Wednesday, October 1, 2025

Mensa Tetrahedron Solution


 The 4 vertices of a particular tetrahedron are also half the vertices of a unit cube.

The base vertices are one unit away from the fourth (apex) vertex as shown in the diagram above.   Given this information find the height (H) and volume (V) of the tetrahedron.

Hint: The volume of a tetrahedron is one third the area of the base times the height H.


Solution:

The edges connecting the base vertices are face diagonals of the cube. Therefore, the base of the tetrahedron is an equilateral triangle  of edge length  Ö2
 

The height h of the equilateral triangle can be found from the Pythagorean theorem:

Ö2 / 2)    +   h    =   Ö(2) 2   Þ  (½)   +   h    =   2

   =   3/2  Þ       =   Ö6/ (2)

The distance from a base vertex to the center of the base is: Ö6/ (2)  -   x

Then:

Ö2 / 2)    +   x    =  (Ö6/ (2)  -   x )   x 2


And:

(½)   +   x    =   3/2   -   Ö6 x )  -  x 2


Ö6 x )  =   1
Þ 
  x  =    1  / Ö6  =   ( Ö6/ 6  )

The distance from a base vertex to the center of the base is now:

Ö6/ (2)  -   Ö6/ 6  )  =   2 Ö6/ 6  ) =  Ö6/ 3  )

The height H of the tetrahedron can then be found from  Pythagorean theorem, i.e.:

Ö6/ 3  ) 2  +    H  2  =  1 2

Þ 
 
2/3 +    H  2  =   1    

 2  =   1    -  2/3   = 1/3

H  =   Ö3/ 3  )

The volume  V of the tetrahedron can then be found from  

 V  = 1/3 Area of the base x height H

Then:

V = 1/3    [½] (Ö2) (Ö6/ 2)   (Ö3/ 3  )

V =  1/6  cubic units

Correcting A Journalist's Theological Misperceptions: Trump Is Not A Christian In Any Form

                        

                              
'God made Trump' image circulated by NAR


"Donald Trump is Satan. The Satan in question is perhaps the greatest literary character in the finest epic poem in the English language, John Milton’s Paradise Lost. .. Trump as Satan: you might think it’s absurdly over-the-top, or want to celebrate it as a (literally) epic takedown, but this is not a political statement hurling invective. It’s an analytical statement describing what is happening – what will happen – and why, aided by the Miltonic intervention. Great literature helps us understand the world. Both politicians and literary critics do their jobs best when they’re willing to use literature to think about life."-  Jeffrey R. Wilson, Spectator USA , June 2020
                                    

It appears WaPo columnist Shadi Hamid believes "there are two forms of Christianity pulling American politics in opposite directions". At least according to his article "Two Versions Of Christianity Battle For American's Soul'.  Hamid being a Muslim, might be forgiven for mixing up performative Christianity with the real McCoy. Also, given he may never have studied Christianity's own revealed texts - like in the New Testament- he may think simply making quasi -religious claims qualifies a person as religious or even Christian. 

Let's go to his actual words in his WaPo piece: 

 "Oddly enough, the moment that might define the faith’s political future happened not in a church but in a football stadium. At Charlie Kirk’s memorial service, tens of thousands of mourners witnessed something extraordinary: the explicit theological division of the American right. On one level, they saw faith calling for forgiveness and loving one’s enemy. But they also saw a faith that weaponizes politics through righteous anger — and the chasm between them might be unbridgeable."

He was referencing Erika Kirk's words, on the one hand channeling Yeshua's from the cross, delivering the service’s most powerful moment,

Father, forgive them, for they know not what they do.” 

Then, applying them directly to her husband’s killer.

 “That man — that young man — I forgive him. I forgive him because it was what Jesus did and is what Charlie would do.”

Which is an echo of genuine Christianity, not merely the Right's version. Ericka Kirk - at that moment-  could be said to be actually applying her Savior's own words and to that extent embodied authentic Christianity.

Hamid's error was in judging Trump's response to her words to be equally authentic and Christian - when they were nothing of the sort. Again, in Hamid's words from his piece:

When Trump appeared onstage afterward, he looked out at that same crowd and rejected Erika Kirk’s Christian witness: “[Charlie] did not hate his opponents; he wanted the best for them. That’s where I disagreed with Charlie. I hate my opponents, and I don’t want the best for them.”

Hamid is correct when he writes that "here was a disagreement over political strategy".  But he is dead wrong when he adds:

" It was also much more than that. This was one of Christianity’s deepest theological questions playing out in real time: Can one turn the other cheek and still engage in the combative work of democratic politics? And more to the point: Why do so many Christians seem indifferent to one of Christ’s core teachings: of elevation of the weak over the strong, of virtue over power."

Elevating Trump's hate response to the equivalent of a theological counterpoint is misplaced and shows a deficient grasp of Christianity. (Yes, I am an atheist but took 3 years of theology at Loyola University!) 

