Tuesday, August 31, 2010

Yom Kippur and the Cosworth brigade

Many Formula 1 histories make the critical error of treating motorsport as if it is an activity hermetically sealed from the rest of the world. A far more interesting approach is to understand the coupled-evolution of Formula 1 with more general economic, technological, political, environmental, sociological and cultural trends.

As a case in point, consider Formula 1 in the 1970s (and early 1980s). This was an era in which the major manufacturers were largely absent from the sport, and consequently it was an era in which a small group of people could design a chassis, lease some Cosworth engines, and go racing. Ferrari were knocking about to disturb the Cosworth hegemony, but the racing was extremely competitive, with nine different drivers winning in the 1975 season, and four different drivers from four different teams winning the championship between 1978 and 1981.

But why were the manufacturers so manifestly absent from the sport during this decade? Why did the 1960s not culminate with the influx of manufacturer investment and television coverage ultimately seen in the early 1980s?

The one factor which dominates the explanation of all seventies phenomena was the economic stagnation. This clearly deterred manufacturers from involvement in glamorous forms of expenditure, and the slow rate of economic growth retarded the progress of all technology, including the telecommunications technology which Formula 1 would ultimately depend upon.

To be specific, one can attribute the competitiveness of Formula 1 in the late 1970s to the Yom Kippur war of 1973. This commenced on 6th October 1973, the holiest day in the Jewish calendar, when Syria and Egypt jointly attacked Israel. It was only the support of the United States, which sent $2 billion of arms, that enabled Israel to repel its assailants. However, the support that the United States provided to Israel triggered OPEC, the Arab oil-production cartel, into raising oil prices and cutting oil production, and this caused the worldwide oil crisis of 1973-1974.

Prior to the crisis, the average rate of economic growth in the West had been around 5 percent; after the crisis, growth reduced to zero, and inflation rose to around 10 percent. The late 1970s, then, was an era of stagflation, and without this period of stagflation, the influx of major manufacturers, eventually seen from about 1982 onwards, may well have commenced circa 1976.

As a postscript, one might ask whether four different drivers from four different teams have won the World Championship over four consecutive years any time since. They have indeed, the years in question being 2006-2009...

Sunday, August 29, 2010

Lewis Hamilton's second Spa victory

"I'm losing my efficiency." Michael Schumacher, August 29th 2010.

The most crucial contribution to the result of today's Belgian Grand Prix was the lobbying that McLaren and Mercedes made to the FIA four weeks ago.

To recall, the subject of their angst was the degree of front-wing deflection clearly visible under aerodynamic load on the cars of Red Bull Racing and Scuderia Ferrari. The FIA were at the time testing for such aeroelasticity by merely requiring that each front-wing endplate deflect by no more than 10mm under a load of 50kg. For this weekend's Belgian Grand Prix, however, the requirement was increased to a deflection of no more than 20mm under a load of 100kg. In other words, the FIA were testing for non-linear deformation of the front-wing under loads greater than 50kg.

"The front wings we have here are the same as we had in Budapest," claimed Red Bull Racing Team Principal Christian Horner on Thursday. "I can't say [the new test] will affect us any more or less than any other team."

Well, raw pace is generally best judged by the lap-times set in Q2 on Saturday, when the cars carry a low fuel load, and before they adopt race-distance settings in Q3. In Q2 at the Hungaroring, the previous race, Lewis Hamilton was 1.3 secs behind the Red Bulls; in Q2 at Spa, Lewis was 1 second faster than the Red Bulls. In other words, despite a two-week factory shutdown, there was a net 2.3 second swing between McLaren and Red Bull. Ferrari's Fernando Alonso, meanwhile, the closest challenger to the Red Bulls in Hungary, was down in tenth place after qualifying.

