What an Engineer thinks of Zeno's paradoxes.

I have at some or other times thought about Zeno's paradoxes.

Here is what I have come up with lately [1] / [2] . There are some possible answers to Zeno's paradoxes

in the Google world, but they don't sound entirely convincing to me. So I decided to roll my own.

Most of the material for the following discussion is in the foot-notes which cover about 2/3 of the text.

Please look there if you want to get some of the finer points. Zeno's paradoxes touch some of the most

delicate foundations of human thinking, and cannot be dealt with in a curt manner.

Even if I approach the matter with a certain style of robustness, I have to make provision that there

exists some philosophical balance. Because I am also a philosopher, not only an engineer.

For the engineer, [3] Zeno's paradoxes are a prime example of a certain style of thinking

that the ancient Greek philosophers as a group excelled in:

Extremely tricky and brilliant in the details, but entirely impractical. [4]

There were many accomplished practical engineers in ancient Greece.

but beyond the engineers there were only philosophers that are well-known today. [5] / [6] / [7] / [8] / [9]

This ancient thinking style is also called confusionism, or illusionism, or Hegelism. [10]

Zeno's problem was the idea of continuum in the Universe

The underlying natural-philosophical problem of Zeno was the idea of the continuum.

Continuum is just an idea, with nothing corresponding to it in the Real Universe.

To add some more confusion, the mathematicians called Continuum Numbers the Real Numbers.

Zeno's paradoxes are based on these assumptions:

1) that the Real Universe is a continuum.

3) that movement takes place in the above continuum of time and place.

Here we deal with Zeno's paradoxes on the level of engineering and science. What all the scientific

exploration in the last 400 years or so has shown, is that there is no such thing as a continuum.

Everything in the Real Universe comes in quanta, even time. [11]

But let us note that these quanta are not the quanta of "quantum theory" . [12]

A more common physical term to use would be the word "particle" , but there is no particle of time.

the quantum of light is the photon. [13] / [14]

the quantum of electricity is the electron.

the quantum of time is the clock-tick. [15]

the quantum of movement or (loco-)motion is the diameter of an atom.

There still exist some incompatibilities: [16] / [17]

A short excursion into illusionism in mathematics

Since the modern science of mathematics is based on the ancient Greek work, we can find many examples

of illusionism in mathematics. Real numbers of the irrational kind exist only as a concept. [18]

A good example is Pi. [19] Some more examples are in the footnotes: [20] / [21]

The calculus works with this illusionist assumption: That one can go down to "infinitesimal" ,

and then go up to "infinite" . The reasoning is, that infinite cancels out infinitesimal, [22]

and so, the calculus works and gives practical results.

The engineer has the following calculations to make

Let us start with Achilles and the tortoise:

This shall be a real assignment for an engineer, and one does best to disregard all the above

philosophical logisticisms, and especially Zeno's own misleading prescriptions.

One must make certain assumptions and base some calculations on these.

First, we must set a certain distance of the race course. The curse of the modern metric system of the

French Revolution is that it correlates to nothing of the human proportionality, nor the human ability,

which was the base of all the ancient measuring systems. [23]

The modern metric system is a procrustean [24] measure of the kind: One size fits no-one.

This makes it very difficult to visualize an actual travelling or traversing process,

for which we are seeking an engineering solution in a computer simulation.

The best [25] standard to use for a practical evaluation of the Zeno paradox is the Roman army standard.

So we set the distance of the race course at the Roman army standard: [26] 100 strides of human legs. [27]

The ancient measure of stride is conserved in the roman stride: [28]

gradus = single stride = 2½ pes = 740.880 mm = 0,740880 m = 74,0880 cm.

passus = double stride = 2 gradi = 1,48176 m = 5 feet

The most widely used ancient distance constant was the Roman mile. This is the constant distance which

a Roman legion traversed with 1000 double strides or: mille passuum [29]

mille passus = mile = 1000 passus = 1,48176 km = 5000 feet.

The milia passuum is a very constant measure, and it is recorded that all the Roman legions

traversed exactly this distance with 1000 passuus (or double strides), when travelling on even ground. [30]

This was so accurate, that all the geography of the Roman empire depended on it. Some inaccuracies are

introduced by the factors of terrain and weather: [31] When the march goes steeply uphill, the stride will

shorten. But to compensate, when it goes downhill, the stride will lengthen.

The Romans tended to build straight roads, probably to ease their distance computations. [32]

But in ordinary terrain, roads are winding more or less, so on a winding road, the real distance of x miles

travelled, computes to somewhat more than the absolute distance of two points on the map. [33]

The latter is also called the bird-flight or straight distance. [34] From the Roman measure derives the english

or american mile of today. [35] The Roman legion marching speed was variable, depending on terrain and

urgency, from 3 to 4 miles per hour. [36]

3 miles per hour computes to 4,8280 km/h. [37]

4 miles per hour computes to 6,4373 km/h. [38]

For the example we count here in single strides (gradi).

