We want to divide: z1 = -2 + 2i by z2 = -2 - 3i
Monday, June 30, 2025
Solutions To Complex Numbers In Polar Form (Part 2) Problem
Friday, June 27, 2025
Some Ruminations on Max Tegmark's Multiverse Types (And "Alternate" Universes)

One person who has tried to provide categories and clarity is Max Tegmark of M.I.T. He has suggested a fourfold classification scheme, but only three of them are relatively comprehensible to most ordinary folk without advanced physics backgrounds. (It is those I will deal with in this post.)
Type 1:
Thus, if light takes 13.8 billion years to travel to the maximum distance we can actually see (assuming space is static) that turns out to be 13,8 billion LIGHT YEARS. (One light year being the distance light travels in one year.)
In fact, this is a simplification because space or rather space-time isn't static. Because of its expansion immediately following the Big Bang the actual radius of the cosmos is 42 billion light years or some 28.2 billion LY greater than the telescopic limit. Assuming physical reality, i.e. the universe, exists beyond the actual Hubble radius then all permitted arrangements may exist - and in infinite numbers.
In effect one finds separate "cosmi", cut off from each other by their own individual Hubble radii. These would be like separate compartments or "bubbles" cut off from each other. The key point is that the laws of physics in one "bubble" are the same in those in all the others because in the end the universe - despite the disparate "bubbles- is one entity.
While the Type 1 version is based on the cosmological principle, so the laws of physics are the same in all the separate "bubbles" with their own Hubble radii, in this Type 2 case they can vary from one universe to another. The value of G, the Newtonian gravitational constant may be G as we know it (6.7 x 10-11 Nm2/kg2) in our universe, but 1.1G in another in the Type 2 Multiverse, and 0.98G in another. The result would be separate universes remarkably different from each other.
As I noted in previous blog posts, the genesis of the Type 2 Multiverse is distinct from the Type 3 which is really Hugh Everett's "Many worlds" quantum-based theory (which we will get to.) In the Type 2 all the universes in the Multiverse were spawned as a result of cosmic inflation immediately following the Big Bang.
Regarding inflation, most current standard theories propose inflation starting at about 10-35 s and doubling over a period of anywhere from 10-43 to 10-35 s after the initial inception. Estimates are that at least 85 such 'doublings' would be required to arrive at the phase where entropy rather than field resident energy dominated. The initial size (radius) of our universe would have been likely less than a proton's - maybe 1 fermi (fm) or 10-15 m, by the time the doubling process began. By the time it ended (after 90 'doublings') it would have been around 1.25 x 1012 m. This is roughly eight times the distance of Earth from the Sun.
In effect, the role of inflation is to give cosmic expansion a huge head start or boost, without which our universe would be much smaller. If such an "inflationary field" could spawn our universe it could spawn many others (up to an infinite number). Further, there is no reason why these offshoot universes from inflation should have the same laws of physics as any of the others.
This is a delightful conclusion since it disposes at once of the "specialness" of the cosmos that too many invoke as a cosmological argument to demand a deity or "Creator". However, if universes are commonplace, and the physical laws that govern each vary, then the need for a "human-friendly" creator vanishes. It is no longer a fluke that one universe has just the right conditions for life if gazillions of them don't.
Type 2 universes, then, aptly deal with the annoying fine tuning problem that religionists endlessly invoke.
Type 3:
The Type 3 "Multiverse" is in reality a product of Hugh Everett's Many worlds quantum interpretation, which was devised to counter the Copenhagen Interpretation's strange ramifications. In the Copenhagen Interpretation, any observer's consciousness is theoretically capable of "collapsing" the wave function, yielding one and only one eigenstate or final observation, i.e. observed state. Everett, to his credit, argued that rather than dealing with one wave function for whatever observed entity (particle, universe, cat in a box - subject to release of cyanide if a cesium atom decays triggering the release device) one might let ALL possible outcomes occur.
In this case, the universe is constantly undergoing a kind of multiple "fission" of reality into umpteen daughter universes where different events unfold from the one we're in. To fix ideas, in one of them Lee Oswald is a published Professor of History at Tulane, not an accused assassin. In the same or other universe, LBJ's plan to have JFK killed is exposed before the executive action and the SOB is tried for treason. In another the Challenger disaster never occurs, it goes off perfectly because NASA took the time to solve the O-ring problem. In yet another, there is no Indonesian tsunami that killed 200,000 in December, 2004 - but there is a massive ocean asteroid strike that kills just as many in SE Asia. You see what I mean?