Thus, whilst the high-speed requirements of Spa were expected to suit McLaren to a greater degree, and whilst their F-duct clearly provided an advantage here, it's fair to say that the new deflection tests also seemed to play a significant role. McLaren Team Principal Martin Whitmarsh duly noted that "we can all see the wings are in a different stiffness and positional domain than they had been in previous races."

As Craig Scarborough explains, at 70mph - the speed of the slowest corner on an F1 track - each half of the front wing is already producing 70kg of downforce. Hence, one might hypothesise that prior to Spa, the Red Bulls and Ferraris were using non-linear front-wing aeroelasticity, triggered from about 70kg of downforce onwards.

Nevertheless, Lewis Hamilton's victory was no walk in the Ardennes, for it included an off-track "excursion" through the Rivage gravel track when rain began falling once again in the late stages of the race. As Lewis slid into the gravel, it was instantly clear that he needed to retain sufficient speed not to bog down, yet also faced the exigency of rubbing-off sufficient speed not to hit the tyre wall. For fractions of a second, either seemed possible, and there were momentary echoes of both China 2007, when he ignominiously bogged-down in the pit-lane gravel-trap, and Spain this year, when a puncture took him at a shallow angle into the wall. On this occasion, however, he just brushed the canvassed tyre barrier before rejoining, still in the lead.

A short time later, after a pit-stop for intermediate tyres, a couple of laps behind the safety car, and some relatively uneventful final laps, Lewis crossed the line to win at Spa Francorchamps for the second time.

The end of the race may have been uneventful, but the start was anything but, poleman Webber suffering clutch difficulties, and being instantly swamped by those around him. Into La Source it was Hamilton first, with Button attempting to snatch second-place on the inside of Kubica. Jenson, however, couldn't quite get the traction he needed on the tighter line, and although he got his left-front alongside Kubica's right-rear, he had to briefly lift and cede second on the drop to Eau Rouge. Vettel had been caught on the outside of Kubica and Button into La Source, and was fourth, with Massa fifth, and Webber sixth. On the Kemmel straight Sutil's low-drag Force India drove clean down the outside of Webber, but Adrian outbraked himself somewhat, and Webber swiftly reclaimed the place through the left-handed component of Les Combes.

As the train of cars tore a sinuous path down through the valley, light rain was clearly falling, but not to the degree which the drivers encountered when the reached the heavy braking zone at the end of the lap. Here, almost the entire field appeared to run wide under braking into the bus-stop chicane, Hamilton first, then Button and Kubica as one, Jenson initially attempting to out-brake the Renault down the inside. The McLaren and Renault ran wide in parallel, but here Button crucially snagged his left-front endplate on Kubica, a small detail which was to have serious repercussions for the Briton later in the race.

As all hell broke loose, Jenson was just about the only driver who decided to double-back on himself and rejoin the road at the apex of the corner, the others taking the same expeditious short-cut which had got Lewis Hamilton into so much hot water back in 2008. By so doing, Kubica, Vettel and Massa all rejoined ahead of Button.

At about this moment, Rubens Barrichello demonstrated the benefit of 300 Grands Prix worth of experience, by pile-driving his Williams into the right-rear of Fernando Alonso's Ferrari, who was at the time carefully traversing the perimeter of the track on the outside of Sutil. Whilst Rubens cleanly severed the left-front of his car from the chassis, it appears that the driveshaft and rear suspension of the Ferrari are made of carbon nano-tubes, for Fernando's car was unscathed, and he merely trundled into the pits to attach a set of intermediates.

Button, meanwhile, retook Massa down the start-finish straight, and slipped inside Vettel into La Source. Whether this was due to excessive caution on the part of Massa and Vettel, or because they were voluntarily ceding their places back, is unknown. Whatever, Jenson was back into third, and as the cars tip-toed through Eau Rouge in the damp conditions, Kubica had a tank-slapper through the left-kink on the incline, opposite-locking his Renault onto the tarmac run-off. As Robert lifted off and rejoined, Button was past into second. Vettel then tried to follow the McLaren through round the outside of the right-kink onto Kemmel, only to find Kubica putting him onto the grass. Sebastian lifted off, and whilst still partially alongside the Renault, furiously gesticulated at Kubica.

