Oct 28, 2011

Cold Fusion

Energy of sun and stars

First comments and results from Andrea Rossi on their 1MW E-Cat plant test:
Andrea Rossi
October 28th, 2011 at 10:37 AM
Andrea Rossi
October 28th, 2011 at 5:33 PM

Since the late 80s, when first reported by Fleischmann and Pons, cold nuclear fusion could not be reproduced; a room-temperature reaction when two nuclei fuse together becoming a bigger one, and steaming a large amount of energy in the process. But last January, Andrea Rossi and Sergio Focardi (University of Bologna), demonstrated it is indeed possible by means of their LENR (low-energy nuclear reactions) device, which produced subatomic particles (neutrons) out of nickel and hydrogen (Ni58 + p → Cu59, copper), getting 12400W of heat power in exchange to just 400W consumption. That means a power gain of 12400 / 400 = 31, a potentially cheap and clean energy source:

Specifics of Andrea Rossi’s “Energy Catalyzer” Test
University of Bologna, January 14, 2011
Marianne Macy

On January 14, 2011, Andrea Rossi submitted his “Energy Catalyzer” reactor, which burns hydrogen in a nickel catalyst, for examination by scientists at the University of Bologna and The INFN (Italian National Institute of Nuclear Physics). The test was organized by Dr. Giuseppe Levi of INFN and the University of Bologna and was assisted by other members of the physics and chemistry faculties. This result was achieved without the production of any measurable nuclear radiation. The magnitude of this result suggests that there is a viable energy technology that uses commonly available materials, that does not produce carbon dioxide, and that does not produce radioactive waste and will be economical to build.

The reactor used less than 1 gram of hydrogen, less than 1,000 W of electricity to convert 292 grams of water per minute at ~20°C into dry steam at ~101°C. The unit was turned ON and began producing some steam in a few minutes, and once it reached steady state continued producing steam until it was turned OFF. The amount of power required to heat water 80°C and convert it to steam is approximately 12,000 watts. Dr. Levi and his team will be producing a technical report detailing the design and execution of their evaluation.

A representative of the investment group stated that they were looking to produce a 20 kW unit and that within two months they would make a public announcement. He declared that their completed studies revealed a “huge, favorable difference in numbers” between the cost to produce the Rossi Catalyzer and other green technologies. “We had a similar demonstration six months ago with the same success we’ve had today. We are almost ready with the industrialized product, which we think is going to be a revolution. It is a totally green energy.” The representative offered that the company was called Defkalion Energy, named for the father of the Greco Roman empire, and was based in Athens.

Giuseppe Levi, PhD in nuclear physics at the University of Bologna and who works at INFN, offers exclusive comments on the test, which he deemed “an open experiment for physicists. The idea was like a conference: to tell everybody what was going on and eventually to start new research programs on that topic.”

The first measurements Levi described were energy measurements to determine the input of energy inside the reactor and the output of energy of the reactor. “I don't have conclusive data on radiation but absolutely we have measured ~12 kW (at steady state) of energy produced with an input of about just 400 watts. I would say this is the main result. We have seen also this energy was not of chemical origin, by checking the consumption of hydrogen. There was no measurable hydrogen consumption, at least with our mass measurement.” By measuring with a very sensitive scale, within a precision of a 10th of a gram, Levi measured the weight of the hydrogen bottle before and after the experiment “If the energy was of chemical origin you would have expected to consume about 100 to 600 more than the sensitivity of the scale. You measure the bottle before and after and then you see in your measurements there was almost no hydrogen consumed.”

The workings of the Rossi reactor was, Levi explained, unknown to them because of “industry secrets.” He said: “What we've done is to measure the water in the flux and we are heating and making steam for that water. We are measuring the water flux and carefully checking that all the water was converted into steam, then it is easy to calculate power that was generated. You are measuring the power that was going in the system by quite a sensitive power meter. Initially the system started up and we had 1 kW of input and then we reduce the input to just 400 W. The output energy was constant at about 12 kW.”

The flow rate, Levi continued, was measured with a high precision scale. “The flow rate was 146 g in 30 seconds. Using a simple measurement gives a simple result. There was a pump putting in a constant flux and what I have done is – with the reactor completely off take measurements – we spent two weeks of the water that flowing through the system to be certain of our calibration. After this calibration period I have checked that the pump was not touched and when we brought it here for the experiment it was giving the same quantity of water during all the experiment. The water was coming from an Edison well and the pump was putting it in the system. Then we were releasing the steam into the atmosphere; there was not a loop.”

To determine if the steam was coming out dry and at atmospheric pressure, Professor Gallatini, a specialist in Thermochemics and a former head of the Chemical Society of Italy, verified that all the water came out as steam. “There was no water in the steam,” Levi certified. “The outer temperature measured was 101° centigrade at atmospheric pressure.” The instrument he used was a Delta OHM # HD37AB1347 Indoor Air Quality Monitor. Gallantini inserted the probe inside the exit pipe with the steam.

Levi was asked: How did you compute the thermal energy production by the Energy Catalyzer (ECat)?

He responded, “The calculation is very, very simple. Because you know the number of grams of water per second delivered to the ECat you know you must raise the water to 100°C, this is the transient phase of operation. Once the water is at 100°C the energy is used to make the water into steam. It takes 2272 joules per gram to convert water at 100°C to steam. Because the ECat provided more energy the steam became hotter, 101°C. So our conservative estimate of the steady state thermal output of the ECat, neglecting thermal radiation and other losses, is just 2272 joules per gram multiplied by the 4.9 grams per second = 11, 057 joules per second or Watts. When you realize that you have to add the energy to raise the temperature of the water you get by about 80°C and the steam by another 1°C the total thermal power the ECat releasing is about 12,400 Watts. These are not our refined estimate but they indicate that the input electrical power of 400 W produces using an amount of hydrogen less than a gram in a couple hours of operation we are seeing a system with a power gain = 12,400/400 = 31.”

Jan 21, 2011

NASA Astronomy Picture of the Day

Belt of Orion
(c) Sergi Verdugo Martínez (

Alnitak, Alnilam, and Mintaka, are the bright bluish stars from east to west (left to right) along the diagonal in this gorgeous cosmic vista. Otherwise known as the Belt of Orion, these three blue supergiant stars are hotter and much more massive than the Sun. They lie about 1,500 light-years away, born of Orion's well-studied interstellar clouds. In fact, clouds of gas and dust adrift in this region have intriguing and some surprisingly familiar shapes, including the dark Horsehead Nebula and Flame Nebula near Alnitak at the lower left. The famous Orion Nebula itself lies off the bottom of this colorful star field. Recorded last December with a modified digital SLR camera and small telescope, the well-planned, two frame mosaic spans about 4 degrees on the sky. — NASA

Congratulations Sergi ( for this great image.

Sep 28, 2010


Planetary ecology

The thing the ecologically illiterate don’t realize about an ecosystem is that it’s a system. A system! A system maintains a certain fluid stability that can be destroyed by a misstep in just one niche. A system has order, a flowing from point to point. If something dams the flow, order collapses. The untrained miss the collapse until too late. That’s why the highest function of ecology is the understanding of consequences. — Pardot Kynes in "Appendix I: The Ecology of Dune" (Dune, Frank Herbert, 1965)

Despite the common "political" similarities that arise from Frank Herbert's Dune,
I conceived of a long novel, the whole trilogy as one book about the messianic convulsions that periodically overtake us. Demagogues, fanatics, con-game artists, the innocent and the not-so-innocent bystanders-all were to have a part in the drama. This grows from my theory that superheroes are disastrous for humankind. Even if we find a real hero (whatever-or whoever-that may be), eventually fallible mortals take over the power structure that always comes into being around such a leader.

