Complexity, organization, evolution, and constructal law

During the 20th century, statistical thermodynamics, quantum mechanics, information theory, and computer science have changed the scientific discourse on everything, from science itself to what life is.¹ Instead of terms and images that did not require an advanced education, today it seems that legitimacy on this topic comes from speaking a language of disorder, uncertainty, scale, emergence, chaos, entropies of many types, and, above all, “information.” The fact that few seem to understand this kind of talk is going unnoticed, obviously, because the world does not speak jargon.

This does not have to continue this way. In this article, we go against this movement and draw attention to a simple truth: words have meaning. We review the key words of the discourse and start with the observation that information is not knowledge.² 

The computer “is not” because it is nothing more than an extension of the human who uses it to move (to live) more easily. It is one artifact among very many. On the other hand, you “are,” with or without that artifact. With what you are, you make decisions (purposeful choices and changes), and as a consequence you and your group move (live) more easily and with longer lasting power.¹ 

Those whose mother language is not English have to learn English, and along the way they acquire the habit of checking the dictionary. We did this ourselves, as students and now while writing this article. Here is the meaning of some of the key words that do not require advanced education:

Information is a universal term, like geometry and energy: it is expressed by the same word in many languages. It comes from the Latin verb informo-informare, which means to give form and shape, to form, to fashion. In English, it means something told, news, intelligence (as in spying), facts, data, text, and figures. These days, information also means data that can be stored in or retrieved from a computer.

In modern languages all over, information means a sign, a signal: seeing 24 June instead of 23 June on the headline of today's newspaper, or seeing 01 vs 10 in a computer text. What you, the observer, decide to do with the sign is you. What I do with it is I. Both are physical changes (design changes in you and me, with time direction). They are dynamic. They are the action that knowledge is and are not to be confused with “information.”

There was an entire landscape of “information” chiseled on obelisks in Egypt, and it meant absolutely nothing, no design, no change, no action, no embodiment into anything. It stayed that way until Jean-François Champollion taught his contemporaries how to discern signs from obelisks. What you read and do now with Egyptian information is you, not somebody else.

Change has a time direction, an arrow of time called evolution.¹ The ability to affect design change is an integral part of the moving thing that morphs and evolves in order to move on earth more easily, farther, and longer in time. The “ability” of the moving system comprises many physical features: freedom to change, access to information, memory of changes that facilitate movement versus those that do not, and so on, on the staircase of better and better organization over time. This applies to everything, inanimate and animate.^3–5 

To progress beyond information, we now review the meaning of two words that refer to more subtle and more complicated notions.

Design is a plan, a scheme, a project with purpose, an intention (aim) for an outcome. Design is the arrangement of parts, details, form, and color, so as to produce a complete unit that has purpose. Design is not “pattern.” The tree-shaped architecture of so many things that flow (river basin, lung, etc.) is a changing design because of its purpose, which is to facilitate flow between a point and an area, or between a point and a volume. Pattern is a static, regular, mainly unvarying arrangement of form, parts, or elements. The tiles on the bathroom floor and the atoms locked in the crystal lattice have pattern but not evolving design, because they are dead. In them, there is no flow, no change, no freedom, no time direction, and therefore no life and evolution.

Organization is a consolidated group of live (flowing) elements, as in an organ. Organization is a systemized whole, the organs connected and flowing together in the moving animal body, the river channels and wet interstices in the drainage basin, and the components moving together in the vehicle on the highway. In society, organization is the live group assembled for a specific purpose (activity, movement), such as club, union, political party, executive structure in business, university, or government. Design is living organization, not dead pattern.

A picture is worth a thousand words. This is why all these notions, from information (sign, form) to design and organization, are images in the mind, before they are spoken as words. This is also why the most important keyword to understand is the image, which, coming from the human hand, is a drawing.