 No, it was not. It was a vile, noisome reaction from a vile, noisome and hateful excuse for a man. A repugnant would-be bearer of the "word" but in fact a viral disseminator of lies, division and chaos. This mistaken position of Hamid's is clear when he writes:

"The second Christianity, embraced by Trump and amplified by other speakers, described Charlie Kirk’s assassination as part of an epic battle between forces of light and darkness."

But this isn't Christianity it's a perversion of it based on Old Testament "eye for an eye" othering mixed with channeling Trump's own pestilence. The New Apostolic Reformation (NAR) is also at the heart of this bogus theology. The latter, is "rooted in a long standing alliance of charismatic worship and grievance politics dating back to post-WWII Pentecostal Revivals."  Those, we are informed, "fused nuclear- age anxiety" ( i.e. over the commie or 'radical leftist' threats) "with the spiritual values" of free market capitalism."

In its new revival NAR  has "come under the influence of a Texas oilman and marketeer named Lance Wallnau who proposes to short circuit the old model of evangelical political influence in favor of directly annexing the national government, together with other cultural and civic institutions to revive the American Christian nation

But it is not a true Christian nation, but rather a false mutation based more on Old Testament ‘eye for an eye’ thinking.

Skeptical readers are well advised to check out the book White Evangelical Racism, by University of Pennsylvania Religious Studies chair Anthea Butler, who warns (ibid.): 

"All the talk you hear about demonic stuff, about violent stuff - people should take that very seriously."

She was referring to the militant and existential cast embedded in the NAR's theological politics: i.e. an essential painting of Trump's Democratic opposition as "Satan's minions".  If they are Satan's minions then they are de facto 'othered' i.e.  cannot be treated as humans but as perverse demons in disguise, who must be removed by any means possible. A preview of the type of potential violence that is spawned by such demonization occurred back in 2016 with the Pizzagate conspiracy, e.g.

Crackpot "Pizzagate" Tale Promoted By Michael G. Flynn Almost Led To D.C. Tragedy

 In that case tragedy was narrowly averted when a brain-jacked lunatic (Edgar Lynch) was barely stopped from shooting up a D.C. restaurant called 'Comet Ping Pong'.  This after being incited by fables of Demos holding child sex slaves there and sacrificing them to Satan.  Looped into insanity this crackpot loaded his rifle and drove nine hours from North Carolina to liberate the "hostages  to Satan."

Jeff Sharlett also pulls no punches in his 2020 Vanity Fair piece:

https://www.vanityfair.com/news/2020/06/inside-the-cult-of-trump-his-rallies-are-church-and-he-is-the-gospel

Showing the warp and woof of what Trump has done to debase Christianity, turning it into a- quasi religious personality  cult.  As he writes in the article:

"Many Trump followers deploy a familiar Christian-right formula for justifying abuses of power, declaring Trump a modern King David, a sinner nonetheless anointed, while others compare him to Queen Esther, destined to save Israel—or at least the evangelical imagination of it—from Iran. Still others draw parallels to Cyrus, the Old Testament Persian king who became a tool for God’s will. ...Lance Wallnau, a founding member of Trump’s evangelical coalition, dubs him “God’s chaos candidate”: “the self-made man who can ‘get it done,’ enters the arena, and through the pressure of circumstance becomes the God-shaped man God enables to do what he could never do in his own strength.”

And even more unhinged:

"In Trump’s case, divine backing is more about smiting than healing. When Rep. Elijah Cummings died last October shortly after sparring with Trump about Baltimore, Peterson declared on his radio show, “He dead”—like Trump enemies John McCain and Charles Krauthammer, Peterson noted. “That’s what happens when you mess with the Great White Hope. Don’t mess with God’s children.”

This is not exaggeration, and we've seen then how Trump has become deified, especially after surviving the July 2024 assassination attempt. In an analogous way Jim Jones became deified in the 1970s, finally culminating in the horrific Jonestown massacre:

Jim Jones, like Trump, was also a sociopathic narcissist possessing divine delusions. Like Trump, his accumulation of power also led him to think of himself in god-like terms.  For Trump, deification basically followed his assassination survival and raised fist performance:


With many pundits now agreeing it tilted the election to him even before Kamala's campaign was underway. "He had to have been chosen by God, but wait, maybe he even was a god! He's indestructible"

  Crazy? Histrionic?  Only if you didn't recall Trump's Truth Social burps back in March.  Especially:

He who saves his Country does not violate any Law!

Then compounding this blatherskite to the extent of claiming (according to NY Times piece: 'Trump Suggests No Laws Are Broken If He' Saving Country') that surviving two assassination attempts was evidence of "divine backing to enforce his will."  With the author adding:

"He has brought a far more aggressive attitude toward his use of power to the White House in his second term than he did at the start of his first. The powers of the presidency that he returned to were bolstered by last year’s Supreme Court ruling that he is presumptively immune from prosecution for any crimes he may commit using his official powers."

