crash-landing in aunt effie’s cabbage patch: calculating the probability, speed and timing of the...

The Random Evolutionary Drift of Inventions

1

Crash-Landing in Aunt Effie’s Cabbage Patch: Calculating the Probability, Speed and Timing of the Random Evolutionary Drift of Inventions

Gilbert Moore, PhD 8/21/2015

Path Integration. Path integration is one of the weirdest calculation methods ever invented. Nevertheless, Richard Feynman demonstrated its fundamental correctness. In QED: The Strange Theory of Light and Matter Feynman invites us to imagine the path of a photon emitted from a light source, reflecting off the surface of a mirror and being detected by a photon detector. He asks: What path would the reflected photon travel in reaching the detector? For hundreds of years classical optics had correctly predicted that the photon would travel the path in which the angle of incidence equals the angle of reflection, and this path would be the path requiring the least time for the photon to reach the detector (See diagram 1). This was not just a theory, it had been confirmed experimentally hundreds of times.

In view of this, what was so strange and new about Feynman’s path integration?

Its premises.

Feynman’s path integration method does not assume a single correct path for the photon, but a superposition of all paths, and this is deeply disturbing to the logical mind. Paths traveling to distant corners of the universe are being accounted for in the path integration of a single photon

Crash-Landing in Aunt Effie’s Cabbage Patch

2

traveling, let’s say, six feet from its source to the mirror and then to the detector. How can that be? Feynman answered: By summation of the probabilities for each path we arrive at an amplitude and path for the photon corresponding exactly to the path predicted and confirmed experimentally by classical optics: Summing across all its paths, travel time for the photon is minimized to the least time and its angles of incidence and reflection are equal.1

While justifiably puzzled by this, let’s nevertheless construct an analog to photon path integration to estimate the speed and timing of evolutionary drift. In the one case of primary interest to us -- the path of descent of ancient inventions from throw sticks and mancala to dice and modern chess --, the drift in form and function of the common ancestor to Xiangxi in China and Chatranga in Persia is unobserved, on a time scale of thousands of years tens of thousands of years ago, and happening with a poverty of evidence. What could be worse experimentally? Despite this, if the Nash equilibrium model motivating our hypothesis is valid, the path integration analog should apply equally well to both that instance and to more recent instances, that is, to inventions in the last hundred years on a time scale of 10 to 20 years or less. And this invariance across time could give us clues to solving the puzzle of primary interest.

Let’s first review the Nash equilibrium model of invention we outlined in the Appendix to The Mysterious Origin and Strange Descent of Chess and then construct a path

1 R. Feynman, QED: The Strange Theory of Light and Matter; also see: R. Feynman and A. Hibb, Quantum Mechanics and Path Integrals

http://bluelogic.us/pages/2-1055.php

http://bluelogic.us/pages/2-1055.php


3

integration analog for testing the model experimentally.

A Nash equilibrium model. How does the model work?

1) It begins in a state of Nash equilibrium for inventions: Imagine a colony with several inventors in it. Not everyone in the colony is an inventor. Just these men and women. Constantly in communication, they share their strange vision of things with each other all the time. What this or that really is, how to make it work better, why any of it works at all, etc. Prototypical science-engineer-mystics. They see and invent strange things for the colony all the time. Little trivial things, a few significant things. Like a water-resistant dye for cave drawings or flint for making fires or a water-wheel.

Sometimes, however, there is no incentive in any one inventor to invent anything because none sees any benefit in it for the colony over what has been invented already or over what others are thinking about inventing. The incentive to invent = 0, the incentive not to invent = 1, the incentive to invent index = sum(incentives to invent)/sum(incentives not to invent)+1 = 1. The inventors are in a state of Nash equilibrium. This state can last a very very long time, thousands of years in fact, especially in the absence of long distance communication of any kind.

2) Despite this, whenever any one or more inventors sees some benefit in an invention for the colony, the state of Nash equilibrium is broken, and there is an incentive to invent. The incentive to invent = 1, the incentive not to invent = 0, the incentive to invent coefficient = sum(incentives to invent)/sum(incentives not to invent)+1 > 1. The inventors


4

have entered a state of Nash disequilibrium.

3) In a state of broken Nash equilibrium for invention, collectively or independently and alone, inventors would be at work making something work better or differently for the colony. Unlike Athena out of the head of Zeus, however, inventions do not spring ready-made and functioning out of the heads of their inventors. Time would nearly always be a limiting factor. It might have taken months, years, even several generations before the tiny spark of an original incentive became a finished, practical, and more or less zero-defect working invention for the colony. Let’s call this the time to invention delay factor, or simply the elapsed time to invention.