Into Les Combes on lap two, then, it was Hamilton, Button, Kubica, Vettel, Massa and Webber. As the rain continued to fall, Webber took Massa on the outside going into Rivage, and was right behind team-mate Vettel when the pace car was triggered, without, it seems, any clear justification.

The slick-shod field gratefully accepted the opportunity to spend a lap behind the pace-car while the rain passed, and as the race re-started on lap 4, Kubica was forced to take a tighter line into La Source to defend his place from Vettel. The Renault slid marginally wide at the apex, however, and Vettel was into third.

The rather intemperately tempered Mr Vettel was thence able to close on the McLaren of Jenson Button, who was suffering a handling imblance due to his first-lap front-wing damage. Sebastian struggled for some laps to find a passing opportunity, hitting the rev-limiter in the McLaren's slipstream on the climb to Les Combes. Finally, however, on lap 16, Jenson was rather tardy through Stavelot, and Sebastian was tucked indecently close to the McLaren's rear-end through Blanchimont, preparing to make his move into the chicane.

When the moment came, however, Button predictably covered the inside line, and after briefly contemplating a move permissible only in Batman Begins, Vettel tried whipping across to the outside, only to loose control "over a bump". Presumably, this was the same bump over which Vettel had lost control trying to pass Webber in Turkey, for the Red Bull slewed sideways in a similar manner, punching a hole through the flank of the McLaren, which steamed into retirement. Vettel, meanwhile, plunged into a spiral of drive-through penalties, pit-stops, punctures and post-race inquisitions.

Beyond the main storyline, we also had an exciting battle between the Mercedes drivers, Rosberg and Schumacher. Rosberg had led initially, but after losing an encounter with Vitaly Petrov on lap 11, found his team-mate slicing across his bows, removing a small part of the left front-wing in a trail of golden sparks. On the final re-start, Rosberg outbraked Schumacher into Les Combes, and as Schumacher tried to hang on into the left-hander, Rosberg calmly drove him off the road. Tasty.

The final word, however, goes to Michael, who in comparing the fourteen places he made up in today's race, to the sixteen he recovered to win the 1995 race, reflected that "I'm losing my efficiency."

Losing My Efficiency was, of course, the less well-known B-side to REM's 1991 hit, Losing My Religion.

Friday, August 27, 2010

The orogenesis of Spa-Francorchamps

One of the more surprising conclusions to be drawn from modern geology, is that the greatest motor racing circuit on the current Formula 1 calendar, is the combined result of orogenesis, localised plastic strain, and periglacial erosion.

Spa-Francorchamps is a rich seam of asphalt found within the Ardennes Massif, a region created by tectonic uplift between 345 and 225 million years ago. Counter-intuitively, perhaps, the Ardennes was not covered by ice during the last ice age (a reassuringly decimal multiple of 10,000 years ago). It was, however, on the border of the glaciated region, and by definition was therefore affected by the less dramatic contribution of periglacial processes, which seem largely to involve permafrost and the creation of scree.

Thus, the Ardennes is characterised by deep-sided valleys, not because of glacial erosion, but because the tectonic uplift raised all the rivers high above their base level, and under such conditions downcutting will take place faster than lateral erosion, creating steep, V-shaped valleys.

During orogeny (mountain-building), the sedimentary rocks in the Ardennes underwent substantial compressive loads. Now, under protracted load, rock tends to suffer irreversible 'plastic' deformation. Jean-Pierre Burg emphasises the role of plastic strain instability within the Ardennes, arguing in Tectonophysics 309 that shear instabilities "appear necessary to explain complex, even contradictory relationships between cleavage, folds, thrusts, coeval normal faults and a downward decreasing strain and displacement."

