Esteban, now 23 years old, lay awake on a mattress in a walk-in closet. Thunder cracked and rain beat the gabled roof. Wind blustered through the gaps in the windows.

Esteban could hear his hosts, five young gay Filipinos, laughing downstairs. He was here because he had decided, after a year of working himself to the ground to pay for two classes, to quit and leave.
“It never went through my mind to really move west,” Esteban said. “It was more like a rebellion against my circumstances.” He had found his hosts on a couch-surfing online forum.

Though he’d originally hoped to adventure on a Greyhound from North Carolina to New Orleans, and then north to California, he decided flying would eliminate risks of hitting migrant checkpoints. He arrived in Los Angeles on the first morning of spring and what would be California’s wettest day in 2011, a fluke for The Golden State. He got drenched at the bus station. That day there were also flash floods, mudslides, power outages. Stubborn marathon runners were rushed to emergency rooms for hypothermia.

Of course, after his cross-country flight from North Carolina, the pressing problem was not weather but whether this new land would also strangle his dreams.

The day before, two dozen white supremacists had rallied in Claremont, CA. Bald young men held up swastika flags and wore battle uniforms. The southwest regional director of the National Socialist Movement, Jeff Hall, decried California’s apparent friendliness to illegal immigrants.

Back in Chapel Hill, Helen Martyn, an elderly Los Angeles native, came to Mediterranean Deli for dinner one day. With glittering eyes she said Mexicans crossed over just to demand free services and hate her country.

“They come here to spit on us,” she said.

So any distance away from home, the anger shined from sea to shining sea. Esteban would leave for Seattle tomorrow.

Motherland — 2014

Esteban’s parents were now in Peru, irrevocably, and he thought of them. He recalled his far-flung childhood, when anything seemed possible, when he didn’t know he was poor or an alien.

He remembered the high, dry, thin air of his rural birth village in Jauja Province, 11,000 feet above sea level. Fruiting Blue Gum eucalypts with silvery bark and tassels of flowers. Gnarled and squat, red-barked Queñua evergreens that broke through crags.

He remembered chewing coca leaves just like European tourists did to combat altitude sickness.

From a distance, he could still see crop lines on Jauja’s flat fields, their underbellies clotted with potatoes. The sky was wide and the Andes visible on the low, distant horizon. He remembered summer hail and rain rattling the tin roofs of adobe houses. The incense of earthen hearths blazing village-wide.

When he thought of the Mercado Municipal, Jauja’s open-air bazaar, he smelled the rust and iron of congealing blood. From morning till dark, raw lamb heads hung on hooks, men and women in traditional garb stirred up lamb head stew and vendors roasted lamb chops and suckling pigs. Herds of livestock bleated, brayed and shed their sour, loamy funk. Timber and native crops like quinoa and potatoes bumbled along on the backs of donkeys.

He marveled at the potatoes. He said he ate 15 varieties, small and rich as egg yolks to graceless as Russets from Idaho. He remembered the olluco, an indigenous tuber with the consistency of radish but long, yellowish and flecked red. He laughed to recall its tastiness.

Fourteen years and 3,278 miles after those market aromas, his memories persisted. Peru was the last place he felt like a visible human being. Even though he had lived in America for most of his life, it had never seen him as its own.

 

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If everybody in the world had one dollar’s worth of Bitcoin

The simple math is $6 billion ($1 of BTC for each person) divided by say around 13 million, the number of the coins currently mined. So many people live on less than or equal to a dollar per day.

Do you want to own a dollar of Bitcoin or $245.00 of it?

Never before in human history has there been a global currency that could be immediately transferred across the globe.

Never before in human history  has there been the opportunity to transfer wealth into the hands of the people and away from oligarchs bankers and the elite.

However, we must act now to secure our share in the future. That means spreading the word to all the people of the world.

The next Bitcoin bubble will either be created by a mass revolution of all the people of the planet fighting for the small share of that coin, or banks will seize the opportunity first.

 

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The real deal on DOGE – Double pan and handle LTC and USD

Get DOGE today. A pan and handle is forming on Doge in terms of LTC and USD over different periods of time. The next couple weeks will tell, but if DOGE outperforms these two over the next couple weeks, look for a massive surge in Doge price.
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https://coinreport.net/cryptsy-debitway-partner-foster-altcoin-trading/

 

News like THIS does not hurt either. With the alt coin market gaining traction for trading more regularly and with more liquidity, I’m gunna go ahead and say that DOGE is a buy.

The risk that doge will decrease is substantial, but with Bitcoin’s bouncing of the 266 low, you could look for stability there, and gain faith in the entire alt coin market.

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Cryptography for Starters

 

  Some Bitcoin for Thought


You will want to read the following for starters on understanding bitcoin \
cryptography.
https : // en.bitcoin.it/wiki/Secp256k1
http : // en.wikisource.org/wiki/NIST_Koblitz _Curves _Parameters
The naive way to break an encryption algorithm is to brute –
force the key.The complexity of that attack is 2 n,
where n is the key length.All cryptanalytic attacks can be viewed as \
shortcuts to that method.And since the efficacy of a brute –
force attack is a direct function of key length,
these attacks effectively shorten the key.So if, for example,
the best attack against DES has a complexity of 239,
that effectively shortens DES⎟ s 56 –
bit key by 17 bits.That⎟ s a really good attack, by the way.Right now the \
upper practical limit on brute force is somewhere under 80 bits.However, \
using that as a guide gives us some indication as to how good an attack has \
to be to break any of the modern algorithms.These days, encryption algorithms \
have, at a minimum, 128 –
bit keys.That means any NSA cryptoanalytic breakthrough has to reduce the \
effective key length by at least 48 bits in order to be practical.
http : // en.wikipedia.org/wiki/Elliptic_Curve _DSA
http : //
www.reddit.com/r/Bitcoin/comments/1 nvsn7/
elliptic_curve _secp256k1 _vulnerability/
http : // blog.ezyang.com/2011/06/the – cryptography – of – bitcoin/
http : //
bitcoinmagazine.com/6021/bitcoin – is – not – quantum – safe – and – how –
we – can – fix/
http : // en.wikipedia.org/wiki/Cryptographic_hash _function
http : // mathworld.wolfram.com/EllipticCurve.html
We can liken the finding of bitcoins to someone firing at a target, and they \
are force to fire at the target while wearing a blind fold.