In a way, Hugh Everett's "Many worlds" interpretation is actually a theory of alternate universes, at least at the level of potential quantum states. It is more compellingly described this way than as a third type Multiverse, in my opinion. Especially the Type 2 comprising actual physical universes incepted from the selfsame primordial vacuum state (via inflation) as our own universe. Thus, an actual primordial vacuum - not a human observer or consciousness making observational choices- is the source of the real set of universes. Thus, all putative parallel universes plausibly emerged from the primordial vacuum the way ours did, e.g. from the Big Bang.
I repeat this is an idealized model which assumes that N-cosmi were incepted at equal intervals of time - as manifested by the equal spacing in Θ.
In principle, we don't know a priori how "close" (e.g. in complex time) another universe may be to our own. When one uses the assumption of "equal time intervals" between inceptions in our idealized multiverse, one isn't stating what those times are, and so they could be minuscule - and the smallest time unit imaginable is the unit tau, τ. (About 10-43 s, and note Θ = f(τ).)
If we specify an exact parallel time displacement we might be able to show how one universe can be "mapped" topologically onto an adjacent one. As an example, let two parallel universes be distinguished by a 1-τ difference in fundamental time parameter, viz. [1 + 2τ] and [1 + 3τ], then we would require for connection, a mapping such that:
(Universe 'A'): f:X -> X = f(Θ,φ) = (Θ, 2φ)
(Universe 'B'): f:X -> X = f(Θ,φ) = (Θ, 3φ)
which means the absolute coordinate φ is mapped onto itself 2 times for [Universe A] and mapped onto itself 3 times for [Universe B]. Clearly, there’ll be coincidences for which: f(Θ,2φ) = f(Θ,3φ) wherein the two universes will 'interweave' a number of times.
For example, such interweaving will occur when φ = π/2 in [A] and φ = π/3 in [B]. The total set or system of multiple points obtained in this way is called a Synchronous temporal matrix. The distinguishing feature of this matrix is that once a single point is encountered, it is probable that others will as well. If one hyperspace transformation can occur linking adjacent universes, A and B, then conceivably more such transformations can occur, linking A and C, D and E etc.
What if both absolute toroidal coordinates (Θ,φ) map into themselves the same number of times? Say, something like:
f:X -> = f(Θ, φ) = (2Θ, 2φ): Universe A
f:X -> = f(Θ, φ) = (3Θ, 3φ): Universe B
For example, given the previous conditions for coordinate φ, now let 2Θ = 3Θ for discrete values of Θ (e.g. 2π). For all multiples of 2π, the same toroidal cosmos will be experienced - if the absolute time coordinates are equal (e.g. π/2 = φ in A, and π/3 = φ in B) then we will have: Universe A = Universe B.
This isn't necessarily poppycock. Stephen Freeney of Imperial College, London has surmised that two adjacent universes in a Type 2 Multiverse could conceivably 'butt up' against each other and leave "imprints" in each other's space. He reasons that these imprints would likely show up in the cosmic microwave background radiation, generating 'splotches' in the radiation field or differing energy density signatures. As yet no such signal has been found, but in truth we may not yet possess the instruments needed to identify such signatures.
If and when we do at least one of Max Tegmark's propositions will have been validated. Until then it's nice to speculate.
Wednesday, June 25, 2025
Three Basic Mensa Logic Problems
1.The top equilateral triangle shown in Fig. 1(a) is rotated clockwise through 60 degrees, then 60 degrees counterclockwise,
R1: Rotation clockwise by 90 degrees
R2: rotation clockwise by 180 degrees
R3: rotation clockwise by 270 degrees
M1: Mirrors corners exactly through the midline of the square
M2: Mirrors two opposing diagonal corners: from top left to bottom right
M3: Mirrors two opposing corners in the opposite sense to M2
If we apply the sequence of operations to the square in the order identified below:
R2 M1 M3
Then predict the result - assuming the square in Fig. 2 is given by ABCD.
Tuesday, June 24, 2025
Newsflash - If Iran has followed a playbook of nuclear deception, it was written first (in JFK Era) by Israel.
Few Americans, even dedicated history buffs, are aware that David
Ben-Gurion, Israel’s first prime minister, decided in the mid-1950s that Israel
needed a nuclear weapon as an insurance policy against its Arab neighbors. In
the 1950s and 1960s, Israel secretly acquired the technology and material to
build nuclear weapons, frequently misleading the U.S. government (and other
governments) about its intentions. The reason for the duplicity was clear: The then U.S. government (namely JFK's) was dead set against an Israeli bomb.
