Why there was a useful plausible analogy between geodesic domes and spherical viruses

215Hist. Phil. Life Sci., 28 (2006), 215-236

1 See, for example, Hesse 1966; Holyoak and Thagard 1995; Bailer-Jones 2002.

Why there was a Useful Plausible Analogy betweenGeodesic Domes and Spherical Viruses

Gregory J. Morgan

Department of PhilosophySpring Hill College4000 Dauphin St.

Mobile, AL 36608, USA

ABSTRACT – In 1962, Donald Caspar and Aaron Klug published their classic theory ofvirus structure. They developed their theory with an explicit analogy between sphericalviruses and Buckminster Fuller’s geodesic domes. In this paper, I use the spherical virus-geodesic dome case to develop an account of analogy and deductive analogical inferencebased on the notion of an isomorphism. I also consider under what conditions there is agood reason to claim an experimentally untested analogy is plausible.

KEYWORDS – Context of Discovery, Spherical Virus, Geodesic Dome, TomatoBushy Stunt Virus, Quasi-equivalence, Buckminster Fuller, Aaron Klug, DonaldCaspar, Francis Crick, James Watson, Isomorphism, Analogical Reasoning, History ofStructural Biology, Philosophy of Biology

Introduction

The history of science is rife with analogies: Gilbert’s analogybetween the earth and a small magnet; Huygen’s analogy betweenlight and sound waves; Lavoisier’s analogy between respiration andcombustion; Carnot’s analogy between heat and fluid flow; Darwin’sanalogy between natural and artificial selection; Kekulé’s analogybetween benzene and a snake; Morgan’s analogy between chromo-somes and beads on a string; and Luria’s analogy between mutationand a slot machine, to mention just a few of the more famous. Tothis list, one may add Donald Caspar and Aaron Klug’s analogybetween geodesic domes and spherical viruses, a case little knownto historians and philosophers of science.

Scientists’ use of analogy raises a number of questions for thephilosopher of science.1 I am interested in two: what is the nature of

© 2006 Stazione Zoologica Anton Dohrn

216 GREGORY J. MORGAN

scientific analogy? And what roles do analogies play in scientificinference? Analogies appear to function in a number of differentways in science: they may be used pedagogically to increase students’understanding, they may be used to help explain a novel phenome-non, or they may be used to formulate novel testable hypotheses, etc.For reasons of space, I will focus only on the use of analogy to inferplausible novel hypotheses that may be pursued further by empiricaltesting. In other words, I will focus on what is called analogical infer-ence or analogical reasoning in the context of discovery. UsingCaspar and Klug’s analogy between spherical viruses and geodesicdomes, I will sketch an account of analogy and defend a deductiveaccount of analogical inference.

In the first section, I narrate the events that led up to DonaldCaspar and Aaron Klug’s presentation of their theory of virus struc-ture. They developed their theory using explicit analogies betweengeodesic domes and virus structure. In the second section, I developan account of analogy based on the notion of an isomorphism. In thethird section, I show that once an analogy is made explicit, an ana-logical inference is a deductive inference. Finally, I consider infer-ences to the plausibility of a useful analogy between two systems.

The Genesis of the Caspar-Klug Theory of Virus Structure

In June 1962, at the Cold Spring Harbor Symposium on ‘BasicMechanisms in Animal Virus Biology’, Donald Caspar presented theseminal paper, co-authored with Aaron Klug, ‘Physical Principles inthe Construction of Regular Viruses’ (Caspar and Klug 1962).Among other things, they proposed that the protein shells (capsids)of spherical viruses are structured like microscopic geodesic domes.The ideas they presented were widely accepted until the early 1980swhen the first exception to their theory was discovered (in Caspar’slab). The Caspar-Klug theory of virus structure extended ideas ofFrancis Crick and James Watson who in the mid 1950s hypothesizedthat the small ‘spherical’ viruses must have cubic symmetry. Cubicsymmetry requires at least four three-fold rotational axes. For exam-ple, all the platonic solids – the icosahedron, the dodecahedron, thecube, the octahedron, and the tetrahedron – have cubic symmetry.Imagine a cube: if you look down the body-diagonal, you are lookingdown a 3-fold rotation axis and there are four such axes.

In 1955, Caspar was able to confirm Crick and Watson’s ideas bydiffracting x-rays through single crystals of Tomato bushy stunt virus

217GEODESIC DOME-SPHERICAL VIRUS ANALOGY

(TBSV). He showed that there were ‘spikes’ of intensity in the diffrac-tion pattern indicating the virus possessed five-fold rotational symmetryand thus very probably icosahedral symmetry (called 532 symmetry bycrystallographers), one of the three cubic symmetries. (Mirror symme-tries are ruled out by the chirality of protein.) With the knowledge ofCaspar’s result, Crick and Watson rewrote an article for Nature, inwhich they argued that ‘a virus possessing cubic symmetry must nec-essarily be built by the regular aggregation of smaller asymmetricalbuilding bricks, and this can only be done in a very limited numberof ways’ (Crick and Watson 1956, 474). A virus with cubic symmetryis necessarily made up of a multiple of 12 and a maximum of 60 iden-tical equivalently-situated ‘subunits’. Caspar’s experimental resultsfor TBSV were published in Nature immediately following Crick andWatson’s article (Caspar 1956).