We can make the statistical probability of hitting the target greater if we \
either

1. Take off the blind fold (very hard)
2. Make the target bigger (easier)
3. Create a net so that we hit the target and drag it back to us so that it’ \
s easier to hit.
Making target bigger :
Say we know what the elliptical curve that bitcoin uses, we can can create \
more facets to it.
We can embed different functions within it; reduce it to other variable \
entirely.
Then, we have better ways of converting to potential solutions to the \
equation.
We need to know what the field characteristic of the bitcoin elliptical curve \
is.
http : //
bitcoin.stackexchange.com/questions/21907/what – does – the – curve – used –
in – bitcoin – secp256k1 – look – like
http : // www.secg.org/index.php?action = secg, docs_secg
Actually secp256k1 is defined over a Galois field, not a ring of integers \
modulo a prime.Now, it turns out that the secp256k1 field is a prime field \
and therefore isomorphic to a ring of integers modulo a prime, but this is \
not true for all ECDSA curves– in fact, the “sectXXXyZ” curves (for which \
much faster hardware exists than the “secpXXXyZ” curves) cannot be described \
using rings of integers.See this page for an explanation of why every finite \
field has a GF representation but only the prime fields have a a Z/
pZ representation :

en.wikipedia.org/wiki/Finite_field # Statement
See : Fp2 (via quadratic residues modulo p) –
Not wholly true. Not necessarily correct.
Here, phenomenological velocity can be used.
Please research what is the relationship between Fp2 and bitcoin secp256k1?
http : //
www.reddit.com/r/Bitcoin/comments/1 nvsn7/
elliptic_curve _secp256k1 _vulnerability/
“On their protocol specification wiki they say that in their scripts they \
provide hexidecimal decompressed x,y coordinates (though these are really r,s \
values)”
So, if x and y are actually r and s, then we can boil either down to only one \
variable. We can actually force r to be purely in terms of s by definition.
But what are r and s?
http : //
bitcoinmagazine.com/7781/satoshis – genius – unexpected – ways – in – which –
bitcoin – dodged – some – cryptographic – bullet/
Thus, elliptic curve cryptography uses an elliptic curve with two \
modifications.First,
the equation is now y2 =
x3 + ax + b +
kp, where k can be any integer and p is some large prime number (a \
parameter of the curve alongside a and b).Second, x and y must be \
integers.Although the resulting set is hardly a ⎥curve⎠, surprisingly enough \
the same math still works, and the restriction to integers avoids rounding \
errors.
y^2 = x^3 + ax + b + kp
In general, however, the curves fall into two categories :
⎥pseudorandom⎠ curves and Koblitz curves.In a pseudorandom curve, the \
parameters a and b are chosen by a specified algorithm (essentially a hash) \
from a certain ⎥seed⎠.For secp256r1,
the standard 256 –
bit pseudorandom curve, the seed is \
c49d360886e704936a6678e1139d26b7819f7e90, giving rise to the parameters :
Vitalik Buterin : Fortunately, Bitcoin does not use pseudorandom curves;
Bitcoin uses Koblitz curves.In Bitcoin⎟ s secp256k1, the parameters are :
p = 11579208923731619542357098500868790785326998466564056403945758400790883467\
1663
a = 0
b = 7

y^2 == x^3 + 0 x + 7 +
k 11579208923731619542357098500868790785326998466564056403945758400790883467\
1663
This is the base equation for bitcoin elliptical curves from what Vitalik \
says, and Vitalik is the top notch #1 programmer in the bizzz.                \
\

Theta BASE
y = height of cone = h = Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π)
x = base of cone = Sqrt[(r^2 – η^2)] = (2 π r – r θ)/(2 π)
y^2 == x^3 + 0 x + 7 +
k 1157920892373161954235709850086879078532699846656405640394575840079088346\
71663 =

(Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2 == ((2 π r – r θ)/(2 π))^3 +
0 (2 π r – r θ)/(2 π) + 7 +
k 11579208923731619542357098500868790785326998466564056403945758400790883467\
1663
(4 π r^2 θ – r^2 θ^2)/(4 π^2) ==
7 + 1157920892373161954235709850086879078532699846656405640394575840079088346\
71663 k + (2 π r – r θ)^3/(8 π^3)
Solve[(4 π r^2 θ – r^2 θ^2)/(4 π^2) ==
7 + 115792089237316195423570985008687907853269984665640564039457584007908834\
671663 k + (2 π r – r θ)^3/(8 π^3), r]
{{r -> (2 (4 π^2 θ – π θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) + (4 2^(
1/3) (4 π^2 θ – π θ^2)^2)/(3 (8 π^3 – 12 π^2 θ + 6 π θ^2 –
θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6)^2))^(1/3)) + (-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413253317\
30501069312 k π^9 + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239759951\
91503207936 k π^8 θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549699939\
89379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033133293\
26252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887424984\
97344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774849969\
9468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145708308\
289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132\
5331730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397\
5995191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496\
9993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331\
3329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874\
2498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748\
499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457\
08308289079208 k π^3 θ^6)^2))^(
1/3)/(3 2^(1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3))}, {r -> (
2 (4 π^2 θ – π θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) – (2 2^(
1/3) (1 + I Sqrt[3]) (4 π^2 θ – π θ^2)^2)/(3 (8 π^3 – 12 π^2 θ +
6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6)^2))^(
1/3)) – ((1 – I Sqrt[3]) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6)^2))^(1/3))/(6 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3))}, {r -> (
2 (4 π^2 θ – π θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) – (2 2^(
1/3) (1 – I Sqrt[3]) (4 π^2 θ – π θ^2)^2)/(3 (8 π^3 – 12 π^2 θ +
6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6)^2))^(
1/3)) – ((1 + I Sqrt[3]) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6)^2))^(1/3))/(6 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3))}}
But what is zero? Is it really so nothing?
Usefulness : 0 = 2 π r – 2 π x – θ r
0 = 2 π r – 2 π x – θ r
So, let’ s try that again :
(Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2 == ((2 π r – r θ)/(2 π))^3 +
0 (2 π r – r θ)/(2 π) + 7 +
k 11579208923731619542357098500868790785326998466564056403945758400790883467\
1663
Solve[(4 π r^2 θ – r^2 θ^2)/(4 π^2) ==
7 + 115792089237316195423570985008687907853269984665640564039457584007908834\
671663 k + (2 π r – r θ)^3/(8 π^3), r]
(Sqrt[4 π r^2 θ – r^2 θ^2]/(
2 π))^2 == ((2 π r – r θ)/(2 π))^3 + (2 π r – 2 π x – θ r) (2 π r – r θ)/(
2 π) + 7 +
k 11579208923731619542357098500868790785326998466564056403945758400790883467\
1663
(4 π r^2 θ – r^2 θ^2)/(4 π^2) ==
7 + 1157920892373161954235709850086879078532699846656405640394575840079088346\
71663 k + (2 π r – r θ)^3/(8 π^3) + ((2 π r – r θ) (2 π r – 2 π x – r θ))/(
2 π)
Solve[(4 π r^2 θ – r^2 θ^2)/(4 π^2) ==
7 + 115792089237316195423570985008687907853269984665640564039457584007908834\
671663 k + (2 π r – r θ)^3/(8 π^3) + ((2 π r – r θ) (2 π r – 2 π x – r θ))/(
2 π), r]
{{r -> -((2 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3))) – (2^(
1/3) (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 –
θ^3)))/(3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6 +
,((-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 –
30720 π^10 θ^2 – 20736 π^8 x θ^2 – 69120 π^9 x θ^2 +
242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6)^2 +
4 (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 –
θ^3))^3))^(1/3)) +
1/(3 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413253317\
30501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239759951\
91503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549699939\
89379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033133293\
26252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887424984\
97344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774849969\
9468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145708308\
289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6 +
,((-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132\
5331730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397\
5995191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496\
9993989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331\
3329326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874\
2498497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748\
499699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457\
08308289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6)^2 +
4 (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3))^3))^(
1/3)}, {r -> -((2 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 –
θ^3))) + ((1 +
I Sqrt[3]) (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)))/(3 2^(
2/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6 +
,((-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 –
30720 π^10 θ^2 – 20736 π^8 x θ^2 – 69120 π^9 x θ^2 +
242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6)^2 +
4 (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 –
θ^3))^3))^(1/3)) –
1/(6 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (1 – I Sqrt[3]) (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641325331\
730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923975995\
191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154969993\
989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103313329\
326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288742498\
497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577484996\
99468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814570830\
8289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6 +
,((-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413\
25331730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239\
75995191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549\
69993989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033\
13329326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887\
42498497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774\
8499699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145\
708308289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6)^2 +
4 (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3))^3))^(
1/3)}, {r -> -((2 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 –
θ^3))) + ((1 –
I Sqrt[3]) (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)))/(3 2^(
2/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6 +
,((-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 –
30720 π^10 θ^2 – 20736 π^8 x θ^2 – 69120 π^9 x θ^2 +
242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6)^2 +
4 (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 –
θ^3))^3))^(1/3)) –
1/(6 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (1 + I Sqrt[3]) (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641325331\
730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923975995\
191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154969993\
989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103313329\
326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288742498\
497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577484996\
99468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814570830\
8289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6 +
,((-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413\
25331730501069312 k π^9 – 8192 π^12 – 18432 π^11 x + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239\
75995191503207936 k π^8 θ + 12288 π^10 θ + 24576 π^11 θ + 9216 π^9 x θ +
55296 π^10 x θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549\
69993989379009920 k π^7 θ^2 – 6144 π^8 θ^2 – 27648 π^9 θ^2 – 30720 π^10 θ^2 –
20736 π^8 x θ^2 – 69120 π^9 x θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033\
13329326252673280 k π^6 θ^3 + 9216 π^7 θ^3 + 24576 π^8 θ^3 + 20480 π^9 θ^3 +
18432 π^7 x θ^3 + 46080 π^8 x θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887\
42498497344752480 k π^5 θ^4 – 4992 π^6 θ^4 – 10752 π^7 θ^4 – 7680 π^8 θ^4 –
8064 π^6 x θ^4 – 17280 π^7 x θ^4 + 18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774\
8499699468950496 k π^4 θ^5 + 1152 π^5 θ^5 + 2304 π^6 θ^5 + 1536 π^7 θ^5 +
1728 π^5 x θ^5 + 3456 π^6 x θ^5 – 1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145\
708308289079208 k π^3 θ^6 – 96 π^4 θ^6 – 192 π^5 θ^6 – 128 π^6 θ^6 –
144 π^4 x θ^6 – 288 π^5 x θ^6)^2 +
