Friday, July 3, 2026

Mensa Angle Geometry Problem Solution

 

Redrawn with highlighted circles, angles

Solution:

Reconstruct the geometry as shown in the diagram above, which includes the more detailed information given below. Then draw a line from the center of each circle to the nearest  45 vertex AND to either point of tangency with the triangle (A/B, a/b).   These lines and the sides of the triangle make right triangles with angles: 22.5 o , 67.5 o, 90 o  (22.5 o  is the 45 o  vertex  bisected.)

Therefore, the intercepted minor arcs between the points where the circle touches the triangle (at AB  and ab) measure 135 o (67.5 o + 67.5 o).  

Now, draw a diameter through the center of each circle to the point of tangency with the circle's hypotenuse.  Let these diameters be AC and ac. Since they are perpendicular to the same line they are perpendicular to each other.  The measure of arcs AC and ac are both  180 o  so: BC = bc = AC - AB = ac - ab =

180 o -   135 o  =  45

Now, draw a line between the centers of both circles. It passes through the point of tangency between both circles (X), which is also the vertex of <x. Note the line joining the centers of both circles intersects parallel lines AC and ac. By the alternate angle theorem, the pairs of alternate interior angles (u and v) are congruent. Hence, u + v = 180 o .

Note BOX and boX are both isosceles triangles. Let the interior angles of BOX be: t,  v + 45 o , and t. Let the interior angles of triangle boX be: s,  u + 45 o , and s.

We have:

 t+ v + 45 o  + t = 180 o ® 2t  + v  = 135 o 

s+ u + 45 o  + s = 180 o ® 2s  + u  = 135 o 

Adding both equations together yields:

2t + 2s + v + u =   270 o 

Substitute  180 for v + u:   2t + 2s + 180  =   270 o 

Then:  2t + 2s =  90  ® t  + s  = 45 o 

Since the line connecting the centers of the circles (O and o) is straight:

t + x + s =  180 o

Substituting 45 o for t + s:    45 o + x =  180 o  

Then:  x =  180  - 45 o  = 135 o     




WSJ Guest Contributor Exposes The Mirage Of "Trump Accounts" (Money Taxed Coming In & Going Out)

 

"I make my money the easy way with crypto, the rest of you can take your chances!'


The problem we Americans now face — one that citizens of modern nations have faced in various parts of the world for centuries — is that our government, having been seized and now run by psychopathic predators, has become predatory itself rather than protective.- Thom Hartman, Predators Always Need Prey, smirkingchimp.com

Five months ago I warned, in a blog post about Trumpers trying to create incentives for more young people to have kids : 

"Those specialized, tax-deferred, long-term Trump 530A investment accounts ("Trump accounts") for children born between January 1, 2025, and December 31, 2028- won't help . Created under the Working Families Tax Cuts Act, these accounts provide a $1,000 initial Treasury-funded deposit, invested in stocks to build wealth."

But what I didn't know then, and only learned recently in a WSJ piece from Adam Michel -the Director of Tax Policy Studies at Cato Institute- is that these misbegotten accounts are unlikely to build wealth even in the long term. Why? Double taxation.

As Mr. Michel puts it:

"My 1-year-old son qualifies for a Trump Account, and I’ve opened it to claim the $1,000 government deposit. But I won’t be putting any of my personal after-tax wages in it, and neither should most parents. Normal investments get taxed twice. You pay taxes when you earn the money as wages, and then any profit you make on the investment gets taxed again at the lower capital-gains rate. Nearly every tax-advantaged investment account—401(k)s and individual retirement accounts, 529 plans for education—eliminates either the wage tax on the way in or the capital-gains tax on the way out.

A traditional IRA lets you make pretax contributions and taxes the withdrawals. A Roth IRA flips it.  The result is similar: tax the money going in or tax it going out. Trump accounts do both. The accounts accept after tax dollars from parents and other authorized individuals, but when the child turns 18, they convert to the traditional IRA for retirement.

Adding:

"That means any gains, along with the original $1,000, are taxed at withdrawal as ordinary income rather than at the lower capital gains rate. (Which would have been applied if the investment weren’t in a Trump account.) Instead you pay taxes on the front end and the high rate on the back end – no deductions, no capital gains rate, no flexibility."