A path integration analog. Imagine if we could go behind the timeline of evolutionary drift and observe the mechanism causing it. What would we see? A simple setup looking on the surface very much like our photon detector model: At one end a fixed source of inventions representing the moment an inventor’s Nash equilibrium to remain in a state of inertia and not invent is broken, secondly, a moving time-sensitive thin film strip registering the elapsed time between the breaking of Nash equilibrium and the actual time of invention. Both events emit a photon whenever each happens. Thirdly, an observer of both events at the other end seeing and recording the elapsed time between the two light signals (see diagram 3) – first seeing one and then waiting for the other.

Besides the obvious difference that the photon detector is only light-sensitive and its thin-film analog is time-sensitive as well, the main difference is that the evolutionary drift


5

analog captures vital information on the probability of the incentive to invent happening or not happening in the first place. Draw a line from the photon emitted at A when the inventor’s incentive to invent first happens to the second photon emitted later at the time of invention on the moving thin film strip at B (see diagram). This line forms an angle alpha to the thin-film strip. The ratio alpha/90 gives the probability p of the second photon being emitted. A simple Bernoulli probability, p is registering the probability of success at inventing anything. The steeper the angle of descent alpha, the greater the probability of success p and the shorter the elapsed time between photons T. At alpha = 90, p = 1, and T = 0.

The observer sees both photons -- a photon signaling the time Nash equilibrium exists and a second photon signaling the time Nash equilibrium is broken and measuring the probability of a break thru. The elapsed time is simply the time Nash equilibrium exists minus the expected value of the time it is broken where the expected value equals the product of its Bernoulli probability of happening p and the time Nash equilibrium is first recorded To. Likewise with the probability of a succesful invention.

In shorthand, we have:

To = time of Nash equilibrium inertiaTN = time Nash equilibrium is brokenTI = elapsed time to inventionpN = Bernoulli probability of breaking Nash equilibriumpI = Bernoulli probability of inventing


6

TN = elapsed time from zero incentive to some incentive = To(1 - pN)TI = elapsed time from Nash break through to time of invention = TN(1 – pI) = To(1 - pN) (1 – pI)H(p) = -plog2p - (1-p)log(1-p)= binary entropydH(p)/dp = -log2(p/(1-p)) = rate of change of binary entropy (speed of invention)

8/16/2015

Some examples. Let’s start with a few inventions from our information-rich modern times. Here Shannon uncertainty is at a minimum but still exists. That is because even though we know with more or less accuracy the time of an invention TI, we know much less about when the Nash equilibrium inertia for it was broken TN, and next to nothing about either the probability of breaking this equilibrium pN or the probability of inventing the device afterwards pI. Because of this we will have to reverse engineer them:

We will start by estimating the time of Nash equilibrium inertia TN being broken from bits and pieces of biographical details about the inventors -- mainly their recorded visionary light bulb moments --, then back-track with this time estimate and the actual known time of invention to the unknown probabilities of success at doing this. Finally we will calculate the amount of Shannon uncertainty or binary entropy in our estimates. Having trained ourselves with this analog to look into the information rich present, we will then turn its strange mathematical lens on our primary target, the distant, information-poor prehistoric beginnings of throw sticks and mancala.


7

Leonardo da Vinci, the Wright Brothers and Kitty Hawk. 497 years separate Leonardo’s visionary 1506 designs for a human-powered airplane with flapping wings and a screw-powered helicopter from the first Kitty Hawk flights of the Wright Bothers in 1903. But da Vinci’s remarkable designs did not directly inspire the Wright brothers. That came later from a different source: the rubber band.

Patented in 1845 by Stephen Perry, the rubber band made its debut as the power source of choice for the first successful twisted rubber-band powered helicopter in 1871 -- a model helicopter designed and built by a 20-year aeronautical engineering genius named Alphonse Penaud, who was himself inspired by da Vinci’s drawings. Seven years later 7-year old Orville Wright and his 11-year brother Wilbur played in fascinated amazement with a toy replica of Penaud’s helicopter model -- paper, cork and rubber band to twirl its rotor. They played with the toy helicopter so much that it broke, and they built their own. In that moment the Nash equilibrium of inertia to invent an airplane was broken in them.