4 (-4 (8 π^4 – 4 π^2 θ – 8 π^3 θ + π θ^2 + 2 π^2 θ^2)^2 –
24 (2 π^4 x – π^3 x θ) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3))^3))^(
1/3)}}
Solve[(4 π r^2 θ – r^2 θ^2)/(4 π^2) ==
7 + 115792089237316195423570985008687907853269984665640564039457584007908834\
671663 k + (2 π r – r θ)^3/(8 π^3) + ((2 π r – r θ) (2 π r – 2 π x – r θ))/(
2 π), k]
{{k -> (-56 π^3 – 16 π^4 r^2 – 8 π^3 r^3 + 16 π^4 r x + 8 π^2 r^2 θ +
16 π^3 r^2 θ + 12 π^2 r^3 θ – 8 π^3 r x θ – 2 π r^2 θ^2 –
4 π^2 r^2 θ^2 – 6 π r^3 θ^2 +
r^3 \
θ^3)/(926336713898529563388567880069503262826159877325124512315660672063270677\
373304 π^3)}}
Solve[(4 π r^2 θ – r^2 θ^2)/(4 π^2) ==
7 + 115792089237316195423570985008687907853269984665640564039457584007908834\
671663 k + (2 π r – r θ)^3/(8 π^3) + ((2 π r – r θ) (2 π r – 2 π x – r θ))/(
2 π), k]
{{k -> (-56 π^3 – 16 π^4 r^2 – 8 π^3 r^3 + 16 π^4 r x + 8 π^2 r^2 θ +
16 π^3 r^2 θ + 12 π^2 r^3 θ – 8 π^3 r x θ – 2 π r^2 θ^2 –
4 π^2 r^2 θ^2 – 6 π r^3 θ^2 +
r^3 \
θ^3)/(926336713898529563388567880069503262826159877325124512315660672063270677\
373304 π^3)}}
c := 2.99792458*(10^8)
Solve[(Sqrt[r Sqrt[1 – (v)^2/c^2]] Sqrt[θ/Sqrt[1 – (v)^2/c^2]] Sqrt[
4 π r – r θ])/(2 π) == r Sin[β], v]
{{v -> -((
1. Sqrt[-1.12941*10^18 θ + 8.98755*10^16 θ^2 + 3.54814*10^18 Sin[β]^2])/
Sqrt[-12.5664 θ + θ^2 + 39.4784 Sin[β]^2])}, {v ->
Sqrt[-1.12941*10^18 θ + 8.98755*10^16 θ^2 + 3.54814*10^18 Sin[β]^2]/
Sqrt[-12.5664 θ + θ^2 + 39.4784 Sin[β]^2]}}
Solve[(4 π r^2 θ – r^2 θ^2)/(4 π^2) ==
7 + 115792089237316195423570985008687907853269984665640564039457584007908834\
671663 k + (2 π r – r θ)^3/(8 π^3) + ((2 π r – r θ) (2 π r – 2 π x – r θ))/(
2 π), k]
(Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2 == ((2 π r – r θ)/(2 π))^3 +
0 (2 π r – r θ)/(2 π) + 7 +
k 11579208923731619542357098500868790785326998466564056403945758400790883467\
1663
0 = (2 π r – 2 π x – θ r) =
0 = θ r – (2 π r – 2 π Sqrt[(r^2 – (Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2)]) =
0
(Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2 == ((2 π r – r θ)/(2 π))^3 +
0 (2 π r – r θ)/(2 π) + 7 +
k 11579208923731619542357098500868790785326998466564056403945758400790883467\
1663
θ r – (2 π r – 2 π Sqrt[(r^2 – (Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2)])
Solve[(Sqrt[4 π r^2 θ – r^2 θ^2]/(
2 π))^2 == ((2 π r – r θ)/(
2 π))^3 + (θ r – (2 π r –
2 π Sqrt[(r^2 – (Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2)])) (
2 π r – r θ)/(2 π) + 7 +
k 1157920892373161954235709850086879078532699846656405640394575840079088346\
71663, k]
{{k -> (-56 π^3 + 16 π^4 r^2 – 8 π^3 r^3 – 8 π^3 r Sqrt[r^2 (2 π – θ)^2] +
8 π^2 r^2 θ – 16 π^3 r^2 θ + 12 π^2 r^3 θ +
4 π^2 r Sqrt[r^2 (2 π – θ)^2] θ – 2 π r^2 θ^2 + 4 π^2 r^2 θ^2 –
6 π r^3 θ^2 +
r^3 \
θ^3)/(926336713898529563388567880069503262826159877325124512315660672063270677\
373304 π^3)}}
Solve[(Sqrt[4 π r^2 θ – r^2 θ^2]/(
2 π))^2 == ((2 π r – r θ)/(
2 π))^3 + (θ r – (2 π r –
2 π Sqrt[(r^2 – (Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2)])) (
2 π r – r θ)/(2 π) + 7 +
k 1157920892373161954235709850086879078532699846656405640394575840079088346\
71663, r]
{{r -> (2 (4 π^2 θ – π θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) + (4 2^(
1/3) (4 π^2 θ – π θ^2)^2)/(3 (8 π^3 – 12 π^2 θ + 6 π θ^2 –
θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6)^2))^(1/3)) +
1/(3 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413253317\
30501069312 k π^9 + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239759951\
91503207936 k π^8 θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549699939\
89379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033133293\
26252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887424984\
97344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774849969\
9468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145708308\
289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132\
5331730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397\
5995191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496\
9993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331\
3329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874\
2498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748\
499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457\
08308289079208 k π^3 θ^6)^2))^(1/3)}, {r -> (2 (4 π^2 θ – π θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) – (2 2^(
1/3) (1 + I Sqrt[3]) (4 π^2 θ – π θ^2)^2)/(3 (8 π^3 – 12 π^2 θ +
6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6)^2))^(1/3)) –
1/(6 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (1 – I Sqrt[3]) (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641325331\
730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923975995\
191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154969993\
989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103313329\
326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288742498\
497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577484996\
99468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814570830\
8289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413\
25331730501069312 k π^9 + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239\
75995191503207936 k π^8 θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549\
69993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033\
13329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887\
42498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774\
8499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145\
708308289079208 k π^3 θ^6)^2))^(1/3)}, {r -> (2 (4 π^2 θ – π θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) – (2 2^(
1/3) (1 – I Sqrt[3]) (4 π^2 θ – π θ^2)^2)/(3 (8 π^3 – 12 π^2 θ +
6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6)^2))^(1/3)) –
1/(6 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (1 + I Sqrt[3]) (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641325331\
730501069312 k π^9 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923975995\
191503207936 k π^8 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154969993\
989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103313329\
326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288742498\
497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577484996\
99468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814570830\
8289079208 k π^3 θ^6 +
,(-256 (4 π^2 θ – π θ^2)^6 + (-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413\
25331730501069312 k π^9 + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239\
75995191503207936 k π^8 θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549\
69993989379009920 k π^7 θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033\
13329326252673280 k π^6 θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887\
42498497344752480 k π^5 θ^4 + 18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774\
8499699468950496 k π^4 θ^5 – 1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145\
708308289079208 k π^3 θ^6)^2))^(1/3)}, {r -> (
2 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 + 4 π^2 θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) + (4 2^(
1/3) (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 + 4 π^2 θ^2)^2)/(3 (8 π^3 –
12 π^2 θ + 6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 + 245760 π^10 θ^2 +
242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 – 163840 π^9 θ^3 –
91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6 +
,(-256 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 +
4 π^2 θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 +
245760 π^10 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 –
163840 π^9 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6)^2))^(
1/3)) +
1/(3 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413253317\
30501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239759951\
91503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549699939\
89379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 + 245760 π^10 θ^2 +
242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033133293\
26252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 – 163840 π^9 θ^3 –
91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887424984\
97344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774849969\
9468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145708308\
289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6 +
,(-256 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 +
4 π^2 θ^2)^6 + (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132\
5331730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397\
5995191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496\
9993989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 +
245760 π^10 θ^2 + 242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331\
3329326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 – 163840 π^9 θ^3 –
91488 π^5 θ^4 –