A drawing is an image that can be discerned by the eye and understood by the mind. “Understood” means that the image is connected to other images in the brain, in ways that make the storage of images more compact and their retrieval more rapid.⁶ 

A drawing has three essential features, which are self-standing, i.e., independent of each other. The maker of a drawing can attest to the fact that one feature is chosen independently of the others:

• The drawing has size. Large or small, the size is represented by the length scale of its frame, sheet of paper, canvas, or computer screen. The maker of the drawing selects the size of the image.

• The drawing has a meaning (a message) that is conveyed to the viewers. The message is knowledge when the recipients act based on the received message. The message spreads naturally,⁷ from those who know to those who need to know. The meaning is represented by one or more features: shapes, structures, aspect ratios (proportions), and the organization of all these features on the viewed plane. Each feature is distinct. The round shape of the cross section of the blood capillary is not to be confused with the bifurcation of the blood vessel, or the pairing of smaller blood vessels into a larger vessel. All the features are organized in a particular way, and as a consequence, they convey the message. A simple line drawing of your face today displays the same organization as a drawing made 20 years ago, but a few features are different. The drawing morphs with age, but the message remains: the portrait is yours.

• The drawing has svelteness, which is a measure of the relative thinness of the lines used to convey the message (b). The svelteness is a dimensionless number defined as⁸ 

The drawing made with relatively thin lines has a large Sv value (Fig. 1). The same drawing drawn with a thicker pen or brush, or copied on a poor copy machine has a smaller Sv. A water color rendition of the original line drawing has an even smaller Sv. A good forgery has a different Sv than the original. The Sv value belongs to the artist, to one brush, and one style of brush strokes, and it distinguishes the original artist from the forgerer.

Amazing, even the discussion of a single drawing is complicated! Then, what is complexity?

Complexity is a difficult concept, like turbulence. In the beginning, when people knew a lot less than what we know today, complexity meant difficulty, fuzziness, headache, and why bother. Its Latin origin betrays this feeling of defeat: “complex” (cum + plex, i.e., twisted forms together) comes from the same Latin observation as “perplex” (through + twisted). As science progressed, people began to see organization and message in the said complexity. As the thinking became sharper and deeper, the organization of complexity gave birth to theory, which is the design of the mind with which to predict the observed complexity. Once understood, complexity becomes easier, and we call it architecture, weave, tissue, design, organization, and many more names that are a lot less puzzling.

It happened this way with turbulence, which as a science evolved from fuzziness in the late 1800s (complications effaced intentionally through a time-averaged description, thanks to Reynolds⁹), to the “large scale structure” of turbulence in the 1970s and to the evolution of the structure, which is now predictable.¹⁰ For example, a flat jet or plume always evolves into a stream with round cross section (Fig. 2).¹¹ The reverse is not true: round jets and plumes do not evolve into streams with flat cross sections. This holds true for turbulent and laminar jets and plumes.

It will happen the same with complexity. The language of complexity will be replaced by geometrically precise notions such as size (i), organization (ii), and svelteness (iii).

We often read that the fractal dimension (D) of a geometrical object (called drawing in this paper) is important because it accounts for the complexity of the figure.¹² Had this been true, we would have seen by now a ranking of drawings according to their complexity, because newly calculated D values appear in the literature unabated. Whether the fractal dimension adds anything to the quantitative description of complexity is doubtful.

Consider the drawing made in Fig. 3. An isosceles triangle is divided in half by the bisector of its 90° angle. The construction is repeated n times. The bisectors form a toothy line to which we add the left side (L) of the original triangle. The total length of the toothy line is

where r = 2^−1/2= 0.707, and G is the smallest length scale of the drawing, G = rⁿL. In the limit $n → ∞$⁠, the total length approaches

The total length L_n can be expressed in dimensionless form as

where $L̃n=Ln/L∞$ and $ε=G/L<1$⁠. The fractal dimension (D) of the toothy line is defined as¹³ 

In view of Eq. (4), D depends on $ε$ and r. In the limit $ε → 0$⁠, $n → ∞,$ D approaches the irrational number

As $ε$ decreases (or as n increases), the D value approaches $D∞$ from above. For example, when n = 2, $ε$ is 0.5 and $D ≅ 1.379$⁠. When n = 4, the D value furnished by Eq. (5) drops to 1.338.