100 strides * 74,0880 cm [39] results to rounded: 74 meters.

This is the distance of the race course that both Achilles and the Tortoise must traverse.

Now come some more running speed/time/distance calculations: [40]

Lets assume that Achilles doesn't really run but takes the accelerated Roman marching speed of about: [41]

6 km/h. See the speed calculations. [42] This results to:

6 km/h = 6.000 m/h = 100 m/min = 1,6666 meters/sec

We also need to know how many human strides (of 74,0880 cm) Achilles makes per sec.

This is about: 2,2521 human strides per sec. The time needed to traverse 100 strides or 74,0880 m

at 6 km/h is 44.4 sec. Also the time step or clock needs to be quantified. One second is an arbitrary

measure, a better one would be the average human heartbeat when running leisurely.

But we don't need such precision, and we stay with the Second.

Now let us assume that the tortoise has a starting advance of 37.04 meters or 50 human strides,

which is half the running distance. Then let us assume that the tortoise has a speed of: [43]

0,6 km/h = 0,1666 meters/sec = 32,8083 feet/minute

We need to take into account that a tortoise has a different stride than a human.

Let us assume the tortoise stride is 7,40880 cm.

Now we can set up a time/distance computing program. This is can for example be done with two

concurrent processes, one for Achilles, and one for the tortoise. At each clock count, each process

communicates with the other which absolute distance from the start both have travelled.

We can also make a graph, exactly determining the position of Achilles at each clock second.

Likewise we can make a graph, exactly determining the position of tortoise at each clock second.

To write a program for this, I leave this as an exercise to computer science students.

I will not go into more programming details, but it is clear, that in 1 clock second:

The tortoise has traversed: 0.1666 meters

To traverse the 37 meters of tortoise's starting advance, Achilles needs 22.20 secs. But since we have a

computer simulation stepping with a fixed rate, we have to look at clock cycle 23 secs.

This makes 38,3 m distance traversed. [44]

At clock cycle 24 secs Achilles is at 40 m from start.

At clock cycle 24 secs the tortoise is at 37 m + 4 m = 41 m.

At clock cycle 25 secs Achilles is at 41,6 m.

At clock cycle 25 secs the tortoise is at 37 m + 4,1 m = 41,1 m.

To visualize, we can take both time/distance graphs, overlay them and see immediately, when Achilles

has overtaken the tortoise. No matter what Zeno thinks of the matter. Achilles will have overpassed the

tortoise at a very fixed point of distance and clock time. This is given by natural logic, and can be shown

with the diagram. Because both Achilles and the tortoise proceed by strides, not by infinitesimal

The problem is the wrong application of mathematics, and Zeno's impossible prescription. If you make an

impossible prescription, the outcome is also impossible. This is the GIGO [46] principle of computing.

When we make an infinitesimal approximation, as Zeno does, it is computationally very difficult to

calculate the exact split second when Achilles will have reached the tortoise.

This correlates to the level of precision that we want to achieve.

This is a quite well known computing problem, since computers only simulate real numbers with floating

point arithmetic, and they can never go to infinitesimal, but they must iterate or recurse n times. [47]

This must stop at some level n because there is only a limited amount of RAM or disk space

available to keep track of the process. Thus, it is quite easy to calculate the exact clock step at which

Achilles will have surpassed the tortoise. When one applies some basic engineering and computer

knowledge, Zeno's paradox dissolves into nothing.

To repeat the reasoning: it makes the matter more transparent when we consider that both

partners move in strides, which means fixed distance quanta that cannot be subdivided into

A human stride is 74,0880 cm, and the running distance is 74,0880 meters or 100 strides.

So the winner is clear. Believe Zeno or not. It is the computer simulation.

The Arrow Paradox: Mass, Size, Speed, Time and Brownian Motion

What Zeno didn't know, is that the speed of an object in the mesocosmic [48] world

of our daily experience is something entirely different from the microcosmic world.

As said above, there is no measurable movement on the subatomic level, [49]

everything in the mesocosmic world must move by n times the quantum of movement,

So, it is actually quite difficult to figure exactly how an arrow moves. For the engineer, we have the next

best explanation: An arrow moves just like a caterpillar on a quantum level at quantum speeds.

This is also known as longitudinal oscillation. [50] It means: By each universal clock-tick, the arrow

contracts for some nano- or pico- or femto- meter in some nano- or pico- or femto- seconds,

and then it expands again. When it has expanded, it anchors its tip at the new position and pulls up its tail

by some nano- or pico- or femto- xy- meters. [51] Since we cannot see nor measure at the quantum level,

the exact nature of the movement of an arrow remains a mystery. In this, Zeno is right.