Twenty one years later in 1899 Wilbur Wright was writing a letter to the Smithsonian requesting information and publications about aeronautics. That same year the brothers began their mechanical aeronautical experiments on fixed-wing, 3-axis human-controlled flight. Four years later, in 1903 at Kitty Hawk, after inventing the first stabilizing 3-axis control system for aircraft, Wilbur Wright flew Flyer I, a fixed-wing gasoline=powered double-decker airplane, 120 feet in 12 seconds.


8

In this first test of an analog to Feynman’s path integration method we will isolate four major time to invention events, starting with Leonardo da Vinci’s visionary designs. The events are listed below and shown in Tables 1 thru 3:

1506 - clock time for da Vinci’s visionary airplane and helicopter drawings

1871 - evidence of Alphonse Penaud’s Nash incentive to invent an airplane in his rubber-band powered model aircraft -- first a model helicopter and then a model airplane

1899 - evidence of the Wright brother’s Nash incentive to invent an airplane in their letter to the Smithsonian and the beginning of their experiments that year

1903 - clock time for the flight of Flyer I at Kitty Hawk.

Table 1 tells an interesting story: An elapsed time of about 364 years separates Penaud’s constructions from da Vinci’s designs. Anchoring Penaud’s Nash incentive to invent between two objective time clocks over which we have no control -- the date of Leonardo’s drawings and the date of Penaud’s invention of rubber-band powered model aircraft -- we begin back tracking and summing over possible probabilities of success at breaking Nash equilibrium to find the average probability of success: 0.7578 with a standard error of +/- .1113 and a binary entropy of .7987.


9

What does this probability 0.7578 +/- 0.1113 mean?

One interpretation: If in fact Leonardo da Vinci’s drawings were the visionary source inspiring Penaud’s rubber-band powered hrlicopter and our analog coupling these two events is correct, the latter event -- Penaud constructing his model at 20 -- could only happen if his Nash incentive to invent was moderately high and no higher (or lower). than the upper and lower limits of 0.7578. Any probability of success lower than this and the event would have happened later, which it didn’t. Any probability higher than this and the construction would have happened earlier, which it also


10

didn’t. At 0.7987 the amount of Shannon uncertainty in the estimate is relatively high, meaning with this estimate there is some advantage, but not much, to having it give us an estimate of the true probability of success. Stated differently, the estimate may, in fact, be misleading.

Tables 2 tells a similar story. This time anchoring the


11

Wright brothers incentive to invent an airplane between the objective time clock of Penaud’s incentive (1871) and the time clock of their own incentive (1899) -- an elapsed time of about 28 years -- we again back track and sum over possible probabilities of success to find an average probability of .9849. Once again, any probability of success lower than this and the event would have happened later, which it didn’t, and any probability higher than .9849 and Wilbur’s letter confirming their inertia to inventing an airplane had been broken would have happened earlier, which it also didn’t. This time, however, the amount of binary entropy in the estimate is much less: 0.1132, meaning with this estimate one gains significant and valuable insights into the true probability of success.

Table 3 tells the fascinating 5 year time-to-invention story from Dayton to Kitty Hawk numerically. This time the anchoring time clocks are the date of Wilbur’s letter to the Smithsonian suggesting he and Orville were pretty fired up about putting an airplane in the air (1899) and the date Flyer I lifted off the ground at Kitty Hawk and traveled its historic 128 feet in air (1903). What happened in those five years? For one thing, the issue of the moment was no longer the probability of success at breaking thru inertia but the simple probability of success at inventing a fixed wing airplane that could fly without crashing. Perhaps thousands of manhours of effort later, the Wright brothers did this for 12 seconds. Back tracking computationally from 1903 to 1899 and then forward again and summing probabilities of success with each backtrack, we arrive at an average probability estimate of 0.9974. The estimate couples the time clock 1899 to the time clock 1903 probabilistically. Also at 0.0262 the


12

estimate’s binary entropy is at a minimum computationally, meaning the estimate is telling us just about the most an estimate can ever tell us about the true probability of success at inventing anything without degenerating into an omniscience probability = 1 wth an information value of zero.