150066547651561789268947996571259528577837900126670170995137028874\
2498497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748\
499699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457\
08308289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6)^2))^(
1/3)}, {r -> (2 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 + 4 π^2 θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) – (2 2^(
1/3) (1 + I Sqrt[3]) (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 +
4 π^2 θ^2)^2)/(3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 + 245760 π^10 θ^2 +
242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 – 163840 π^9 θ^3 –
91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6 +
,(-256 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 +
4 π^2 θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 +
245760 π^10 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 –
163840 π^9 θ^3 – 91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6)^2))^(
1/3)) –
1/(6 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (1 – I Sqrt[3]) (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641325331\
730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923975995\
191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154969993\
989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 + 245760 π^10 θ^2 +
242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103313329\
326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 – 163840 π^9 θ^3 –
91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288742498\
497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577484996\
99468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814570830\
8289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6 +
,(-256 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 +
4 π^2 θ^2)^6 + (-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413\
25331730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
48021295248499772566063358902803049144908128040534454718443849239\
75995191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549\
69993989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 +
245760 π^10 θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033\
13329326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 –
163840 π^9 θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887\
42498497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774\
8499699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145\
708308289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6)^2))^(
1/3)}, {r -> (2 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 + 4 π^2 θ^2))/(
3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) – (2 2^(
1/3) (1 – I Sqrt[3]) (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 +
4 π^2 θ^2)^2)/(3 (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3) (-96768 π^9 –
160070984161665908553544529676010163816360426801781515728146164132533\
1730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
480212952484997725660633589028030491449081280405344547184438492397599\
5191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
600266190606247157075791986285038114311351600506680683980548115496999\
3989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 + 245760 π^10 θ^2 +
242944 π^6 θ^3 +
400177460404164771383861324190025409540901067004453789320365410331332\
9326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 – 163840 π^9 θ^3 –
91488 π^5 θ^4 –
150066547651561789268947996571259528577837900126670170995137028874249\
8497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
300133095303123578537895993142519057155675800253340341990274057748499\
699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
250110912752602982114913327618765880963063166877783618325228381457083\
08289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6 +
,(-256 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 +
4 π^2 θ^2)^6 + (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641\
325331730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923\
975995191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154\
969993989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 +
245760 π^10 θ^2 + 242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103\
313329326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 –
163840 π^9 θ^3 – 91488 π^5 θ^4 –