We still don't really know what (loco-)motion is. [52]

It is interesting to note that we can observe the transversal oscillation of an arrow with a high speed

camera. The arrow bends up and down and sidewise in a sine-wave pattern. [53] This is quite amazing, since

one ordinarily thinks that the arrow moves stiff and straight to its target. In Real nature, there is never

anything stiff and straight like in mathematics or logics. It all comes in oscillations or resonances.

Now some additional reasonings. An arrow is a momentum machine. Speed or movement in air is a

function of the momentum of an object restricted by the resistance of the surrounding medium, which is

air in this case. In water, the resistance is so much higher, and there torpedoes are the weapon of choice.

The power of an arrow results from its relatively high mass, as compared to its diameter. This is also the

reason for its transversal oscillation, since its mass center moves around outside the straight line of a

The increase in length of factor x results in an increase of Mass as x**3. Mass and momentum [54] are

proportional. The diameter parameter is the prime factor of air resistance. The less diameter, the less air

resistance. Obvious. On the downside, a decrease in length of factor x results in a decrease of mass as

x**-3. If you decrease length by about x= (5-10) factors of magnitude, mass decreases at x**-(5-10).

Which comes to almost nothing. (Infinitesimal).

Let us assume for simplicity that an arrow traverses 60 m in 1 sec. [55]

This results to: [56] 60 m/sec = 216 km/h

The Zeno Arrow Paradox constructs an infinite regress, reducing length and traverse to infinitesimal.

The catch is the same as with the Achilles paradoxon. If the arrow has a length of 1 m, this means its

stride is 1 m. [57] But this is more difficult to visualize, because the arrow is at point x by some clock n, and

at point x+1 m by some clock n + 1/100 sec. If we set the clock cycle to 1/1000 sec, the arrow has then

moved by 10 cm in each cycle. We can set the clock cycle to any 10**x fraction, the arrow always moves

by 10**-x m. So its stride still remains 1 m per 1 second clock.

Infinitesimal is nice in mathematics, but not in Reality, and also not on a computer. In the Real Universe,

there is no infinitesimal entity. [58] At some deeper microscopic nanometer level, all resolves to quantum

mechanics. At the microscopic nanometer level, any particle that is suspended in air, cannot move as it

will, since its motion is inhibited by a phenomenon called Brownian Motion. It has so many collisions

with air molecules of the same mass, that it cannot move at all. So it will not move anywhere. Thus, an

arrow reduced to the length of an air molecule cannot move at all.

This is the dis-(ill- /sol-)ution of the Zeno Arrow paradox. Zeno has cleverly constructed an illusion for

us, and the world has fallen prey to this for almost 2300 years now.

(URL) http://www.google.de/#hl=d......sclient=psy-ab&q=Zeno+pa,or.r_gc.r_pw.r_qf.&fp=273aa0d90065244e&biw=1124&bih=715

(URL) http://www.learner.org/courses/mathilluminated/.../04.php

(URL) http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

(URL) http://de.wikipedia.org/wi......illes_und_die_Schildkr%C3%B6

(URL) http://plato.stanford.edu/entries/paradox-zeno/

(URL) http://mathworld.wolfram.com/ZenosParadoxes.html

This gives also one of the shortest refutation of Zeno's paradoxes. Quoted from the above:

The dichotomy paradox leads to the following mathematical joke.

A mathematician, a physicist and an engineer were asked to answer the following question.

A group of boys are lined up on one wall of a dance hall,

and an equal number of girls are lined up on the opposite wall.

Both groups are then instructed to advance toward each other

by one quarter the distance separating them every ten seconds

(i.e., if they are distance d apart at time 0, they are d/2 at t=10,

When do they meet at the center of the dance hall?

The mathematician said they would never actually meet because the series is infinite.

The physicist said they would meet when time equals infinity.

they would be close enough for all practical purposes.

[2] (URL) http://www.stamoulis.de/index_de.html

[3] Since I am equally well versed in Software Engineering and Philosophy and Etymology,

I can (if I will) take the hat of a philosopher, and like a a philosopher, I can think in/of an imaginary world

of infinite impossibilities. That is nice, and it puts me in the same rank with a theologian, if I should want

to. A theologian can think of God anything s/he wants to, because God is beyond any limits. But most of

the times, I prefer my other hat: That of the (computer-) engineer. With this hat, I impose on myself some

thinking restrictions: I then can think only in the limits of a Real Universe of Finite Actualities.

And this is the base of the present discussion.

[4] The reason why ancient Greek philosophers were not too concerned with practical matters is that manual

work was considered not befitting to a Greek Gentleman.

(This reminds us vividly of present-day Greek problems.)

Manual work was reserved for slaves. Therefore, the ancient Greek philosophers didn't learn the practical

experiences that one can only gain by dirtying one's hands with solid, messy, dirty, and sturdy matter.

Aristoteles was the exception, because in his profession as a doctor, he had to deal with messy matters all

the time, especially blood and guts, sperm, saliva, and other such un-nicieties.