Caspar and Klug began their collaboration in 1958. RosalindFranklin had been scheduled to present her recent work on virusstructure at a plant pathology meeting organized to celebrate the50th anniversary of the American Phytopathological Society.Tragically, she lost her battle with cancer in April of 1958, fourmonths before the meeting. Under these unfortunate circum-stances, Caspar was invited by the committee to speak in Franklin’splace and he invited Franklin’s colleague Aaron Klug to be a co-author. They wrote a review of the x-ray diffraction of viruses andadded her as first author posthumously (Franklin, Caspar and Klug1959).

In June of 1959, Klug and his colleague John Finch publishedtheir results on poliovirus. Newspapers popularized their results.The Manchester Guardian printed a half-page story, entitled ‘TheArchitecture of Viruses’, on June 30, 1959. One of BuckminsterFuller’s intellectual admirers, the proto-pop artist John McHale,read one of the newspaper reports and noticed the potential similar-ities between viral structure and Fuller’s geodesic domes. He wroteto Klug and Finch to inform them of the possible connection.Consequently, Fuller met with Klug and Finch on a visit to Londonlater in 1959.

Klug and Caspar obtained copies of Robert Mark’s book, TheDymaxion World of Buckminster Fuller (1960), and spent some timelooking at the many images of geodesic domes and their construction.The variety of enclosures designed and called geodesic domes by Fullermake it difficult to define the term. One can however describe an exem-plary case. The exemplary geodesic dome is constructed from manyapproximately equilateral triangular faces. (See Figure 1) It approxi-


mates a tessellated icosahedron projected onto a sphere, although inmost cases not all the triangular subunits are identical. Many of theedges of the faces, when projected onto a sphere follow arcs of great cir-cles, i.e., geodesics. Perhaps viral shells and geodesic domes were con-structed in a similar manner. The possibility of a rich analogy betweendomes and viruses suggested that there might exist a one-to-one map-ping between the protein subunits of a spherical virus and the subunitsof a geodesic dome (Fig. 1).

Under these circumstances, Caspar and Klug came to their famousidea of ‘quasi-equivalence’, an important concept in the emerging fieldof structural biology. Their insight was to see that if the inter-subunitbonds can vary a little in their angles and lengths, structures with morethan 60 identical quasi-equivalently bonded subunits could be built.Consider the triangular ‘subunits’ in the geodesic domes depicted inFigure 1. The subunits lie in at three quasi-equivalent positions. Thereare subunits in ‘pentamers’ (groups of five; shaded black in Figure 1)and the two types in ‘hexamers’ (groups of six). Notice how the relativeangles between the neighbors differ slightly for the pentamers and hexa-mers. The relaxation in the number of allowed subunits was importantbecause there was increasing evidence that many viruses had more than60 subunits. Caspar derived a formula that predicted which multiples of

Fig.1 - A T=3 geodesic dome (left) and Tomato bushy stunt virus (TBSV) at 2.9 Å resolution (right).Each shell consists of 180 subunits. The subunits on five fold axes are colored black. The double-headed arrows represent a sample of the one-to-one mapping between the subunits of the geodesicdome and virus coat proteins. Image of TBSV rendered using results from Steve Harrison’s group andthe Virus Particle Explorer Database (VIPERdb) (Harrison et al., 1978; Shepherd 2006).


60 subunits could build a closed icosahedrally symmetric biologicalassembly. The formula was remarkably simple: T = h2 + hk + k2 whereh and k are non-negative integers. Spherical viruses have 60T subunitsand can be classified by their respective T-numbers.

Caspar began to write a manuscript that explicated the idea of quasi-equivalence. In the meantime, Klug was invited to speak at the 1962Cold Spring Harbor meeting, a good venue to present their ideas. Theresulting paper has become a citation classic with more than 1300 cita-tions. (see Morgan 2003; 2004; and 2006 for more details on the historyof early structural virology)

Analogy as Isomorphism

How exactly did Caspar and Klug use their analogy between virusesand geodesic domes to develop their theory of virus structure? Whatdid the analogy allow them to do? Did it justify their subsequent devel-opment of the analogy? Before one can begin to answer these questionsproperly, one must clarify the nature of the analogy.

As Bailer-Jones (2002) points out, the role of analogy in science isclosely related to the roles of models and metaphors in science. Forexample, Hesse (1966) claims that scientific models are analogues toaspects of the real world. And Black ([1977] 1993) claims that allmetaphors ‘may be said to mediate’ an analogy (Black ([1977] 1993, 30).Rather than look at the relationships among models, metaphors, andanalogies, I plan to examine one important species of analogy directly.The term ‘analogy’ is sometimes used broadly as a synonym for ‘similar-ity’. Thus some accounts of analogy begin by defining a similarity met-ric to capture the degree of similarity between objects (Niiniluoto 1988).However, like phenetic approaches to biological classification, thisapproach is fraught with difficulties – which aspects of the respectiveobjects do you select and how? How do you place relative weights onthe different aspects? For these reasons, I will be more concerned witha narrower conception of analogy.