13

Nell, V-2 Rockets, and the Apollo Moon Mission. In 1865 the visionary science fiction writer Jules Verne wrote and published de la Terre a Lune (From the Earth to the Moon), inspiring generations of young science fiction readers born thereafter to imagine spaceflights to the Moon. One of these young readers was H.G. Wells, who became a visionary science fiction writer himself, writing War of the Worlds in 1897. The second was Konstantin Tsiolkovsky who read Jules Verne’s classic in 1870, and the third was Robert Goddard, a precocious mathematical genius, who read both Jules Verne and HG Welles.

In 1897 inspired by de la Terre a Lune, Tsiolkovsky formulated his famous rocket equation: u = vln(Mo/M)+uo where u = final velocity, v = velocity of exhaust gases. Using this equation in his most important work, Exploration of Outer Space by Means of Rocket Devices (Russian: Исследование мировых пространств реактивными приборами 1903), Tsiolkovsky calculated that the horizontal speed required for a minimal orbit around the Earth is 8,000 m/s (5 miles per second) and that this could be achieved by means of a multistage rocket fueled by liquid oxygen and liquid hydrogen. The first break through had been made.

In 1899 two years after reading H.G. Wells classic and while climbing a cherry tree behind the family barn, a 16-year boy named Robert Goddard became transfixed looking up into sky and thinking about a spacecraft lifting itself off the ground by the centrifugal force of a flywheel spinning at an accelerating velocity around a metal shaft. In that moment the inertia to building a spacecraft was broken a second time. In 1913, using Tsiolkovsky’s rocket equation and


14

experimental data of his own, Goddard the trained applied physicist calculated the position and velocity of a rocket in vertical flight, given the weight of the rocket and weight of the propellant and the velocity of the exhaust gases. In 1926 after generous funding by Lindberg, the Guggenheims and the Smithsonian, using his earlier calculations and others published in his groundbreaking study A Method of Reaching Extreme Altitudes, Goddard launched the first liquid-fueled (gasoline and liquid oxygen) rocket in Auburn, Massachusetts. Later dubbed the Nell, the rocket rose just 41 feet off the ground and travelled 184 feet in the air, crashing-landing in Aunt Effie’s cabbage field 2.5 seconds later. The first liquid-fueled spacecraft had been invented. Eleven years later, in 1937, Goddard launched the L-13, a much improved version of Nell. It reached an altitude of 8,900 feet.

The fourth reader of de la Terrre a Lune was a young German boy named Herman Oberth. At 11 years of age in 1905 Oberth started reading Verne’s classic, and the inertia to inventing a rocket was broken a third time, this time in a boy still in puberty. Three years later at the age of 14 Oberth designed and built his first model rocket. Twelve years later in 1920, after reading Goddard’s pioneering study, Oberth published his own contribution Die Rakete zu den Planetenraumen. No science fiction, this visionary engineering treatise on the theory and engineering of rocketry stunned the German scientific/engineering community, inspiring among others a young man named Wernher von Braun. Von Braun became Oberth’s apprentice. Working together on rocket engine designs copied from Goddard’s liquid-fueled static firing experiments, in 1929, on the eve of the Great Depression, the pair designed, built and did a static firing of their own


15

liquid-fueled rocket engine, the Oberth-von Braun rocket engine.

Thirteen years later in 1942 with Germany, Europe and the rest of the world engulfed in war, now a Nazi and SS member, von Braun invented the V-2 Retribution Rocket, using pirated component designs from Robert H. Goddard’s L-13 rocket. On the eve of being defeated, Hitler authorized the firing of thousands of V-2 rockets into London. Unable to invade his Island Nemesis with German troops, Hitler settled on terrorizing its capital’s popultion with van Braun’s invention. Some 9,000 London residents, mostly civilians, died in the V-2 bombings, and at least an equal number of war prisoners died in the Mittelbau-Dora concentration camp assembling the V-2 rockets under brutalizing conditions of slavery. 2

In 1945, with Germany defeated and Hitler assassinated, to avoid capture by the Russians, von Braun surrendered 2 . Jaroff, Leon (March 26, 2002). “The Rocket Man’s Dark Side”. Time. Retrieved June 29, 2008. Ernst Stuhlinger; Frederick Ira Ordway (April 1994). Wernher von Braun, crusader for space: a biographical memoir. Krieger Pub. p. 42. ISBN 978-0-89464-842-7. Retrieved December 18, 2011. Roop, Lee (October 4, 2002). “Aide says von Braun wasn’t able to stop slave horrors; Objection would have gotten rocket pioneer shot, Dannenberg says”. The Huntsville Times. Archived from the original on October 26, 2002. Biddle, Wayne (2009). Dark Side of the Moon: Wernher von Braun, the Third Reich, and the Space Race. W. W. Norton & Company.:124–125 Michael J. Neufeld (Feb., 2002) “Wernher von Braun, the SS, and Concentration Camp Labor: Questions of Moral, Political, and Criminal Responsibility”, German Studies Review, Vol. 25, No. 1, pp. 57-78