Platon was the exact opposite. He was an aristocrat of the old classes, and he had probably never touched

any messy base matter in his life. It is known that he had practically no sexual life, and so he didn't have

to deal with the messy and dirty aspects of this part of human existence.

For these and other reasons, Platon's philosophy was useless and sterile for any practical matters.

Unfortunately, philosophers down the ages had overlooked this glaring problem of Platon's philosophy.

Philosophy as a grand human adventure is therefore eternally confused by the god-like enshrinement of

the Great Grand Master Platon. But the ancient Greek philosophers were only a small group

even though they were disproportionally influential on the thought of humanity.

This is because of the peculiarities of the literature traditon of the christian monks who propagated these

works. If there existed any technical writings by ancient Greek engineers, the medieval monks couldn't

understand these, and they couldn't copy them, and so these were lost to posterity.

Vitruvius is one of the few whose technical works were preserved. But without the technical diagrams.

Plinius (Pliny) the elder was also one writer who had technical skill.

(URL) http://ancienthistory.about.com/.../a/PlinyPompeii.htm

(URL) http://en.wikipedia.org/wiki/Pliny_the_Elder

His work was one of the few preserved down the ages. Unfortunately without the drawings.

Greek civilization could not have survived, if it had had no skilled engineers. There were many very

practical ancient Greek engineers and technicians, who excelled in productions for the Greek war

machine. (See Alexander's conquests) Of these, not much is recorded in the history books.

(URL) http://www.google.de/searc......suvius+pompei&hl=de&

[5] The best known of these was Archimedes, then comes closely Heron of Alexandria.

Unfortunately, they didn't succeed to start the industrial revolution 2300 years ago.

There was no need for technical improvement when slave labor was abundant.

(URL) (URL) http://www.google.de/searc......rew+archimedes&hl=de&

(URL) (URL) http://en.wikipedia.org/wiki/Archimedes

(URL) (URL) http://de.wikipedia.org/wiki/Archimedes

Another good example: The engineers who built the Antikythera, 2300 years ago, were very accomplished

practical engineers. They invented something, that after these 2300 years, one would call a computer.

(URL) http://de.wikipedia.org/wiki/Mechanismus_von_Antikythera

(URL) http://en.wikipedia.org/wiki/Antikythera

(URL) http://www.spektrum.de/alias/technikgeschichte/.../1065416

(URL) http://www.spiegel.de/wissenschaft/...

But the technical people had a lesser social status because they had to work with their hands.

[7] Bronze casting in ancient Greece 2300 years ago was at a technical level that was probably never

surpassed anywhere and anytime later. A possible exception were the cannons of Urban or Orban,

but only for sheer mass, not for intricacy. (See: the conquest of Byzantium).

(URL) http://en.wikipedia.org/wiki/Fall_of_Constantinople

(URL) http://en.wikipedia.org/wiki/Dardanelles_Gun

(URL) http://en.wikipedia.org/wiki/Byzantine%E2%80%93Ottoman_Wars

The unsurpassed examples of the ancient skill are the Riace bronzes, also called Riace Warriors.

(URL) http://en.wikipedia.org/wiki/Riace_bronzes

They have survived the ravages of a lesser medieval humanity because they had been

Most of the other ancient greek bronze statues had been molten down for their metal.

There is one more exception from history:

The horses of Venice, stolen from Byzantion in the raid of 1204.

(URL) http://en.wikipedia.org/wiki/Horses_of_Saint_Mark

In Germany, we have the Bavaria Statue in Munich,

which seems to be the only specimen that comes close to the ancient Greek technology.

(URL) http://en.wikipedia.org/wiki/Bavaria_statue

The statue of Liberty in New York cannot compete at all with this,

for technical reasons that would take another book to explain.

(URL) http://en.wikipedia.org/wiki/Statue_of_Liberty

I am an engineer, and I know this stuff. You can believe me or not.

[8] Also well known in mathematics is Euklid for his foundation of geometry, but his theorems are entirely

impractical. He had just re-formulated the ancient knowledge of the Egyptian engineers so that absolutely

no-one could understand it. This could be called a master-piece of reverse-engineering.

[9] One more paradoxical Gedankenexperiment: If one would accelerate some atomic matter to the speed of

light, which is impossible, but only here for us to think of: Then this matter would have infinitesimal

diameter, and also infinitesimal mass. But, because of its momentum, it would have infinite mass.

And therefore I believe that Einstein didn't understand some of the very basics of this universe.

Most of the rest of his followers didn't understand anything more nor else.

And for this, there are so many Nobel Prices awarded. What a waste.

[10] This is for Hegel's (in)famous exclamation:

"If the facts contradict the theory, then this is too bad for the facts"

[11] (URL) http://en.wikipedia.org/wiki/Second

A good example is that with every advance in measuring technology, the clock time step becomes

shorter. But it never beomes infinitesimal.

