Turbulence Riddle Solved by AI as Automated Computational Mathematical Modeling ACMM

The (super)human intellect of Euler (1707-83) formulated Euler's equation for fluid flow.

The information society is based on computational mathematical modeling with Automated Computational Mathematical Modeling ACMM now emerging as a form of Artificial Intelligence AI.

The FEniCS Project is software for ACMM in a setting of (partial) differential equations (mathematical models of physical systems) offering automation of discretisation and computational solution. 

Unicorn/FEniCS is a unified solver for solid and fluid dynamics, which offers a solution to the major unsolved problem of mathematical modeling of turbulent fluid flow in the form of an automated turbulence model as the product of best possible computational solution of Euler's equations for fluid flow. 

This is presented by Johan Jansson in edX courses on High Performance Finite Element Modeling Part I and Part II. Take the courses and see yourself! This is a high-light of the educational program DigiMat carrying from basic school to university bringing ACMM to the people!

The turbulence model is the result of AI in the form of a computational procedure for finding a best possible solution to a set of partial differential equations formed by the Human Intelligence HI of Euler as the sharpest mind of the scientific revolution. Here HI sets the goal and ACMM is the factual process to reach the goal.

The solution of the riddle of turbulence thus comes out from a combination of HI and AI in a setting where HI showed to be too weak to give an answer, as witnessed by 

Werner Heisenberg:

• When I meet God, I’m going to ask him two questions: why relativity? And why turbulence? I really believe he’ll have an answer for the first.

Horace Lamb:

• I am an old man now, and when I die and go to heaven, there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. About the former, I am really rather optimistic.

Turbulent flow (video) around a jumbojet in landing as best possible solution of Euler's equations.

We here view AI as an extension of HI and not (merely) as a replacement as in self-driving cars,
recalling that after all driving a car does not need much intelligence. But predicting turbulent flow does and requires both HI and AI.

Notice that AI is beyond full understanding of HI since AI is a self-learning system and not a system taught by HI. In particular the automatic turbulence model offered by ACMM is beyond full understanding by HI as a the result of a self-learning adaptive computational procedure, generating the turbulent viscosity which showed to be evasive for HI alone.

Notice further that once turbulent fluid motion can be simulated by ACMM, computational fluid environments can be set up for testing of airplane designs, training of pilots or self-learning of flying vehicles as a combined vehicle-environment application of AI.

DigiMat = Ny Start för Matematik-IT

Matematik-IT gör nu en nystart under namnet DigiMat med den nya web-editorn p5.js för JavaScript lanserad av Processing som ersättning/komplement till den tidigare använda plattformen Codea.

DigiMat kommer att lanseras som en ny matematikutbildning för skolan/högskolan relaterad till FEniCS Project som open source plattform för automatiserad matematisk simulering.

DigiMat är en konkret realisering av den nya läroplanen för grundskolan med programmering som en  integrerad del av matematikämnet. DigiMat är det digitala samhällets matematikutbildning.

Huvudprincipen i DigiMat är att alla matematiska objekt konstrueras/beräknas genom exekvering av datorprogram. DigiMat är konstruktiv matematik med datorberäkning som motor.

DigiMat utvecklas av internationellt ledande forskare vid högskolan enligt de principer som driver den främsta forskningsfronten framåt (edX 1 och edX 2).  DigitMat ger en sammanhållen utbildning över alla stadier från förskola till högskola förankrad i en topp-position och ger därmed nytt innehåll och mening åt matematikämnet, i dagens digitala samhälle.

DigiMat erbjuder:

• begriplig, meningsfull och intresseväckande matematik för alla elever
• flexibel bredd och djup efter olika elevers förutsättningar     
• high tech för elever med speciellt intresse för matematik och naturvetenskap
• plattform för spelutveckling för hackers.

New Alarm: Light Bulbs Causing Dangerous Global Warming!

To the old alarm of global warming by CO2 emissions from burning of fossil fuel is now added a new alarm from the emission of light from light bulbs, issued by group of concerned climate scientists supported by UN.  Above is a map showing the light emitted by light bulbs during night over Europe. You see that there is a lot!

Basic physics tells according to the scientists that the light from light bulbs is reflected back from space onto the surface of the Earth with warming effect which is 100 times more intense than that 

from CO2. 