Personal observation has convinced me that in the power area of politics/economics and in their logical consequence, war, people tend to give over every decision-making capacity to any leader who can wrap himself in the myth fabric of the society. Hitler did it. Churchill did it. Franklin Roosevelt did it. Stalin did it. Mussolini did it.

This, then, was one of my themes for Dune: Don't give over all of your critical faculties to people in power, no matter how admirable those people may appear to be. Beneath the hero's facade you will find a human being who makes human mistakes. Enormous problems arise when human mistakes are made on the grand scale available to a superhero. And sometimes you run into another problem.

It is demonstrable that power structures tend to attract people who want power for the sake of power and that a significant proportion of such people are imbalanced-in a word, insane.

That was the beginning. Heroes are painful, superheroes are a catastrophe. The mistakes of superheroes involve too many of us in disaster.

Yes, there are analogs in Dune of today's events-corruption and bribery in the highest places, whole police forces lost to organized crime, regulatory agencies taken over by the people they are supposed to regulate. The scarce water of Dune is an exact analog of oil scarcity. CHOAM is OPEC. — Frank Herbert, "Dune Genesis" (Omni Magazine, 1980)

the utter message has to do with ecology.
I went to Florence, Oregon, to write a magazine article about a US Department of Agriculture project there. The USDA was seeking ways to control coastal (and other) sand dunes. I had already written several pieces about ecological matters, but my superhero concept filled me with a concern that ecology might be the next banner for demagogues and would-be-heroes, for the power seekers and others ready to find an adrenaline high in the launching of a new crusade. — Frank Herbert, "Dune Genesis" (Omni Magazine, 1980)

The plot takes place mostly on a desert planet called Arrakis (Irak?, from the Arabic name "al-'Iraq", meaning "fertile land", as Mesopotamia had been known), aka Dune. The CHOAM Corporation (OPEC) controls and regulates the flow and trade of spice (oil) commodity, essential for (interstellar) transport (monopoly of the Spacing Guild).

OPEC's 50th Anniversary - The Economist

Amongst, with a feudal empire where the main noble houses (G-7?) are ruled under the Landsraad (Scandinavian word for a "Land Council"), the first-born of the Atreides House ("the son of -mythological- Atreus", directly descendant from Agamemnon, a hero from Greek Homer's The Iliad) would become the Kwisatz Haderach (from the Hebrew term "K'fitzat Haderech", which means "a jump forward along the path") as the final product of Bene Gesserit's (Catholics?) sisterhood (from the legal Latin expression "quamdiu se bene gesserit", meaning "as long as he/she shall have conducted himself/herself well"; who are trained, in what they call "the Voice", to control others merely by selected tones of the voice) eugenics (breeding), and the long awaited messiah of the native Fremen (free-men?, Jewish from Israel?).

Besides these winks (and much others), in the backstage lies the long-term aim to terraform a whole arid desert planet into a new ecosystem, starting with moisture trap, water recycle and poverty grass and flora planting, until rain falls and turns it into a paradise. Though there's a drawback, that would mean the end of the valuable spice currency; and whose social, religious, political and economic (and military) consequences will drive them all to a Jihad (holy war).

Albeit, in John Herbert's own words:
I refuse, however, to provide further answers to this complex mixture. You find your own solutions. Don't look to me as your leader. And when someone asks whether you're starting a new cult, do what I do: Run like hell. — Frank Herbert, "Dune Genesis" (Omni Magazine, 1980)

The film you will never see

The Chilean filmmaker and writer Alejandro Jodorowsky had originally planned on filming Dune in 1975. To accomplish this he contacted Jean Giraud "Moebius" and H.R. Giger for visual design, and surrealist painter Salvador Dalí to play the Emperor, along with Pink Floyd to compose the soundtrack. Even discovered Dan O'Bannon in an amateur sci-fi festival through his film Dark Star. But
The project was sabotaged in Hollywood. It was French and not American. Their message was 'not Hollywood enough'. There were intrigues, plundering. The storyboard circulated among all the big studios. Later, the visual aspect of Star Wars strangely resembled our style. To make Alien, they called Moebius, Foss, Giger, O'Bannon, etc. The project showed Americans the possibility of making science-fiction films for big shows, outside of the scientific rigor of 2001: A Space Odyssey. The project of Dune changed our lives. — "Dune, le film que vous ne verrez jamais" (Alejandro Jodorowsky, Métal Hurlant 170)

Later, Jodorowsky and Moebius created the comic books series The Incal (with their character John Difool, since 1980 onwards; and The Metabarons saga from 1992, illustrated by Argentinian Juan Giménez), where they put in much of their previous artwork and plot broad strokes from Dune. The movie was later achieved by David Lynch in 1984.

Sep 6, 2010

Astrophotography, light years away

Andromeda galaxy (M31)
(c) Sergi Verdugo Martínez (

A good photo depends basically upon an adequate illumination, that is, the flux of light that comes from the object towards the camera for a certain period of time.

The photometric relation of this Flux that involves 2 surface elements (i.e., a nebula of extension S, and the pupil or a camera CCD of area S') would be dF = L · dS · domega · cos(theta), where L means the object's luminance, omega the falling solid angle differential ( = dS' / r^2, being r the distance between the nebula and the lens of the camera), and cos(theta) applies for the cosine component of the surface, its orientation (Lambertian cosine law).

Luminance is defined from the spectral radiance emitted by the object (see Greenhouse effect post), but weighted to the average sensitivity curve of the human eye to wavelength, which drives to a "brightness perception", the amount of light the eye would perceive from a particular viewpoint.

There are 2 different shifted curves for sensitivity due to both types of eye photo-receptors (cones and rods). Photopic vision affects cones, which are composed of three separated photo pigments to enable color perception; and the scotopic vision implies rods, that are more sensitive to light and less to color. Photopic responses under normal lighting conditions. This curve peaks at 555 nanometers, so the eye is most sensitive to a yellow-green color (oddly human eye has evolved to match Sun's). At low light levels, near to darkness, the eye response fits the scotopic curve, and peaks at 507 nm, closer to blue-violet.

Thus, the incoming flux of light entering the pupil, or the lens of a telescope, of diameter D would be dF = L · dS · domega · cos(theta) = L · dS · [ pi·(D/2)^2 ]/r^2 · cos(theta). In fact the solid angle may subtend another cos(theta') due to the orientation of S' surface, but from now onwards we will consider both theta and theta' angles as 0, so their cosines are 1.

The illuminance, that is the received illumination on surface S', related as well to the irradiance, is just E = dF / dS' = L · dS/dS' · pi·D^2/4·r^2, measured in [lux] (or [lumens/m2]).

Now to determine the dS/dS' areas ratio we must consider the focal length and the (linear) magnification equation of a lens.

