Solving the Riddle of Zeno's Paradox

The ancient Greek Zeno posed a number of paradoxes on motion. In the dichotomy or the racetrack, Zeno argued that a runner will never reach the stationary goal line on a straight racetrack. The reason is that the runner must first reach half the distance to the goal, but when there he must still cross half the remaining distance to the goal, but having done that the runner must cover half of the new remainder, and so on. If the goal is one meter away, the runner must cover a distance of 1/2 meter, then 1/4 meter, then 1/8 meter, and so on ad infinitum. The runner cannot reach the final goal, says Zeno. There is also another version which states the inverse, namely that the runner cannot even take his first step because the first step can be infinitely divided up. 

Aristotle, in Physics, said of the Dichotomy that it is possible for a runner to come in contact with a potentially infinite number of things in a finite time provided the time intervals become shorter and shorter. Aristotle said Zeno assumed this was impossible, and that is one of his errors in the Dichotomy. However, it was not until the invention of calculus that a mathematical theory, which enabled the computation of the finite amount of time, could provide the detailed solution and it is calculus that provides the detailed Standard Solution to Zeno’s Paradox today. The sum of the series of path lengths or segments ½+¼+⅛+.... converges to 1 and not infinity. 

A far simpler solution could in fact have resolved the Paradox even without calculus. The runner is also composed of infinite units, so infinity divided by infinity equals 1. This means the runner does not have infinite space to run but has absolute space of equal parts in the same proportion to the runner’s sprint that sums to 1. At each point that his foot touches the ground, the same infinite units that Zeno recognises in the ground exist in the athlete’s foot and therefore the distance to be covered at each step is no longer Zeno’s infinitesimal decimal units but one unit. This allows the runner to complete the distance to the finish line within a certain amount of time depending on his sprint. Infinities are cancelled out by the existence of infinities within both the runner and the track. 

Infinity is the background noise to the universe, it is the commonality for everything. Zeno went wrong because he attributed infinitesimal qualities to only one part of matter and space and then set up a false Dichotomy. Because it is a commonality, the infinite quality of matter can be ignored, and what remains is the proportional relationship between objects and between objects and space i.e. ratios.

So the answer to Zeno’s Paradox is that infinity also lies within the athlete's foot. 

Some Christmas Reading

 

Electricity in the USA is already half the price it is in the EU. Therefore, by 2050 in the EU, it is highly probable we just won’t have any viable jobs and disposable income left to spend, as it will have all gone to the energy bill to eliminate the fossil fuels currently forming 72% of the energy mix. 

Charles MacKay in his 1841 book ‘Extraordinary Popular Delusions and the Madness of Crowds’ pointed out: “Men, it has been well said, think in herds; it will be seen that they go mad in herds, while they only recover their senses slowly, one by one”. How on earth did we get to this situation, in a so called intelligent modern society? Were laws broken? Is there something wrong with the weather or do we just have a political crisis with respect to incompetent management? If for example you are paying serious amount of hard earned money for pollution, which is not occurring, can you get your money back or is there going to be even more of the same? [Pat Swords - 2020]

Pat Swords has written a new book on energy and other related matters which can be downloaded for free here :

 The Polluter Pays, but to Whom, How Much and On What Basis – Science or the Cult of Witchcraft ? 

It takes a look at the EU's mad panic to reduce carbon emissions and the negative consequences of doing so without conducting a proper assessment. 

I have also written a book but not energy related, about motion and perception, which includes a collection of articles on Einstein, Descartes, Newton, Empiricism, Mathematics, Ancient Greeks and Galileo and attempts to solve some of the paradoxes of motion and perception that they presented. It will also be free and if you want a copy send me an email. 

The Secret to Kepler's Harmony of the Planets

This article is somewhat of a departure from the usual article on this blog, anyone interested in physics, astronomy or mathematics might find it interesting. 

by Owen Martin

The solar system is based around circular motion , but as I will now demonstrate, the mechanism by which it's bodies move in relation to the Sun is based on the triangle. Astronomer and mathematician, Johannes Kepler, stumbled upon the idea that a trianglular mechanism governed how the planets moved in 1595 while in the middle of a lecture on the conjunctions of Saturn and Jupiter. Each set of three points he wrote on the blackboard corresponded to a conjunction and formed almost  precise equilateral triangles. These triangles rotated around the board to form two circles, with the radius of one half that of the other one, approximating to the distances of Jupiter and Saturn. 

Thus, a triangular pattern determined the distances between two planets moving with a circular motion. This may have something to do with the fact that a circle is constructed using three points just like a triangle. He was then inspired by this discovery to attempt to fit the five Platonic polyhedra solids into an arrangement that corresponded to the known distances at the time between the six known planets. His completed model, published in Mysterium Cosmographicum, was magnificent but a little bit on the complex side (and proven to be a failure over time when greater measurements were found). He actually missed out on something much simpler and even more remarkable. He could have realised this after 1619, when he discovered his third law, published in Harmonices Mundi, the Harmony of the Planets,  but at that stage his focus had shifted to the musical harmony of the planets. 

Keplers Third Law, arguably his greatest discovery, states that the ratio of the squares of the orbital periods of two planets is proportional to the ratio of the cubes of their average distances from the Sun. This applies to all bodies in the solar system, including the moons on Jupiter. It became an accepted law of physics and Isaac Newton later used it to derive his laws of gravitation. It means that the planets do not move in an arbitrary way, but rather according to a finely tuned mathematical formula. 