Mr. Michel then goes on to advise that if you plan to invest after tax money for your children "a 529 plan is the better plan for education savings."

Of course, because the 529 is a rational plan, not a grifter's bait and switch. I mean what more can you want given a sensible option for which money goes in after tax and qualified withdrawals are tax free? Besides which, as Michel points out, the 529s have been made even better by recent rules changes, i.e. by allowing up to $35,000 to be rolled over into a Roth IRA.  The plus? The child's funds are liberated from the education restrictions.

Michel advises that the Trump accounts can be "fixed", but that assumes the deranged, grifting lunatic sitting in the Oval Office is ok with that happening. I doubt it. Trump, to remind readers, is only out for grifting for the benefit of his own odious clan as even a recent WSJ editorial notes, e.g.

The Trump Family and ‘Honest Graft’ - WSJ

So no one should buy into any codswallop pertaining to "change" for anything Dotard has incepted, and that includes finally returning the reflecting pool to what it was before the orange fungus turned it into a swamp.

See Also:

Trump busted for self-dealing as MAGA says enough! Ari reports the receipts

And:

by John Feffer | July 2, 2026 - 4:58am | permalink

— from Foreign Policy In Focus

The Trump administration concluded a recent mineral deal with Kazakhstan that, not surprisingly, enriches not only Trump’s own family but that of his secretary of commerce, Howard Lutnick. Trump’s two eldest sons, part owners of Dominari Securities, are set to profit from the Kazakh tungsten deal. So is Cantor Fitzgerald, the investment firm run by Lutnick’s two sons.

As The New York Times pointed out in its investigation of the scheme, “Their sons were soon doing business with partners in a deal that their fathers were negotiating, continuing a pattern of self-enrichment in the second Trump administration that has few precedents in American history.”

» article continues...

And:

by Ellen Brown | July 1, 2026 - 4:34am | permalink

As Americans prepare to celebrate the 250th anniversary of the Declaration of Independence, few are paying attention to a bill moving through Congress that could seriously impinge on our financial independence.

The Clarity for Payment Stablecoins Act, H.R. 4766, is slated to make privately issued stablecoins a major component of the U.S. monetary system. Supporters see stablecoins as a way to strengthen the dollar’s global role while creating a vast new market for U.S. Treasury securities. Critics see the rise of programmable private money that can be monitored, frozen, or restricted by its issuers. Banks fear the loss of the deposits that are essential to advancing affordable credit. What appears to be a debate about digital tokens has thus become a battle over the future of banking itself and finance.

» article continues...

And:

Donors were misled by Trump-backed Freedom 250, House Democrats allege - The Washington Post

Excerpt:

Some donors who intended to give money to a bipartisan effort to celebrate the nation’s 250th anniversary were, instead, steered to a White House-backed initiative under false pretenses, House Democrats allege in a report released Thursday morning, citing whistleblower interviews and newly obtained documents.

The donors meant to give money to America250, a congressionally chartered initiative to celebrate the nation’s semiquincentennial, according to Democrats on the House Natural Resources Committee. They instead were given routing and account numbers that directed their funds to Freedom 250, which President Donald Trump established last year to organize anniversary events,

And:

Trump Made $1 Billion on Crypto Deals While His Fans Lost a Fortune - WSJ

Excerpt:

Morten Christensen made a big bet on digital tokens sold by the Trump family’s World Liberty Financial last year, hoping that a surge in value might be enough to help him retire.

Instead, the value of those tokens tanked. While Christensen and many like him lost big, the president made a fortune, netting $800 million from that crypto project, according to a financial disclosure he filed this week. It has been clear for some time that Trump's forays into the crypto world have been lucrative, earning him some $1.4 billion last year.  This underscored the different reality Trump is living in compared to many of the investors who embraced digital assets alongside him. 

While Trump raked in cash by issuing new assets, World Liberty tokens, and meme coins, those who bought them at high prices had to suffer as their value went belly up. Political followers and crypto true believers who bought into the Trump brand were left holding the bag.


