Alikhani Ma,b, Sangsuwon Ca, Teixeira CCc
Figure 1: Entropy. Assume we confine the molecules of a gas to the corner of a box (A). As soon as we remove the barrier confining them to that corner, the gas molecules escape in every direction and fill the whole box (B). In other words, the probability that they disperse in every direction is very high (high entropy), while the probability that the molecules stay together in the corner of the box after removing the confining barrier is very low (low entropy).
Figure 2: Entropy of form. Form in this example is the arrangement of the balls on the floor. If we drop three balls on the floor, they can produce endless forms, with the overwhelming majority of them being a triangular arrangement (high entropy) (A). They may produce a linear form (B) on extraordinarily infrequent occasions (low entropy).
Figure 3: Entropy of the form does not represent disorder. When we repeatedly drop the pieces of a broken plate on the floor, the pieces (units) of the broken plate can take an infinite number of positions in relation to each other. All configurations of the pieces on the floor have a probability of occurrence, including when they gather in the shape of a plate. We can categorize all the possible distributions of the pieces on the floor into two general patterns: one pattern is the shape of a plate, and the other pattern is anything but the shape of the plate. The probability of the broken pieces staying away from each other is much higher (infinity) than staying together as in the original shape of the plate (almost impossible).
Figure 4: Randomness and Entropy of the form. In the above experiment, balls (units) collected in the basket randomly take one path leading to one of the boxes A to D. When a ball gets to a box, regardless of which ball arrived, the box automatically creates a form (forms A through D). While the unit distribution is random, at the level of form, probabilities arise, with some forms having a higher probability of appearing than others. In this example, form B is more likely to occur than the other forms. Therefore, the probability of creating forms A to D is not random. One can say form B has higher entropy than form D. In this figure, the balls’ paths represent constraints that limit how the units can move and interact with other units.
Figure 5: Emergence of functional form. Mass form is an additive form of all its units without a specific organization (A). In contrast, the functional form reflects the emergence of a new organization of its units (B). Therefore, the functional form has specific new characteristics or functionality that did not exist in the individual units before introducing new constraints. The functional form of this micro-state (in B) becomes the units for the next micro-state organized into a new functional form (C).
Figure 6. Emergence and trajectory of the form. In this example, two different forms are developing in 3 stages. In the first example (example 1), a change in the intermediate micro-state (micro-state A and micro-state B) does not change the shape of the final macro-state (C). Therefore, the form trajectory does not diversify. In the second example (example 2), different intermediate microstates (A or B) change the possible form of the final macro-states (C or D) and, therefore, diversify the form trajectory.
Figure 7: Form is controlled by physical and chemical laws. Both surface tension and gravitational force play a role in the form of a drop of water (A). A small drop of water stays spherical due to tensional forces (B), while the shape of a larger drop of water on a hard surface is flattened due to gravitational force (C).
Figure 8. Emergence and predictability of complex forms. If we add soap to water, the interaction between the hydrophilic head and the hydrophobic tail of the soap spontaneously creates a complex micelle form that can be explained by the physical and chemical properties of both soap and water molecules. This process is an example of a predictable complex form created by emergence.
Figure 9: Spontaneous formation of complex protein forms by emergence. Proteins are made up of amino acids. The chemical and physical interactions between different amino acids and their surroundings produce four forms of functional proteins: primary, secondary, tertiary, and quaternary structures, corresponding to different micro-states of protein folding.
Figure 10: The tertiary structure of proteins. The 3D form of a polypeptide chain is created spontaneously by emergence due to a favorable reaction between an aqueous environment and non-polar and polar amino acids. The protein spontaneously fold so that the non-polar amino acids are located internally in the protein structure, while the polar amino acids are found in the external portion of the protein structure
Figure 11: Different forms of proteins are created by emergence. Through the emergence process, the final macro-state of protein folding results in Globular (A), Fibrous (B), or Membrane (C) protein forms. These three groups of proteins have different shapes, solubility, and function.
Figure 12: Scale-dependent complexity. If we look at the tooth and its surrounding alveolar bone at lower magnification (A), we may not recognize the complexity of the organization that can be observed at higher magnification (B). One can say that the structure in “A” mostly represent emergence into a much later micro-state, while in “B” we can appreciate the interaction of entropy and emergence at very earlier micro-states.