Take a spheric mirror (simpler than a lens -no refracting indexes involved- and will do as well) of radius of curvature r. Geometrically, the image point P' at distance s' can be deducted from the different angle relations. Notice the beta angle (at the center of the mirror sphere) is the sum of alpha (drawn from the object P) and theta (the reflecting angle). Similarly gamma = alpha + 2·theta. Combining both equations and removing theta we obtain 2·beta = gamma + alpha.

As these angles are considered "small" (sinus ~ angle), they can relate the P object distance alpha ~ sin(alpha) = l/s; the P' image distance gamma ~ sin(gamma) = l/s'; and the C center of curvature distance beta ~ sin(beta) = l/r. So, finally, 2·l/r = l/s' + l/s, that is 2/r = 1/s' + 1/s. When the distance to the object is quite larger (at the "infinite", so incoming rays are parallel -paraxial-) than the mirror radius of curvature, 1/s is negligible, and s' = r/2. This s' distance is then the focal length f of the mirror (or lens), and P' the focal point: 1/f = 1/s' + 1/s.

The (linear) magnification m quantifies the apparent ratio between the image and the object sizes m = h'/h. Notice that according to the lens, h/p = h'/q, or the paraxial rule h/(p-f) = -h'/f, thus m = h'/h = -(p-f)/f.

In our case, the dS/dS' areas ratio is proportional to (h/h')^2 (squared because we are talking about surfaces). Replacing in the illuminance equation E = dF / dS' = L · dS/dS' · pi·D^2/4·r^2 = L · [ -(r-f)/f ]^2 · pi·D^2/4·r^2 = pi/4 · L · [ (r-f)/r ]^2 · [ D/f ]^2. For astronomical observation (r much greater than f), illuminance can be approximated to E = pi/4 · L · [ D/f ]^2.

The D/f amount stands for the relative aperture (adjustable with the camera diaphragm or the pupil iris), and its inverse for the diaphragm number N = f/D. The aperture limits the incoming amount of brightness (pupil or diaphragm size) to the eye or camera. But same apertures will produce equal illuminance even varying focal length f and diameter D.

Apertures are commonly expressed as fractions of the focal length, called f-numbers or f-stops, and each stop represents half the light intensity from the previous one, that is f/1 = f/sqrt(2)^0, f/1.4 = f/sqrt(2)^1, f/2 = f/sqrt(2)^2, f/2.8 = f/sqrt(2)^3 and so on (root-squared because it is lately squared-powered in the illuminance equation to halve the incoming light). Then lower f-numbers denote greater apertures, which means more light to the camera sensor. Maximum aperture (or minimum f-number) defines the (lens) speed: The greater the aperture, the faster the lens, as it lets in more light (higher illuminance). So the shutter speed will be faster as well.

Bearing this in mind, astrophoto may require different focal ratios (relative apertures) according to the astronomical objectives to be shot. Among others, they can be divided into planetary or deep sky.

The apparent (or angular, or visual) magnification M of the (refracting) telescope is determined by the ratio of tangents of the angles under which the object is seen with (beta(i), apparent field of view) and without (beta(s), true field of view) the lens, respectively.

Thus, tangent( beta(i) ) = h / fe = (D/2) / (fo+fe), where fo is the objective focal length and fe the eyepiece's, h the image height, and D the objective lens diameter; and tangent( beta(s) ) = h / fo = (d/2) / (fo+fe), here d means the eyepiece lens diameter. So the telescope magnification power M = tangent( beta(i) ) / tangent( beta(s) ) = fo / fe = D / d.

For planetary observation a telescope with greater magnification power is worth (longer focal, usually a refractor -dioptric- telescope, which uses lenses), and for deep sky higher illumination is needed (wider diameter, mostly a reflector -catoptric- telescope, with curved mirrors), as nebulae or galaxies are extended objects and their apparent magnitude is distributed over a wider angle than planets or stars.

The magnitude of an object is a logarithmic measure of its relative brightness. Relative to the star Vega, which has a (almost) 0 magnitude. Sun has a -26.74 magnitude (brighter), Moon -12.74 (less brighter), or Mars ranges from -2.91 (brighter than Vega) to 1.84 (fainter than Vega). So, even the M42 nebula (Orion) has a 4.0 magnitude, it is less visible than a star of the same apparent magnitude because its dimensions are 65x60 arcminutes.

Or even a catadioptric (lens and mirror) telescope for a combination of both planetary and deep sky observation. But telescopes are not perfect, as paraxial optics laws applies strictly to light rays that are infinitesimally displaced from the optical axis of a system, and a series of optical imperfections (aberrations) must be considered.

Refracting telescopes suffer from chromatic aberration, a distortion by which the lens cannot focus all colors at the same (converging) point. This dispersion is caused by different refractive indexes depending on light wavelengths.

Chromatic aberration

It can be partially fixed adding more lenses (achromatic Fraunhofer doublet, the sum of a convergent -crown- lens plus a divergent -flint-; or apochromatic, adding more lenses, better focus correction of wavelengths), or minimized with greater quality lenses (lower dispersion, made of fluorite).

But also reflecting telescopes do have aberrations. They gather light with a mirror, and it is primarily parabolic, not spheric, to avoid spherical aberration, where light at the edges of the mirror focus closer than that reflecting from the center. This is corrected with a parabolic mirror instead, as in Newtonian telescopes.

Spheric mirrorParabolic mirror

But parabolic mirrors trouble with coma aberration, that's a change of magnification for incoming light closer to the edges (off-axis) of the curved mirror. It can be partially fixed closing the aperture 1 or 2 stops, along with an increase of exposure time to photograph.

Coma aberration

This lack is better solved in Schmidt-Cassegrain (and Maksutov-Cassegrain) catadioptric telescopes, which combine a correcting lens with a primary spherical mirror and a secondary parabolic convex, that multiplies the focal length, thus getting a compact telescope with high magnification power and wide angle, optimal for both planetary and deep sky.

Because of all these side effects, and also due to diffraction, the image of a point becomes a spot (an Airy disc). The angular resolution (or power resolution) of a telescope is a measure of the minimum angular separation between distinguishable objects in an image, according to the Rayleigh criterion sin(theta) = 1.22 · lambda/D, where 1.22 is nearly the first zero of Bessel function, angle theta is measured in [arcseconds], and lambda (light wavelength) and D (aperture diameter) in same units (i.e. [mm]).

Since theta will be a "small" angle, the expression can be approximated by sin(theta) ~ theta = s / f, being s the separation of both objects in the image (focal) plane and f the focal length. Thus s = 1.22 · lambda · f/D = 1.22 · lambda · N, where N is the diaphragm number.

Once mounted the telescope on an equatorial platform (i.e. GEM -German Equatorial Mount-, much better than alt-azimuthal for shooting, easier following position movement of celestial objects), one just needs a camera (attaching it to the telescope as primary focus) to begin with astrophoto. Even a webcam will do, though a digital CCD is highly recommended.

Orion and Running Man nebulae (M42 and NGC1977)
(c) Sergi Verdugo Martínez (

Unlike it is commonly believed about the dutch origin of the telescope around 1608 credited to Hans Lippershey, the oldest reference about its existence is a noble's inheritance written legal document dated as of April/10/1593, and his inventor was the catalan optician (from Girona) Joan Roget, as published in a book authored by Girolamo Sirtori in 1609.

Acknowledgement to Sergi Verdugo Martínez ( for his awesome images.

