Monday, June 29, 2026

Scorching European Heat Wave Shocks American Tourists & Bears Out Forecast Of A Climate Tipping Point

 

Current European heat wave relative to earlier periods
Parisians desperate to cool off on a 46 C day
Londoners cope with a 35 C day in UK heat wave

Thanks to a new climate analysis we now know the heatwave scorching western Europe is the most severe and widespread ever and is only possible due to the climate crisis driven by fossil fuel burning. This, of course, introduces the role of tipping points.

Prof. Gunter Weller (formerly of the Univ. of Alaska-Fairbanks Geophysical Institute) as long ago as 1985-86 forecast warming of the Arctic leading to a global tipping point in temperatures. He predicted that by the 2010s heat waves occurring in limited planetary regions (say Europe, India or western U.S.) would be 3-4 C warmer than heat waves experienced by those regions in the 1970s. In fact this has been found to be the case.

                                              Prof. Gunther Weller (UAF), 1987

 A new analysis by scientists from the World Weather Attribution (WWA) consortium shows how rapidly extreme heat is worsening regional heat waves exactly as Weller predicted. Weller's forecasts in 1986 were based on the increase rate of CO2 concentration build up in the atmosphere (it was 347 ppm in 1986). Warning that concentrations in excess of 410 ppm would unleash heatwaves 3-4C hotter than those in 70s.

 Coincidentally, as recently as 2003, a heatwave like the current one in Europe would have been 2C cooler due to the lower level of global heating at the time - and the lower CO2 concentration (376 ppm). In 1976, another famous heatwave year, it would have been 3.5C cooler. (I.e. 3.5 C warmer now).  All this would have led Weller to remark: "I told you so!".

In line with Weller's tying CO2 concentrations to more significant regional heat waves half of Europe’s 850 largest cities are enduring their worst ever heat stress.  This stress represents a combination of temperature and humidity. Such muggier conditions mean sweating is less effective at cooling the body, making heatwaves even more dangerous.

The analysis comes as the UK recorded its hottest ever June temperature on Thursday, 36.7C (98.06F) in Somerset, and much of western Europe recorded a sharp rise in medical emergencies, including some deaths. In summer 2022, more than 60,000 people died due to a heat wave in Europe. The statistical analysis needed to assess the impact of the current heatwave will take time to complete. Nonetheless, the heatwave is certain to exact a heavy toll and is also disrupting lives and livelihoods, with schools closed, hospitals struggling and rail and air journeys cancelled across the continent.

Just as concerning are the sweltering night-time temperatures currently about 100 times more likely today than in 2003. Prof. Weller, in his 1986 forecasts warning a significant component of his tipping point would be increasingly higher night time temperatures coinciding with higher daytime temperatures. A recipe for increasing heat deaths in regions like Europe where barely 25% have air conditioning.

The WWA scientists, meanwhile, have warned that without urgent climate action, future heat conditions would get even more extreme and the current summer could seem relatively cool in retrospect. This again aligns with Weller's predictions of 3-4 week long heatwaves by the 2030s with temperatures in the 40- 50C range.

Dr. Theodore Keeping, an extreme weather research associate at Imperial College London and part of the WWA team, has noted:

This is the most severe and widespread heatwave to have ever affected this large a region of Europe. We found that in the last 50 years, during which time the planet has warmed by 1.1C, the chance of a heatwave like this has changed immensely. This event would not have been possible in June without climate change. "

He added that many capital cities were experiencing not only their hottest recorded three-day period in June but the hottest three-day period at any time of year. At least 100 million people in Europe were expected to face temperatures above 35C on Thursday.

The scientists used wet bulb globe temperatures to assess the additional impact of high humidity. Prof. Keeping noting:

 “It accounts for the ability of the human body to cool itself down. With the worst conditions ever experienced in 45% of cities over 50,000 people, the health impacts of this heatwave are likely to be extremely high. The speed of change is startling.

Commenting on the WWA analysis, Simon Stiell, the UN’s climate chief - quoted in The Guardian, said: 

Climate change is running rampant, caused by the world’s addiction to burning coal, oil and gas. But the solutions are equally clear: a faster shift to clean energy – which is now much cheaper than fossil fuels – as well as protecting forests and building climate resilience.”