One might, in fact, represent the stresses and strains of the Ardennes Massif with exactly the same type of Finite Element Analysis software, also used to design the conveniently deformable Formula 1 cars which will caress its skin this very weekend.

Red Bull and multiphysics simulation

Multiphysics simulation software enables the user to represent the interaction between physical systems which, in isolation, would be represented by quite different mathematical models. For example, to model Fluid-Structure Interaction (FSI), such as that involved in aeroelasticity, it is necessary to employ software which uses fluid mechanics to model the airflow, and solid mechanics to represent the (hyper-)elastic deformation of appendages interacting with the airflow. In principle, it is possible for a multiphysics solver to contain Finite Element Analysis (FEA) and Computational Fluid Dynamics (CFD) code in the same package.

In March 2009, Steve Nevey, Business Development Manager and Technical Consultant at Red Bull Racing, said that Red Bull Racing were looking to develop a multiphysics simulator. In December of 2009, Racecar Engineering interviewed Steve Nevey about Red Bull's technical partnership with MSC Software, purveyors of, amongst other things, multiphysics simulation software. Steve was asked where he saw the greatest potential for future development, and this is the answer he gave:

The use of composite materials continues to dominate much of the F1 car, and although the computational modelling is already sophisticated, further improvements in defining lay-ups, representing material performance, and modelling failure analysis are anticipated over the coming seasons. Multi-physics or multi-discipline simulation is another important area in which we anticipate further progress. Using MSCs latest MD (multi-discipline) software versions of Nastran and Adams, we already combine mechanism and deformable finite element simulations. We also increasingly use aerodynamic output directly from CFD analysis to generate more accurate loads for the structural simulations. There are rule restrictions to limit this, but multi-physics coupling of these effects allows us to legally enhance the performance of deformable components, for example to optimise down-force and drag characteristics for flexible wing components. Chaining the various analyses stages is time consuming and prone to error.

All of which may explain the following in August 2010:

Amid mounting speculation that the lower front wings observed on the Red Bull Racing and Ferrari cars are being put into use through clever flexing of the car's floor...the FIA is to introduce extra tests on the underside area of the car...From the Italian Grand Prix it is understood that the skid block can comprise of no more than two pieces, and that no piece of the skid block can be less than one metre long. A number of teams are understood to use several sections of skid block on the underside of their floor. To further ensure teams are not deflecting the floor, from Monza all joints, bearing pivots and any other form of articulation must also be fixed. Autosport.com, 26th August 2010.

Tuesday, August 24, 2010

Initial thoughts on combustion

The internal combustion engine in a Formula 1 car is basically just a heat engine, which uses a venerable thermodynamic process called the Otto cycle to extract useful work from the release of chemical internal energy. The most interesting part of this process, however, is the combustion stage, and to understand this we need to understand some thermodynamics and some shock physics. In particular, we need to understand a 2-dimensional surface in 3-dimensional thermodynamical space, called the Equation of State, and a curve in 5-dimensional shock space, called the Hugoniot curve.

Thermodynamics is something of a curious subject, for in many ways it deals directly with only the equilibrium states of macroscopic systems, rather than the dynamical processes which interpolate between them. Recall that the equilibrium states of a system are the states in which the system is spatially homogeneous, and the pressure, density and temperature of the system can be characterised by single values. Here we need to recall also the distinction between microstates and macrostates:

A microstate is the complete, detailed description of a system, which specifies the components of position and the components of momentum, for all the particles in the system. A system consisting of n particles has a 6n-dimensional state space, called the phase space, and each microstate corresponds to a point in this 6n-dimensional space.