Aug 21, 2010

Paragliding aerodynamics

Once you have tasted flight, you will forever walk the earth with your eyes turned skyward, for there you have been, and there you will always long to return. — Leonardo da Vinci

A paraglider are just a couple layers of fabric (nylon) connected internally in such a way to form a row of cells, closed in the rear edge and open in the front so that incoming air keeps the wing inflated and shapes cross-sections as airfoil teardrops.

Dune du Pyla

Flight can be depicted in terms of weight (gravity), lift, drag and thrust.

Aero forces

When flying, an upward force on the wing, referred to as Lift, has to be generated from the (relative) airflow around the wing balancing downward gravitacional force, and is perpendicular to relative wind; while the air resistance to motion, or Drag, should be balanced by forward thrust, parallel to relative wind.

Lift is produced as the airfoil deflects the airflow downward. For a given angle of attack, the airfoil produces more lift as the air speed increases. The amount of lift created is proportional to the square of the air speed Lift ~ velocity^2·angle-of-attack. For instance, if the air speed doubles and the angle of attack is held constant, the lift increases by a factor of four.

As the angle of attack increases, the incoming air is deflected more, resulting in increased lift. The amount of lift created at a constant air speed is proportional to the angle of attack. For example, if you double the angle of attack while holding the airspeed constant, you double the amount of lift.

In other words, as the velocity increases, the angle of attack decreases. Conversely, as the air speed decreases, the angle of attack must increase if the lift is to remain constant.

As the glider cannot generate thrust, descent hover at a certain forward speed will fit. Then the Lift/Drag ratio, which represents flight performance, may be captured by Lift·sin(descent_angle) = Drag·cos(descent_angle), where the descent angle relates descent direction and horizon. Thus, Lift / Drag = cos(descent_angle) / sin(descent_angle) = 1 / tg(descent_angle) = horizontal_distance_advanced / vertical_height_lost. An average paper plane may achieve a 3:1 ratio, a paraglider should get to at least 10:1.

The drag force can be expressed as Fd = 1/2·density·velocity^2·Cd·area = dynamic_pressure·Cd·area, where density is the fluid's (air), velocity is referred to the object (wing) speed through the fluid, Cd is the drag coefficient (that will depend on the shape of the wing, and angle of attack), and the projected area. Similarly, the lift force Fl = 1/2·density·velocity^2·Cl·area, being Cl the lift coefficient. Thus, Lift / Drag ratio = Cl / Cd.

Lift/Drag ratio and coefficients vs angles-of-attack

Usually, it is assumed that the projected area and air density are constant. In a paraglider, however, the area is not constant, and not just because of wing span or internal air pressure changes affecting aerofoil shape.

One of the secrets of flight is to exchange resistance to lift. To quantify, drag coefficient measures shape exposure to air. A Cd = 1 is given by a 1m^2 plate.

The paraglider moves through a fluid (relative air) at a slight angle of attack, between the chord of the aerofoil (line connecting the leading and trailing edges) and the direction of the airflow, creating 2 different flows (above and under the wing) not simmetric any more.

Airfoil terms

The angle of attack will affect both the lift and drag forces. Zero angle implies minimum drag, but no lift as well. As seen, lift increases along with the angle of attack, until a critical angle is reached (around 20º), when drag increase exceeds lift decrease and the glider stalls, and eventually acts like a parachute in free fall. When the airfoil drops off, airflow is no longer deflected, so the airfoil does not produce nearly as much lift. The drag also increases dramatically. Dangerous.

Lift/Drag ratio

An airfoil is more likely to stall at low air speeds, because in order to create the required lift, the airfoil must operate at a high angle of attack. The critical angle of attack is determined primarily by the shape of the airfoil. However, it can also be affected by the surface condition of the airfoil.

That wings should be curved on top (air forced to move a little quicker, as it has a longer path to the trailing edge) and flatter on the bottom is related with the misconception that air spends the same time to travel above and below the wing.

Airflow vs angles-of-attack

However, even the air flowing above the wing has a longer path, it gets to the back edge earlier than the one below. And as soon as it reaches the trailing edge, then quickly returns to its original speed, so this change in relative position will be permanent, and the air that flowed under the wing will never catch up the one on top.

According to Bernoulli's principle, which states that faster (air)flow (over the top) means lower pressure or energy, and slower (air)flow (underneath) implies higher fluid's pressure, the result of that pressure difference is then an upward force, lift.
Airflow and pressure

Notice that the air ahead of the wing, at a certain angle of attack, is redirected upwards (upwash). Also, the trailing air will move downwards (downwash). Depression above sucks air and produces 2/3 of lift force, lower wing surface pushing air is responsible for the remaining 1/3, with a resulting lift/drag quotient over 10.

But Bernoulli’s principle apply to inviscid (no viscosity, no friction, thus no resistance) incompressible fluids along a streamline, where there is a steady flow.

Viscosity is a physical property that quantifies how much adjacent molecules "stick" together. Air has viscosity. Air flows around objects, and the air in touch with the aerofoil is actually carried along, thus it (air) has zero speed with respect to the surface (shear stress).

Boundary layer

At the surface of the object (wing) air is stagnant due to friction. Velocity increases from the surface to a maximum in the upstream of the airflow. This region of velocity profile in the airflow, due to the shear stress, is the boundary layer. Commonly, the thickness of this boundary layer extends up to the point where speed reaches 99% of the free-flow stream; the remainder of the airflow is known as the external inviscid flow.

Adverse pressure gradient

The above figure shows the formation of the separation process. The fluid (air) accelerates from the leading edge to turn round the sphere (or aerofoil), as it has further to go than the surrounding fluid (when there is a pressure decrease in the direction of flow, the fluid will accelerate and the boundary layer will become thinner, equally as a narrowing pipe). It reaches a maximum velocity, still with positive pressure gradient (pressure drops with distance, dp/dx>0) and lift force occurs.

As air travels over the surface and gets to the other side of the sphere, fluid will immediately start to slow down (when pressure goes up, so does the potential energy of the fluid, leading to a reduced kinetic energy and a deceleration of the fluid, in a similar way as a widening pipe), hence the thickness of the slower layer increases (smaller velocity gradient) and becomes less stable, and eventually turbulent. When the pressure rises in the direction of flow (adverse pressure gradient, downstream side of the sphere/aerofoil), fluid outside the boundary layer has though enough momentum to overcome this pressure that is trying to push it backwards, but the fluid within the boundary layer has so little momentum due to friction that it will be quickly brought to rest, and likely reversed in direction. When this reversal occurs, the fluid flow becomes detached from the surface of the object at some point before the trailing edge, and instead takes the forms of eddies, whirls and vortices (flow separation and vortex shed), and forces the flowfield into an unstable wake, with the flow being redirected downwards according to the angle of attack. This happens because fluid to either side is moving in the opposite direction.

Vortices result in very large energy losses in the flow. This has quite important consequences in aerodynamics, as flow separation significantly modifies the pressure distribution along the surface and hence the lift and drag characteristics. Such flow separation causes a large increase in the pressure drag (pressure differential between the front and rear), since it greatly increases the effective size of the wing section, which also result in loss of lift and stall.

Flow separation

Whether or not the boundary layer separates against an increase in pressure (adverse gradient) depends upon the character of the boundary layer: laminar (viscous forces higher, lower drag) or turbulent (higher inertial forces, higher drag).