The present scientifically established clock tick is the oscillation of a caesium atom. See:

(URL) http://en.wikipedia.org/wiki/Atomic_clock

A good discussion is in Prof. Peter Janich's works about the inter-connection of technology and theory.

But there still remains a time step. Believe Einstein or not. There is no such thing as a continuum.

Especially not in time. There is always some illusionism, even in science:

(URL) http://www.spiegel.de/wissenschaft/...

[12] Actually, this is called "quantum mechanics". But this is another discussion and would lead too far from

[13] Here comes in quantum theory, because light can also be considered

(URL) http://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality

(URL) http://www.spectroscopyonline.com/...

Zentimeter cm 10-2 10 mm Veraltet (um 1900): Centimeter

Millimeter mm 10-3 1.000 µm 10 Millimeter sind 1 Zentimeter. Entspricht einem

Mikrometer µm 10-6 0,001 mm Veraltete Bezeichnung: Mikron, im technischen

Sprachgebrauch auch als µ (sprich mü) bekannt.

Nanometer nm 10-9 1000 pm Entspricht einem Milliardstel Meter (einem Millionstel

Ångström Å 10-10 100 pm gebräuchlich in der Atomphysik und in der Kristallographie

Pikometer pm 10-12 0,001 nm Entspricht einem Billionstel Meter (einem Milliardstel Millimeter).

[14] To show where the conceptions of motion are going wrong, we quote from the above article.

In the Real Universe out there, NOTHING moves in a straight line. It all moves in ellipsoid spiralloids.

It should be obvious that matter — anything that has mass and takes up space — is ultimately particulate. In fact,

matter defines what it is to be a particle. Particles come in pieces that can be broken into smaller pieces, and

smaller pieces can be fused together to make larger pieces. No two pieces can occupy the same exact point in

space, a concept that applies at the atomic and molecular level for matter that is a solution, a liquid, or a gas. Matter

in motion tends to move in a straight line unless some force acts on it, and matter in motion tends to follow

Newton's laws of motion (which can be reformulated into Lagrange's or Hamilton's laws of motion). Moving matter

can be blocked by the presence of other matter, much like an umbrella blocks the paths of raindrops falling on it.

Moving matter also can bounce off of surfaces, as demonstrated by many a basketball player.

[15] Actually, time is another illusionist thing. This is because time is a measure of changes that can be

observed, versus observed quasi-universal recurrences and quasi-universal constants. The day and the year

are such quasi-universal recurrences, and the star constellations of the night sky are quasi-universal

constants, because they haven't changed perceptibly in many millennia, too slow for human observation.

But, if nothing changes, no difference can be observed, and there might as well have been no time

involved. Since in the Real Universe, things are changing all the time, humans got the impression, that

there is something like time, and they simply invented the concept of time. Because modern

instrumentation can detect differences at an ever finer scale, therefore the clock-tick becomes ever finer.

(URL) http://en.wikipedia.org/wiki/Detection_limit

[16] Movement or (loco-)motion can only be observed / measured down to the level of atoms. So the lowest

delimiter of movement of matter is the diameter of an atom. There is no such thing as infinitesimal

[17] Of course, there is movement at the subatomic level, of all kinds of xyz-ons, like phot-ons or electr-ons

or neutr-ons or neutrino's. But it cannot be measured, and the only practical use is in particle accelerators

(like Cern), but not in everyday households. By relativity, the faster they move, the less diameter they

have, and at the speed of light, they have infinitesimal diameter. Therefore, what works for photons,

doesn't work for atomic matter. This is a different ballgame, as one would say colloquially.

It is impossible to accelerate atomic matter to the speed of light.

(URL) http://en.wikipedia.org/wiki/Real_number

Here we are focussing on the irrational numbers among the Real Numbers. There are no irrational

numbers found in Reality (meaning in the Real Universe out there). The simple reason why an irrational

number is unreal (or illusionistic), is: An irrational number has an infinite number of non-repeating digits

behind the decimal point, or comma (german vs. english notation). This means, if we wanted a Real

representation of this number, like as a computer value, or on a piece of paper, we would have the

problem that the universe does not hold enough atoms to represent it. This means, only finite

approximations to irrational Real Numbers can exist. The practical limit of approximation is the largest

number of bytes by which a computer stores its floating point numbers.

[19] Let us take Pi, the best known irrational Real Number.

(URL) http://www.wolfram.com/mathematica/...

(URL) http://www.google.de/searc......+circle&hl=de&prmd=i

Pi is defined as the relation of the circumference of a perfect circle to its radius.

This is mathematically nice, but unfortunately impossible in practice. The first problem is that there exist

no perfect circles in the Universe. The ancient Greek philosophers preferred to think in/of perfect circles

for their aesthetic reasons. But this caused serious problems, for example in cosmology. In the "Real

Universe out there", everything spins around something, because of Gravitation. And the spinning

movements are not in circles, but in ellipsoids and other curvoids, mostly spiralloids.