Immediate steps have been taken by the Swedish government to initiate a total reconstruction of Sweden into light-free society in a process to be completed by 2030. By a successive reduction by 50% per year of the energy spent on illuminating light bulbs, a completely dark society can become reality in so little as 10 years, says concerned politicians determined to do something: With this action Sweden will take the lead in a global transformation into a light-free world.  Of course that includes the light emitted by mobile and laptop screens. Sorry.

To survive the long dark winters in Sweden, light from burning wood in open fireplaces will be admitted during a period of transformation. On the other hand, in Summer time Sweden, in particular above the Polar Circle, will have an advantage of full light though the night which will give a much wanted boost to Swedish industry. It will also give Africa as an already dark continent an advantage.

How Far Can a Battery Fly?

Electrical vehicles of all kinds are currently being investigated with electrical airplanes as an novel option attracting eager investors. Let us seek an answer to the key question of the possible range of such a device powered with lithium-ion batteries.

The energy density of lithium-ion battery is about 1 MegaJoule per kilogram. Equipped with weightless wings with a lift-to-drag ratio of 15 a battery of weight 1 kp = 10 N would be able to fly a distance S determined by the equation

• 10/15  x S = 10^6 

that is S = 1500 km. Not so bad. But that is the theoretical maximum which will be reduced  adding weight for passengers, airplane structure and engine efficiency. How much? 

Let us compare with kerosene fuel with energy density 40 MJ giving a jumbojet with again lift-to-drag of 15 a maximal range of 40 x 1500 km = 60 000 km, which we compare with the real range of about 10 000 km , thus with a reduction factor of about 6 to account for passenger+structure+engine efficiency. 

The practical maximal flying distance of a battery driven airplane could then be about 250 km.  In contrast to surface travel, the energy consumption for subsonic flight is independent of speed.

With a lift-to-drag of 30 the maximal distance would be up to 500 km, which may be what is needed

to make electrical flight economical. For a more information on flight take a look at The Secret of Flight.

Weinberg Still (at 85) Unhappy with QM

Lubos reports on yet another talk by the famous physicist and Nobel Laureate Steven Weinberg, where Weinberg confesses that he is not happy with Quantum Mechanics QM:

• In  a graduate course on QM, which eventually turned into a book, I found that I was unable to explain the foundations of QM in a way that I found (entirely) satisfactory.

Weinberg recalls that there are two interpretations of Schrödinger’s wave functions as solutions of Schrödinger’s multi-dimensional wave equation as the basic model of QM:

• instrumentalist (wave functions predict outcomes of experiments).
• realist (wave functions describe all possible worlds = multi-versa).

The instrumentalist interpretation says nothing about the world we live in, only something about certain experiments. 

In the realist or rather multi-realist interpretation, multi-d wave functions describe all possible worlds or multi-versa and thus the (quite special and particular) world we are living in gets little attention, if any.

With only these alternatives, it is not difficult to understand that Weinberg cannot make sense of QM. Weinberg also recalls that physicists agreeing that QM is correct, do not agree on what is correct.

Of course Lubos claims as usual that he can make sense to QM, or rather that there is no reason for him to do that, because this was done long ago. Not by Schrödinger or Einstein, but by Born who invented a probabilistic interpretation, which abhorred Schrödinger and Einstein.  

Weinberg has a Nobel Prize while Lubos runs a blogg. Weinberg says there is a problem with QM, which is denied by Lubos. What do you think? Is there a problem or not? Weinberg is at the end of his career and can afford to tell the truth.

In a travesty to a statement by Clark (Someone saying that something is impossible, is mostly wrong. Someone saying that something is possible, is mostly correct), one might say:

• Someone strongly claiming that there is a serious problem, is mostly correct.
• Someone shouting that there are no problems whatsoever, is mostly wrong.

If you think there is a problem with QM in its current form, you may be motivated to browse Real QM offering a new realist interpretation, where the objective is the world we are living in and not all possible worlds. Take a look!

One may say that the current deep crisis of theoretical physics witnessed by so many, is rooted in the status of QM as a theory which the expert cannot make sense of. So to get out of the crisis it is necessary to come up with a QM that makes sense. Right?

There are several new books sending the message that QM does not make sense therefore is not understood by physicists:

• Beyond Weird by Philip Ball
• What's Real by Adam Becker
• Through Two Doors at Once by Anil Ananthaswamy.