16

himself and his scientific/engineering staff to US Army troops occupying Germisch, Germany. Soon a captive guest of the US government in El Paso, Texas, with his Nazi and SS past bleached out of his files, von Braun spent the next 24 years during the 1950s and 1960s building in succession the Redstone ICBM, the Jupiter series of rockets propelling the Explorer satellites into Earth orbit (1958) and the 3-stage Saturn V launching its payload of Apollo astronauts, and orbit and landing spacecrafts to the Moon (1969).

The key time-to-invention events in this dark and sinister tragedy:

1865 - Jules Verne writes and publishes de la Terre a Lune, his vision of interplanetary space travel

1897 - Inspired by Jules Verne’s classic, HG Wells writes and publishes War of the Worlds. In the same year, also inspired by Verne, Tsiolkovsky formulates his rocket equation: u = vln(Mo/M)+uo where u = final velocity, v = velocity of exhaust gases

1903 -- Tsiolkovsky publishes Exploration of Outer Space by Means of Rocket Devices with his rocket equation being used to calculate the horizontal speed required for a minimal orbit around the Earth is 8,000 m/s (5 miles per second) and that this could be achieved by means of a multistage rocket fueled by liquid oxygen and liquid hydrogen.


17

1905 Oberth reading Jules Verne science fiction From the Earth to the Moon at 11 builds his first model rocket 3 years later at 14.

1919 - The Smithsonian published Goddard’s groundbreaking work, A Method of Reaching Extreme Altitudes. The report describes Goddard’s mathematical theories of rocket flight, including Tsiolkovsky’s rocket thrust equation, his own experiments with solid-fuel rockets, and the possibilities he saw for exploring the Earth’s atmosphere and beyond.

1920 Herman Oberth’s publishes his Die Rakete zu den Planetenraumen

1926 - Goddard launched the first liquid-fueled (gasoline and liquid oxygen) rocket on March 16, 1926, in Auburn, Massachusetts - The rocket, which was later dubbed “Nell”, rose just 41 feet during a 2.5-second flight that ended 184 feet away in Aunt Effie’s cabbage field.

1929 -- Von Braun, Oberth and their engineering team in Germany pirate Goddard’s patents and designs for a liquid fueled rocket and use them to build the Oberth-von Braun liquid fueled engine and perform a static firing of it.

1937 - Goddard’s L-13 rocket reached an altitude of 2.7 kilometers (1.7 mi; 8,900 ft), the highest of


18

any of Goddard’s rockets.

1942 German V-2 Retribution Rocket invented by Wernher von Braun -- Three features developed by Goddard for the L-13 rocket appeared in the V-2: (1) turbopumps to inject fuel into the combustion chamber; (2) gyroscopically controlled vanes in the nozzle to stabilize the rocket until external vanes in the air could do so; and (3) excess alcohol fed in around the combustion chamber walls to protect the engine walls from the combustion heat with a blanket of evaporating gas

1946 In Germany at the end of WWII the US Army seizes R-1, a German V-2 – the Soviet Army seizes R-7, another V-2

1953 R-1, renamed the Redstone ICBM, is modified to carry a nuclear warhead. In Russia the Soviet Union modifies R-7 to carry a nuclear warhead and rename it the Semyorka ICBM.

1957 Spuitnik launched atop the Semyorka ICBM

1958 - Explorer I launched atop the Redstone ICBM 1965 - von Braun supervise design and building of the 3 -stage Saturn V rocket

1969 Apollo Moon Mission launched atop a Saturn V


19

Calculations. Tables 4,5 and tell the honeymoon part of the tragedy, no pun intended. Table 4 shows an elapsed time of about 32 years separating Jules Verne’s visionary novel from Tsiolkovsky’s break through rocket thrust equation and two more years separating it from Goddard’s centrifugal force flywheel vision. Never mind the incorrect physics underlying a 16-year old boy’s vision of lift, this was a life-defining break through. Anchoring Tsiolkovsky and Goddard’s Nash incentives to invent between two objective time clocks -- 1865 for the publication date of From Earth to the Moon, and 1897 for Tsiolkovsky’s mathematical break thru -- we begin


20

back tracking the algorithm and summing over possible probabilities of success at breaking Nash equilibrium to find the average probability: 0.9827 with a standard error of +/- ..0066, a binary entropy of .1257, and a rate of change in binary entropy of -1.74.

