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Unfortunately, however, there is another serious catch. Theory dictates that such discoveries must occur at an increasingly accelerating pace; the time between successive innovations must systematically and inextricably get shorter and shorter. For instance, the time between the “Computer Age” and the “Information and Digital Age” was perhaps twenty years, in contrast to the thousands of years between the Stone, Bronze, and Iron ages.
It’s as if we are on a succession of accelerating treadmills and have to jump from one to another at an ever-increasing rate.
like organisms and cities, companies also scale as simple power laws. Equally surprising is that they scale sublinearly as functions of their size, rather than superlinearly like socioeconomic metrics in cities. In this sense, companies are much more like organisms than cities. The scaling exponent for companies is around 0.9, to be compared with 0.85 for the infrastructure of cities and 0.75 for organisms.
For organisms, the sublinear scaling of metabolic rate underlies their cessation of growth and a size at maturity that remains approximately stable until death. A similar life-history trajectory is at work for companies.
In their youth, many are dominated by a spectrum of innovative ideas as they seek to optimize their place in the market. However, as they grow and become more established, the spectrum of their product space inevitably narrows and, at the same time, they need to build a significant administration and bureaucracy.
As they grow companies tend to become more and more unidimensional,
Cities, on the other hand, become increasingly multidimensional as they grow in size.
Galileo asked what happens if you try to indefinitely scale up an animal, a tree, or a building, and with his response discovered that there are limits to growth.
if the shape of an object is kept fixed, then when it is scaled up, all of its areas increase as the square of its lengths while all of its volumes increase as the cube
Thus, when an object is scaled up in size, its volumes increase at a much faster rate than its areas.
most heating, cooling, and lighting is proportional to the corresponding surface areas of the heaters, air conditioners, and windows. Their effectiveness therefore increases much more slowly than the volume of living space needed to be heated, cooled, or lit, so these need to be disproportionately increased in size when a building is scaled up.
Successive powers of ten, such as these, are called orders of magnitude and are typically expressed in a convenient shorthand notation as 101, 102, 103,
for every order of magnitude increase in length, areas and strengths increase by two orders of magnitude, whereas volumes and weights increase by three orders of magnitude.
This kind of scale where instead of increasing linearly as in 1, 2, 3, 4, 5 . . . we increase by factors of 10 as in the Richter scale: 101, 102, 103, 104, 105 . . . is called logarithmic.
logarithmic scale allows one to plot quantities that differ by huge factors on the same axis,
Using the totals of these three lifts from the weight lifting competition in the 1956 Olympic Games, Lietzke brilliantly confirmed the ⅔ prediction for how strength should scale with body weight.
Also well known is the related concept of an invariant quantity, such as our pulse rate or body temperature, which does not change systematically with the weight or height of the average healthy individual.
much of the research is conducted on so-called model animals, typically standard cohorts of mice
Of fundamental importance to medical and pharmaceutical research is how results from such studies should be scaled up to human beings in order to prescribe safe and effective dosages or draw conclusions regarding diagnoses and therapeutic procedures.
Until the 1970s and the rise of its popularity, the BMI was actually known as the Quetelet index.
Quetelet’s goal was to understand the statistical laws underlying social phenomena such as crime, marriage, and suicide rates and to explore their interrelationships.
Quetelet brought something new and important to these analyses, namely the statistical variation of these quantities around their mean values, including estimates of their associated probability distributions.
How much freedom do we have in shaping our destiny, whether collectively or individually? At a detailed, high-resolution level, we may have great freedom in determining events into the near future, but at a coarse-grained, bigger-picture level where we deal with very long timescales life may be more deterministic than we think.
Quetelet’s body-mass index is defined as your body weight divided by the square of your height and is therefore expressed in terms of pounds per square inch or kilograms per square meter.
The BMI is therefore presumed to be approximately invariant across idealized healthy individuals, meaning that it has roughly the same value regardless of body weight and height. However, this implies that body weight should increase as the square of height, which seems to be seriously at odds with our earlier discussion of Galileo’s work where we concluded that body weight should increase much faster as the cube of height.