One of the first to document the Israeli subterfuge (and later enmity with JFK's government) was former British Intelligence officer Col. John Hughes-Wilson – whose seminal work on the Kennedy assassination I’d posted about in Aug. 2016:
One Of Top Ten Books On The JFK Assassination - From Former British Intelligence Officer
Hughes-Wilson has perhaps done the most to dig into the Israeli- dodges about their nuclear program and how it also may relate to the JFK assassination. He goes into details on p. 91:
The starting point was that Israel wanted a bomb and JFK wanted to stop them from getting one. Ben-Gurion lied to both his countrymen and to JFK. The ageing Zionist claimed to the Israeli parliament that the Dimona plant was a scientific research facility "investigating the problems of fauna in arid regions."
All of which meshes with Glenn Kessler's piece yesterday in the Washington Post:
How Israel deceived the United States about its nuclear weapons program
Wherein we read:
When U.S. intelligence discovered the secret facility at Dimona deep in the desert late in the 1950s, Israeli officials lied to the American Embassy and said it was only a textile plant. When that turned out to be false, Israeli officials offered another explanation: It was purely a metallurgical research installation that did not contain the chemical reprocessing plant needed to produce nuclear weapons.
In December 1960, Ben-Gurion revealed the facility in a speech in the Knesset, saying the 24-megawatt reactor at Dimona would not be completed for four years. It was, he said, “intended exclusively for peaceful purposes.”
Newly elected U.S. President John F. Kennedy, alarmed about the potential spread of nuclear weapons, pressed Israeli officials for regular inspections of Dimona. A 1961 team concluded the site lacked the necessary facilities — such as plutonium reprocessing — needed for a weapons program. Yet U.S. officials wanted regular inspections so they could assure Arab nations, especially Egypt, that Israel did not have a secret bomb program.
The diplomatic record, obtained by the National Security Archive at George Washington University, shows that Israel put off or delayed additional inspections until Kennedy sent a blunt message in July 1963 to a new Israeli prime minister, Levi Eshkol. (Originally the letter was drafted for Ben-Gurion but he resigned before it could be delivered.)
JFK assassination suspect Clay Shaw was also involved with Permindex
through its subsidiary, Centro Mondiale
Commerciale (CMC). Years later, with release of files under the JFK
Records Act, it would become evident that Shaw was indeed a CIA Contract Agent.
As CIA Doc. (JFK 1993: 6.28.16.07.26.560280) notes:
A memorandum marked for files says that J. Monroe Sullivan, #280201, was granted a covert security approval as of 10 December 1962 so he could be used in Project QKCHANT [Clay L. Shaw has #402897]
In effect, New Orleans' DA Jim Garrison’s tying him to the JFK assassination
as a key coordinator had been validated. (At his 1969 trial in New Orleans Shaw
denied he had an alias or worked for the CIA)
Was Mossad involved in the JFK assassination? Hughes-Wilson doesn't go that far, but he does note (ibid.):
"Mossad's link with North America became aware in the summer of 1963 that Mafia sources warned of a death threat already made against JFK. To Israel and its friends this could only be good news. It would remove the problem of JFK - and should anything happen to him, the successor would be Lyndon Johnson. Fortunately, LBJ was known to be a strong supporter of Jewish interests."
He has also been tagged as the architect of the assassination itself. This was by author Phillip F. Nelson, who leaves no arcane move unturned, in his masterful book:
That includes that LBJ had previously been exposed in the killing of Henry Harvey Marshall, a USDA official - who had discovered Estes' underhanded business dealings with Johnson. See e.g.
Nelson’s book ‘LBJ – The Mastermind of the JFK Assassination’ – is noteworthy for its exacting documented detail. As he observes (Chapter 6: The Conspirators, p. 317):
“The crime could only have been accomplished with at least the acquiescence and foreknowledge of the only man capable of choreographing the massive cover-up which was immediately launched. It is axiomatic that since the cover-up started before the shots were fired, the order for JFK’s assassination could only have come from his successor, Lyndon B. Johnson.”
He goes on to note that no other conceivable person, whether
Santos Trafficante, Sam Giancana, Clint Murchison, H.L. Hunt….or Lee Harvey
Oswald…" had the motive, means, the opportunity, the demonstrated
pattern of previous criminal – even murderous conduct, and the resolve to see
it through.”