In contrast, a macrostate is a collection of empirically indistinguishable microstates. Now, a macrostate can be either an equilibrium macrostate, or a non-equilibrium macrostate. An equilibrium macrostate is uniquely specified by the homogeneous temperature, pressure and density of the system. In contrast, a non-equilibrium macrostate must be specified by something like the Boltzmann distribution function f(q,v), where q denotes position and v denotes velocity. This distribution specifies the velocity profile of the system (the distribution in v), and the distribution of the particles in space. The idea is that q and v should each be partitioned into macroscopically small, but microscopically large cells, and a macrostate of the n-particle system can then be specified by the number of particles in each cell. The equilibrium macrostates then correspond to the special class of Boltzmann distribution functions in which the particles are uniformly distributed in space, and the velocity profile has a Maxwell-Boltzmann distribution. Systems with different equilibrium temperatures simply have Maxwell-Boltzmann velocity distributions with different mean values.

The entire 6n-dimensional phase space of a system can be exhaustively partitioned into non-overlapping macrostates, some of which are non-equilibrium macrostates, and some of which are equilibrium macrostates. The system can also be stratified into a 1-parameter family of 6n-1 dimensional hypersurfaces, each consisting of states with the same internal energy E. Assuming the system is unconstrained in space, then within each constant energy hypersurface there is a unique equilibrium macrostate, and this macrostate will have a volume overwhelmingly larger than any of the other macrostates on that hypersurface. The entropy of a macrostate is proportional to the volume of the macrostate, hence each equilibrium macrostate is the highest entropy macrostate on its respective constant energy hypersurface. Non-equilibrium macrostates tend to evolve towards equilibrium macrostates, in accordance with the Second Law of thermodynamics.

If we loosen the assumption that the system is unconstrained in space, then it is no longer true to say that there is a unique equilibrium macrostate for each value of the system's internal energy E. Instead, for each fixed volume within which the system is bounded, and for each internal energy of the system bounded within that volume, there is a unique equilibrium macrostate.

Now, to return to our main expository objective, the Equation of State P(V,T), is an expression which links the pressure, volume (or density) and temperature of a material's equilibrium macrostates. Each different substance has its own Equation of State. In graphical terms, the Equation of State can be represented by a two-dimensional surface which specifies the pressure corresponding to each possible combination of volume (or density) and temperature. The temperature T determines the (internal) energy E hypersurface of phase space, and the curve across the Equation of State surface at a fixed volume V, specifies the pressure of the volume-V system in an equilibrium macrostate, at any possible temperature.

Here, however, we should add that the Equation of State is not necessarily restricted to the global equilibrium states of a system. Even in a spatially inhomogeneous substance, if a macroscopically small but microscopically large region around every point of the substance has equilibrated, then the system is said to be in local thermodynamic equilibrium (LTE). The thermodynamical relations which apply to global equilibrium states, can also apply to the fields which represent pressure p(x), density ρ(x), and temperature T(x), in a substance out of global equilibrium.

So that's the Equation of State. Now, the release of chemical energy in a combustible gas, can be idealised by representing the region in which combustion occurs to be infinitely thin, and the combustion to be instantaneous. This, however, introduces a discontinuity into the flow, and the discontinuity is called the reaction front, or combustion front. The presence of the discontinuity entails that the equations of conservation of mass, momentum and energy, normally used in gas dynamics, have to be replaced by the Rankine-Hugoniot jump relations.

By combining the jump relations with the Equation of State of the gas in question, we obtain a curve, centred on the initial pressure and volume of the gas, called the Hugoniot curve. This curve doesn't specify the sequence of states of the gas during combustion; rather, each point specifies a possible post-combustion equilibrium macrostate of the gas.

Strictly speaking, the curve exists in a five-dimensional space in which the coordinates are pressure, volume (or density), internal energy, speed of the discontinuity Us, and the particle speed after the passage of the discontinuity up. Typically, however, the curve will be projected onto a plane, such as the P-V plane.

The Hugoniot curve can be divided into two branches, the upper and the lower. The upper branch specifies those states reached through detonation, in which the discontinuity becomes a supersonic shock compression, whilst the lower branch specifies those states reached through a so-called deflagration, in which the discontinuity moves with subsonic speed, via thermal conductivity. It is only the latter process which engine designers consider to be a useful form of combustion.