On my account, analogy is a binary relation between the source system,e.g., a geodesic dome, and the target system, e.g., a spherical virus, that doesnot come in degrees. Typically the source system is better studied or moreeasily studied than the target system that is a given scientist’s ultimate objectof study. Thus for a virologist, the target system is the virus and the sourcesystem is some other purported part of nature the study of which it is hopedwill enlighten the study of viruses. Extending a suggestion of Newell and


2 My definition of structure differs from standard presentations of structure in mathematical logic.For example, Enderton (1972, 79) takes a structure to be a function whose domain is the set of param-eters of a language. My notion of structure is closer to the range of this function.

Simon (1972), my account of analogy focuses on the trans-system counter-parts of relations that hold between the parts of two systems, rather than anaccount that is based on the similarity of attributes (Newell and Simon1972, 851; Gentner 1983). More explicitly:

An analogy exists between two systems, the source and the target, if and only if thereexists an isomorphic mapping between their respective faithful structures.Understanding this definition requires an understanding of the three terms, ‘system’,‘faithful structure’, and ‘isomorphic mapping’.

First, a system is either a set of facts, purported facts, fictional facts, or amixture. Since each fact consists of at least one object and at least one prop-erty or relation in which the object stands, a system can also be thought ofas a set of objects and properties/relations. Systems can be real or imaginary.A real system is a piece of reality. For example, the components of virusand their structural properties/relations is a real system, as is the com-ponents of a geodesic dome and their structural properties/relations. Inempirical science, imaginary systems come in two flavors: the simplifiedsystems of idealizations and the non-existent systems posited by falsetheories. An idealization, unlike an abstraction of a real system,describes simplified relations that do not obtain in reality. Real gasesviolate the ideal gas laws, for example. Any refuted theory posits at leastone false state of affairs. Such falsehoods do not prohibit one fromdrawing an analogy, however. One could, for example, draw an analogybetween a Copernican solar system with circular orbits and a Bohr atomwith circular orbits. Neither real electrons nor real planets have circularorbits, but the analogy may nonetheless obtain between the (imaginary)idealized systems. It is important to keep the following ontological cat-egories distinct: a phenomenon in nature (the real system), a theory of aphenomenon, and a structure that represents what a theory of a phe-nomenon posits.

Second, a structure is a representation of a system.2 It is a set-theoret-ic entity consisting of a set of elements (the domain) and sets of orderedn-tuples of these elements. To obtain a structure one must decomposethe system into parts, represent these parts by elements, and representthe n-ary relations that hold between parts of a system by sets of orderedn-tuples of elements. Since a set of n-tuples is normally taken to be arelation, I will use the term ‘relation’ for both a set of n-tuples and whatis represented by the set of ordered n-tuples, i.e., both the abstract set-

theoretic entity and the posited real n-ary property in the world. A rep-resentation of a system is faithful if and only if all of the features (i.e., theparts of the system and their relations) represented actually exist in thesystem.3 An abstraction is a faithful representation, but an idealization isnot. For the purposes of the paper, I assume that theories are sets of sen-tences, i.e., linguistic entities, that purport to describe the world accu-rately. If our theory of the phenomena is true, then a structure that faith-fully represents the posits of the theory also faithfully represents nature.One may then say that one draws an analogy between two real systems(i.e., parts of nature). On the other hand, if our theory of the phenome-na is false, then the structure that faithfully represents the posits of thetheory might not faithfully represent nature and some faithful represen-tations of the system will represent false states of affairs. The analogybetween the Copernican solar system and a Bohr atom can either beviewed as an analogy between non-existent systems posited by therespective theories, or between two phenomena in possible worlds inwhich the theories of the Bohr atom and the Copernican solar systemare true. In either case, this account of analogy is broad enough to applyto systems that do not obtain in nature.

Structures can differ in their degree of abstraction. For example, con-sider the following system: the 12-person family of Charles Darwinand Emma Wedgwood. This system might be represented by a struc-ture consisting of twelve elements that represent a father, a mother,six boys, and four girls and four sets of ordered pairs that representthe relations, ‘ __is a mother of__’ , ‘ __is a father of__’ , ‘ __is a sonof__’ , and ‘ __is a daughter of__’. Alternatively, a more abstract rep-resentation is given by twelve elements that represent two parents andten children and one set of ordered pairs that represent the relation‘__is a child of__’. To construct the first structure, one requiresknowledge of the gender of the members of the family, whereas in thesecond structure, the genders are abstracted away. Although they dif-fer in the degree of abstraction, neither structure introduces any false-hood about the Darwins. It is true that the Darwin family consistedof a father, a mother, six sons and four daughters. It is also true thatthe family consisted of two parents and ten siblings. On my account,this faithfulness to the system is a necessary condition on a structurerepresenting the system. A vital component of a scientist’s develop-ment of an analogy consists in determining the degree of abstractionof the two structures which yields the most informative isomorphism,i.e., maximizes the number of relevant relations shared by both sys-


3 Note that this definition differs from the meaning of faithful representation in Group Theory.


4 For a related discussion of this issue, see Sellars 1965, § III

tems without introducing relations which have no counterpart in theother system.4 Note that differences between the two systems, e.g., rela-tions that hold in one system with no counterpart in the other, will notappear in either of the two abstract structures that represent the two sys-tems if there exists an isomorphism between the two structures.