This is even more critical given that wider use of air conditioning can't be depended upon  - either in the U.S. or Europe. How will communities adapt to constant power grid breakdowns arising from electricity overuse - say during prolonged 3 to 4 week 35- 40C  heat waves? The answer isn't clear apart from utility companies rationing the 'juice'. This may mean doing without electricity for extended periods each day in 40C + heat waves, perhaps up to 12 hours or more. Can Americans adapt to such rationing? Most power grids as currently designed will no longer work 24/7 but have to be shut down because of the lack of capacity as millions try to stay cool -  overtaxing the system.

And we haven't yet gone to the threat of transportation and infrastructure damage such as expansion of rails for rails systems. This led Deutsche Bahn and other rail companies to warn against all nonessential travel over the weekend. Meanwhile, as temperatures hit 104 F outside Berlin, the concrete of the A2 highway burst because of the insanely high temperatures.

Let's be clear the WWA team research is not fanciful or 'scare mongering' any more than Dr. Weller's 1986 work at the Geophysical Institute.  Both used observed and reliable forecast temperature data (ice cores in the case of Weller) to render predictions, In the WWA case it meant analyzing the hottest three-day period across a large area of western Europe, sitting under a “heat dome”. 

Using peer-reviewed methods, they unequivocally found that climate change was the driving force behind the severity of the heat. In Prof. Weller's case, he unequivocally found from the increased CO2 concentrations found in ice cores that climate change was definitively behind future heat waves in disparate regions - from India, to Europe to the USA.

Both the WWA and Weller ruled out natural variability of the weather.  In Weller's case from solar variability (thanks to inputs from the Space Physics Dept.) In the WWA's case discounting particular influences from the El NiƱo event that began in the Pacific Ocean 2 months ago. The current weather pattern, a blocked high-pressure system trapping hot air over Europe and drawing warm air up from the Sahara, is not unusual in summer, the scientists said. Instead, the level of heat has been supercharged by global heating. Prof. Gunther Weller would concur.

The UK government’s official adviser, the Climate Change Committee, said in May that the country’s infrastructure was “built for a climate that no longer exists”.

That advisor is spot on correct. Most illuminating has been a weekend (June 27-28) WSJ article ('Americans Get Humbled In Heat Wave', p. 1A), noting: 

"Americans in Europe have been getting the shock of their lives this summer as temperatures soar to new records across the continent. Highs have approached 100 degrees Fahrenheit in the UK this weekFor the tourists from some of the muggiest U.S. states there is bewilderment: How has a lifetime of sweltering heat not prepared them for this?"

With a common refrain quoted from one American from Texas (similar to others from FL, NC):

"Being a Texan, it kind of feels embarrassing to be in London, and be like, 'I can't handle the heat here'"

Or as another Yank from NC put it:

"These heat waves are different. You can't get any reprieve from it. Once you do get hot, you stay hot."

Even more sobering, as the article puts it:

"Heat waves are becoming more common in Europe - this is the second in two months - and deadly."

So one almost felt sorry for the Americans visiting London, many for the first time, who expected "mostly overcast skies and cool temperatures in the mid-60s or low 70s."

Well, they would have had to have visited London in July, 1978 when we did, having to wear light jackets (or sweaters) most of the time, e.g.

Near River Thames, London, July 1978
           Me in Hyde Park, Janice at Greenwich Observatory, July 1978

For now these Yank tourists can take some solace in the observation (WSJ, ibid.):

"But extreme temperatures aren't yet a fact of every day life."

However, if Prof. Weller's forecasts hold they soon will be. Needless to say, we are glad we visited London (10 days) when we did.

See Also:



And:


And:


And:

Friday, June 26, 2026

All Experts Redux: Orbital Period Error Corrections For A Fictional Alien Planet In 1st Novel

 Question: 

I am writing a science fiction novel set on an alien planet in another solar system.  The orbital period of my planet is 438 Earth days or 10512 hours. Since there are 36 hours in one of my planet's days then there are only 292 planet days in its year. 

By using 292/365 or 0.8 years, in my calculation, I came up with a smaller semimajor axis (7/8 AU). By using 438 Earth days, I arrive at 1.129 AU. How will this affect the (axial) tilt I need, say of 11 degrees? 