As of 2021, Israel is believed to possess 90 nuclear warheads for delivery by aircraft, land-based ballistic missiles and possibly sea-based cruise missiles, according to an estimate by researchers at the Federation of American Scientists.
Those who lose sleep worrying about Iran's nuclear program might want to worry a bit more about Israel's - possessing enough warheads to incinerate the entire Middle East.
See Also:
And:
Kennedy, Dimona and the Nuclear Proliferation Problem: 1961-1962 | Wilson Center
And:
Dealing With The Basics Of Complex Numbers (Part 2): Division In Polar Form
Let's say we want to divide:
z1 = Ö2(cos(-45) + isin(-45)) = Ö2 cis(-45)by
z2 = 3.6(cos(56.3) + isin(56.3)) = 3.6 cis(56.3)
In all such cases of complex division we require that the z, r in the denominator
not be zero.
Thus:
(z1/z2) = (r1 cis(q1)/ r2 cis(q2)) = (r1/
r2) cis (q1 – q2)
Now: (r1/ r2) = (1.414/ 3.6) = 0.39
And we saw previously:
(q1 – q2) = arg(z1) – arg(z2) =
(-45) – (56.3) = -101.3
Thus, the basic procedure for division entails dividing the lengths (r’s) and subtracting the angles (q1 – q2).
So:
(z1/ z2) = 0.39 (cos (-101.3) + isin(-101.3))
= 0.39((-0.195) + i(-0.98)) = -0.07
+ 0.38i
What about? (1 +
i) ¸ Ö3 – i
The
first order of business is to get dividend and divisor each into polar form,
specifically as a (cis) function:
Then (1 + i) = z1 = x1 + iy1, so arg(z1 ) =
arctan (y1/x1)
Further:
arctan (y1/x1) = arctan (1/1) = arctan (1) so q1 = 45 deg
What about r1? r1= [12 + 12]1/2 = Ö2 = 1.4
so z1 = 1.4 [cos (45) + isin(45)] = 1.4 cis(45)
Now: z2 = Ö3 – i So:
arg(z2) = arctan(y2/x2) = arctan(-1/ Ö3) so q2 = (-30 deg)
And for r2: r2 = [(Ö3)2
+ (-1)2]1/2 = Ö4 =
2
Then: z2 = 2[cos(-30) +isin(-30)] = 2cis(-30)
We divide: (z1/z2)
Which means dividing the r’s first:
r1/r2 = Ö2/
2
Then subtract angles: [(q1 – q2) ] = {(45 deg) – (-30 deg)} = 75 degrees
So the end result of the division is:
(z1/z2) = Ö2/ 2 cis(75) = Ö2/ 2 {cos(75) + isin(75)}
= 0.707{cos(75) + isin(75)}
Since cos(75) = 0.258 and sin(75) =0.966, we have:
(z1/z2) =
0.707[(0.258) + i(0.966)] = 0.183 + 0.683i
Problem:
Use the polar form of complex numbers to divide:
z1 = -2 + 2i by z2 = -2 - 3i
Monday, June 23, 2025
Re-engineered Alien Technology (in B-2 Bombers) Used In Trump's Onslaught? Probably (& Other Voices Weigh In On Attacks)
It’s been known at least since 1947 that special tech teams (mainly at Wright-Patterson Air Force Base) have been dedicated to re-engineering the crashed Roswell craft. But toward more earthly designs given a decided lack of tech savvy. Aircraft like the B-2 Spirit Stealth bomber - seven of which were used to dump 14 "bunker buster" bombs on three Iranian nuclear sites over the weekend. One might well be immediately drawn to this link: highlighting a B-2 landing at a UK airfield.
Seeing B-2 stealth bombers land is like seeing alien UFOs come to Earth : r/aviation

This elicits the question of what - if anything - my late AF vet brother Jerry:
Last weekend was a lollapaloosa, wasn't it? Sidestepping the Constitution again, Trump & Co. personally declared war on Iran with what is purported to be the biggest airstrike in our nation’s history. It wasn’t supposed to happen based on his say-so, but it did.