Third, two structures are isomorphic if and only if there exists a one-to-one mapping from all the elements of the source structure to all theelements of the target structure such that all the n-ary relations have animage in the target structure. There can be finite or infinite numbers ofelements. I impose no constraints on the identity of the meaning of theterms one uses to refer to the represented relations under a mapping. Ifa relation R holds between a and b in the source structure, then in anisomorphic structure there will exist a relation R* that holds between a*and b* (where a* is the image of a and b* is the image of b). However,the terms which refer to the relations represented by R and R* may dif-fer in their meaning. Typically people consider analogies where theseterms have similar meaning, but it is not part of the definition of an iso-morphism nor an essential part of my account of analogy. Higher orderformal properties such as the transitivity (or its lack) of each of the 2-aryrelations also have an image in the target structure. In fact, as Sellars(1965) points out, both source and target structures often exhibit thesame higher order properties (Sellars 1965, 346). An isomorphism is asymmetric relation between two structures. Thus on my account, ifthere is an analogy between system A and system A*, there is also ananalogy between system A* and system A. If two structures are fullydetermined, it is an a priori question whether there exists an isomor-phism between them. It is possible to have an analogy between two verydifferent systems if one structure is sufficiently more abstract thananother. In other words, while isomorphism is symmetric, asymmetriesbetween the systems can often be offset by asymmetries in abstraction.

In maturing empirical sciences, often the question is whether there isa useful analogy between two real systems. This empirical question hastwo components: (1) whether there exists an analogy between two realsystems and (2) if there exists an analogy, whether it is useful. First,while it is an a priori question whether there is an isomorphism betweentwo structures, one must consult the world to determine whether eitherstructure faithfully represents a real system. Thus, whether there is ananalogy between two real systems is empirical. While a faithful structureof a system, as I have defined it, may abstract detail from a system, itmust not introduce elements that do not exist, or relations that do not


hold in the system represented. If the system is real, one must knowsome things about it before one can know whether a structure has intro-duced relations that do not hold in the system. Second, the usefulnessof an analogy is relative to the scientist and to the state of knowledge inquestion. Scientists wish to be able to infer useful hypotheses, the test-ing and refining of which would constitute the continuance of theirresearch program. Let us call an analogy whose structures contain therelations in which a scientist is interested, a useful analogy. Further, ifthe useful analogy will sustain a research program, let us call it a richanalogy. A goal of constructing an analogy is to determine what degreeof abstraction best captures the relations in which the scientist is inter-ested. If the structures are too abstract, the relations in which the scien-tist is interested will not be represented. If the structures are notabstract enough, there may not be an isomorphism since differences willstill be represented or the isomorphism may be too complex to be prac-tical. Further, one must add the adjective ‘useful’ because at somedegree of abstraction, everything is analogous to everything else. Onedimension of usefulness is the inferential power of the analogy. (I willsay more about inferential power in the next section.)

The case study sheds some light on how the correct degree of abstrac-tion is obtained in practice. In this case, the level of abstraction was con-strained by the goal that the scientists wished to achieve. Caspar andKlug had a well-defined problem that they desired to solve. Their prob-lem was, given that many viruses have more than 60 protein subunits,how are the subunits of these viruses arranged to form a closed con-tainer to protect the viral nucleic acid? The analogy with geodesicdomes was drawn in order to provide a hypothesis about the way inwhich the viral subunits might be arranged, i.e., a means to achievingtheir goal. Any structure that did not represent the low-resolution three-dimensional arrangement of subunits would be too abstract to be usefulto Caspar and Klug. Russell (1989) calls this goal-directed use of analo-gy the ‘backwards’ direction of analogical inference as opposed to the‘forwards’ data-driven direction in which a new similarity is inferredfrom known features of the source without the teleological constraintson what the similarity can be (Russell 1989, 9).

One structure that faithfully represents a geodesic dome consists of adomain of n elements that represent n triangular faces and the set ofordered pairs that represent the binary relation ‘ __is adjacent to__’ . (Ifthe ordered pair <a, b> is a member of the set of ordered pairs repre-senting the adjacency relation let us abbreviate this ‘ aAb’ .) Intuitively,two elements are adjacent if and only if the faces they represent share anedge. If xAy and yAz and x≠y and y≠z, then the faces x and z represent


share a vertex. The triangularity of faces is captured in the structure byspecifying that for each element, it is adjacent to exactly three others,i.e., ∀x∃y∃z∃ω(xΑy & xAz & xAw & x≠y & y≠z & z≠w & ∀q(xAq ⊃(q=y ⁄ q=z ⁄ q=w))). One can specify some high-order relations in termsof the 2-ary relation ‘__A__’ . For example, elements are clustered intogroups of five or six, what are called pentamers and hexamers. An ele-ment x is a member of a pentamer if and only if ∃y∃z∃ω∃q(xAy & yAz& zAw & wAq & qAx & x≠y & y≠z & z≠w & w≠q). An element x is amember of a hexamer if and only if ∃y∃z∃ω∃q∃p(xAy & yAz & zAw &wAq & qAp & pAx & x≠y & y≠z & z≠w & w≠q & q≠p). One way to re-present other relations, such as the symmetry properties of the virus, isto use a one-to-one function from the set of elements onto itself. A sym-metry property exists when such a function preserves all the adjacencyrelations (i.e., is an automorphism). Each distinct symmetry will have adistinct function. For example, a function, f(x), that captures a five-foldrotation symmetry will have the property that ∀x(f(f(f(f(f(x)))))=x)whereas a function, g(x), which captures a mirror symmetry will havethe property ∀x(g(g(x))=x). On this account, symmetries can be thoughtof as analogies of a system with itself.