Answer:

First, there is only ONE correct orbital period, not two. The error you made here was in confusing the assigned actual 'Earth days' for your planet's orbit (which is the actual parameter to be used in the Kepler eqn.) with an artificial value you deduced by using a longer rotation period, 36 hrs.  By doing this you erroneously converted a rotation period into a false period of revolution, ending up with the smaller (erroneous) semimajor axis of 7/8 AU. By way of comparison, for example, though Jupiter's day is much shorter than Earth's ( »   10 hrs.) that doesn't mean you change it to 2.4 Earth days! (24 h/ 10) Its period of revolution is still measured in standard EARTH years. 

The crucial point is the value of the orbital period of your fictional planet to make one circuit is: 438/365 = 1.2 yr. =  P'

To get its semimajor axis (call it a') we use the general form for the Kepler 3rd law equation given the period  P' (1.2 yrs.): 

(P/ P' )2 = k(a'/ a)3

Where P, a are the standard Earth values (period, semimajor axis) and P', a' are your own model values.

You assigned:  P' = 438 d 

= 438d/ 365.25 d/yr = 1.199  »   1.2 yrs.

Which goes into the Kepler eqn., viz.

(P/ 1.2 yr)2 = k(a'/ 1 AU)3

So we now just need to find a':

Using basic algebra to solve for a':

a' = {[P']2}1/3  

a' =  {[1.2 yr]2}1/3  

=  {(1.44 yr)}1/3  

Or, on solving for a' (and remembering the units used):

a' = {1.44 }1/3  = 1.129 AU »   1.3 AU


The '292 days' is erroneous in any case - as I showed- so is irrelevant in working out the axial tilt - which can't be computed from any  Kepler equations. Kepler's Laws and equations are applicable entirely to calculating a planet’s orbital mechanics (e.g., speed, position in space, and orbital period). 

Because axial tilt relates to the orientation of the planet’s spin axis relative to its orbital plane, it is completely independent of these laws. Calculating a planet's axial tilt requires entirely different methods, such as observing how the apparent position of the central star changes in the planet's sky over the course of a year. I.e. measuring the planet’s Sun's maximum altitude at different seasons or tracking long-term shifts in the planet's gravitational pull.

The main thing about Kepler's 3rd is always to bear in mind the units are AU and Earth Years. This never alters for any supposed solar system work, so you can't just lengthen the rotation period to obtain a different period of orbital revolution. 

 Re: axial tilt, consider from the diagram below the axial tilt of the Earth at 23 ½  degrees,  which is also the tilt of the ecliptic with respect to the celestial equator (determined from Earth)


Since you demand 11 degrees axial tilt for your planet this means the ecliptic angle with respect to its celestial equator will be 11 deg not 23 ½  degrees.  Comparing again to our Earth system and orientation - say for defining the seasons,



So for the total planet year - or 438 day period -  and 11 degrees being the maximum tilt (i.e. at its summer solstice, call it day 0) we see how it moves to its first equinox (say from day 0 through day 110), whence it then moves back to the equator (day 111-221), then 11 degrees south, and so on and so forth. Basically we have + 11 degrees at your planet's solstices, analogous to + 23.5 degrees for Earth's

 Thus all the same behavior applies as to our Earth -- except now you re-do this for the total 438 day period, and 11 degrees is the maximum moved north (say from day 0 through day 110, e..g.  A to B), whence it then moves back to the equator - to get to your autumnal equinox.  Then B to C (day 111-221), to get to  maximum 11 degrees south (for your winter solstice), and so on and so forth.

Assuming you construct a similar latitude and longitude system things will also be different.  Mainly the longitudes, given latitudes will remain the same, except 12.5 deg less tilt means more solar illumination at the summer solstices (more daylight hrs.) and less at the opposite (winter) solstices.

 Back to longitude. On Earth longitude meridians are separated by the fact it turns through 360 deg in 24 hrs, so:

360 deg/ 24 h  = 15 deg/ h   

So each longitude meridian is 15 degrees apart.  But for your planet, since you assign a 36 hr day:

360 deg/ 36  h  = 10 deg/ h   

You will end up with 36 longitude zones (time zones) of 10 degrees each compared to 24 for Earth (of 15 degrees each).

Given your designated axial tilt (11 degrees) the Sun for your planet will also assume different observed azimuth values for different dates.

Consider, on Earth we designate the Tropic of Cancer at 23.5 N. For your planet it would be 11 N.  Similarly, on Earth the Tropic of Capricorn is at 23.5 N. For your planet it would be 11 S.