Thursday, July 2, 2026

How Our Understanding of the Black Death Has Been Advanced By Genomic Evidence - & A Study Of Medieval History


14th-century painting with Belgian citizens burying the victims of the Black Death

            St. Sebastan Chapel in Garmisch, Germany, showing Black Death striking villagers


In a recent (June 12)  WSJ Review piece (Black Death: Shadow Over the World)  by Kyle Harper, we read:

"In the immediate aftermath of the 14th-century Black Death, a highly educated Venetian wrote a chronicle claiming that the plague had wiped out a third of his city’s population. Another account from Venice, composed around the same time, claimed that more than half were lost in the pandemic. 

"A remarkable inscription in the Scuola Grande di Santa Maria della Carità (a kind of 14th-century Knights of Columbus) carved right after the outbreak put the city’s death toll at two out of three. Yet another Venetian chronicler claimed seven out of 10. And by the reckoning of one observer, the mortality carried off three out of every four inhabitants.

Welcome to the challenges of writing the history of the Black Death, the great pandemic of bubonic plague that ravaged the late medieval world. "

Ravaged is correct. And we saw some of the lasting residue when we traveled to Garmisch, Germany in 2013 and beheld the detailed art work on St.Sebastian Chapel wall, not far from our guest house. 

After ten minutes or so in the chapel, noting the fetid odor, we exited at the rear and walked outside to find a green expanse of about an acre. On entry, we saw a small notice that this hallowed ground (extending behind the chapel)  formed a "cemetery". Oddly, there were no grave markers. Evidently, at least 1, 100 plague victims were simply buried in unmarked graves. Likely these victims made up the faithful of the small congregation, while others ..e.g. unbelievers, witches, or Jews who perished, simply had their bodies consigned to huge funeral pyres -  actually manifested throughout Europe during the Black Death. (Europe lost an estimated one-third of its then population.)

                                       Plague mask displayed in Oberammergau

                                  Plague reaper in Oberammergau town square


A few days later, our German friends Reinhardt and Elli drove us to Oberammergau.  There they pointed out  engraved iron works displaying other unsettling images of the Plague "Reaper".  These were often accompanied by prayers or other pleas to spare the town. Reinhardt related that the plague toll was fearsome with so many corpses piled up in the streets and few places to put them.  Most had to be carted away to be incinerated in open pits.

In regard to the WSJ reviewer's issues, here is the problem: there were multiple waves of the Black Death - with different intensity or 'ferocity' if you will. What he focuses on is the 14th century version. But what we beheld in our German visit were the inescapable residues of the 17th century version.  What gives?  

The 14th-century Black Death and later 17th-century outbreaks (like the Great Plague of London) were later found to be distinct, centuries-separated waves of the same continuous pandemic. They were also found to be genetically related, driven by the same bacteria (Yersinia pestis), but which differed significantly in how rapidly it spread and how society responded in each era.

The target flea for the Yersinias pestis bacterium is Xenopsylla cheopsis. Once it infects this flea, Yersinias' tendency is to keep on multiplying until it eventually totally blocks the flea's oesophagus making the critter chronically thirsty. The flea must then seek blood sources, i.e. rodents, humans. Since rats are the most prolific, and especially at the times of the plague bred relentlessly because of horrific hygienic conditions, they were the primary vectors.


The Black Death (1347–1351) marked the devastating start of the "Second Plague Pandemic." (the first occurring over 527- 565, during the reign of the Roman Emperor Justinian, and hence called 'the Plague of Justinian') This wave did not end in the 1300s; rather, it repeatedly recurred across Europe, the Middle East, and North Africa for over 300 years, culminating in major 17th-century events like the Great Plague of London (1665–1666).

Despite the 300-year gap, medical knowledge barely advanced. In both the 14th and 17th centuries, people did not know about germs and attributed the plagues to divine punishment, the alignment of the stars, or "miasma" (bad air).

During the 14th-century Black Death, the Italian city of Venice originated the concept of isolating victims. By the 17th century, these practices had evolved into highly organized systems. In 1665, London strictly enforced quarantines by boarding up infected households and marking their doors with a red cross.