To calculate whether a particular flow is laminar or turbulent use the Reynolds number. The Reynolds number (Re) is a dimensionless equation measuring the ratio of inertial forces (rho·v^2/l) to viscous forces (mu·v/l^2), that allows for the cross referencing of aerodynamic characteristics (including drag coefficient) between objects of different sizes that may move at different rates of speed. In sum, two different types of objects going at different speeds that experience the same Re will have the same drag coefficient.

Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion; while turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities:

Re = (rho·v^2/l) / mu·v/l^2 = rho·v·l/mu

where v is the mean fluid velocity; l the traveled length of fluid; rho the density of the fluid [Kg/m3]; and mu the dynamic viscosity of the fluid [N·s/m2=Kg/m/s].

Turbulent boundary layers tend to be able to sustain an adverse pressure gradient better (delaying flow separation and keeping it attached for as long as possible) than an equivalent laminar boundary layer. The more efficient mixing which occurs in a turbulent boundary layer, transports kinetic energy from the edge of the boundary layer to the low momentum flow at the solid surface, often preventing the separation which would occur for a laminar boundary layer under the same conditions. Thus, although the skin friction (parasitic drag) is increased, overall drag is greatly reduced, and performance likewise gained, while the usable angle of attack can be larger, thereby dramatically improving lift at slow speeds. This is the principle behind the dimpling on golf balls, or the fuzz on a tennis ball.

Smooth surfaceRough surface

Air is nonlinear, with no simple math solutions that explain turbulence or flow-attachment, obtained by computation of the basic model of fluid dynamics: the Navier-Stokes/Euler equations, with force boundary conditions. The Reynolds number can be obtained when one uses the nondimensional form of the incompressible Navier-Stokes equations:

rho·(dv/dt + v·Delta(v)) = −Delta(p) + mu·Delta^2(v) + f

where Delta is the gradient (rate of change with position); p is static pressure; and f other body forces.

By elementary Newtonian mechanics, upward lift must be accompanied by downwash with the wing redirecting air downwards. The main issue of flight is how a wing may generate substantial downwash; with downwash there is lift. Real wings fly because of vortex-shedding, and are lifted upwards as they fling air mass downwards.

Air flows all over the wing, not just above or under, but also transversally to both side ends and backwards, due to wing displacement, in the intrados (pressure surface below), and over the extrados (above) flow is inverted from sides towards the centre of the wing, affecting negatively its performance (increase of induced drag). Flow tend to neutralize pressure difference between intrados and extrados, and ends up forming the usual vortex on the ends of the wing that increase its wake.

A way to reduce the induced drag is placing vertical appendices (stabilos) in both ends, whose additional advantage is an increase upon the stabilizer effect on the rotating axis. Likewise, the larger the wingspan (keeping the same surface) and the farer the ends, the lesser influence they will have upon aerodynamic performance.

On the other hand, profile characteristics (torsion) are modified along the wingspan, through aerodynamic torsion while changing profile shapes keeping the angle of incidence; and geometric torsion varying the incidence of those profiles positively, avoiding that way aerofoils run into very low or negative incidence when central angle of attack changes.

Polar curve falcon vs paraglider

The wing is depicted by a range in velocities and configurations: maximum and minimum velocities, best glide ratio, minimum sink rate, etc, that are graphically plotted in a speed (both horizontal/straight and vertical/level) polar curve. Each wing has a different one.

Dune du Pyla

Aug 7, 2010

The Star Diaries

The Ijon Tichy chronicles

Seventh Voyage. Part 1. This is Idiotic

It was on a Monday, April second --I was cruising in the vicinity of Betelgeuse-- when a meteor no larger than a lima bean pierced the hull, shattered the drive regulator and part of the rudder, as a result of which the rocket lost all maneuverability. I put on my spacesuit, went outside and tried to fix the mechanism, but found I couldn't possibly attach the spare rudder --which I'd had the foresight to bring along-- without the help of another man. The constructors had foolishly designed the rocket in such a way, that it took one person to hold the head of the bolt in place with a wrench, and another to tighten the nut. I didn't realize this at first and spent several hours trying to grip the wrench with my feet while using both hands to screw on the nut at the other end. But I was getting nowhere, and had already missed lunch. Then finally, just as I almost succeeded, the wrench popped out from under my feet and went flying off into space. So not only had I accomplished nothing, but lost a valuable tool besides; I watched helplessly as it sailed away, growing smaller and smaller against the starry sky.

After a while the wrench returned in an elongated ellipse, but though it had now become a satellite of the rocket, it never got close enough for me to retrieve it. I went back inside and, sitting down to a modest supper, considered how best to extricate myself from this stupid situation. Meanwhile the ship flew on, straight ahead, its velocity steadily increasing, since my drive regulator too had been knocked out by that blasted meteor. It's true there were no heavenly bodies on course, but this headlong flight could hardly continue indefinitely. For a while I contained my anger, but then discovered, when starting to wash the dinner dishes, that the now-overheated atomic pile had ruined my very best cut of sirloin (I'd been keeping it in the freezer for Sunday). I momentarily lost my usually level head, burst into a volley of the vilest oaths and smashed a few plates. This did give me a certain satisfaction, but was hardly practical. In addition, the sirloin which I threw overboard, instead of drifting off into the void, didn't seem to want to leave the rocket and revolved about it, a second artificial satellite, which produced a brief eclipse of the sun every eleven minutes and four seconds. To calm my nerves I calculated till evening the components of its trajectory, as well as the orbital perturbation caused by the presence of the lost wrench. I figured out that for the next six million years the sirloin, rotating about the ship in a circular path, would lead the wrench, then catch up with it from behind and pass it again. Finally, exhausted by these computations, I went to bed. In the middle of the night I had the feeling someone was shaking me by the shoulder. I opened my eyes and saw a man standing over the bed; his face was strangely familiar, though I hadn't the faintest idea who this could be.

Polish sci-fi writer Stanislaw Lem began, catching in a space-time loop his amusing character Ijon Tichy, a set of "philosophical" short stories in The Star Diaries (1957; later there was Memoirs of a Space Traveler, in 1982; and The Futurological Congress, in 1971). Best known for his novel Solaris (1961).

Always word playing, like in The Cyberiad (1967), or did he meant Siberia, that known Russian and Soviet Union's penal colony for criminals and dissidents; or characters as Dr. Harry S. Totteles (named after Aristotle?), Boels E. Bu or professor L. Nardeau de Vince.

Wrote about virtual reality, nanotechnology, robots psychologically dysfunctional or the nature of intelligence, but above all, he was absolutely critical and despair of mankind, communication and understanding. For instance, in The Futurological Congress, Ijon Tichy finds himself in a dystopian world where hallucinogenic drugs have replaced reality.

No doubt outer space would be much more boring without satirical Ijon Tichy.

Jul 30, 2010

Marathon run

(reduced to figures)

1896 Marathon start
Despite the unrealistic belief of Pheidippides route from Marathon battlefield to Athens in 490BC that supports its legend, marathon run has written its own history. Neither the first modern Olympic games in 1896 were actually the first, as they had already been held since 1859. But the marathon race event did start in 1896 in Greece. Though the current distance of 42.195 Km was agreed in memory of Dorando Pietri's race in 1908 Olympics, that first one was originally of 40 Km, and then Spiridon Louis its first ever finisher, not Pheidippides, in 2h58'50".