Consider the azimuth for the respective suns (Earth's and yours) at summer solstice. The Sun's azimuth (A) for a given latitude (lat) on Earth for that date can be computed from:  

cos (A) = sin (23.5)/ cos (lat)

Since 23.5 degrees is Earth's axial tilt.

For your planet is would be:

cos (A) = sin (11)/ cos (lat)

Since 23.5 degrees is Earth's axial tilt.

Referencing Earth, and taking London's latitude of 51.5 N  we'd get:

cos (A) = sin (23.5)/ cos (51.5)

Or  »  49 degrees  North of due east

But for your planet, using the same latitude of 51.5 deg:

cos (A) = sin (11)/ cos (51.5)

Or A =  72.1 deg N or E. 

  Hopefully, these corrections and elaborations will help you in developing your fictional planetary model which - for good science fiction - always needs to be faithful to actual physical laws!

Thursday, June 25, 2026

Mensa Angle - Geometry Problem

 


Two circles are inscribed in an isosceles right triangle, as shown in the diagram above. They are tangent to the sides of the triangle and to each other.

An angle x is formed between two chords, facing the right angle of the triangle. The chords meet at the point of tangency between two circles and extend to the points where the circle touches a leg.

Find the angle x.

Hint: Use the information to re-construct the diagram, incorporating the particulars of the geometry described.


     

Wednesday, June 24, 2026

Steven Spielberg's 'Disclosure Day': A Timely UFO Film My Air Force Brother Would Have Loved

 

Jerry - relentless in insisting govt was hiding alien facts
UFO discussion in 1990: Jerry showed us Twining Memo

From the opening scenes in the new Steven Spielberg movie, Disclosure Day, I could tell my late brother Jerry (who died ten years too early) would have been riveted to his seat. As Janice and I certainly were, when we went to see the 9 a.m. showing of the film yesterday. (We had tried earlier, on Saturday at 12:25 - but all the seats had been taken except for the 2 rows closest to the screen. A deal breaker.)

I can imagine Jerry's reaction now seeing the opening scenes unfold, as cyber specialist Daniel Kellner made off with a tranche of stolen UFO-alien evidence with former nun novitiate, Jane. Both hotly pursued by the baddies from Wardex - the corporate bad guys charged with keeping the aliens secret - along with the Pentagon, FBI and other powers that be (e.g. NSA)

Jerry would have howled - if seeing it with us: 

"See! That's what I've been talking about! The bastards will stop at nothing to keep the truth from coming out."


And that is how the film unfolds for the next two hours, showing the efforts of Kellner, his girlfriend Jane, TV meteorologist Margaret Fairchild and former cyber security guy Hugo Wakefield, to escape the government goons and get the word out.  Well, Jerry certainly tried - by word of mouth based on his own first hand experience - but with no actual evidence in hand like the movie's Kellner.

For years, hell decades, Jerry recited his involvement with recovered alien artifacts - and corpses - while working in "the Blue Room" of Wright-Patterson Air Force Base.  (He also pointedly noted the corpses were not described as "aliens" but as EBEs - extraterrestrial biological entities.)

When prompted he'd go into exhaustive detail about what he saw and the stringent security measures he'd have to endure (Could only wear t-shirts, trousers, shoes - no clothing with pockets, no note books, cameras etc. or other paraphernalia- in or out, searches all the time). 

His bottom line take: there was a Roswell crash in 1947, but the debris (and four bodies) were conveyed to then Wright -Patterson Army Airfield, not to Area 51 as the current mythos claims.  One of the bodies was "badly dismembered" in a botched initial autopsy attempt while three others remained intact and were preserved. Again, in Wright -Patterson's Blue Room 

Jerry quickly added that the Air Force's  disinformationists have included in their targets the name 'Blue Room' itself - dismissed as "urban myth".  But the  official designation of record for Blue Room operations, is NASIC - National Air and Space Intelligence Center and "it's as real as a heart attack."  

He was adamant that the assorted rooms: Hangar 5 (not Hangar 18), which later became the Bldg. 18 complex -  included Bldgs. 18A, 18B, 18C, 18D, 18E, 18F, 18G - the next to last with 4 cold rooms where the corpses of EBEs were stored.  The complex was formally operated under NASIC  (formerly Foreign Technology Division or FTD) and basically it had oversight concerning disposal and storage of craft wreckage, EBE bodies and artifacts.