For example: Planets travel around their home suns in ellipsoid spiralloids. Also their home suns move in

their galaxies in some spiralloid fashion, and their galaxy centers move also from somewhere to

somewhere. So far, science has established that everything moves from somewhere to something.

Then, the earth is not a perfect sphere, and it turns around at about:

1,670 km per hour at the equator, which makes 0.46 km per sec.

(URL) http://geography.about.com/od/learnabouttheearth/...

(URL) http://geography.about.com/library/faq/blqzearthspin.htm

So, when you start drawing a circle on a presumably solid earth ground, this will not be a perfect circle

when you have finished. Assuming that it took you 1 sec for the drawing, the end of the circle is at

0.46 km distance from the start, if you were at the equator. By the time you end drawing your circle,

your planet has moved so much in a spiralloid manner. First by rotating around itself, then by travelling

See: (URL) http://curious.astro.cornell.edu/question.php?number=356

This is at about 100,000 km/h, making it 27,777... km/sec.

And then, our own sun has travelled from somewhere in the galaxis to somewhere else.

Therefore, Pi, although mathematically nice, does not exist in this Real Universe.

(URL) http://en.wikipedia.org/wiki/Circle

(URL) http://en.wikipedia.org/wiki/Circumference

(URL) http://www.mathopenref.com/radius.html

(URL) http://www.mathgoodies.com/lessons/vol2/circumference.html

[20] That an irrational Real Number has no correspondent in the Real Universe, is a matter of clumsy

naming only, and one can still work with the concept. But this style of mis-naming is one reason why so

many normal people find it hard to work with higher mathematics.

Another good example of illusionism in mathematics is the name of "imaginary numbers". Irrational Real

numbers are as imaginary as imaginary numbers. That naming scheme doesn't distinguish them in any

meaningful sense. Of course this doesn't inhibit a trained mathematician to work with them successfully.

The problem is again that common-sense lay people tend to get confused.

[21] And there are many more counter-intuitive terms in the sciences, like in physics. The term "energy" is

entirely misleading ( for common-sense lay people), because en-ergia in ancient Greek means "the

potential for work". But this is called "free energy" in physics.

The term "entropy" poses a similar problem, because en-tropia in ancient Greek means "the potential for

change". In physics, it is defined as exactly the opposite. The wording: "unavailable for doing useful

work" makes it sound like it comes from a lawyer. See this quote:

(URL) http://en.wikipedia.org/wiki/Entropy

"Entropy is a thermodynamic property that is the measure of a system’s thermal energy per unit temperature that is unavailable for doing useful work."

(URL) http://en.wiktionary.org/wiki/entropy

[22] "Illusionist" means here that one makes an impossible assumption, but somehow gets away with it.

(URL) http://www.learner.org/courses/mathilluminated/.../04.php

The idea of a limiting value to an infinite process is at the heart of calculus.

So, infinity became a tool that could be used, as long as one didn't look too closely at exactly how it worked.

Mathematicians came to accept that one could indeed have a finite limit to an infinite sum. This concept made it

possible to arrive at a finite magnitude by summing an infinite number of infinitely small pieces. Such pieces, which

became known as infinitesimals, have, in some disturbingly vague sense, arbitrarily small but non-zero magnitudes.

The great Newton, one of the fathers of the calculus, the revolutionary new theory of the 1600s that described

motion both in the heavens and on Earth, at first based his ideas on these troublesome infinitesimals. It wasn't until

the 1800s that Augustine Cauchy turned matters around and developed a sound base for the subject by speaking of

limits. This had a profound effect on the concept of "number," for Cauchy also found a consistent way to give

meaning to irrational quantities, essentially defining them as limits of sequences of rational quantities. For example,

let's return to the mysterious quantity 0.101001000100001000.... Cauchy would define this as the limit of the

sequence of rationals, 0.1, 0.101, 0.101001, 0.1010010001, .... This shift of perspective represented a marrying, of

sorts, of the potential and the actual infinite, and it brought some logic to the concepts of the infinity of irrationals

and the infinite sums that arise in calculus.

As calculus began to assume a larger and larger role in both math and science, the need to understand infinity

became greater. This quest for understanding ultimately required a shift in thinking, away from looking at whole

numbers and magnitudes, toward thinking about sets. In the next section, we will see some of the fundamental

[23] (URL) http://www.metric-conversions.org/

Interestingly, even though people come in many different sizes, their bodily proportions are much more

uniform. Proportionality is one of the keys for understanding nature. The golden Section and the

Fibonacci Series play an essential role there.

[24] (URL) http://en.wikipedia.org/wiki/Procrustes

(URL) http://www.zeit.de/2012/30/Puritanismus

(URL) http://www.zeit.de/lebensart/partnerschaft/...

[25] "Best" is a matter of preference. This is entirely based on experience.