What does all this mean? The drawing of Fig. 3 is a “fractal object” strictly in the limit $ε → 0,$ when the construction algorithm would be repeated an infinite number of times, and the number of lines used in making the drawing would be infinite. To the artist who attempts to make the drawing, this would mean that the fractal object would have infinite complexity. This is the physical reason why the fractal object is impossible to draw (and, even less, see), and why the fractal images that populate the literature are not fractal. They are Euclidean (cf. Mandelbrot,¹³ page 39), because the algorithm assumed in making the drawing is intentionally stopped (cut off) at a small length scale that is sufficiently large so that the drawing can be made, printed, viewed, discerned, and discussed.

In view of this, review the fractal dimension calculated during the construction of Fig. 3. First, the drawing with infinite complexity has the fractal dimension $D∞=1.3071$⁠, which is finite, not infinite. Second, the Euclidean drawings with complexity (finite n and $ε$⁠) have D values greater than $D∞$⁠. All the drawings that the reader can see in Fig. 3 are decidedly less complex than the fractal drawing, yet, their calculated dimension D is greater than the fractal dimension $D∞$ of the drawing with infinite complexity. When D decreases, complexity increases. The conclusion from these two observations is that the fractal dimension is not a measure of complexity.

The “fractal object” described throughout fractal geometry is not an object. According to its definition, the word “object” means a thing that can be seen or touched, or a person or thing to which action, thought, or feeling is directed. The original Latin word, objectus, means something thrown in front of you, a thing that appears (from the verb objicere, where ob means toward, for, before, and jacere means to throw, from which the word “jet” in Romance Languages and English).

The fractal object is, at best, a thought “in the limit,” never achievable, never palpable, never to be seen, like Sadi Carnot's reversible heat engine. Yet, there is a big difference between the two thoughts in the limit: the reversible engine springs in the mind because of physics (the laws of thermodynamics), whereas the fractal object is as arbitrary as the algorithm chosen by the mathematics artist, or the paint and brush chosen by the painter.

An icon is a simple drawing that conveys the same message as numerous and much more complicated drawings. Making drawings is a means of communication that evolved under certain rules, which constitute the discipline of graphic design.^14,15 The rules are stricter when the icon is about human safety. For example, the design of the pedestrian “Walk” and “Don't Walk” sign on street corners¹⁶ (Fig. 4) must meet traffic regulations. These signs evolved with the technology available for communicating messages visually.

Today, most of the world has adopted images of a walking person (pictogram) and a raised hand (ideogram) or standing person (pictogram), respectively, to indicate when to cross the street and when not to cross.^17,18 The raised hand is an icon because it represents the hand gesture of a traffic policeman. These signs have a characteristic size: not too big, not too small. There is a size range that works for most pedestrians in most circumstances (Fig. 5). The larger size is easier to see, but it is more expensive to make. The smaller sign is cheaper but it is more difficult to see and understand. The aspect ratio of the sign plays an important role on the speed with which the message is scanned by the two human eyes.¹⁹ 

The icon represents an individual in the most general sense (male, female, tall, short, old, and young). The drawing is an organization of simple elements that resemble the head, torso, legs, and arms (Fig. 6). By itself, each element is meaningless. Key is their organization. If the elements are organized in other ways, the pedestrian is puzzled, and the message does not flow from the sign to the viewer.

Svelteness is an essential property of the icon. In Fig. 7, the svelteness of each icon was calculated by considering the effective outline as inner characteristic length, and the square-root of the lit area (white) as the external characteristic length in Eq. (1). Icons with higher svelteness are newer designs. The icon with the highest svelteness appeals to the knowledge of the pedestrian, that is, it conveys information as a set of lit dots, and the mind of the pedestrian creates the image and then interprets it.