Interestingly, he also maintained that no political honchos, Senators or even presidents - including JFK, FDR, Truman  - have been allowed themselves into the Blue Room or the Building 18 complex. In other words, it's regarded as uber top secret.  (The last one to try was Sen. Barry Goldwater in 1964   - and for his request was "reamed out" by Air Force Gen. Curtis LeMay.)

When I brought up the matter of the bodies, debris etc. being mistaken for Project Mogul balloon wreckage - and dummies - e.g.  from a June 23, 1997 TIME magazine issue on 'The Roswell Files',

Jerry scoffed at the claim.  "That's total bullshit!"  Adding: 

"Don't forget an actual physicist based at the New Mexico Institute of Science and Technology splattered that crap!"  

He was referring to Prof. C.B. Moore who came out soon after the attempted debunking (by Gen. Roger Ramey)  asserting no balloon would leave a debris field as wide as two hundred yards by  three - fourths of a mile.  

Jerry then made clear:

"The military powers that be scarfed up those bodies and the debris within hours and had them shipped off to Wright Patterson Army Airfield  near Dayton Ohio."

He added that from what he could discern from the photographic- film materials he handled, there were four categories of interest for what was recovered and put in the Blue Room i) Sections of light, thin aluminum colored metal which were 'unbendable', ii) I-beam structures with weird symbols "like hieroglyphics" embossed on them, iii) Tiny wiring that resembled today's nanofilament wires, iv) thin aluminum cloth like metal that could be scrunched up then released and recovered its original shape.  

When I mentioned UFO debunker Philip Klass asserting in the same TIME article that "the real Roswell crashed saucer coverup was by UFO book authors" he fairly exploded.  

"Klass couldn't find his asshole from a hole in the ground!  He never went to the places I was or saw what I saw so he doesn't know jack shit!"  

 Before he died (of an aortic aneurysm) in August, 2016 Jerry maintained "more of these craft will soon be upon us, in our skies"  He didn't live to see the videos from the U.S. Navy pilots or their eye -opening interviews from '60 Minutes'. Where the Nimitz pilots reported craft diving into the water at 13,000 mph, e.g.  see link below:

Navy pilots describe encounters with UFOs - YouTube

Those Nimitz pilots, of course, were also victims of hyped-up secrecy given the radar tapes they expected to be released from their encounter never were. So Lt. Cmdr. David Fravor and his fellow pilots  (from the Nimitz 'Black Aces') were left out there hanging in the breeze with only their oral accounts.

Which, like the theme of the film, brings up the question of why so much secrecy? Chris Impey, an astronomer at the University of Arizona,  quoted in a 2023 WaPo article had noted:

The U.S. has always tended toward secrecy. Only in the last year or so has there been a push for transparency, but the backdrop of that was flat-out denial or secrecy.”

The article itself noting:

"Whether a country shares UFO information often comes down to military interests. The United States has often been less willing to publicize or publicly engage on questions about UFOs — even going so far to spread misinformation in the 1950s — for fear of ceding a strategic advantage to adversaries and jeopardizing national security."

All of which supported Jerry's 1990 explanation for why such secrecy shrouding UFO files was/is a given, and then leading to his acid observation: 

"Hey, if there's nothing really there and all this UFO stuff is about wackos and fruitcakes just seeing things - why hide it?"

And we now know - from the same WaPo piece- this is not done in other nations, e.g.

In U.S., most UFO documentation is classified. Not so in other countries.

Which elicits the question: Why in the U.S.  mount an extraordinary decades long effort to undermine every reported UFO, including kneecapping the reputations of those doing the reporting?  Well, because in the U.S. recovered craft are re-engineered and re-purposed in order to obtain a military advantage - in novel weaponry - that other foreign rivals (China, Russia) don't have.  

What gives?  In the words of Brazilian Rodolpho Santos, a historian at the Federal Institute of Minas Gerais:

"It’s comparatively easy to get this information here. And the variety of records is good and considerable.

It’s all in public records. There are no security clearances such as demanded in the U.S. (Where even Ike, we were informed by Jerry, was told he didn't have a clearance high enough to see the crashed Roswell craft, after taking the reins as president in 1953).  But totally different in Brazil. No heavily redacted documents. Anyone can access the files — the military reports, the videos and audio recordings, the grainy unverified photographs — and thousands of people have.