1500665476515617892689479965712595285778379001266701709951370288\
742498497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577\
48499699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814\
5708308289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6)^2))^(
1/3)) –
1/(6 2^(
1/3) (8 π^3 – 12 π^2 θ + 6 π θ^2 – θ^3)) (1 + I Sqrt[3]) (-96768 π^9 –
1600709841616659085535445296760101638163604268017815157281461641325331\
730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +
4802129524849977256606335890280304914490812804053445471844384923975995\
191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
6002661906062471570757919862850381143113516005066806839805481154969993\
989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 + 245760 π^10 θ^2 +
242944 π^6 θ^3 +
4001774604041647713838613241900254095409010670044537893203654103313329\
326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 – 163840 π^9 θ^3 –
91488 π^5 θ^4 –
1500665476515617892689479965712595285778379001266701709951370288742498\
497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
3001330953031235785378959931425190571556758002533403419902740577484996\
99468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
2501109127526029821149133276187658809630631668777836183252283814570830\
8289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6 +
,(-256 (16 π^4 + 4 π^2 θ – 16 π^3 θ – π θ^2 +
4 π^2 θ^2)^6 + (-96768 π^9 –
16007098416166590855354452967601016381636042680178151572814616413\
25331730501069312 k π^9 + 65536 π^12 + 290304 π^8 θ +