Your performance in such a race is determined by concepts like VO2max or the lactate threshold.

1. VO2max is the aerobic (moderate intensity for long intervals) capacity, the maximum amount of oxygen you can consume, in ml/Kg body weight/min. That is, as you make more effort, your body increases its need of oxygen, but it just can hold up to an upper limit.

After training your body regenerates muscles and the number of cells increase, thus raising your rate of oxygen uptake level. For instance interval training.

You can estimate your VO2max peak according to Jack Daniels and Jimmy Gilbert formula (Oxygen Power. Performance Tables for Distance Runners, 1979):

VO2max = (-4.60 + 0.182258 * velocity + 0.000104 * velocity^2) / (0.8 + 0.1894393 * e^(-0.012778 * time) + 0.2989558 * e^(-0.1932605 * time))

where velocity is in meters/minute, and time in minutes as well. For instance, Spiridon Louis VO2max, according to his 40 Km marathon finish time, would be at 50.5 ml/Kg/min.
In 1979, Jack Daniels and Jimmy Gilbert published "Oxygen Power. Performance Tables for Distance Runners". This series of tables predicted all-out racing times for virtually every racing distance. Each performance time in the table is related to a VO2max index, called VDOT. The tables were generated using two regression equations: (1) relating oxygen consumption with velocity, and (2) predicting the amount of time one can run at a given percentage of VO2max. By combining these two equations, substituting VDOT indices, and looking for convergence for Newton-Raphson curve fitting analysis, one can then mathematically match up a predictable racing time expected at a given distance for someone having a particular VDOT index. The validity of these tables is strongly supported by looking at the known VO2max scores of some world record holders and their respective record times.

Inversely, from a known VO2max, racing and training paces can be infered.

2. Lactate threshold is the anaerobic (high intensity in a short interval) level at which it (lactate, or lactic acid) is faster produced by muscles than metabolized, thus starts accumulating in the blood. It is usually between 90-95% of your maximum heart rate. It can also be increased with the appropriate workout, like farleks.

Better training will improve your limits. And that also means better programmed, plus a balanced diet.

Slower run paces (i.e. at 75% of your speed in VO2max) metabolize fat better than faster ones. Quick efforts need quicker energy availability, that is glycogens. Glycogen produces glucose, which reacting with oxygen produces carbon dioxide plus water, and energy. When no more carbohydrates are available, fat is metabolized instead. A slower process, driving to decreasing performance.

Only a small proportion of your workout should be run at fast paces. Anaerobic does not use oxigen, therefore it is less efficient, and performance decreases faster.

Panathinaikon stadium

Jul 2, 2010

Greenhouse effect

A rough numerical estimation

If we consider the Total Solar Irradiance (radiation intensity) reaching Earth as an averaged 1365.5 [Watts/m2] (known as the Solar constant), according to the Stefan-Boltzmann law (that is infered from Planck's law of spectral radiance), a "black-bodied" (absorbs all incoming radiation) Sun's emittance (energy flowing per unit surface area and time) depends on its temperature at a epsilon·rho·T^4 rate, where epsilon=1 for black bodies, and rho states for the Stefan constant of proportionality, whose value is 5.67·10^-8 [W/m2·K4].

Thus, the total energy radiated (radiation power) by the Sun is Wt = Ws · (4·pi·radius^2), being Ws the Stefan-Boltzmann law applied on the surface of Sun. As an isotropic radiation (does not depend on direction), at Earth's distance Wt = We · (4·pi·distance^2), being We the solar irradiance measured on Earth (decreases with distance).

Combining both, the Sun's temperature is derived from Ws = rho·Ts^4 = We · (distance/radius)^2, then Ts = (We/rho)^0.25 · (distance/radius)^0.5 = (1365.5 [W/m2] /5.67·10^-8 [W/m2·K4])^0.25 · (149.5978707M [Km] / 696K [Km])^0.5 = 5775.44 [Kelvin] (actually 5780 K, or 5507º Centigrades), where the Sun-Earth distance is approx. 150 million Kms (that is 1 AU, astronomical unit), and Sun's radius is around 700 thousand Kms.

Wien's displacement law denotes that temperature as a function of the shorter wavelength (lambda) at which radiation is mostly emitted: lambda · T = 0.2898 [cm·K]. The Sun radiates its peak at lambda close to 500 nm (10E-9 m), that corresponds to the yellow. Graphically:

In order to deduce Earth's temperature, we must take into account its albedo, or rate of reflected radiation (what makes Earth visible from space), due to atmosphere (mostly clouds 23%, and air 4%), and Earth's surface (snow, sea, 4%), that may average to 30% as a whole.

Furthermore, Earth's surface average insolation (incoming solar radiation) is approximately Ws/4, averaged. This can be explained by the projection of Earth's "spheric" surface (4·pi·radius^2) as a circle of area pi·radius^2. Because any unit of area, due to Earth's rotation, is not always being insolated. At any time, half the Earth is on the dark side. And the rest depends on the direction of the incident radiation: that is, when they are perpendicular, you get the whole radiation; in any angle, you just get the cosine component (Lambertian cosine law).

The solid angle is then integrated as a function of sin(theta)·d(theta) from 0 to pi/2 (half insolated sphere); do not forget to multiply it by cos(theta), and by d(phi) from 0 to 2·pi (the whole circumference). The latter is just 2·pi; and the former (considering u=sin(theta) and du=cos(theta)·d(theta)) sums 1/2; thus totals 2·pi/2 = pi. So finally get the same 1/4 ratio, as the whole sphere subtends a 4·pi angle.

Then, Earth's temperature might be deducted from the former black body relationship Te = ((1-albedo)·(Ws/4) / rho)^0.25 = ((1-30%)·(1365.5/4) [W/m2] / 5.67·10^-8 [W/m2·K4])^0.25 = 254.79 [Kelvin] = -18.21º [C]. But that's obviously untrue, because of Greenhouse effect.

Incoming solar radiation energy is 95% in the 300 to 3000 nm wavelength range: 10% in the UV (ultraviolet, 300-400 nm); 40% in the visible (400-700 nm), and 50% in the infrared (700-3000 nm). Along with the albedo, not all incoming radiation is transmitted through atmosphere to the ground because of:

- Clouds (3%) and atmosphere (21%) partially absorb it, then converted into heat and causing the emission of their own radiation in all directions. Basically shortwaves are by H2O vapour, CO2 carbon dioxide, methane CH4, nitrous oxide N2O; and ultraviolets mostly by ozone O3, and nitrogen N2 and oxygen O2; also aerosols CFCs contribute. These are the main Greenhouse gases.

- Scattering takes place when small particles and gas molecules diffuse part of the incoming solar radiation in random directions, even back to space; this causes the sky bluish color (Rayleigh's dependance on wavelength^-4) and white clouds (Mie's on wavelength^-1).

Earth emittance is 90% in the range from 3 to 30 microm, with its peak around 10 microm. It corresponds to a black body temperature around 300 K, close to the 288 K average surface temperature.