 

 It seems that in Brazil and much of South America, there has been a more relaxed attitude toward the inexplicable, the public’s right to know and the limits of scientific explanation. In South America, at least four countries — Uruguay, Argentina, Chile and Peru — have public government programs that study and investigate UFO activity. Argentina and Chile regularly release reports on identifying aerial objects. And in Uruguay, which has passed UFO details along to the United States since the 1970s, the military runs the Commission for the Reception and Investigation of Complaints of Unidentified Flying Objects.

 Meanwhile, legislators in Washington push for the same transparency that other parts of the world have enjoyed for years,  even holding House hearings to make the case, featuring former Air Force vets.  And yet what do they get in return? More stonewalling amid torrents of disinfo from the military, intel agencies. And even the esteemed "fourth estate" chipping in its two cents,  heaping scorn  on testifying military senior personnel such as David Grusch and David Fravor, e.g.

UFO congressional hearing: Whistleblower says US concealing 'multi-decade' UFO program | AP News

Jerry?  He wasn't surprised in our 1990 talks and he wouldn't be surprised now.   As he put it when the conversation ended:  "The fuckers will never come clean until and unless one of them lands right in the middle of D.C.  I just hope I am alive to see it."

Alas, he didn't make it, dying from a burst abdominal aneurysm on Aug. 6, 2016. A damned pity he didn't live to see Spielberg's movie, as he would have been in 7th heaven seeing the ending.

See Also:

Disclosure Day | Official Trailer

And:

disclosure day final trailer - Google Search

And:

The Age of Disclosure - Official Trailer | IMDb

And:

UFO whistleblower calls for expedited disclosure in D.C. news conference | Mystery Wire

And:

   1947 Twining Memo On Unidentified Flying Discs Paved The Way For First FOIA Release Of UFO Docs

And:

JPL-NASA Director Put On Spot About UAP Alien Origin - Doesn't Leave Us Impressed. 


And:



And:


Navy pilots describe encounters with UFOs - YouTube


And:



And:


Neil deGrasse Tyson (on Real Time) Embarrasses Himself With Strawman Objections To Military- Recorded UFOs 


And:


And:

And:

Gov't Whitewashing OF UAP-UFOs Is Still Based On Myth of Human Sovereignty..




Tuesday, June 23, 2026

Solutions To P-Adic Tutorial Problems

 The  problems again:

1) For the triangle in Fig. 2 (June 19 post), 

use 7-adic absolute values applied to the sides of the triangle, thereby compute: A, B and C and show it is isosceles


Solution:

[A ] 7   =  [14 - 1/7] 7   =  abs {[2 x 7 ] 7   -  1/[1 x 7] 7 }

=   abs [1/7 - 7]  =   6 6/7   =  48/7

[B ] 7   =  [1/7 -  21] 7   =   abs {1/ [1 x 7 ] 7   -  [3 x 7] 7 }

=   abs [1/7 - 7]  =   6 6/7   =  48/7

[C ] 7   =  [21  -   14] 7   =  abs {[3 x 7 ] 7   -  [2 x 7] 7 }

= abs[1/7 - 1/7]  7 =    [0 ] 7   = 1

From these calculations of  (7-adic) absolute values we find side A = side B = 48/7.  Hence in the p-adic format the triangle is isosceles.

2) Find the value of the sum S for:

 S = 1 + 7 +  (7) 2    + (7) 3   + (7) 4  + (7) 5  + .......

 Rewrite multiplying both sides by 7:

7 S =  7 +  (7) 2    + (7) 3   + (7) 4  + (7) 5  + (7) 6  + .......

Subtract the last form from the previous one:

    S = 1 + 7 +  (7) 2    + (7) 3   + (7) 4  + (7) 5  + .....

- 7 S =  7 +  (7) 2    + (7) 3   + (7) 4  +    (7) 5  +  ..

S - 7S = 1 (All other top and bottom terms cancel)

Whence:

-6S = 1 and therefore, S = -1/6

3)  Using (2) as written, "invent" a new irrational number based on the p-adic form.

The most direct way to invent a new p-adic is to employ the same 7-adic form and simply increase powers in a slightly different way..

Then let the new irrational be S'  where S' is a variation of S such that:

S'  =   (7) 3   + (7) 6  + (7) 9  +   + (7) 12  + (7) 15  +.....

Hence, the new irrational S'  is obtained merely by changing the sequence of the powers of 7.