48021295248499772566063358902803049144908128040534454718443849239\
75995191503207936 k π^8 θ + 49152 π^10 θ – 196608 π^11 θ – 362880 π^7 θ^2 –
60026619060624715707579198628503811431135160050668068398054811549\
69993989379009920 k π^7 θ^2 + 12288 π^8 θ^2 – 110592 π^9 θ^2 +
245760 π^10 θ^2 + 242944 π^6 θ^3 +
40017746040416477138386132419002540954090106700445378932036541033\
13329326252673280 k π^6 θ^3 – 18432 π^7 θ^3 + 98304 π^8 θ^3 –
163840 π^9 θ^3 – 91488 π^5 θ^4 –
15006654765156178926894799657125952857783790012667017099513702887\
42498497344752480 k π^5 θ^4 + 9984 π^6 θ^4 – 43008 π^7 θ^4 + 61440 π^8 θ^4 +
18336 π^4 θ^5 +
30013309530312357853789599314251905715567580025334034199027405774\
8499699468950496 k π^4 θ^5 – 2304 π^5 θ^5 + 9216 π^6 θ^5 – 12288 π^7 θ^5 –
1528 π^3 θ^6 –
25011091275260298211491332761876588096306316687778361832522838145\
708308289079208 k π^3 θ^6 + 192 π^4 θ^6 – 768 π^5 θ^6 + 1024 π^6 θ^6)^2))^(
1/3)}}
Solve[(2 π (2 π r – 2 π Sqrt[r^2 – (η)^2]))/θ^2 == (4 π r^2 – 2 r^2 θ)/(
2 Sqrt[4 π r^2 θ – r^2 θ^2]), r]
0 = (2 π (2 π r – 2 π Sqrt[r^2 – (η)^2]))/θ^2 – (4 π r^2 – 2 r^2 θ)/(
2 Sqrt[4 π r^2 θ – r^2 θ^2]) = 0
Solve[(2 π (2 π r – 2 π Sqrt[r^2 – (η)^2]))/θ^2 – (4 π r^2 – 2 r^2 θ)/(
2 Sqrt[
4 π r^2 θ – r^2 θ^2]) == (θ r – (2 π r – 2 π Sqrt[(r^2 – η^2)])) + (1 –
Sqrt[4 π θ – θ^2]/(2 π Sin[β])), r]
{{}}
Solve[(2 π (2 π r – 2 π Sqrt[r^2 – (η)^2]))/θ^2 – (4 π r^2 – 2 r^2 θ)/(
2 Sqrt[
4 π r^2 θ –
r^2 θ^2]) == (θ r – (2 π r –
2 π Sqrt[(r^2 – (Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2)])) + (1 – Sqrt[
4 π θ – θ^2]/(2 π Sin[β])), η]
{{η -> -1/(
32 π^3) (,(512 π^4 r θ^2 – 1024 π^5 r^2 θ^2 +
512 π^4 r Sqrt[r^2 (2 π – θ)^2] θ^2 – 64 π^3 r^2 θ^3 +
512 π^4 r^2 θ^3 – 64 π^2 θ^4 + 256 π^3 r θ^4 + 48 π^2 r^2 θ^4 –
512 π^4 r^2 θ^4 – 128 π^2 Sqrt[r^2 (2 π – θ)^2] θ^4 +
256 π^3 r Sqrt[r^2 (2 π – θ)^2] θ^4 – 128 π^2 r θ^5 – 4 π r^2 θ^5 +
512 π^3 r^2 θ^5 – 128 π^2 r Sqrt[r^2 (2 π – θ)^2] θ^5 – r^2 θ^6 –
128 π^2 r^2 θ^6 – (r^2 θ^7)/(4 π – θ) +
256 π^4 r θ Sqrt[r^2 (4 π – θ) θ] –
64 π^3 r θ^2 Sqrt[r^2 (4 π – θ) θ] –
64 π^2 θ^3 Sqrt[r^2 (4 π – θ) θ] –
16 π^2 r θ^3 Sqrt[r^2 (4 π – θ) θ] +
128 π^3 r θ^3 Sqrt[r^2 (4 π – θ) θ] –
64 π^2 Sqrt[r^2 (2 π – θ)^2] θ^3 Sqrt[r^2 (4 π – θ) θ] +
16 π θ^4 Sqrt[r^2 (4 π – θ) θ] – 4 π r θ^4 Sqrt[r^2 (4 π – θ) θ] –
96 π^2 r θ^4 Sqrt[r^2 (4 π – θ) θ] +
16 π Sqrt[r^2 (2 π – θ)^2] θ^4 Sqrt[r^2 (4 π – θ) θ] +
4 θ^5 Sqrt[r^2 (4 π – θ) θ] – r θ^5 Sqrt[r^2 (4 π – θ) θ] +
8 π r θ^5 Sqrt[r^2 (4 π – θ) θ] +
4 Sqrt[r^2 (2 π – θ)^2] θ^5 Sqrt[r^2 (4 π – θ) θ] +
2 r θ^6 Sqrt[r^2 (4 π – θ) θ] + (4 θ^6 Sqrt[r^2 (4 π – θ) θ])/(
4 π – θ) – (r θ^6 Sqrt[r^2 (4 π – θ) θ])/(4 π – θ) + (
4 Sqrt[r^2 (2 π – θ)^2] θ^6 Sqrt[r^2 (4 π – θ) θ])/(4 π – θ) + (
2 r θ^7 Sqrt[r^2 (4 π – θ) θ])/(4 π – θ) –
256 π^3 r θ^2 Sqrt[(4 π – θ) θ] Csc[β] +
64 π θ^4 Sqrt[(4 π – θ) θ] Csc[β] –
128 π^2 r θ^4 Sqrt[(4 π – θ) θ] Csc[β] +
64 π Sqrt[r^2 (2 π – θ)^2] θ^4 Sqrt[(4 π – θ) θ] Csc[β] +
64 π r θ^5 Sqrt[(4 π – θ) θ] Csc[β] +
32 π θ^3 Sqrt[(4 π – θ) θ] Sqrt[r^2 (4 π – θ) θ] Csc[β] –
8 θ^4 Sqrt[(4 π – θ) θ] Sqrt[r^2 (4 π – θ) θ] Csc[β] – (
8 θ^5 Sqrt[(4 π – θ) θ] Sqrt[r^2 (4 π – θ) θ] Csc[β])/(4 π – θ) –
64 π θ^5 Csc[β]^2 + 16 θ^6 Csc[β]^2))}, {η ->
1/(32 π^3) (,(512 π^4 r θ^2 – 1024 π^5 r^2 θ^2 +
512 π^4 r Sqrt[r^2 (2 π – θ)^2] θ^2 – 64 π^3 r^2 θ^3 +
512 π^4 r^2 θ^3 – 64 π^2 θ^4 + 256 π^3 r θ^4 + 48 π^2 r^2 θ^4 –
512 π^4 r^2 θ^4 – 128 π^2 Sqrt[r^2 (2 π – θ)^2] θ^4 +
256 π^3 r Sqrt[r^2 (2 π – θ)^2] θ^4 – 128 π^2 r θ^5 – 4 π r^2 θ^5 +
512 π^3 r^2 θ^5 – 128 π^2 r Sqrt[r^2 (2 π – θ)^2] θ^5 – r^2 θ^6 –
128 π^2 r^2 θ^6 – (r^2 θ^7)/(4 π – θ) +
256 π^4 r θ Sqrt[r^2 (4 π – θ) θ] –
64 π^3 r θ^2 Sqrt[r^2 (4 π – θ) θ] –
64 π^2 θ^3 Sqrt[r^2 (4 π – θ) θ] –
16 π^2 r θ^3 Sqrt[r^2 (4 π – θ) θ] +
128 π^3 r θ^3 Sqrt[r^2 (4 π – θ) θ] –
64 π^2 Sqrt[r^2 (2 π – θ)^2] θ^3 Sqrt[r^2 (4 π – θ) θ] +
16 π θ^4 Sqrt[r^2 (4 π – θ) θ] – 4 π r θ^4 Sqrt[r^2 (4 π – θ) θ] –
96 π^2 r θ^4 Sqrt[r^2 (4 π – θ) θ] +
16 π Sqrt[r^2 (2 π – θ)^2] θ^4 Sqrt[r^2 (4 π – θ) θ] +
4 θ^5 Sqrt[r^2 (4 π – θ) θ] – r θ^5 Sqrt[r^2 (4 π – θ) θ] +
8 π r θ^5 Sqrt[r^2 (4 π – θ) θ] +
4 Sqrt[r^2 (2 π – θ)^2] θ^5 Sqrt[r^2 (4 π – θ) θ] +
2 r θ^6 Sqrt[r^2 (4 π – θ) θ] + (4 θ^6 Sqrt[r^2 (4 π – θ) θ])/(
4 π – θ) – (r θ^6 Sqrt[r^2 (4 π – θ) θ])/(4 π – θ) + (
4 Sqrt[r^2 (2 π – θ)^2] θ^6 Sqrt[r^2 (4 π – θ) θ])/(4 π – θ) + (
2 r θ^7 Sqrt[r^2 (4 π – θ) θ])/(4 π – θ) –
256 π^3 r θ^2 Sqrt[(4 π – θ) θ] Csc[β] +
64 π θ^4 Sqrt[(4 π – θ) θ] Csc[β] –
128 π^2 r θ^4 Sqrt[(4 π – θ) θ] Csc[β] +
64 π Sqrt[r^2 (2 π – θ)^2] θ^4 Sqrt[(4 π – θ) θ] Csc[β] +
64 π r θ^5 Sqrt[(4 π – θ) θ] Csc[β] +
32 π θ^3 Sqrt[(4 π – θ) θ] Sqrt[r^2 (4 π – θ) θ] Csc[β] –
8 θ^4 Sqrt[(4 π – θ) θ] Sqrt[r^2 (4 π – θ) θ] Csc[β] – (
8 θ^5 Sqrt[(4 π – θ) θ] Sqrt[r^2 (4 π – θ) θ] Csc[β])/(4 π – θ) –
64 π θ^5 Csc[β]^2 + 16 θ^6 Csc[β]^2))}}