The source structure consisting of n elements and the adjacency rela-tion abstracts away features of real geodesic domes. The structure doesnot represent the way in which the subunits are held together, for exam-ple. The structure representing a spherical virus consists of m elements,which represent what Caspar and Klug called structure units, and whichbiologists now know are (usually) single protein chains, and the set ofordered pairs that represent the real relation ‘__is bonded to__’. Thisstructure abstracts away the three-dimensionality and the chirality ofprotein chains. Caspar and Klug’s claim that there is an analogy betweena spherical virus and a geodesic dome is the claim that there is a one-to-one mapping between the n dome elements and the m viral elements(where n = m), such that for any two dome elements, x, y, which repre-sent adjacent faces, their images in the virus structure, x*, y*, representviral subunits bonded to one another. This mapping preserves the icosa-hedral symmetry of dome structure in the virus structure.

Deductive Inference from an Analogy

One important role of analogy in science is to allow scientists to inferhypotheses about the target system. These inferences, I will call infer-ences from an analogy or simply analogical inferences. One also caninfer that it is plausible that a useful analogy exists between two systems.


I call this second type of reasoning, inferences to an analogy. In the nextsection, I will consider inferences to an analogy and in this section I willconsider inferences from an analogy. I will show that the analogybetween viruses and domes differs from frivolous analogies that do notallow for useful novel inferences.

Rather than characterize analogical inference as an amplitive infer-ence, I follow Weitzenfeld (1984) and propose that analogical infer-ence is actually a deductive inference. The inference from the exis-tence of an n-ary relation R in the source structure to the existence ofa counterpart n-ary relation R* in the target analog is an ellipticalinference. The suppressed premise is that there exists an isomorphismbetween the two structures. If one makes the suppressed premiseexplicit, the inference has the following form:

1. There exists an isomorphism between structure A and structure A*2. Relation R exists in structure A∴ There exists a relation R* in structure A*A narrower conception of analogy can be defined if one restricts

the meaning of the term which refers to the counterpart relationrepresented by R* to have the same meaning as the term whichrefers to the relation represented by R, but this is not required inthe more general definition. All that the isomorphism requires isthat if the relation R holds among source components a1, ... , an, thenthere exists some relation R* which holds among the target compo-nents a1*, ... , an*, where the mapping takes aj to aj*. Without con-straints on the meanings of R and R* there is the possibility ofinducing counterpart relations under arbitrary mappings, but thesead hoc analogies will not tend to produce useful inferences and thuswill usually be examples of frivolous analogies. (I will discuss frivo-lous analogies further below.)

Inferences made by Caspar and Klug were more concrete than thescheme above since the nature of the isomorphism is easily madeexplicit. Because there are only a finite number of elements in thedome source and viral target structure the mapping can be enumerat-ed. It consists of a set of ordered pairs, i.e., {<dome element a1, viruselement a1*>, <dome element a2, virus element a2*>, ... , <dome ele-ment an, virus element an*>}. For an example of a more concrete infer-ence consider the following: suppose one names two adjacent domeelements, a1 and a2. It follows that a1Aa2. From this fact and the enu-merated isomorphism, one can deductively infer that a1*A*a2*. As men-tioned in the last section, the image of the relation ‘_A_’ , i.e., ‘_A*_’represents the relation ‘_is bonded to_’ , thus the conclusion is that a1*and a2* represent subunits bonded together in the virus structure.

One can also draw inference to the cardinality of the elements in thetarget structure. For example,

1. There exists an isomorphism between structure A and structure A*2. Structure A consists of n elements∴ Structure A* consists of n elements.Caspar and Klug made inferences of this form. They were interested

in the number of subunits that made up a viral shell. By determining thenumber of elements in the faithful structure of a given geodesic dome,they could infer the number of subunits in the analogous virus. This pre-diction/hypothesis could then be tested biochemically.

Some analogies have little inferential power. Inferential power is ameasure of the number of useful inferences that can be drawn weightedby their relative utility. I call analogies with little inferential power friv-olous analogies. One species of frivolous analogies contain few usefulrelations in the structures that represent the systems. For this reason,these analogies do not allow one to infer a number of useful novelhypotheses needed to sustain a research program.5 Frivolous analogiesare not rich analogies. The ability to sustain a research program is,ceteris paribus, proportional to the inferential power of the analogy. Onmy account, frivolous analogies do not differ in kind, but only in degree,from non-frivolous analogies. The difference between frivolous andnon-frivolous analogies that support equally useful inferences is simplythe number of novel inferences one can make. For example, JamesClerk Maxwell compared the molecules of a gas to a swarm of bees: ‘Ifwe wish to form a mental representation of what is going on among themolecules in calm air, we cannot do better than observe a swarm ofbees.’ (Maxwell [1873] 1952, 368) One symptom of the frivolousness ofthe analogy is the absence of a mathematical treatment of bee swarmsthat if appropriated for the study of gases would allow Maxwell to inferuseful novel hypotheses about gases. Most mathematical modelsembody numerous useful relations that, if present in an analogoussource system, would allow one to make useful inferences about the tar-get system. Rather, Maxwell merely wanted us to make a single infer-ence: conclude that if one could see gas molecules, they would appearto be a relatively stationary morass of individual small masses ‘flyingfuriously’. Often frivolous analogies are used pedagogically or in popu-larizing scientific ideas to the non-mathematical layperson.