In fact, a detailed analysis of the combustion fronts in deflagration reveal not an infinitely thin layer, but a two-zone structure, with a leading pre-heat zone in which the temperature is raised by conduction, and a trailing reaction zone. The portion of the flame ahead of the reaction zone is typically five times as thick as the reaction zone itself.

All of which constitutes enough for now.

Sunday, August 15, 2010

Max's trip to the doctor

Thursday March 10th, 2005

Confound that little hairy mongrel excuse for a man! Stoddart has had the temerity to call for my resignation! Well, we'll see about that; pawns can be troublesome, but their alliances are easily broken.

Meanwhile, time to meet my new GP, a tryst postponed for some months now by the indulgent maternity leave granted to those on the public sector payroll. My new purveyor of potions, prescriptions, pharmaceutical products and placebos, is a slim woman ('Annabel'), with shoulder-length blonde hair, and a low-cut top, who greets me with the type of weary professional smile reserved for 4pm on a Thursday afternoon. Glossy inkjet photos of her newly-minted progeny festoon the desktop wall.

Check my blood-pressure? Why, of course doctor, let me roll my shirt-sleeve up.

Annabel bends over me to wrap the sphygmomanometer's cuff around my upper arm, and velcros it in place with a firmness that suggests the good doctor is not unfamiliar with the joys of bondage. As she does so, I am unavoidably presented with a scopophilic view of this medical expert's shameless mammaries, swaying in pendulous harmony like a couple of tan-coloured, water-filled balloons, the perimeters of these bosom buddies framed within her deep-cut neckline, like a pair of white-wine-glasses in an Argos catalogue.

Perhaps in years to come, some medical statistician's meticulous, retrospective study will discover an inexplicable upward bias in the blood pressure of Dr Annabel's male patients, and much Bayesian analysis will thence be devoted to the deconvolution of this anomaly. For now, however, the only manipulations are those applied by Annabel's healing fingers, massaging the sphygmomanometer's fleshy bulb, pumping the tightly-wound cuff into a tumescent state, from which we both find the nearby couch and curtain offer the only possible relief. God bless the Hippocratic oath!

Friday, August 13, 2010

Adrian Newey's warp drive

According to relativity, nothing can move faster through empty space, than the local speed at which zero-mass particles move through empty space. Light is composed of zero-mass particles called photons, hence the local limit on the speed of motion is expressed in terms of the local speed of light.

However, on a non-local basis, the time taken to travel between two spatial locations is dependent upon the geometry of the path traversed between those two locations. By creating or choosing the appropriate path, the journey can be completed before a beam of light taking a different path. This fact was ably demonstrated by Miguel Alcubierre's 1994 proposal for a warp drive space-time.

The basic idea of Alcubierre's warp drive is that a spaceship, residing within a bubble of flat space, could surf a wave of distorted space-time geometry, and reach its destination faster than light travelling outside the bubble. This trick would be achieved by contracting the space in front of the bubble, and expanding it behind. The spaceship reaches its destination rapidly because it travels a very short distance, not because it exceeds the local speed of light. The Einstein field equations, however, suggest that such a space-time geometry would require negative energy fields, and whilst the experimentally verified Casimir effect seems to point to the existence of such fields on small scales, doubts remain about their viability over macroscopic length-scales.

Meanwhile, in Formula 1, the debate continues over how Adrian Newey's Red Bull RB6 is able to run its front wing so close to the ground. The simplest explanation is that the front wing possesses a certain non-linear aeroelasticity, and given that the Red Bull's rear diffuser is less sensitive to ride height variations than most, the car can also be set-up with a large degree of rake. One might say that the Red Bull is surfing a wave of distorted airflow, the front of the car contracting the aerodynamic streamlines, the rear allowing them to expand again, whilst the driver resides between in a bubble of stable serenity. This, then, is Adrian Newey's warp drive, and it enables the Red Bull RB6 to reach its destination faster than any car designed outside Milton Keynes.