Solve[1 – Sqrt[4 π θ – θ^2]/(2 π Sin[β]) ==
θ r – (2 π r – 2 π Sqrt[(r^2 – (Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2)]), r]
Solve[1 – (2 π Sin[β])/Sqrt[4 π θ – θ^2] ==
θ r – (2 π r – 2 π Sqrt[(r^2 – (Sqrt[4 π r^2 θ – r^2 θ^2]/(2 π))^2)]), r]

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Bitcoin Fat Finger Part Deux

http://www.reddit.com/r/Bitcoin/comments/2ib5hm/possible_mass_market_manipulation_by_chinese/

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Bitcoin Fat Finger August 1st 2014 (8-1-14) or Massive Buy? Dark Pool Settlement? Bitcoin Trades over $4300.00 (BTC $7000+ eye witness)

Is Bitcoin poised for a massive gain or was the event on August 1st 2014 just a fat finger accident. Did massive amounts of bitcoin exchange hands at massive prices around 22:00 hours UST? Did Bitcoin hit an all time high on August 1st 2014 of $7328.00 with an average price per BTC of $4417.90 on extra heavy trading? At around 5:00 in the afternoon, I got an alert on my phone from BTC Avg http://www.bitcoinaverage.com triggered by Bitcoin passing its all time nominal high of $1280.00 per each on Bitcoinaverage.com. I looked at my phone, and I couldn’t believe it, bitcoin had just traded at over $7300.00. I then went to Bitcoinaverage.com for actual confirmation of the volume. I couldn’t believe it. The average BTC price had jumped to $4300.00 across exchanges. There was double the volume in a single trade for the whole day. Somebody had exchanged over 17,000 bitcoins. Bitstamp was not reporting totals to the bitcoinaverage.com. A dark pool settlement occurs mostly outside of an exchange, or if provided by an exchange, is supposed to remain secret. It looks as though somebody may have leaked just one of these kinds of transactions intentionally or by accident.

If it was an accident, which exchange reported the data? Why is nobody investigating this? Why am I, a little blogger, the only one covering this at all so as to really get to the bottom of what happened?

I knew this wouldn’t last for long if it were an accident, and I knew someone would try to cover their tracks if it wasn’t an accident. I captured the screenshot from my phone while it still lasted. Moments later, this phenomenon vanished.

See:

Bitcoin Price Spike Hits All Time High from Revealed Dark Pool?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Could dark pools be trading BTC on dark wallets and dark exchanges, unnoticed by the greater trading community? Yes, they could be. Could they slip up and let us know the real price of bitcoin when you need to buy 17,000 of them? Yes, they could have.

Could this just be a glitch? Perhaps, let’s examine the evidence.

If you needed to buy 17,000 BTC, how would you get it? Who would be there to sell it to you? Prices seem to have settled back after the massive, seemingly unnoticed spike in the price, but perhaps most traders did not notice the slip up from the dark pool, where Bitcoin could be trading at massive all time highs due to the real volume. I have seen no other mention of what happened today with the price, so I felt obliged to post. Could a major player in the financials be getting into bit coin, resulting in a massive spike?

Local BTC was trading at over $900.00 at the time, and the other clue is that Bitstamp was not reporting figures at the time of the trade, but somehow, the feed kept going into Bitcoinaverage.com. Many, many Local BTC traders peg their pricing to the BitStamp.net spot. This leads me to think that Bitstamp settled a massive trade for a big player at a high BTC cost relative to the market, and the market was left undisturbed. People really didn’t notice, because it was masked relatively well on most exchanges. The order books would have gone out of control entirely and spiked the price of BTC permanently on a trade like that, creating massive volatility in the markets. Here, we see the principle of phenomenological velocity, the canceling of the volume while the trade actually takes place. If traders are looking for more BTC velocity, then look no further. Buy bit coin now.

Flat all day compared to the massive spike

Flat all day compared to the massive spike

With the bitcoin price’s exceeding $4300 per each bitcoin, some even settling for over $7000.00 each, the relative volatility for the rest of the day looked flat. You can probably still see this if you download the iphone app, btcReport on iPhone. We can see from the chart, that the bitcoin average price actually hit $1261.2 today. https://bitcoinaverage.com/markets#USD still reports that Bitstamp has been unreachable for a long time.

Furthermore, we can see that as noted, at 22:00-22:04, a flash spike in BTC took place. No data is published during that time, as it was just exactly between the cracks of reported trading on the official ledger at bitcoinaverage.com.

 

 

 

 

Here, we see how the official ledger does not mention the trade of BTC, some of which cost over $7300.00 (this is only my eye-witness account, because before I had a chance to think about capturing it on my phone, the price had slipped down under $4300.00).

How would the math work out here? What kind of formula could have been used to determine the cost of bitcoin for a “dark pool” settlement. With an average cost of bitcoin reported in the above example (exhibit), let’s assume $1261.00 for the average cost of bitcoin including the “fat finger” BTC or, “dark pool” exposed bitcoin sale.  The reported number of bitcoins sold was about 17800 BTC. 17800 * 1261 = $22,445,800, but the “real” reported average BTC cost is assumed to be $600.00, or about $10,680,000 daily volume. The difference is $11,765,800. So, we know the price of the dark pool bitcoins was between $1200-$7300.00 each. Let’s assume the average of those two prices as the cost per bitcoin. That would be $4250.00 per BTC. This number is somewhat evidenced, because after the $7300.00 figure, the price came down to ~$4400.00 and then even lower to around $1280.00. This tentatively indicates that only about 2768.42353 bitcoins were sold in a lump sum.

Conclusion: Hold your bit coins. Buy more bitcoins. We could see this price revealed to the entirety of the BTC market within the coming months and at that point, at what price will Dark Pools be trading bit coin?

 

Was this leaked by a dark pool or just a fat finger?

Dark Pool trade settlement or fat finger?

 

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