I am concerned only with non-frivolous analogies, which is not to say


5 Kekulé’s analogy between benzene and a snake is a frivolous analogy as I have defined it. Therewas only one inference: Kekulé inferred that the benzene molecule was circular and so began the fieldof aromatic organic chemistry. However, aromatic chemists did not refer to snakes to make furtheradvances in their field and so the analogy did not guide a research program.


that there are not frivolous analogies in the early history of structuralvirology. For example, Crick and Watson (1956) suggest that under highmagnification, a ‘spherical’ virus may appear ‘like a rather sphericalmulberry’ (Crick and Watson 1956, 474). Horne and Wildy (1961) sug-gest that the asymmetrical building blocks of viruses may be bundledtogether like a ‘faggot’, i.e., a bundle of sticks (Horne and Wildy 1961,357). Their idea is that an individual stick is asymmetrical, but a bundlehas approximate rotational symmetry. Clearly, no one is suggestingthat to study viruses one needs to look at berries or bundles of sticks,and rightly so. Rather the inferential power of frivolous analogiestends to be quickly exhausted and they do not allow for the forma-tion of a research program aimed at further development of the anal-ogy.

The analogy between domes and viruses is not frivolous. It allows oneto infer numerous useful novel hypotheses that are able to guide aresearch program in structural virology: it is a rich analogy. In fact,Caspar and Klug implicitly draw an analogy for every type of sphericalvirus. Consider the following inferences:

1. The faithful structure representing Tomato bushy stunt virus (TBSV) is isomorphic to the faithful structure representing a T=3 geodesic dome.

2. A T=3 geodesic dome has 180 subunits.∴ TBSV has 180 subunits.

Note that the conclusion can be about the virus itself and not merelyabout the structure representing the virus, if the structure is a faithfulrepresentation of the virus. What is true of the faithful structure is alsotrue of the virus.

1. The faithful structure representing Adenovirus is isomorphic to thefaithful structure representing the T=25 geodesic dome.

2. The structure representing the T=25 geodesic dome has 240 hexamers.

∴ Adenovirus has 240 hexamers.

1.The faithful structure representing TBSV is isomorphic to thestructure representing the T=3 geodesic dome.

2.The faithful structure representing the T=3 geodesic dome hasthree types of quasi-equivalent elements.

∴ TBSV has three types of quasi-equivalent elements.These analogies can be generalized to apply between a generalized geo-desic dome and a generalized spherical virus:

1. The faithful structure of the T=n geodesic dome is isomorphic tothe faithful structure of a T=n spherical virus.

2. The faithful structure of the T=n geodesic dome has 60n


elements, 60(n-1) elements arranged in rings of 6, and 60 elementsarranged in rings of 5.

∴ A T=n spherical virus has 60n elements, 60(n -1) elements arranged in rings of 6, and 60 elements arranged in rings of 5.

In sum, there is a non-frivolous rich analogy between geodesic domesand viruses since one can represent viruses and domes faithfully withisomorphic structures and the isomorphism allows for many usefulhypotheses to be made about viruses. These inferences proved worth-while for virology because they sustained a research program and savedCaspar, Klug, and others additional time and effort that would havebeen necessary if they had developed a theory directly from the obser-vation of viruses.

Inferences to a Plausible Analogy

On my account, it is an empirical question whether there exists auseful analogy between two real systems, such as between a geodesicdome and a spherical virus. Therefore, to justify a claim that there isan analogy between two systems will require empirical evidence thattwo isomorphic structures faithfully represent the target and sourcesystems. However, this is not to say that there cannot be reasons forthinking that a given useful analogy is plausible before one beginsdesigning experiments to test whether the analogy actually exists. By‘plausible’ I mean that the probability of the analogy given the back-ground knowledge is greater than some significantly non-zero thres-hold value k, i.e., p(analogy/B) ≥ k. I take plausibility to be a strongerthan mere physical possibility, but a weaker than high probability.The exact numerical value of k is difficult to specify and varies fromcontext to context.