D³ 𝞪 T²

So if the distance of a planet from the Sun is doubled,

2³ = T²

T² = 8

T = √ 8

The change in the orbital period is the square root of eight or 2.82. The planet will take 2.82 times longer to travel around the Sun.

Now we can work out the change in velocity. Remember the formula, speed equals distance over time from school ?


V = D / T

V = 2 / √ 8

V = 2 / 2.82

V = 0.707

This means the velocity of planet B will be around 70% that of planet A, if B's distance is double that of A's from the sun. In the solar system, Mars is approximately double the distance of Venus, and Uranus is double the distance of Saturn from the Sun and we see that their respective orbital periods and velocities differ by the above ratios.


So, to sum up when:

Distance = x 2

Orbital Period = √ 8

Velocity = 0.707 


The great beauty of these numbers is that they are represented perfectly by Pythagoras's famous right angle Triangle. Once again, remember from school, the square on the hypotenuse is equal to the sum of the squares on the other two sides. If the two sides are both two, then the longer side (hypotenuse) is equal to the square root of  2² + 2², i.e. √ 8, thus representing the change in orbital period.

The connecting line between the 90 degree angle and the centre of the hypotenuse is equal to half of √ 8 or 1.41.

1.41 divided by the side representing distance 2, equals 0.707. Thus this centre line represents the change in velocity.

Also, interesting to note that Sine 45 degrees also gives 0.707 ( sine equals the opposite line over the longer line, the hypotenuse).

So, the solar system is not based on a complex polyhedral system as Kepler believed, but is actually based around the much simpler Triangle. The secret to the Harmony of the Planets and the Mysterium Cosmographicum lies not with Plato and his solids but with the first known Greek mathematician and philosopher, Pythagoras of Samos, and his right angled triangle.

This means that the mechanics of celestial motion has been fine-tuned in accordance with the dimensions of a right angled triangle. The builders of the pyramids (which Pythagoras may have visited) would surely have been satisfied.

Astronomy Ireland Lecture: The Search for Other Life in the Universe and Why Fermi asked the wrong question

A bit of a diversion on Irish Energy Blog to a physics/astronomy topic that's quite popular now - Exoplanets and the search for ET life. Astronomy Ireland held an excellent lecture in Trinity College last night which was given by Dr Duncan Forgan. Its quite exciting to hear about the new telescopes that will be launched in the coming years as successors to Hubble and the new techniques that are been used to find life in the universe. However, the question as to why we still haven't made contact with life out there is still puzzling physicists.

It was Italian physicist Enrico Fermi who first put the search for life into context by asking the simple question "Where is Everybody?". If there are billions of stars in the galaxy, and some of these stars are billions of years older than the Sun, then alien life should have arrived at Earth by now, given that it should only take them about 10 million years to travel across the galaxy.  This became known as Fermi's Paradox.

Why Fermi asked the wrong question

There are various theories which attempt to answer Fermi's Paradox which Dr Forgan explained but it seems to me that a different question needs to be asked.

If we consider how our own relatively advanced civilization came about, it was the discovery of fossil fuels that allowed us to "get off the land" and "into space". Fossil fuels had such high Energy Return on Investment (EROI), due to the energy intensity contained in them, that they allowed humans to get relatively easy access to huge levels of energy enabling us to move away from a largely agricultural based society and build the modern society which you see around you and eventually rockets and spacecraft. The problem is that the Fossil Fuel era is only a short blip in the Earth's history and indeed mankind's lifespan. The only reliable alternative to fossil fuels is nuclear power and indeed it even has a higher EROI than fossil fuels (see here). However, discovering nuclear power is by no means a certainty for any intelligent life (as opposed to discovering coal or gas) but we did it - and guess who was the architect behind it, none other than Enrico Fermi (pictured above). Note that the ESA used solar power to power their spacecraft to nearby comets but NASA used nuclear power to send spacecraft to Pluto and beyond the Solar System. When fossil fuels eventually run out (or become too uneconomical to drill for) nuclear power will be the main energy source for future civilizations until they can come up with another alternative when that runs out. But that is by no means certain either and who knows, the human race may eventually return to the pre-industrial age. Should this happen the modern advanced civilization that we know (and will know) will be simply a blip in man's history. And a very tiny footnote in Earth's history.

So where does this leave us ? Well, given the number of stars and planets in the universe, its an impossibility that we are alone as Fermi pointed out. But communicating to and receiving communication from other life out there depends completely on the stage of evolution of the other life, which depends on the quantity of fossil fuels available to them, and indeed whether they have discovered nuclear power or something similar that can allow their advanced civilization survive after fossil fuel stocks have been depleted on their own planet. This period of advanced alien life is likely to be just a blip on most of the habitable planets in our galaxy. Perhaps only a tiny few can extend well beyond their "fossil fuel age", if any at all.

So this then narrows down the number of habitable planets to those planets that currently hold advanced intelligent life. There may have been million of these planets in the past and there may be many more million in the future long after the human race have returned to the pre-industrial age or indeed become extinct. But they need to be on the same timeframe as us for us to make contact with them or vice versa. Imagine an alien life sending radio signals during the Renaissance - they would have unfortunately fallen on dead ears. "Sorry, I missed your call" on an intergalactic scale !

There is also the added problem of the time difference given the huge distances between us and habitable planets. I think a distance of 1 million light years equates to 1 year time difference. So, say an alien life is sending radio signals at a distance of 100 million light years away. This would reach us 100 years later at which point they may be unable to receive a return signal from us which would reach them another 100 years hence.

Therefore, the question we should be asking is not "Where is Everybody?" but "When is Everybody?"