So how in fact does weight scale with height for human beings? Various statistical analyses of data have led to varying conclusions,
To understand why this might be so, we have to remind ourselves of a major assumption that was made in deriving the cubic law, namely that the shape of the system, our bodies in this case, should remain the same when its size increases. However, shapes change with age,
In addition, shapes also depend on gender, culture, and other socioeconomic factors that may or may not be correlated with health and obesity.
alternative definition of the BMI has been suggested in which the BMI is defined as body weight divided by the cube of the height; it is known as the Ponderal index
Growth and the continual need to be adapting to the challenges of new or changing environments, often in the form of “improvement” or increasing efficiency, are major drivers of innovation.
in order to build larger structures or evolve larger organisms beyond the limits set by the scaling laws, innovations must occur that either change the material composition of the system or its structural design, or both.
Modeling is now so commonplace and so taken for granted that we don’t usually recognize that it is a relatively modern development.
Models of various kinds have been built for centuries, especially in architecture, but these were primarily to illustrate the aesthetic characteristics of the real thing rather than as scale models to test, investigate, or demonstrate the dynamical or physical principles of the system being constructed. And most important, they were almost always “made to scale,”
Nowadays, every conceivable process or physical object, from automobiles, buildings, airplanes, and ships to traffic congestion, epidemics, economies, and the weather, is simulated on computers as “models” of the real thing.
Failure and catastrophe can provide a huge impetus and opportunity in stimulating innovation, new ideas, and inventions whether in science, engineering, finance, politics, or one’s personal life.
who was Isambard Kingdom Brunel, and why is he famous? Many consider him the greatest engineer of the nineteenth century, a man whose vision and innovations, particularly concerning transport, helped make Britain the most powerful and richest nation in the world.
They worked together when Isambard was only nineteen years old building the first ever tunnel under a navigable river, the Thames Tunnel at Rotherhithe in East London.
In 1830 at age twenty-four Brunel won a very stiff competition to build a suspension bridge over the River Avon Gorge in Bristol.
Brunel subsequently became the chief engineer and designer for what was considered the finest railway of its time, the Great Western Railway, running from London to Bristol and beyond.
One of his most fascinating innovations was the unique introduction of a broad gauge of 7 feet ¼ inch for the width between tracks. The standard gauge of 4 feet 8½ inches, which was used in all other British railways at that time, was adopted worldwide and is used on almost all railways today. Brunel pointed out that the standard gauge was an arbitrary carryover from the mine railways built before the invention of the world’s first passenger trains in 1830. It had simply been determined by the width needed to fit a cart horse between the shafts that pulled carriages in the mines.
The parallels with similar issues we are facing today regarding the inevitable tension and trade-offs between innovative optimization and the uniformity and fixing of standards determined by historical precedence, especially in our fast-developing high-tech industry, are clear.
He realized that the volume of cargo a ship could carry increases as the cube of its dimensions (like its weight), whereas the strength of the drag forces it experiences as it travels through water increases as the cross-sectional area of its hull and therefore only as the square of its dimensions.
Thus the strength of the hydrodynamic drag forces on a ship relative to the weight of the cargo it can carry decreases in direct proportion to the length of the ship.
In other words, a larger ship requires proportionately less fuel to transport each ton of cargo than a smaller ship.
the Great Eastern was not a technical success either. She was ponderous and ungainly, rolled too much even in moderately heavy waves, and, most pertinent, could barely move her gargantuan mass at even moderate speeds.
there are key examples like that of the Great Eastern where the major reason for failure was that they were designed without a deep understanding of the underlying science and of the basic principles of scale.
Even at the time of the Great Eastern, there was very little, if any, such “real” science in shipbuilding.
The tried and tested process of simply extrapolating from previous design worked well when designing and building new ships, provided the changes were incremental.
the Navier-Stokes equation, arises from applying Newton’s laws to the motion of fluids, and by extension to the dynamics of physical objects moving through fluids, such as ships through water or airplanes through air.
Although the Navier-Stokes equation describes fluid motion under essentially any conditions it is extremely difficult, and in almost all cases impossible, to solve exactly because it is inherently nonlinear. Roughly speaking, the nonlinearity arises from feedback mechanisms in which water interacts with itself.