Ultimately, while the 14th-century Black Death was a sudden, apocalyptic shock to a completely naive population, 17th-century outbreaks were part of a long-established, evolving disease cycle within a society that was actively learning how to isolate and manage the spread.  

The WSJ reviewer specifically zooms in on Thomas Asbridge and his take in a recent book, noting:

"One contemporary Florentine writer considered it the worst natural disaster since the biblical flood. In “The Black Death: A Global History of Humanity’s Most Devastating Pandemic,” a vivid and sweeping new history of the outbreak, Thomas Asbridge concurs, judging it “the worst natural disaster in recorded history.

The survivors who wrote about the plague leave no doubt that the Black Death was catastrophic. And as Mr. Asbridge points out, thanks to the spread of the new technology that we call paper, there was rather more contemporary writing about the Black Death than previous disasters. Consequently, we know a tantalizing amount about the plague. But the predicament we face in deciphering the Venetian statistics is hardly unique. Modern historians must try to reconstruct the nature, extent and consequences of this horrific calamity from anguished and often contradictory reports.

Mr. Asbridge’s strategy is to explore “a series of representative case studies (or what might be termed micro-histories).” He surveys the general course of the pandemic in Venice but also focuses, for example. In the example of Venice, Mr. Asbridge breezily concludes that the plague took up to two-thirds of the population, without making a compelling case for this figure among the many credible possibilities."

But 'breezily' claimed or not, author Sean Martin, 'The Black Death' (2007) correctly notes what we now refer to as 'the Black Death' was actually the second pandemic of the plague. Also up to two-thirds of the population was wiped out. How? Because there was no effort at serious isolation of the infected or any attempt to target the actual sources (rats), rather giving in to superstitious rot.

Given existing knowledge had not yet accounted for bacterial existence, blame was most often based on "supernatural" or "spiritual" attributes. I.e. blaming people who were likely not pious enough, or committed foul deeds, or were miscreants in some other fashion, e.g. "witches", atheists, heretics or ....Jews. In many regions, alas, Jewish populations were blamed for the plague's spread, part and parcel of the misbegotten German Völkisch tradition.  

Harper, the WSJ reviewer, Harper goes on:

"Anyone bold enough to attempt a new history of the Black Death today deserves our empathy. The topic is unwieldy, and fundamental mysteries abound. Did the plague hit China? How did the plague stick around for nearly half a millennium after the initial wave, causing repeated outbreaks? Why did Europe rebound from demographic catastrophe while much of the Near East stagnated?"

He basically answers these questions himself in subsequent paragraphs, acknowledging Asbridge's  insight into medieval history ( a reader in medieval history at University of London) but giving short shrift to the science. I.e. he was merely guilty of "sidelining the scientific evidence in favor of the written testimony":

"New DNA evidence transformed what we know of the plague even during the eight years that Mr. Asbridge spent researching and writing this book..

Only two decades ago, it was widely doubted by specialists that the Black Death was caused by the bacterium Yersinia pestis, the agent of the bubonic plague. Then technological advances made it possible to sequence DNA recovered from archaeological samples and determine that Y. pestis was indeed the cause. We are also learning more about the evolutionary history of this devastating bacterium, which has haunted human societies since the Stone Age."

This is a critical point and one that can't be emphasized enough.  As Harper elaborates:

The genomic evidence has been critical in pinpointing the reservoir of the plague in Central Asia and in tracing its lightning-fast diffusion along trade routes. The DNA evidence is also helping us understand how the plague stayed around for centuries. (In short, it lurked in local animal reservoirs.) As Mr. Asbridge rightly recognizes, this longevity was a crucial dimension of the bacterium’s historical impact. The Black Death was not a one-off. It was the beginning of a long age of plague."

Indeed, and as I already noted, i.e. this wave did not end in the 1300s; rather, it repeatedly recurred across Europe.  Why? Because the bacterium was endemic.

The final paragraphs of Harper are worth repeating:

"Like any pandemic, the Black Death was simultaneously a biological and a social event—shaped by both the innate characteristics of a microbe and such all-too-human factors as political systems, religious beliefs and public-health responses. "

 Which brings to mind the tens of million of blithering morons in Trump era 1.0 who clamored for "escape" and pursuit of their "freedom" even as Covid-19 deaths soared past 1 million in the U.S. 