What works best, is best. This is one of the prime rules of engineering. It has nothing to do with beauty,

nor elegance, nor originality. Mostly, engineering style is neither beautiful, nor elegant, nor original.

It just works, does what needs to be done. It is only what works time-trusted and proven. And

mathematical beauty and originality are entirely irrelevant. This is the Roman style of thinking, versus the

ancient Greek style. The Romans were always oriented to practical matters, and eventually, because of

this, they conquered the Greeks because of this thinking style. Modern engineers are a consequential

follower group in the spirit of Roman engineers.

(URL) http://www.roman-empire.net/army/army.html

(URL) http://www.roman-empire.net/army/training.html

(URL) http://www.roman-empire.net/army/soldier-practice.html

(URL) http://en.wikipedia.org/wiki/Military_step

(URL) http://library.thinkquest.org/26602/onthemarch.htm

[26] The Greeks had their specific olympic standard measure of the stadion.

(URL) http://en.wikipedia.org/wiki/Stadion_%28unit_of_length%29

(URL) http://www.google.de/searc......eek+stadium&hl=de&pr

But such trifles are not important here. Here the task is to construct a realistic engineering solution to the

[27] (URL) http://www.ehow.com/how_23......measure-ones-step-stride.htm

[28] (URL) http://de.wikipedia.org/wi......e_Ma%C3%9Fe_und_Gewichte_%28

(URL) http://www.google.de/#hl=d......gs_nf=1&cp=13&gs_id=,or.r_gc.r_pw.r_qf.&fp=273aa0d90065244e&biw=1124&bih=715

(URL) http://www.dl.ket.org/latinlit/mores/measures/measure.htm

[30] The success of Arminius against Varus owes not only to the tactics, but also to the fact that the Roman

legionaries couldn't maintain their usual step frequency because of morass ground. So they literally fell

out of step, and could not fight efficiently.

(URL) http://de.wikipedia.org/wiki/Arminius

[31] When it snows, the stride will shorten to almost nothing. This is called: The Great Slide. No wonder

that the Romans hated snow. All these insecurities and even more, were recorded in the timeless manual

(URL) http://de.wikipedia.org/wiki/Carl_von_Clausewitz

(URL) http://gutenberg.spiegel.de/buch/4072/1

(URL) http://www.clausewitz.com/readings/VomKriege1832/TOC.htm

There is another, also timeless manual of warfare: Sun Tsu/ Sun Tze/ Sun Tzi:

(URL) http://en.wikipedia.org/wiki/Sun_Tzu

(URL) http://en.wikipedia.org/wiki/The_Art_of_War

(URL) http://de.wikipedia.org/wi......_Kunst_des_Krieges_%28Sunzi%

(URL) http://www.aphorismen.de/d......_aphorismen.php?search=9&

[32] (URL) http://www.google.de/searc......man+roads&hl=de&prmd

(URL) http://en.wikipedia.org/wiki/Roman_roads

[33] Some Greek islands have extremely winding roads. There, a bird-fly distance of 40 km can easily

translate to 80 km road distance. To make matters worse: On an extremely winding road, the car travelling

speed will also be reduced to, say: 30 km/h in high gear. This gives an extremely increased fuel

consumption of about 200 %. Also, there are no gasoline stations on the way, and so you might be stuck

in the middle of nowhere because you mis-calculated the time and gasoline for the trip. And this damned

island is only 40 km across. I have experienced it myself. This was hard-won experience.

[34] Not even this is straight, because the earth is a more or less flattened sphere, and: Because of the wind

shift, the bird must compensate for the aberration. This will lengthen the trip a more or less trifle bit.

[35] (URL) http://en.wikipedia.org/wiki/Mile

[36] "Horses and mules can travel at about the same speed as a man, 3 to 4 miles per hours."

(URL) http://www.google.de/searc......e&noj=1&site=webhp&a

(URL) http://garyb.0catch.com/march2_basics/march_basics.html

[37] (URL) http://www.machinehead-software.co.uk/...

[38] (URL) http://www.machinehead-software.co.uk/...

[39] In these calculations, it is important to note that english / american usage for numerical

(URL) http://www.coolrunning.com/engine/4/4_1/96.shtml

(URL) http://www.machinehead-software.co.uk/...

[41] If needed, a legion could maintain this speed for 7 hours a day (42 km), 7 days a week (294 km), and

with 7 weeks marching (2058 km), they would have covered the whole Roman Empire from one end to

the other. This was the strategic calculation rule by which the Romans moved their troops.

This was also not much different from Napoleonic times.

(URL) http://www.roman-empire.net/army/army.html

(URL) http://www.roman-empire.net/army/training.html

(URL) http://www.roman-empire.net/army/soldier-practice.html

(URL) http://www.therthdimension.org/AncientRome/...

An ordinary day's march for the Roman army consisted of 15-18 miles done in 7 of our hours

(URL) http://www.romanarmytalk.com/rat/...