Why is a plausible analogy important for a scientist? An experimen-tal researcher aims to generate plausible hypotheses about her targetsystem since the testing of these would constitute the continuance of herresearch program. Given one can say that there is a reason to think thatan isomorphism between viral structures and geodesic domes is plausi-ble, does it follow that there is a reason to think that the conclusion ofan analogical inference, i.e., the potential hypothesis, is also plausible?Consider the form of the analogical inference:

1. There exists an isomorphism between faithful source structure Aand faithful target structure A*.

2. Relation Ri exists in structure A.


∴ There exists a relation Ri* in the target structure. If there is a reason for thinking that the first premise is plausible and

one determines that the second is true, is there a reason to think that theconclusion is plausible? Yes, since any reason to think that R* does notexist in A* is also a reason to think that the isomorphism does not exist.This can be shown more formally also. A useful analogy is plausible ifand only if p(analogy/B) ≥ k where k> 0 and B stands for backgroundknowledge. One may write p(analogy/B) = p((the appropriate map-ping/B) & (R1* in A*/R1 in A & B) & (R2* in A*/R2 in A & B ) & ... &(Rn* in A*/Rn in A & B)). Since p(X & Y/B) ≤ p(X/B) is a theorem ofthe probability calculus, p(Ri* existing in A*/Ri exists in A & B) ≥ k. IfA* is a faithful structure and that Ri exists in A is part of the backgroundknowledge, then one may infer, p(Ri exists in the target system/B) ≥ k.Thus, if one has reason to think an isomorphism is plausible betweentwo faithful structures, one has reason to take the conclusions of ana-logical inferences using these structures as plausible also. The task thenis to show under what conditions a useful analogy is plausible.

What would constitute a reason for thinking it is plausible that a use-ful analogy exists between two systems? In general, the idea is that onecan justify an analogy’s plausibility by appealing to some relevant com-monality between the two systems and by showing that the differencesbetween the two systems are irrelevant for the analogy. For example, itis plausible there would be an analogy between two copies of the sameissue of The New York Times because common processes produced bothcopies. Thus, a common process of production provides a reason tothink it is plausible that isomorphism exists between two faithful struc-tures that represent products of that process. This is why biologistscommonly assume that an analogy between same-aged members of thesame biological species is plausible. More generally, homologous bio-logical traits are often analogous (in the current sense) because there isa common developmental process.

Unfortunately, domes and viruses do not share a common develop-mental process, so one must look elsewhere for the justification of thisanalogy’s plausibility. I suggest that one can begin by appealing to therespective functions of domes and viruses. Of course, in general, theinference from common function to isomorphic structure, or even theweaker inference from common function to the plausibility of an iso-morphic structure is fallacious. Both inferences depend on the numberof plausible ways the same function can be realized in different struc-tures. Thus, the consideration of common function serves merely as astarting point in our discussion. At one level of description, domes andviral shells have the same purpose. Both have the function of enclosing


6 Many of the important papers are anthologized in Allen 19987 For a defense of this heuristic, see Cosmides et al. 1992, ff. 10.

space so as to create a safer environment inside than outside. Architectsdesign domes to protect the inside from wind, precipitation, fluctua-tions in temperature, etc. Viral shells function to protect the viral nucle-ic acid from external threats to its integrity such as enzymes callednucleases that break nucleic acids into pieces, thereby destroying anyability of the virus to replicate.

There is also, however, an obvious disanalogy between domes andviruses. Domes are man-made structures whereas viruses are naturalbiological agents. Given this disanalogy, one might protest that it isanthropomorphic to attribute functions to natural entities. However,much work in contemporary philosophy of biology has been expendedto show that a useful naturalistic concept of biological function can besuccessfully defended.6 Although there are differences between naturalfunctions and intentional functions, there remains enough similaritythat one can compare human designs with natural designs. A usefulheuristic is to think of viruses and domes as solutions to analogousdesign problems.7 The similarities between the problems that domes andviruses solve provide the beginnings of a reason to think it is plausiblethat an analogy between the respective solutions exists.

There are many ways of skinning a cat, so to speak, and one is notinclined to think that all cat-skinning solutions are isomorphic. Whythen should one think that it is plausible that viruses and domes areanalogous solutions to analogous problems? Given the constraint thatyou must build a structure out of nearly identical subunits that bondtogether in quasi-equivalent ways, geodesic domes are very efficientlybuilt closed structures. Buckminster Fuller made much mileage of thisfact when promoting his domes, often citing the weight-to-volume ratiosof domes, which are orders of magnitude less than conventional build-ings. Given the intense selection pressures and the amount of time virus-es have been evolving, there is good reason to think that viruses haveevolved highly efficient solutions to the problem of enclosing space. Theimplicit assumption is that there are a small number of solutions, possi-bly a unique solution, to the problem of efficiently enclosing space usingnearly identical subunits. Adding the constraint that the solution mustbe efficient excludes structures that inefficiently solved the same prob-lem, but unfortunately it does not tell us how many different (i.e., non-analogous) plausible efficient solutions there might be.

Even with the constraint that the solution must be efficient, it is pos-sible that there are many plausible efficient non-analogous solutions. In


which case it might not be plausible that there is a useful analogybetween two solutions. Both the electric chair and lethal injection areefficient ways of solving the same problem, i.e., performing capital pun-ishment in a reliable quick manner, but there is no useful isomorphismbetween the two processes. Clearly one needs further evidence beforeone can say that it is plausible that viruses and domes solve the sameproblem analogously. Perhaps the most significant piece of informationfor Caspar and Klug was that domes and viruses appeared similar. Thefirst good electron micrographs of viruses showed structures thatappeared like miniature geodesic domes. (See, for example, Brennerand Horne 1959) Furthermore, Caspar and Klug were impressed withthe ease with which domes could be constructed. Often unskilled work-ers assembled the domes by blindly following a small number of simple‘bonding’ rules. Likewise, viruses must assemble their shells without theoversight of an intentional being. With this point in mind, Francis Crickclaimed that the assembly of the viral subunits into the closed shellshould be simple enough that a child could do it. However, even somedegree of similarity added to the knowledge that viruses and domeshave solved analogous problems efficiently is still not enough to say thatit is plausible that the solutions are usefully analogous.