DN Kulturdebatt: Stoppa Fossil Energi med Alla Medel!

Humanekolog Andreas Malm bereds idag i stor plats på DN Kulturdebatt för att predika en extrem klimatalarmism, som presenteras på följande sätt av DN:

• Klimatförändringarna innebär att mänskligheten står i en rulltrappa på väg upp mot temperaturer som saknar slutstation. Det är hög tid att gå från protest till motstånd.

Andreas Malm skriver:

• Lägg ned inrikesflyget. 
• Förbjud försäljningen av fossildrivna bilar. 
• Avlägsna fossilgasen från värmeverken, avbryt kolkonsumtionen i stålverken, stäng oljeraffinaderierna och bygg ut alternativen i den mest rasande takt en modern stat, med alla resurser den potentiellt förfogar över, kan åstadkomma. 
• Föreställningen om att vi tio miljoner individer saknar betydelse måste vändas i sin motsats. Varje ytterligare ton, varje extra kilo koldioxid driver rulltrappan vidare uppåt. Varje utsläppskälla måste täppas till, från matbord till garage, gruva till hamn – och om inte de folkvalda gör det den här gången heller är det upp till alla oss andra. 
• Den här veckan samlas återigen tusentals klimataktivister runt en ångande utsläppskälla, fossilgasfälten i holländska Gröningen, där utvinningen på samma sätt ska stoppas med obeväpnade kroppar. 
• Vi har tagit bussar från avfarter runt om i Europa. Vi har gjort det förr: år 1995 åkte vi till COP-1 i Berlin. På det första klimatmötets sista dag omringade vi konferensanläggningen, kedjade fast oss vid dörrarna och skanderade: ”no more bla bla bla, action now”. Ministrarna smög ut bakvägen. Sedan dess har världens årliga koldioxidutsläpp ökat med mer än 50 procent. 
• Det är hög tid att gå från protest till motstånd – från att önska sig mindre utsläpp till att fysiskt spärra dem – och ta kampen till nästa nivå. 

DN refuserar varje debattartikel som på något ifrågasätter rådande svenska klimatpolitik med mål att leda Sverige in i det fossilfria samhället. Istället basunerar DN ut en extrem klimatalarmism med uppmaning till motstånd mot alla former av fossil energi, uppenbarligen med vilka medel som helst.  Vet DN vad DN gör? 

Andreas Malm är humanekolog och för en humanekolog är det humana underordnat det ekologiska. Utan fossil energi skulle mer än 80% av jordens befolkning stå utan möjlighet till fortsatt existens. För en humanekolog verkar det inte vara något större problem: Om det är ekologins villkor att samhället måste vara fossilfritt, så får det bli så.

Is Pinker Enlightened Believing in AGW?

Steven Pinker's book Enlightenment Now: The Case for Reason, Science, Humanism and Progress, presented by Bill Gates as My New Favorite Book of All Times, has a long chapter on The Environment with focus on Anthropogenic Global Warming AGW from burning of fossil fuels:

• To keep the rise to 2°C or less, the world would, at a minimum, have to reduce its greenhouse gas emissions by half or more by the middle of the 21st century and eliminate them altogether before the turn of the 22nd. The challenge is daunting. 
• Fossil fuels provide 86 percent of the world’s energy, powering almost every car, truck, train, plane, ship, tractor, furnace, and factory on the planet, together with most of its electricity plants. Humanity has never faced a problem like it.
• Anthropogenic climate change (AGW) is the most vigorously challenged scientific hypothesis in history. By now, all the major challenges have been refuted and even many skeptics have been convinced.

Pinker says that the world has to get rid of the fossil fuel that today supplies 86 percent of the world's energy and that this is a daunting challenge. Pinker suggests that this can be achieved in the spirit of enlightened environmentalism:

• An enlightened environmentalism recognizes that humans need to use energy to lift themselves out of the poverty to which entropy and evolution consign them. It seeks the means to do so with the least harm to the planet and the living world. 
• History suggests that this modern, pragmatic, and humanistic environmentalism can work. 
• As the world gets richer and more tech-savvy, it dematerializes, decarbonizes, and densifies, sparing land and species. 
• As people get richer and better educated, they care more about the environment, figure out ways to protect it, and are better able to pay the costs. Many parts of the environment are rebounding, emboldening us to deal with the admittedly severe problems that remain.