A photograph or a moving picture of a person crossing the street would be more realistic than a line drawing, but would it be more effective? Key is the minimum detail that is sufficient to convey a quick, clear, and safe message. The few and simple lines, or the gestalt effect (form) of aligned dots that are lit, are thick but not too thick to convey the message to the persons across the street.

Icons and models share one characteristic, which is their simplicity. Yet, the two are different. The icon is a simple drawing of a mental viewing, observed or imagined. The model is strictly about the observed: it is a manmade simplified facsimile of an object or phenomenon observed in nature. The duck from the wood shop is the model, and the duck on the lake is the observed natural object. The human action of modeling is empiricism, which means observation first and description later. Modeling is the opposite of theory (idea first, comparison with nature later). Modeling is not theory.

Evolution means changes that occur in a discernible direction in time. Evolution, the word, is defined unambiguously at its origin, the Latin verb evolvo, evolvěre, which means to roll out, to roll forth, to spread. Contrary to today's discourse, evolution is a much older and more encompassing concept of physics (of everything) than biological evolution.

Drawings—their veracity and complexity—evolve along with the human ability and technology to describe the mental image. For example, the evolution toward the simplest drawing that still captures the message is illustrated by Picasso in the sequence of lithographs called “The Bull” (Fig. 8). In a sequence of eleven plates, the artist captured the essence of the message that was ultimately expressed in a small set of organized lines with high svelteness.

Evolution is also evident in the accuracy with which a drawing is made or reproduced by a computer, as technology evolves. Figure 9 illustrates this with two examples of circles drawn with pens of two thicknesses. The circles are not perfect, but this is not why they are used as examples. They are used because their svelteness can be calculated [cf. Eq. (1)] as $Sv=p/Ab1/2$⁠, where p is the inner perimeter of the black line and A_b is the area of the black line. To calculate p and A_b, the circle drawings were processed with a commercial code.^20–22 They were digitized with several resolution settings measured as PPI (points per inch), which are plotted on the abscissa in Fig. 9. The scanned drawings were converted into binary (black and white) images in order to calculate their p and A_b values.

Figure 9 shows that Sv is not a constant, unlike the Sv of a mathematical circle with rim of constant thickness. The svelteness of the hand-drawn circle increases monotonically as the scanning and reproduction technology improves toward more PPI. The reason for the increasing trend is the rough and ill-defined edges between the black trace left by the pen and the white paper. The texture of the paper and the force on the pen on paper are features that belong to the particular hand drawing and cannot be reproduced fully, not even in the limit of infinite machine power. Each curve (Sv vs PPI) is like the length of the coast of Britain, which was the calculation that served as starting point of fractal geometry.¹³ 

Along with the claim that nature evolves toward increasing complexity, we often read that the natural tendency is toward greater “disorder,” and that this tendency is commanded by the second law of thermodynamics. This is not correct, as one can see by reading the statement of the second law, made by two of its three original proponents in 1851–1852 (the other proponent was Rankine).²³ 

Clausius: No process is possible whose sole results is the transfer of heat from a body of lower temperature to a body of higher temperature.

Kelvin: Spontaneously, heat cannot flow from cold regions to hot regions without external work being performed on the system.

We often read that the second law states that “entropy must increase,” and that the “classical” laws of thermodynamics pertain to “equilibrium states.” Many even teach that thermodynamics should be called thermo“statics.” Such statements are not thermodynamics. For example, a physicist²⁴ wrote in 2015 that “the second law… applies to closed macroscopic systems consisting of an extremely large number of particles, such as liquids or gases…” This is not true. The second law statements hold for any system (open, closed, isolated, adiabatic, steady state, unsteady state, with configuration, and without configuration).

The second law says nothing about “disorder.” Many confuse the second law with the view that in a box filled with particles the assembly tends toward a larger number of possible energy states.^25,26 This is the core idea of statistical thermodynamics, yet lost in the teaching of it are three important observations.