And:

"The deepest mystery of the plague is why this bacterium—in normal times a pathogen of wild rodents—erupted to cause some of the most world-altering disease outbreaks on record. "

Author Laurie Garrett's sober take, i.e. in her book The Coming Plague (p. 169), may be the best rejoinder to this:

"An individual microbe's world is limited only by the organism's mobility and its ability to tolerate various ranges of temperature, sunlight, oxygen, acidity or alkalinity.  Wherever there may be an ideal soup for a microbe, it will eagerly take hold,  immediately joining to the local microbial ecosystem of  pushing and shoving struggle for survival.... In this fluid complexity human beings stomp about with swagger, elbowing their way without concern into one ecosphere after another. The human race seems equally complacent about blazing a path through a rainforest with bulldozers and arson  - or using an antibiotic 'scorched earth' policy to chase unwanted microbes across the duodenum."

See Also:

 Visits to Plague Towns in Germany - Some Unsettling Sights

And:

What We Learned In Germany Regarding Its 17th Century Black Death Scourge

Wednesday, July 1, 2026

A Math Treat: D.E. Littlewood's Elegant Introduction to Non-Commutative Algebras (Pt. 1)

  In his timeless monograph 'The Skeleton Key Of Mathematics' (Chapter XIII, p. 101, Algebras) D.E. Littlewood gives us one of the best introductions to that domain of mathematics.  He starts in a logical place, Hamiton's 19th century development of the quaternions.  He begins by looking at Hamilton's approach to vector and scalar products, i.e. in 3 dimensions:


Then noting(p. 102) :

 "A vector (e.g. R) with components x, y,z would be denoted by:

 xi + yj + zk"

Adding:

"If for the law of multiplication it is assumed (for the unit vectors):

2 = - I,  j 2 = - I,  k 2 = - I, 

i j =k jk = i, ki = j

ji = -k,  kj = -i, ik = - j

Then the product of two vectors (xi, yj, zk), (x' i, y'j, z' k) will have a scalar part which is equal to the scalar part with the sign reversed, and a vector part equal to the vector product. Thus:

(xi, yj, zk) (x' i, y'j, z' k) =  - (xx' + yy' + zz') + i(y z' - y'z) 

+ j (zx' - z' x)   + k( xy' - x' y)

The quaternions so formed will obey all the laws of numbers except the commutative law for multiplication. Most importantly, they are associative, i.e. three quaternions A, B, C satisfy:

AB (C) =  A (BC)

"If then:

 Qx 0 +  ix 1   jx 2   kx 3

is any quaternion, the quaternion obtained by changing the sign of the vector part Q = x 0 -  ix 1  -  jx 2  -   kx 3 ) is called the conjugate quaternion.  The product of a quaternion with its conjugate is a positive non-zero scalar, i.e.

  ( x 0 +  ix 1   +  jx 2   kx 3 )  (x 0 -  ix 1  -  jx 2  -   kx 3 ) = 

(x 0 2+  x 1 2   x 2 2 +  x 32)

which is called the norm of the quaternion.  It is thus always possible to divide by a quaternion, except of course by zero, e.g.

I/ (x 1 +  ix 1   +  jx 2  +  kx 3 ) =

 (x 0 -  ix 1  -  jx 2  -   kx 3 / (x 0 2+  x 1 2   x 2 2 +  x 32)

Quaternions with real coefficients, or real quaternions, are thus said to form a division algebra.

If complex coefficients are allowed, however, division may fail. This is given that for complex numbers it is possible that:

 x 0 2+  x 1 2   x 2 2 +  x 32   = 0

without all the quantities (x 0 ,  x 1 ,  x 2 ,  x 3  ) being zero.

Therefore complex quaternions do not forma a division algebra.  It can be shown, however, they are equivalent to the algebra of two rowed matrices, if we set:



Thence we have here one of the most elegant - and simplest- presentations of non-commutative algebras. As well as the basis for what we call 'algebras' in general.

Next: Higher dimensional algebras