To decarbonize human civilisation Pinker points to carbon taxes, nuclear energy and climate engineering, however with the following caveats:   

• Carbon taxes, to be sure, hit the poor in a way that concerns the left, and they transfer money from the private to the public sector in a way that annoys the right. But these effects can be neutralized by adjusting sales, payroll, income, and other taxes and transfers. (As Al Gore put it: Tax what you burn, not what you earn.) And if the tax starts low and increases steeply and predictably over time, people can factor the increase into their long-term purchases and investments, and by favoring low-carbon technologies as they evolve, escape most of the tax altogether.
• A second key to deep decarbonization brings up an inconvenient truth for the traditional Green movement: nuclear power is the world’s most abundant and scalable carbon-free energy source. Although renewable energy sources, particularly solar and wind, have become drastically cheaper, and their share of the world’s energy has more than tripled in the past five years, that share is still a paltry 1.5 percent, and there are limits on how high it can go.
• The very idea of climate engineering sounds like the crazed scheme of a mad scientist, and it once was close to taboo.

Does this indicate that Pinker is Enlightened? No, it is rather the opposite. Pinker's thinking is darkened into believing that science says that burning of fossil fuels will overheat the globe. But the enlightened scientist knows very well that the evidence to this effect is very weak, if any. Pinker's mind is further confused into wishful thinking that anything is possible in the hands of politicians and engineers ready to act, and this is rather Dark Age than Enlightenment. Paradoxically, Trump appears to be more enlightened than Pinker.

Pinker shares the AGW alarm of "greenism", but not its ideology in opposition to Enlightenment:

• Starting in the 1970s, the mainstream environmental movement latched onto a quasi-religious ideology, greenism, which can be found in the manifestoes of activists as diverse as Al Gore, the Unabomber, and Pope Francis.
• Green ideology begins with an image of the Earth as a pristine ingénue which has been defiled by human rapacity. 
• The root cause is the Enlightenment commitment to reason, science, and progress. 
• ....unless we repent our sins by degrowth, deindustrialization, and a rejection of the false gods of science, technology...
• As with many apocalyptic movements, greenism is laced with misanthropy, including an indifference to starvation, an indulgence in ghoulish fantasies of a depopulated planet, and Nazi-like comparisons of human beings to vermin, pathogens, and cancer. 
• For example, Paul Watson of the Sea Shepherd Conservation Society wrote, “We need to radically and intelligently reduce human populations to fewer than one billion. . . . Curing a body of cancer requires radical and invasive therapy, and therefore, curing the biosphere of the human virus will also require a radical and invasive approach".

Pinker seems to have forgotten the proverb: When you give the devil a little finger, he'll take the whole hand. The best for Pinker would be to take a serious look at the scientific enlightened evidence of AGW to find that it is not much, and then he could keep his little finger and whole hand as a convinced enlightened AGW skeptic.

Paradoxically perhaps, Trump appears to be more enlightened than Pinker, by understanding that AGW has weak scientific support and therefore leaving the (empty but misleading) Paris agreement.

Time for Real Quantum Mechanics

The new book What Is Real?: The Unfinished Quest for the Meaning of Quantum Physics by Adam Becker, is reviewed by Tim Maudlin in Boston Review under the title The Defeat of Reason with the following summary in the words of Imre Lakatos:

• After 1925 Bohr and his associates introduced a new and unprecedented lowering of critical standards for scientific theories. This led to a defeat of reason within modern physics and to an anarchist cult of incomprehensible chaos.

Yes, this is also my impression that quantum mechanics in the standard commonly accepted text book Copenhagen Interpretation of quantum mechanics, is incomprehensible chaos, and that the alternatives suggested by Bohm, Everett and Bell have not reached any common acceptance at all.

The incomprehensible chaos is handled by a strategy of "shut up and calculate", which does not allow scientific questioning and thus cannot really count as science.

This gives me new inspiration to pursue Real Quantum Mechanics as a theory with physical interpretation. Take a look if you don't like incomprehensible chaos and a dictate of "shut up and calculate".

This is a continuation of an earlier more detailed post on Becker's book stimulated by Maudlin's review.

As usual Lubos Motl enters in defence of standard Copenhagen Interpretation and "shut up an calculate" but it is still incomprehensible chaos.

PS The upcoming book Lost in Math by Sabine Hossenfelder gives more evidence of modern physics as incomprehensible chaos.