What does this all mean?

To begin with, we have worked enough examples to see that time somehow locates the upper and lower limits on the probability of success at breaking through a state of Nash equilibrium and overcoming inertia, and this probability then somehow pinpoints the time of break through – before it happens. How all this happens is a mystery. That it must happen if probability is to play a role has been demonstrated. It doesn’t matter if before the break through, we know neither the probability of success nor the time of break through, objective facts are facts. If, in fact, Jules Verne’s novel was the visionary source inspiring Tsiolovsky and Goddard and our simple probabilistic algorithm coupling these three events is correct, the latter two events could only happen if the probabilities were in the neighborhood of .9827 +/- 0.0066

Any probability of success lower than this and the break throughs would have happened later, which they didn’t. Any probability outside this range and the events would have happened earlier, which they also didn’t. Also at .1257 the amount of Shannon uncertainty in the estimate is low enough to suggest the probability estimate may be close to the true probability of success.

How close?


21

This interesting question brings us to the meaning of -1.74, the estimate’s binary rate of entropy. One way to look at this otherwise counter intuitive rate is that it is measuring in Shannon bits the amount of information lost in moving away from the estimate in either direction by one per cent. In this case, 1.74 bits of information would be lost.

Another way to look at it is tht the binary entropy rate is measuring the speed of the invention.


22

Tables 5 and 6 are telling two different probability of success stories. Table 5 is the story of the probability of Goddard’s successful invention and launching of Nell in 1926, the first liquid-fueled rocket. Estimated to be about 0.9856 +/-0.0062, the probability has a binary entropy of 0.1086 and a rate of information loss of -1.8232. Again, never mind if Goddard’s rocket rose only 41 feet off the ground for 2.5 seconds and crash-landed in Aunt Effie’s cabbage patch. The invention was epoch-making.


23

On the other hand, Table 6 showing the Oberth-van Braun contribution tells a completely different kind of story. It documents the probability of success at pirating someone else’s intellectual property. This time anchoring the probability of success between 1926 and 1929 the probability estimate jumps to about 0.9983 +/- .0007, its binary entropy drops off to 0.0182, and the information loss climbs to -2.7645 bits of information. The elapsed time of 3.3 years between Goddard’s launching of Nell and Oberth and von Braun’s static firing of their own liquid-fueled rocket engine set a speed record no one was even aware of, with the possible exception of the pirates, for the shortest time-to-invention cycle ever. Piracy had its advantages.


24

Table 7 is simply a continuation of the story documented in Table 6, only this time confirming numerically the tragic descent of von Braun and German rocketry engineering into the Hell of Hitler’s obsessive paranoid fantasy of world domination. About 13 years separated the static firing of the Oberth-von Braun rocket engine from the pair’s pirating of Goddard’s L-13 component designs to build the V-2 Retribution rocket in 1942. The probability of a successful invention: 0.9932+/-.0025; the estimate’s binary entropy: 0.0589; the information loss rate: -2.1571.


25

Tables 8 through 11 document von Braun’s re-invention of himself in the U.S. under the watchful eye of the State Department. The time-to-invention cycles documented here reveal how close humanity came to extinction in World War II. The German V-2 represented the delivery mechanism for a nuclear warhead Hitler desperately wanted German scientists to invent. The nightmare never happened in Germany. Instead, it happened in the U.S. Von Braun and his German engineering staff took 6 years to build the Redstone ICBM with its nuclear warhead. The probability of a successful invention: 0.9942; its binary entropy: 0.0518; the rate of information loss: -2.2258 bits of information per one percent change in the probability.


26

Nothing is ever free. That includes the liftoff of a Saturn V in 1969 (documented in Tables 10 and 11) with its payload of Apollo astronauts and their Moon-orbiting and -landing spacecraft. How would one calculate the cost in terms of human suffering for this liftoff and landing of an astronaut on the Moon? Knowing the cost, was it worth it? (to be continued)

G. Moore8/21/2015


27

http://bluelogic.us/

crash-landing in aunt effie’s cabbage patch: calculating the probability, speed and timing of the...

Documents