Marching speed depends, of course, on the terrain covered, the baggage carried and the length of the

march overall, but I believe it's often stated that the Roman legion was expected to cover 20 miles on a

good day - Caesar's forced marches I think did 25. In reality, I would think 15-18 more likely for a

standard legion with all its baggage during normal operations. The baggage train is the real marker for the

speed of the legion as a whole, and Caesar often rushed his troops on ahead of the train (when he was

marching on the Nervii, for example) to pick up the pace.

As for individual speeds, they could be very swift indeed - Suetonius (Caesar, 57) has Caesar travelling

100 miles in one day during a journey from Rome to western Gaul. Plutarch writes that the freedman

Icelus carried the news of Nero's death from Rome to Galba in Spain in only seven days, which would

have required a speed of around 10 miles an hour: certainly possible with regular relays of horses, and

these were available for rapid carriage of important news (or people). Without the rush, journeys would be

much longer - 20 miles a day on horseback is quite tiring enough!

This was in summer, of course - winter could make journeys infinitely protracted, or even impossible. Vitellius' legions took two months to march from the Rhine to northern Italy in the early spring of 69AD.

[44] There is a cognitive trap to easily fall into, since in a second, Achilles makes 2,2521 strides, which is

unreal. Achilles can make only 1 stride per clock time, and the clock cycle should be set to one stride. So

the internal clock would be 0.44 sec's, so that Achilles can make exactly one stride per clock cycle.

[45] At clock cycle 26 secs Achilles is at 43,3 m.

[47] The problem is also known as the floating point rounding error, when we take a floating point number

function, and recurse it many times, feeding the result into the input. After a certain number of back-feeds,

the calculation will consist of rounding errors only. This pattern can be plotted as a fractal, and from there

results the strange self-similarity which is a distinct repeating pattern of float number rounding errors.

(URL) http://books.google.de/boo......6AzR86m780wC&pg=PA89&

(URL) http://books.google.de/boo......PS0LAAAAIAAJ&pg=PA199&am

[49] The electrons in the atom shell, for example, don't move at all. They form a cloud of probability

(URL) http://en.wikipedia.org/wiki/Longitudinal_wave

(URL) http://demonstrations.wolfram.com/...

(URL) http://www.google.de/searc......ngitudinal+oscillation&h

[51] One can also rationalize that as space-time vibrations, since by relativity theory, any moving object

contracts a little, and expands when it is still. By this, the caterpillar-like motion of an arrow can also be

visualized. But the worlds of relativity and of the quantum level can not be computationally translated.

(URL) http://www.thefreedictionary.com/incommensurable

(URL) http://simple.wikipedia.org/wiki/Incommensurability

(URL) http://plato.stanford.edu/entries/incommensurability/

(URL) http://www.learner.org/courses/mathilluminated/.../03.php

[52] (URL) http://wiki.answers.com/Q/What_is_motion

(URL) http://en.wikipedia.org/wiki/Motion_%28physics%29

(URL) http://www.wisegeek.com/what-is-motion-physics.htm

[53] In the literature, this is also known as:

horizontal flexural vibration and vertical flexural vibration.

(URL) http://www.vsgstapelfeld.de/training/...

(URL) http://de.wikipedia.org/wiki/Pfeil_%28Geschoss%29 spine:

Die Anpassung des „spine“ an Bogen und Schütze ist besonders bei den traditionellen Schützen wichtig,

da die Sehne sich beim Schuss genau auf den Bogen zubewegt, der Pfeil sich aber um den Bogen

herumwinden muss. Ein falscher „spine“ bedeuten einen unruhigen Flug oder das (unerwünschte)

Anschlagen des Pfeilschafts an den Bogen. Idealerweise sollten die Schwingungen des Pfeilschaftes nach

(URL) http://en.wikipedia.org/wiki/Momentum

(URL) http://en.wikipedia.org/wiki/Velocity

[55] Google will give you exact numbers for everything. Search for: arrow speed calculator

(URL) http://www.vsgstapelfeld.de/training/...

(URL) http://www.backcountrybowhunting.com/articles/calc/

(URL) http://www.stickemarchery.com/stickemcart/...

(URL) http://www.kyudo-sum.de/ballistik/.../uebersicht.html

(URL) http://de.wikipedia.org/wiki/Pfeil_%28Geschoss%29

(URL) http://www.kyudo-sum.de/ballistik/...

(URL) http://www.campus-lingen.hs-osnabrueck.de/...

(URL) http://www.bogensportverlag.com/.../index.html

The actual speed of an arrow is about 200 - 300 feet per sec.

[57] In reality, the length is about 75 cm which is quite exactly the Achilles stride measure.

[58] (URL) http://de.wikipedia.org/wiki/Meter

(URL) http://books.google.de/boo......Rhi7odTe2BEC&printsec=fr

(URL) http://books.google.de/boo......YMv7jPGUUk0C&printsec=fr