Överklagan till Pressens OpinionsNämnd PON

Jag har nu skickat in nedanstående överklagan till Pressens OpinionsNämnd PON av Allmänhetens PressOmbudsmans POs beslut att avskriva det ärende angående SvDs rapportering från Climate Sense 2018, som jag anmält till PO.

POs beslut har en motsägelsefull undanglidande osaklig motivering, som indikerar att PO aningslöst följer rådande stormediavindar utan vilja eller förmåga till självständigt tänkande. Instruktionen för PO säger att till PO bör utses en person med särskilda insikter inom området för pressetik, som anges omfatta bl följande regler: 

• Massmediernas roll i samhället och allmänhetens förtroende för dessa medier kräver korrekt och allsidig nyhetsförmedling.
• Den som gör anspråk på att bemöta ett påstående ska, om det är befogat, beredas tillfälle till genmäle.

Vilka särskilda insikter har PO beträffande dessa regler, som leder PO att avskriva min anmälan?

PS The Best Schools har ställt samman en lista över ledande alarmister samt skeptiker: The Top 15 Climate-Change Scientists: Consensus & Skeptics som uttryck för följande övertygelse:

• Our position is simple. It is the classical liberal one. Drumroll. Cue the shade of Voltaire (channeled by Evelyn Beatrice Hall): “I disagree with what you say, but I will defend to the death your right to say it.” Why should I defend someone else’s freedom to say what I myself believe to be wrong? Because the truth is one thing, my knowledge of it is something else. And because this means that the essence of rational inquiry is intellectual humility. And also because the slow and painful advance towards truth is best served by the open and honest airing of disagreement. For all of these reasons, we deplore all attempts to use political muscle to shut down academic debate.

Överklagan av POs beslut DNR PO 18146

PO har beslutat att avskriva den anmälan jag gjort angående SvDs rapportering den 3 mars 2018 från den vetenskapliga konferensen Climate Sense 2018. Härmed överklagar jag detta beslut.

Vår tids ödesfråga utbasunerad av politiker och media i Europa påstås vara hotande global uppvärmning från användning av fossil energi. För övriga världen är det ingen stor fråga. Användningen av fossil energi kommer öka under överskådlig tid för att försörja jordens växande befolkning. USA nysatsar på fossil energi och har lämnat Parisavtalet.

Det finns en stor och snabbt växande grupp av välrenommerade vetenskapsmän världen över som kommit till slutsatsen att den vetenskapliga grunden för det påstådda “klimathotet” är synnerligen bristfällig och ingalunda kan motivera den “omställning mot ett fossilfritt samhälle”, som nu förbereds med Sverige som föregångsland och som skulle kunna försörja endast en liten del av jordens nuvarande befolkning. Dessa vetenskapsmän kan beskrivas som “klimathotsskeptiker”, varav en del alltså samlades till Climate Sense 2018 för att presentera aktuell forskning.

SvD rapporterade från klimatmötet den 3 mars 2018 på 3 fulla sidor med stora bilder. SvDs artikel presenterar mötets deltagare med följande karakteristik:

• klimatförnekare av manligt kön, 
• i samhällets utkant 
• personer med auktoritära attityder, låg empati och hög dominans 
• inte längre är i yrkesaktiv ålder 
• kopplade till fossilindustri, auktoritära tider, nationalism och populism 
• gräver sig allt djupare ner i förnekelse ju mer klimatpolitiken skärps med brett politiskt stöd
• byggt hela sina liv på att vara med i verksamheter som släpper ut koldioxid.

Min anmälan avser denna ärekränkande beskrivning av en grupp seriösa forskare inklusive mig själv, en beskrivning i form av förtal som SvD inte låter bemötas i någon form.

PO motiverar sitt avslag med

• Du är utpekad i bild.
• Du associeras i viss mån med uppgifterna om de andra deltagarna och andra klimatförnekare.
• Det är dock i huvudsak allmänna påståenden om klimatförnekare som grupp och Du är inte direkt kopplad till någon av uppgifterna.
• Den publicitetsskada Du förorsakas får därför anses vara begränsad.
• Någon allmän rätt att debattera finns inte.