 Hence, we can't be surprised now all his power has gone to his head as he's violated dozens of norms and laws with each move he makes.  Why be surprised? He's anointed himself a demigod. Worse, too many of his followers have also anointed him as such, as evidenced at the Charlie Kirk memorial.

Let's again make it clear that Trump is no "Messiah" or “divine” entity or even “divinely guided” human. He’s a crass, lying dirtbag traitor and psychopath who has no business in the seat of power.  Which we now see he’s prepared to misuse by wanting to use our cities as “military training grounds.” E.g.

Trump tells a roomful of silent generals to join a ‘war from within’ - The Washington Post

Fomenting war, division and hate in our country is emphatically not the embodiment of any legitimate Christian theology.  Nor is mocking a handicapped NY Times reporter, as Trump did during his 2016 campaign.


It should not take a Mensa level IQ to see this and that  Trump - far from being a Christian in any form -  is as un
qualified as Pete Hegseth as Secretary of Defense. Especially after his loutish fratboy performance in front of 800 military brass at the Quantico Marine complex yesterday, e.g.

Trump's boast of his right to be crowned 'The Dark Lord of Misrule' is no passing fancy.  Indeed, it enjoins every sentient person to see him for what he is, not as a demigod, not as a Christian either, but as a snake - an impostor - that Christ reviled and warned us about, i.e. from Matthew 24: 4-5:

"See that no one leads you astray. For many will come in my name, saying 'I am the Christ', and they will lead may astray."

Or Matthew 7:15:

"Beware of false prophets, who come to you in sheep's clothing, but inwardly they are ravenous wolves".

See Also:

by Robert Becker | September 28, 2025 - 5:18am | permalink

The choice is now Trump vs. Jesus as a profane president sets himself against Christianity’s highest teaching.

A lead historian on belief systems, Joseph Campbell, confirms that Jesus’ most significant if daunting breakthrough instruction was “Love thy enemy.” Not just the difficult “turn the other cheek,” but commandments that sought to transform hateful violence into peace and harmony. Similarly and earlier, paramount virtues in Buddhism are compassion, non-violence and reconciliation: murderous destruction violates the foundation of one’s humanity.

Jesus took non-violent compassion to a higher, humanistic morality. He went beyond “love thy neighbor” to the ideal of "love thy enemy." Jesus per Matthew didn’t say critic or opponent or blasphemer but "enemy"-- then "pray for those who persecute you." This remarkable, transcendent doctrine, per google AI, declares Christians must practice compassion and goodwill toward persecutors. “Instead of seeking revenge or repayment of evil for evil, the command is to do good, bless, and pray for enemies, ultimately aiming to align with the perfection of the heavenly Father.”

» article continues...

And:

Rural WI Voters Are Convinced Trump Is "Anointed By God" - Can They Be Rescued From Their Delusions?

And:

by Maya Boddie | April 8, 2024 - 6:32am | permalink

— from Alternet

Donald Trump's "Messiah complex" perhaps, may have reached its peak last month when the former president announced he was selling $59.99 'God Bless the USA' Bibles.

Laura Brodie noted in a TIME Magazine article the MAGA hopeful's recent antics include telling his supporters "that he’s suffering for their sake, sharing a faux courtroom sketch of Christ at his side, and circulating actor Jon Voight’s bizarre claim that he is being 'destroyed as Jesus.'"

Boston College history professor Heather Cox Richardson, in a Sunday, April 7 op-ed for Milwaukee Independent, submits, "Trump is not the first president to compare himself to Jesus Christ."

And:

by Will Bunch | July 19, 2024 - 6:36am | permalink

— from the Philadelphia Inquirer

I came to the American Heartland to cover a political convention, but all I found was a tent revival, Brother Trump’s Traveling Salvation Show.

The Republican National Convention took just minutes after Monday’s opening gavel to officially nominate its Dear Leader for the third and probably not the last time. The roll call, once the highlight of past conventions, is now an empty ritual. A party platform that was probably written on a Mar-a-Lago cocktail napkin was rammed though with no dissent. RNC schedulers quickly liberated all four nights for the only real purpose they had here in Wisconsin.

The deification of Donald J. Trump.

And:

by Robert Reich | September 29, 2025 - 5:41am | permalink

— from Robert Reich's Substack

Over the weekend, on his Truth Social, Trump shared a video purporting to be a segment on Fox News — it wasn’t — in which an AI-generated, deepfaked version of himself sat in the White House and promised that “every American will soon receive their own MedBed card” that will grant them access to new “MedBed hospitals.”

What?

Believers in the “MedBed” conspiracy theory think certain hospital beds are loaded with futuristic technology that can reverse any disease, regenerate limbs, and de-age people. No one has an actual photo of these beds because they don’t exist.

Trump also posted (again, without any basis in fact) that the FBI “secretly placed … 274 FBI Agents into the Crowd just prior to, and during” the January 6, 2021, riot at the Capitol, during which they were “probably acting as Agitators and Insurrectionists.”

» article continues...