One step further, if Earth emits energy at a T'e=288 K temperature, then yields an emissivity epsilon = (1-albedo)·(Ws/4) / rho·T'e^4 = (1-30%)·(1365.5/4) [W/m2] / 5.67·10^-8 [W/m2·K4] · 288^4 [K4] = 0.61.

Over half of Earth’s emittance lies in the infrared range from 8 to 14 microm. Without this window, the Earth would become too warm to support life.

Let's build a very simplified model to outline this temperature increase.

If we consider Earth as a black body (thus ideal emissivity = 1), and we close that infrared window, then when thermodynamical equilibrium is again reached, power emitted per unit surface outside is all but the selected window: Pem = Integral[ Radiance(frequency, Temperature) dfreq ] avoiding the infrared range.

The power absorbed per unit surface Pab will be the sum of the known total black body radiance rho·Te^4, plus the Integral[ Radiance(frequency, Temperature) dfreq ] over the window range.

Given that in thermal equilibrium net energy flux must be null, Pem = Pab :

Int[ Rad(freq,Te') dfreq ]{freq2..infinite} + Int[ Rad(freq,Te') dfreq ]{0..freq1} = rho·Te^4 + Int[ Rad(freq,Te') dfreq ]{freq1..freq2}

where Te' is the new equilibrium temperature. We can express this in terms of total radiance :

Int[ Rad(freq,Te') dfreq ]{0..infinite} - Int[ Rad(freq,Te') dfreq ]{freq1..freq2} = rho·Te'^4 - Int[ Rad(freq,Te') dfreq ]{freq1..freq2} = rho·Te^4 + Int[ Rad(freq,Te') dfreq ]{freq1..freq2}

So :

rho·Te'^4 - rho·Te^4 = 2 · Int[ Rad(freq,Te') dfreq ]{freq1..freq2}

Solving the integral :

Int[ Rad(freq,Te') dfreq ]{freq1..freq2} = Int[ (c/4) · (8·pi·freq^2·Kb·Te' / c^3) dfreq ]{freq1..freq2}

here Kb refers to the Boltzmann constant 1.38·10^-23 [J/K].

We are using the Rayleigh-Jeans expression instead of Planck's to afford easier calculations. But take into account that this approximation is only valid for long wavelengths, due to this curve tends to infinite as wavelengths become shorter. Planck's, instead, falls exponentially at short wavelengths.

The c/4 factor is because Planck's (and Rayleigh-Jeans) law stands for density (unit volume) of radiant energy, but what is actually measured is radiant emittance or spectral radiance (per unit surface), which depends as well on speed of outgoing radiation.

Finally :

Int[ (c/4) · (8·pi·freq^2·Kb·Te' / c^3) dfreq ]{freq1..freq2} = 2·pi·(freq2-freq1)^3·Kb·Te' / 3·c^2

Thus :

rho·Te'^4 - rho·Te^4 = 2 · [ 2·pi·(freq2-freq1)^3·Kb·Te' / 3·c^2 ] = 4·pi·(freq2-freq1)^3·Kb·Te' / 3·c^2

Rearranging the latter expression after dividing it by rho·Te^4 :

(Te'/Te)^4 = 1 + 4·pi·(freq2-freq1)^3·Kb·Te' / 3·c^2·rho·Te^4 = 1 + ( (4·pi·(freq2-freq1)^3·Kb / 3·c^2·rho·Te^3) · (Te'/Te) ) = 1 + Constant · (Te'/Te)

because Constant is adimensional, and its value is :

Constant = 4·pi·(freq2-freq1)^3·Kb / 3·c^2·rho·Te^3 = 4·pi·((c/8 [microm])-(c/14 [microm]))^Kb·Te / 3·c^2·((1-albedo)·(Ws/4)) = 4·pi·(3.75·10^13-2.14·10^13)^3·1.38·10^-23 [J/K]·254.79 [K] / 3·(299792458 [m/s])^2·238.9625 [W/m2] = 2.841

Constant's value is "almost" small, so we "can" approximate :

(Te'/Te) = (1 + Constant·(Te'/Te))^(1/4) ~ 1 + (1/4)·Constant·(Te'/Te)

Then :

(1 - (1/4)·Constant) · (Te'/Te) ~ 1 ; Te' ~ (1 - (1/4)·Constant)^-1 · Te = (1 / (1 - (1/4)·Constant)) · Te

Here we will make a new approximation, due to the mentioned trend to infinite for short wavelengths. A geometric series can be  expressed as Sum[ c·r^i ]{i=0..infinite} = c / (1-r), being r the ratio. In our case c = 1 and r = (1/4)·Constant. We take just the first terms of the series and discard from the quadratic term onwards, and :

Te' ~ (1 + (1/4)·Constant) · Te = 435.75 [Kelvin]

That way, Earth's temperature would approximately be increased up to 162.75º Centigrades.

Jun 29, 2010

The Matrix

The Matrix : William Gibson is known to be the originator of the word "cyberspace", and he is the genuine creator of the Matrix, in his first novel Neuromancer (1984); and the followings, included within the Sprawl trilogy, Count Zero (1986) and Mona Lisa Overdrive (1988):
A year here and he still dreamed of cyberspace, hope fading nightly. All the speed he took, all the turns he'd taken and the corners he'd cut in Night City, and still he'd see the matrix in his sleep, bright lattices of logic unfolding across that colorless void....
Authoring as well the short story Johnny Mnemonic (1981) -whose film was also featured by Keanu Reeves in 1995-, and scripts for The X-Files, Gibson outlined the characters of Neo (Bobby Newmark in Count Zero) and Trinity (Molly, in Johnny Mnemonic and Neuromancer), and aroused such ideas like direct downloading precise instructions into their minds on how to fly an helicopter or mastering Kung-fu.

The Matrix plays with the concept of Technological Singularity, expression that was spreaded by the mathematician Vernor Vinge to describe the turning point in which social and technological development are due to:

- creation of a computer that surpasses human capacity,
- neural network aware of own consciousness,
- interaction between man and machine that allows to develop its skills,
- or through biological manipulation of human beings

The Matrix also have similarities to Akira (
Katsuhiro Otomo, 1988) and GITS (Ghost in the Shell, by Masanori Ota, under pseudonym Masamune Shirow, hit the screen in 1995), what was actually admited by the Wachowski Bros in The Animatrix. Indeed there are several "borrowed" scenes and story lines. Likewise Dark City (Alex Proyas, 1998) has too many resemblances to The Matrix (even used same sets), as well as The Thirteenth Floor (Josef Rusnak, 1999), that was also premiered before. Both are great movies, despite they have not gotten the same acknowledgment as The Matrix did.

But if there do exists a source from where obvious parallelisms with The Matrix come up, are the comic series The Invisibles, from Grant Morrison: the look of Morpheus and the character King Mob; the famous jump of Neo's training with Morpheus, or Dane Jack Frost and Tom O'Bedlan in the comic; or the interrogation scene between Agent Smith and Morpheus, or Sir Miles and King Mob in The Invisibles, to mention a few. There are even some back references, from Morrison to The Matrix, in retaliation, like the "born" (or awakening) scene of Neo, after deciding to fall into the rabbit hole. In fact, King Mob (Morpheus) bears great resemblance to Morrison.

When Morpheus depicts Matrix to Neo, he sentences "Welcome, to the desert of the real". This same sentence misreads the one in sociologist Jean Baudrillard's book Simulacra and Simulation, concerned about how technology influences social relations, hyperreality denying reality.