Analys av POs skrivning:

PO använder helt ogenerat och aningslöst benämningen klimatförnekare utan citationstecken i sin beskrivning och förefaller helt köpa den karakteristik av denna (förtappade) grupp som ges av SvD.
Att PO inte uppfattar benämningen "förnekare" som varande direkt kopplad till Förintelsen är fullständigt häpnadsväckande. Eller förstår PO kopplingen mycket väl, men anser att den är helt adekvat. 

 
PO medger att jag orsakats "publicitetsskada" genom att jag "i viss mån" associeras med gruppen "klimatförnekare", men PO anser denna skada "är begränsad". Att bli associerad med Förintelsen skulle alltså bara innebära "begränsad" skada. 

PO säger (helt häpnadsväckande) att någon "allmän rätt att få debattera finns inte". Inte heller att kunna få bemöta SvDs beskrivning av klimatmötets alla välrenommerade vetenskapsmän som "klimatförnekare" med förskräcklig karakteristik.

Vi ser en PO som går helt i ledband med svensk stormedia mot målet att utrota alla "klimatförnekare" och få stopp på all debatt som inte stödjer rådande agenda att leda Sverige in det fossilfria paradiset. Det kan man kalla Förintelse.

Jag vänder mig nu till PON med förhoppning att det där skall finnas någon som kan se att SvDs rapportering från Climate Sense 2018 inte motsvarar god pressetik och som förstår nödvändigheten av att stormedia öppnar för belysning av vår tids ödesfråga, där också “klimathotsskeptiker” kan få komma till tals. Det gäller framtiden för vårt samhälle och våra barn.

Vågar PON öppna för debatt, eller lever vi ett samhälle, där som PO säger "någon allmän rätt att debattera inte finns”? Lever vi ett totalitärt samhälle där stormedia som SvD oemotsagt kan trumpeta vilka lögner som helst? Kanske är det så, och då går enda hoppet till PON! 

Nu får vi se vad PON har att säga om SvDs ensidiga inkorrekta rapportering vad gäller  mänsklighetens ödesfråga.

On Beauty in Physics: Euler's Equations

Sabine Hossenfelder asks in a recent post on BackReAction connecting to her upcoming book
Lost in Math: How Beauty Leads Physics Astray

• What do physicists mean when they say the laws of nature are beautiful?

From interviews Sabine has collected the impression that theoretical physicists largely agree that beauty in physics has the following features:

• simplicity
• naturalness
• elegance. 

The title of the book indicates that beauty alone can lead astray, which of course carries some truth, but beauty combined with functionality is irresistible,  following the deep insight that "a beautiful boat is a good boat".

Let me here give an example of ultimate beauty + functionality: Euler's equations for slightly viscous incompressible fluid flow were formulated by the great mathematician Leonard Euler in 1755 as an expression of Euler's Dream:

• My two equations include not only all that has been discovered by methods very different and for the most part slightly convincing, but also all that one could desire further in this science.

The two equations expressed in fluid velocity $u(x,t)$ and pressure $p(x,t)$ depending on a Euclidean space coordinate $x$ and time $t$, take the form in the fluid domain 

• $\frac{\partial u}{\partial t}+u\cdot\nabla u + \nabla p=0$
• $\nabla\cdot u = 0$ 

combined with a boundary condition on a solid wall expressing zero friction as an expression of small viscosity. 

Euler's equations clearly have the features of simplicity, naturalness (they express Newton's 2nd law + incompressibility), and surely they are elegant. 

But are they functional = useful? For 250 years Euler's Dream that his equations contained all that one could desire further in this science was a joke, because Euler's equations admit analytical solutions named potential flow, which cannot be observed, referred to as d'Alembert's paradox.

It took 250 years to understand that potential flow is unstable and thus cannot be observed and until computers were powerful enough to admit computational solution of Euler's equations appearing as turbulent solutions with full agreement with observations. 

This is the revelation of the book Computational Turbulent Incompressible Flow now presented to the world in the edX courses High Performance Modeling 1 and 2, which show that today with the computer Euler's Dream has come true! Everything you may want to know about slightly viscous fluid flow can be found in computational solution of Euler's equations. In particular The Secret of Flight is revealed.

Euler's equations thus combine beauty with functionality and further fulfill Einstein's ultimate requirement of being parameter-free: The specific value of the viscosity is irrelevant as long as it is small. This means that can compute the drag and lift of any body from the form of the body alone.

Euler's Dream: The Niagara Falls captured in Euler's equations with computational solutions giving back the Falls in simulation. Beautiful!