First, to assume a swarm of particles in a closed box is to throw away the “any system” power of thermodynamics. The any-system is the most general system in physics. It is the system with unspecified organization, and, compared with it, the box with bouncing particles is a very special case, with a specified configuration.

Second, no one has seen particles, their disorder, and their tendency toward greater disorder. From such blindness, how can there be a “law of increasing disorder”? This has been a source of confusion in science, because as we look around we are struck by design, self-organization, change after change (evolution), and order out of lack of order.¹ 

Third, decades before statistical thermodynamics, the second law and the first law were stated with reference to systems of unspecified size (e.g., heat engines, animals), not infinitesimal.

In summary, the second law says absolutely nothing about “disorder,” “equilibrium states,” “entropy,” “particles,” “classical,” and “statics.” Important to keep in mind is that “thermodynamics” is the science that brings together two kinds of movement, heating and working, previously seen as separate (caloric theory versus mechanics)²³ (Fig. 10). The only relevant question about the second law statement of 1851–1852 is whether it is correct. The evidence is massively in support of answering “yes,” and it is based on the machines that have been built successfully by relying on the second law of thermodynamics of Clausius, Rankine, and Kelvin. These flow architectures are macroscopic, organized, and evolutionary. Order, not disorder, is their chief characteristic and claim to fame. They are every day futuristic (not “classical”), they are full of life and motion (not in “equilibrium”), and are eminently dynamic (not “static”).

Words have meaning. This is why words matter. This is also why it is necessary to define unambiguously the terms of any discussion about complexity.

The human observation that certain things happen innumerable times the same way represents a distinct one natural tendency, i.e., one phenomenon. To observe the phenomenon is empiricism. A law of physics is a compact statement (text or formula) that summarizes innumerable observations of the same kind. To rely on the law to experience a purely mental viewing of how things should be (i.e., to predict future observations) is theory.

The phenomenon covered by the first law of thermodynamics is the “what goes up must come down.” Today, we recognize this more generally as the conservation of energy, from kinetic to potential when a body is thrown upward, to the energy flow (from heat into work) through a thermodynamic system such as a power plant.

The phenomenon covered by the second law is the “one way” tendency of all flows, such as the flow of water under the bridge. Today, we recognize this natural tendency as irreversibility. Every flow, by itself, proceeds from high to low. Fluid through a duct flows from high pressure to low pressure. Heat through an insulation leaks from high temperature to low temperature. If you do not know beforehand which is high and which is low, then the direction of the flow will tell you. Why, because it is the law, and any thermodynamic system obeys the law.

The phenomenon observed as complexity, organization, design, and the other terms reviewed in this article is natural organization, evolution, and life.^27,28 The occurrence and evolution of freely morphing configurations is present in everything that flows and moves more easily over time.^29,30 This phenomenon is covered by the constructal law.^{1,3–5,24,29–39} Observations of this kind are everywhere: river basin evolution,⁴⁰ lung architecture evolution,⁴¹ city traffic evolution, heat exchanger evolution,⁴² and aircraft evolution.⁴³ These observations reveal the arrow of time² in nature, which points from existing flow configurations to new configurations through which the flowing is easier. Not the other way around. Why, because this is the law, and all systems obey the law.

Marching ahead, the study of complexity can benefit from the example set by the study of thermodynamics. At bottom, thermodynamics is a discipline. It has precise words, rules, and principles. Changing the meaning of the words in mid-course, to benefit the narrative, is not allowed. Complexity and, more generally, organization and evolution in nature are most powerful and useful when pursued as a discipline, with precise terms, rules, and principles.

Professor Bejan's research was supported by the National Science Foundation. Professor Errera's research was supported by CAPES fellowship BEX 9576/11-8 from the Brazilian Federal Government. Helpful discussions with Mr. Matthew Hambro on Graphic Novels are gratefully acknowledged.