Outstanding visual effects, specially John Gaeta's "Bullet Time" technique (even its real owner is Michel Gondry, in a Rolling Stones video), The Matrix has been the film that branded the new science-fiction aesthetics.

Jun 24, 2010

Soccer ball

Theoretical basics : A (classical) soccer ball is made up of 20 hexagons and 12 pentagons, distributed so that 5 hexagons surround each pentagon, and each hexagon is surrounded by 3 pentagons alternated with 3 hexagons. A surface like this cannot be geometrically flat, it will always be curve.

Both pentagons and hexagons have sides of equal length, and this distance is the same of its radius (from any vertex to its center). Given the side length L, we can obtain the radius of curvature R of the soccer ball.

To deduce the radius R of the sphere we use the relation between it and the perimeter P of its equator 2 · pi · R = P, where the pi number is approximately 3.141592. Due to the layout of hexagons over the ball surface, you can see that the perimeter is equal to 15 times the length L. Therefore, if the circle has 360 degrees, then each side L of the polygon corresponds to 360º / 15 = 24º of circumference.

If we take as 1 unit (any) the flat length L' of the polygons' sides (hexagon and pentagon), then the length of its side on the curved surface of the ball (L) will be higher.

Using the expression for trigonometric sine of an angle, we can calculate the radius R of the ball sin(24º) = L' / R, so R = L' / sin(24º) = 1 / 0.4067366430758 = 2.458593335574 units. We can also infer the length L of an arc of circumference, using the same expression as for the perimeter P, since P is proportional to 2 · pi · (360 degrees), L = 2 · pi · (24 º / 360 º) · R = 1.029852953906 units.

Modeling : Let's build the soccer ball upon intersections with a sphere, whose sections and radii of curvature are different, depending on hexagons or pentagons. These caps are generated from revolution curves.

1. First the hexagon, in the Front view, create a circle of radius 1 and 6 sections from a primitive (Objects folder), and place it in the origin (coordinates) with the grid magnet (Alt key). From the Right view, now draw a spline with CVs (Control Vertex): the first point with a magnet on the upper Edit Point of the circle (Ctrl key); the second point is placed with a shift, in relative coordinates, to the position r0 .05 0; the third point of the curve at 0 .05 -.1; the fourth at 0 .1 -.3; the fifth at 0 .1 -.4; and the sixth and last in absolute coordinates at position a0 .3 0. Once we have the spline, we place its pivot in the origin (XForm folder; Pivot icon) with command a0 0 0. Revolution now the curve over the Y axis and generate a surface of 12 sections (Surface folder; Revolve icon). Then template the generating curve and the circle (ObjectDisplay menu; Toggle Template option or Alt+T keys). Now move the generated surface (removing its Construction History) to the relative position r0 2.158593335574, which is the ball radius R minus the height of the sphere cap 0.3. In this position, move the pivot again, now from the surface to the origin: a0 0 0; because when we rotate the cap, we will do it on the center of the ball (the origin of coordinates).

To get the positions of the hexagons that form the soccer ball, we calculate the offset angles with respect to the original position.

Positions of the hexagons near the equator of the ball have their center shifted by an angle, in the X coordinate, proportional to half the apothem a of the hexagon. If the apothem a of the hexagon is, by Pythagoras theorem, a^2 = L'^2 - (L'/2)^2, then a = .8660254037844 units. The angle proportional to the apothem a, with respect to the 24º angle that corresponds to the hexagon's side length L', is a · 24° / L' = 20.78460969083º. Therefore, half the apothem a represents half of this angle: 10.39230484541º.

Rotate (XForm folder; Rotate icon) the surface of the first hexagon, in relative, and over the X coordinate, a r10.39230484541 angle. Duplicate the sphere cap of the first hexagon (Edit menu; Duplicate Object option), and rotate in relative coordinates the copy around the X axis by an equivalent angle to twice (2) the apothem a of the hexagon 41.56921938165.

The position of the third hexagon is shifted (rotated) over the first, and the Z axis, an angle equal to 3/4 the distance between the hexagon's vertexes; that is 1.5 · L = 1.5 · 24° = 36°. And over the X axis an angle corresponding to the apothem a. Let's duplicate then the first surface, and rotate the third copy in relative -20.78460969083 0 36.

The fourth hexagon, duplicate this time the third cap, and rotate this fourth surface -41.56921938165 over the X axis.

The remaining surfaces, we obtain them duplicating (Edit menu; Duplicate Object option) the four we already built, with a rotation over the Z axis by 72º, and a number of 4 duplications. This will generate the rest of sphere caps corresponding to hexagons, and thus closing the soccer ball surface.

2. The process to build the surfaces of pentagons is similar, but must take into account that the initial reference circle radius will be smaller than that of the hexagon. The side L' must be the same for the two polygons. Therefore, the radius r is calculated using sin(36º) = (L'/2) / r, where 36° matches half the angle of the arc that corresponds to a one side of the pentagon (360º / 5 = 72º). Thus, the radius r = (L'/2) / sin(36º) = 0.5 / 0.5877852522925 = 0.850650808352 units.

The pentagon, in the Front view, first create a circle of 5 sections from a primitive, and place it in the origin with the grid magnet. From the Right view, now draw a spline with CVs: the first point with a magnet on the upper Edit Point of the circle; place the second point shifted, in relative coordinates, to the position r0 .1 0; the third point of the curve at 0 .05 -.1; the fourth at 0 .05 -.3; the fifth at 0 .1 -.3; and the sixth and last in absolute coordinates at position a0 .3 0. Once we have the spline, we place its pivot in the origin with command a0 0 0. Revolution now the curve over the Y axis and generate a surface of 10 sections. Then template the generating curve and the circle. Now move the generated surface (removing its Construction History) to the relative position r0 2.158593335574. In this position, move the pivot again, now from the surface to the origin: a0 0 0; because when we rotate the cap, we will do it on the center of the ball (the origin of coordinates).

To get the positions of the pentagons that form the soccer ball, we calculate the offset angles with respect to the original position.

Positions of the pentagons at the poles of the sphere have their center shifted by a 90º angle, in the X coordinate. Duplicate the surface of the first pentagon, rotating it 90º over the X axis.

The position of the rest of pentagons is rotated, with respect to the X coordinate, an angle proportional to the apothem a' of the pentagon, plus one half the apothem a of the hexagon; and over the Z axis an angle equivalent to 3/4 the distance between the hexagon's vertexes; that is 1.5 · L = 1.5 · 24° = 36°. If the apothem a' of the pentagon is, by Pythagoras theorem, a'^2 = r^2 - (L'/2)^2, then a' = .6881909602356 units. The angle proportional to the apothem a', with respect to the 24º angle that corresponds to the pentagon's side length L', is a' · 24° / L' = 16.51658305º.

Rotate the surface of the second pentagon, in relative, and over the X coordinate, a r26.9088879 angle. To generate the remaining surfaces of the pentagons, on the upper hemisphere of the soccer ball, duplicate the second pentagon, with a rotation over the Z axis by 72º, and a number of 4 duplications.

The surfaces of the pentagons regarding to the lower hemisphere, we get them grouping (Edit menu, Group option) and duplicating them, with a Scale of -1 to the Z axis (Mirror).