To infer that an analogy is plausible, in addition to structuressolving analogous problems one must know of no reasons for think-ing that the structures are significantly different. To return to thegeodesic dome/spherical virus case study, a potential differencemight be the difference in scale. Domes are on the order of tens ofmeters in diameter whereas viruses are merely nanometers across.Fortunately, common reasons for thinking that the solutions to theproblem of efficiently enclosing space may be scale-dependent arenot relevant in this case. First, both geodesic domes and viruses aresmall enough that any relativistic effects due to the curvature ofspace-time are insignificant. Second, since viruses are sufficientlymassive (on the order of millions of Daltons where one Dalton = themass of one Hydrogen atom), quantum effects are negligible.Finally, one should consider properties of objects that are scale-dependent such as, surface-to-volume ratio. If one assumes that theweight of an object is proportional to its volume, then the weight ofan object increases to the cube as its surface area increases to thesquare. Since a material structure has to be able to support its ownweight, one might not expect a solution for a small material struc-ture also to be a solution for a larger structure. For example, largeranimals are not merely scaled versions of smaller animals (MaynardSmith 1968, 7). In the case of viruses and geodesic domes, however, this


is a scaling down in size, so these considerations suggest that viruseswould both be relatively stronger than geodesic domes and bettersolutions to the problem. In sum, these three reasons why a solutionto a problem at one scale would not be a solution at another scale arenot applicable to the virus/geodesic dome case. There are no goodreasons in the background knowledge to suggest that viruses anddomes have non-analogous solutions to the problem of efficientlyenclosing space and it was plausible to think that a useful analogyexisted between the two systems.

To generalize the above discussion, I propose the following principle: It is reasonable to infer that a useful analogy between two systems isplausible if they look similar, are efficient solutions to analogous prob-lems, and there are no good reasons in the background knowledge tothink otherwise. This is not to say it is physically impossible that sim-ilar looking solutions to the same problem are not usefully analogous.However, the mere existence of this possibility without any reasonto think it is true should not lead one to think that it is implausiblethat the two solutions are analogous. And this situation is exactlywhat one should expect for empirical science – for if there were aconclusive answer based merely on reflection prior to any experi-mentation, there would be no research program and no experimentsfor scientists to conduct. Most non-frivolous analogies in empiricalscience require testing before one can be justified in believing thatthey exist; nonetheless, there may be reason to think they are plau-sible before any experiments are undertaken, as there was in thiscase. Empirical testing increases the probability that the analogyexists given our background knowledge, if it increases our knowl-edge of isomorphism between the two solutions. For example, bio-chemical experiments showed that viruses had the same number ofsubunits as the postulated analogous domes. This fact increases theprobability that a useful analogy between domes and viruses actual-ly exists.

Conclusion

Historians and philosophers of science have overlooked one ofthe most successful biological theories developed in the 1960s: theCaspar-Klug theory of virus structure. Donald Caspar and AaronKlug developed their theory with explicit analogies withBuckminster Fuller’s geodesic domes. Domes and viruses lookedsimilar and it was reasonable to think of domes and viruses as effi-


ciently solving analogous problems. Furthermore, obvious candidatesfor barring macro-world solutions in the micro-world were not applica-ble. Thus, Caspar and Klug reasonably judged the analogy to be plausi-ble. In general, if two systems resemble one another, are both efficientsolutions to analogous problems and there are no reasons to think thecontrary, then there is a reason to think that a useful analogy is plausi-ble.

Through drawing the analogy between geodesic domes and virus-es, Caspar and Klug were able to infer novel plausible hypothesesabout viruses by first considering geodesic domes. I argued thatanalogical inference is a form of deductive inference. Given that itis plausible that a useful analogy exists between two systems, andthat relation R holds in the faithful source structure, it follows thatit is plausible that R* holds in the target system. Virologists can thendesign experiments to test these plausible novel hypotheses. If theexperiments confirm the hypothesis, the existence of the analogybetween domes and viruses is empirically supported. In general, theexistence of an analogy between real systems is an empirical ques-tion, which does not imply that analogies cannot hold betweenimaginary systems or between a real system and an imaginary sys-tem.

This account of analogy can be applied to other systems.Whenever there is a reason to think a useful analogy between tworelatively unstudied systems is plausible, one can infer plausiblenovel testable hypotheses about the target system. One expects tofind this type of reasoning in science concentrated in cases in whichstudying the target system directly is significantly more difficultthan studying the source system or when studying the target systemdirectly is prohibitive. With different disciplines in science studyingnature at different scales, it would be surprising if the Caspar-Kluganalogy between systems at different scales were the only member ofthis species of useful analogy.

Acknowledgements

I thank Peter Achinstein, Lindley Darden, Robert Rynasiewicz,Stephanie Girard and the two anonymous referees for their commentson earlier drafts of this paper and Joy Striplin for her secretarial assis-tance.


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Why there was a useful plausible analogy between geodesic domes and spherical viruses

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