Sunday, August 08, 2010

Sleep Mechanics

Allow me to introduce myself: I'm Dr Malcolm Byte, a surgeon of the mind, and a connoisseur of the somnolent. No, that's right, you won't have heard of me. A good friend of yours, (and I'm afraid, as one of my ongoing clients, she will have to remain anonymous), contacted me, and told me that you were having difficulties dealing with your feelings about a previous relationship. Well, this is very common, and easily dealt with.

Unfortunately, even when a relationship ends, it leaves hazardous mental residues in its wake. These can either be allowed to fade away over long periods of time, (the default option, it seems, for most people), or more advisedly, can be disposed of immediately by an expert in the field.

The procedure is quite simple and painless. First, we must terminate the mental processes which are running those damaging thoughts and feelings; then we need to delete the memories from which those processes spring (the executables, if you will); and finally we need to remove the registry entries created in the subconscious by the experiences associated with your relationship.

How is this done? Well, don't be alarmed, but the successful completion of the therapy requires me to enter your mind. This is not as difficult or disturbing as it might at first sound. You see, humans have, over time, evolved a clever thing called the empathic mechanism. This is the innate skill most of us possess, to imagine what other people are thinking and feeling. The empathic mechanism gives us the ability to predict how the people around us are going to behave, and in evolutionary terms, that's a distinct advantage.

One application of the empathic mechanism enables you, once you've known someone for a sufficient period of time, to create a subsystem of your mind devoted to that person. In effect, you'll have a simplified version of that individual's mind running inside your own. By this means, you'll be able to anticipate what that person might think or feel about something, without them even being there. This helps with relationships, but it also, in effect, creates a backdoor that practitioners like myself can exploit, to enter your mind, and to alter its contents.

Now, to speed up the process somewhat, I'll need to exploit another loophole in the mind's defences, and this is where my qualification in Sleep Mechanics comes into play. There are, you see, a continuum of mental states between full consciousness and total unconsciousness. People often seem to think that there is a well-defined boundary between the two, with a surface tension, if you like, that prevents the casual exchange of material between the different realms. But the operating space of the human mind is more complex than this. It has many dimensions, and myriad levels of interacting intricacy.

How's your astrophysics? Well, are you familiar with the idea of a black hole? That's fine, you'll only need the general idea. Have you heard the speculative proposal that a new universe is created inside every black hole? Well, minds are rather like such fertile black holes: they suck in perceptions, impressions and sensations from the universe around them, like information sink-holes, but inside each mind a new universe is created, a continuous torrent of thoughts and images, a fountain of the imagination. To get inside your mind, I need to exploit the information flows between your public and private worlds; to be specific, I need to induce a hypnagogic mental state, in which the distinction between the consciously perceived world, and the personal world of memory and imagination has been dissolved.

Are you familiar with that feeling when you're half-awake and half-asleep, the feeling that you can control and direct your dreams at will, sometimes slipping deeper into a dream to enjoy the ride, then pulling back momentarily to re-direct its course? Well, on this occasion, I'm going to be the director.

So lie back, close your eyes, and relax. Let your breathing subside. And remember that warm Summer's day, when a pleasant sense of satisfied exhaustion and comfort blended with the countless shades of green sweeping over your flickering vision...a gentle brook, winding through the verdant meadow as a tinkling harmony of innumerable tiny tintinnabulations...a portly bumble-bee's eager hum, briefly superimposed over the mechanical drone of a lawn-mower in the middle distance...an orange glow silhouetting the trees at sunset, the merest zephyr rustling the highest branches...the colour leaching from the sky, flowing with an inky-blue gradient from a jet black zenith...a jewelbox of silver and golden stars speckled across the velvet sky, twinkling across the lonely lightyears with silent yearning...