According to the Constitution, he was supposed to say “mother, may I?” to get permission from Congress to launch a war, but all he really needed was a) a free pass from the Supreme Court on everything he did as POTUS, b) a cheering section for everything he did from the right-wing propaganda media and damn near every elected Republican, c) a lot of quite easily fooled Americans who passively believed what they were told no matter how preposterous, d) a craven press corps, e) an even more craven bunch of billionaires who put their own interests above all others, f) a great many corrupt corporate CEOs in big tech, big pharma, big oil, big retail, and big "defense" contracting, g) Merrick Garland, who slow-walked justice until it was denied to Americans who weren’t allowed to see the evidence of most of Trump’s crimes, h) thousands of lies, i) Russian interference, j) the electoral college and two very hinky elections we were told he had “won.”
Twenty years ago, one day in June 2005, I talked with an Iranian man who was selling underwear at the Tehran Grand Bazaar. People all over the world want peace, he said, but governments won’t let them have it.
I thought of that conversation on Saturday night after the U.S. government attacked nuclear sites in Iran. For many days before that, polling clearly showed that most Americans did not want the United States to attack Iran. “Only 16 percent of Americans think the U.S. military should get involved in the conflict between Israel and Iran,” YouGov pollsters reported, while “60 percent say it should not and 24 percent are not sure.”
But as a practical matter, democracy has nothing to do with the chokehold that the warfare state has on the body politic. That reality has everything to do with why the United States can’t kick the war habit. And that’s why the profound quests for peace and genuine democracy are so tightly intertwined.
We’ve been here before. It’s never ended well. It’s never ended, period. A few bunker-busters aren’t about to end it either, whether they have Fordow’s Mount Doom in the bag or not. The opposite always happens in the Middle East the moment Israel and the United States substitute barbarism for diplomacy. Always.
There’s not been a single exception to the rule since 1956, the last time the United States intervened to stop Israeli (and British and French) aggression on a neighbor.
It’s not as if certain Israeli and American administrations haven’t known how to negotiate. They’ve had numerous Nobel-worthy diplomatic successes: Israel’s peace with Egypt, with Jordan, with the Palestine Liberation Organization (PLO) and its transformation into the Palestinian Authority, and more recent normalizations with Morocco, the United Arab Emirates, Bahrain and Sudan and what was the almost normalization with Saudi Arabia. Not least, there was the 2015 deal with Iran, which verifiably stopped Iran’s uranium enrichment until the current president recklessly pulled out in his first term, setting up his flim-flanked bombast last night.
In the modern era, it was probably George W. Bush who first said it out loud and then acted on it: When you’re unpopular and losing politically, just start a little war that’s easily winnable and you’ll be back on top.
As he told his biographer, Mickey Herskowitz, in 1999 about his plans for an Iraq war as a strategy to get himself reelected in 2004:
One of the keys to being seen as a great leader is to be seen as a commander-in-chief. My father had all this political capital built up when he drove the Iraqis out of (Kuwait), and he wasted it. If I have a chance to invade Iraq, if I had that much capital, I'm not going to waste it. I’m going to get everything passed I want to get passed, and I'm going to have a successful presidency.
It worked for Bush, although history hasn’t been kind to him as a result. Donald Trump’s second presidency, meanwhile, has been an unmitigated disaster, both in real terms and politically as his approval ratings have slipped so far underwater they’re in late-years Richard Nixon territory:
Friends,
The United States is now at war with Iran.
A single person — Donald J. Trump — has released the dogs of war on one of the most dangerous countries in the world, and done it without the consent of Congress, our allies, or even a clear explanation to the American people.
Anyone who has doubted Trump’s intention to replace American democracy with a dictatorship should now be fully disabused.
I share your despair, sadness, and fear. Just a week ago Saturday millions of us gathered in solidarity against Trump and for democracy, the rule of law, and social justice. Those demonstrations feel as if they occurred years ago.
Last night I spoke with a number of people experienced and knowledgable about American foreign policy and politics. Here, in brief, is what I asked and what I learned.
Followers of the Make America Great Again movement, including vocal supporters of President Donald Trump, are voicing their frustration with the president after he launched a strike Saturday on Iranian nuclear sites.
“We have completed our very successful attack on the three Nuclear sites in Iran, including Fordow, Natanz, and Esfahan,” Trump wrote in a Truth Social post Saturday. “All planes are now outside of Iran air space. A full payload of BOMBS was dropped on the primary site, Fordow. All planes are safely on their way home. Congratulations to our great American Warriors. There is not another military in the World that could have done this. NOW IS THE TIME FOR PEACE! Thank you for your attention to this matter.”
As Financial Times reports, former chief White House strategist Steve Bannon told his podcast listeners “shortly after the strikes” that “an overwhelming majority of the people [in the US] don’t want to get involved in any of this.”