A Connectionist View of Parallel Distributed Processing by David E. Rumelhart; James L. McClelland; PDP Research Group Review by: Keith J. Holyoak Science, New Series, Vol. 236, No. 4804 (May 22, 1987), pp. 992-996 Published by: American Association for the Advancement of Science Stable URL: http://www.jstor.org/stable/1699673 . Accessed: 22/08/2012 13:52

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http://www.jstor.org of which is rendered in the Book of Genesis, all, immaterialfor the resolution of the deep that all mental phenomena are in principle, belongs on the trash-heap of outdated folk mind-body problem of praxis, posed by the explainable by neurobiological theories, just theory. All the same, she concedes that there paradoxical human condition of simulta- as, as a physical chemist, I believe that all will be those who "may tend to see the neously facing the two incommensuratereali- neurobiological theories are, in principle,ex- revision of folk theory and the rise of neuro- ties of science and of ethics, whether psycho- plainable by physico-chemical theories. biological-psychological theory as the irrep- logical theories are or are not reducible to Moreover, I look forward to some progress arable loss of our humanity." But not to neurobiological theories. still being made in the venerable enterprise worry, because Neuroscientists and psychologists do not of reductionist neuroscience. Yet, I doubt need much assistance from philosophers in that a complete reduction is de facto possi- it may be a loss, not of somethingnecessary for their struggle with the mind-body problem, ble. My cardinal hunch is that a significant our humanity,but of something... that, though as it is posed within the context of theoreti- residue of unreduced psychological, as well second nature, blinkersour understandingand cal reason. As Churchland herself points as neurobiological, theory will remain with tethersour insight .... The loss, moreover,may include certain folk presumptionsand myths, out, the controversy regarding neuroscien- us long into the future. that, from the point of view of fairness and tific reduction of psychological theories will Where neuroscientists and psychologists deccncy,we come to see as inhumane. be settled anyhow, in the wash of future do need philosophical help is in fathoming experimental and theoretical developments. not the physical but the metaphysical infra- With the words "fairness,""decency," and My own expectations are those of a member structure of folk presumptions and myths "inhumane," Churchland makes her first of the set styled "boggled skeptics" by and the likely consequences for the human and only reference to the ethical dimension Churchland. We boggled skeptics tend to condition of their abandonment. Church- of the mind-body problem, after having view the human brain as belonging to a class land is not one of the folks who can provide exclusively considered its scientific dimen- of phenomena whose very complexity limits that help. sion. In her earlier discussion of Descartes's the extent to which theories designed to GUNTHER S. STENT contributions she did not mention his moti- explain them can be successfully reduced by Department of MolecularBiology, vation for holding the substance dualist theories developed to explain less complex Universityof California, view, namely the argument that the body, phenomena. As a neuroscientist, I believe Berkeley,CA 94720 being a machine, could not be guided by moral principles; hence the mind, which obviously is guided by such principles, can- not be a physical part of the body-machine. More important, Churchland has provided A Connectionist View of Cognition an inadequate account of Kant's treatment of the mind-body problem, which is gener- realization. Just as a computer program can ally considered to have initiated the Coper- Parallel Distributed Processing. Explorations be described without reference to the partic- DAVID E. nican revolution in philosophy that Church- in the Microstructureof Cognition. ular hardware on which it runs, a functional RUMELHART, JAMES L. MCCLELLAND, and the land thinks is about to be set off by neurosci- analysis of cognition need not directly refer PDP RESEARCH GROUP. MIT Press,Cambridge, to brain processes. entists. By pointing out that we live in two MA, 1986. In two volumes.Vol. 1, Foundations. metaphysically distinct worlds, Kant had xx, 547 pp., illus. $27.50. Vol. 2, Psychological The avowed intent of the authors of Par- replaced Cartesian substance dualism with andBiological Models. xiv, 611 pp., illus.$27.50. allel Distributed Processingis "to replace the an "epistemic"dualism. One of these worlds ComputationalModels of Cognitionand Percep- 'computer metaphor' as a model of mind is that constructed by the theoretical reason tion. A BradfordBook. with the 'brain metaphor"' (vol. 1, p. 75). of science, whose natural objects (including The publication of this massive work is a the brain of Homo sapiens) are governed by Two types of devices are known that can landmark event in for a laws of causal determination. The other support such functions as perception, mem- mixture of scientific and sociological rea- world is that constructed by the practical ory, language, and problem solving. One is sons. The principles of parallel distributed reason of ethics, whose rational human sub- the modern digital computer, programmed processing (PDP), a variety of "connection- jects are governed by laws of freedom that to produce "artificialintelligence" (Al); the ism," challenge the functionalist attitudes of individual free will imposes on each person's other is the human brain, which produces cognitive psychologists, offer a distinct alter- actions. Here practical reason justifies the the natural variety. Given that the latter native to conventional Al techniques, and concept of free will, not on the introspective device seems more intimately connected to suggest representations of linguistic knowl- basis, which, as Churchland justly points the human mind, it may seem surprising edge very different from the rule systems out, cannot claim evidential priority over all that the dominant metaphor for developing typically used by linguists. The volumes other empirical arguments advanced by the- theories of mental processes has been based appear against a backdrop of conferences, oretical reason, but as a logically necessary on the former. Modern cognitive psycholo- workshops, and seminars devoted to the constituent of the intuitive theory of person- gy, which has come of age with the digital PDP approach. Although in hood that governs interpersonal human rela- computer, has been more heavily influenced fact has a long heritage and current models tions. As such, the practical concept of free by computer science than by brain science. have been actively developed over the past will is not, in principle, subject to reduction There are at least two reasons for this. First, decade, the approach has recently acquired by scientific theory, be it neurobiological or we know vastly more about the functioning the vigor of a movement in the first bloom psychological. of computers than of brains. Second, the of youth. The movement has a proselytizing Accordingly, from the perspectiveof episte- basic approach of cognitive has bent, and talk of a Kuhnian "paradigmshift" mic dualism, the neurophilosophical brouha- been predicated on a philosophical position is in the air, accompanied by a spirited mix ha about the reducibility of psychology to known as functionalism, which emphasizes of hype and hope on the part of adherents neurobiology is, to use one of Churchland's that mental functions can be analyzed at an and by expressions of skepticism from vari- phrases, "mere crinkum-crankum."It is, after abstract level separate from their physical ous critics. Parallel Distributed Processing

992 SCIENCE, VOL. 236 provides a focus for this excitement. As one contributor puts it, "The present book offers an alternative paradigm for cognitive sci- ence, the subsymbolicparadigm, in which the na..;stpowerful level of description of cogni- tive systems is hypothesized to be lower than the level that is naturally described by symbol manipulation" (vol. 1, p. 195). The basic tenet of parallel distributed processing is that information is represented solely by patterns of activation over neuron- like "units" linked by synapselike "connec- tions," which can be either excitatory or IN inhibitory. Input units respond directly to [INO [INO [INO [INi [INO [INO features of the environment, and output units represent responses of the system. In "A hardwired processing structure for bottom-up processing of the words IN, NO, ON, and SO presented to any two adjacent letter slots. Note that each letter slot participates in two different word between may lie "hidden" units that perform slots, except at the edges." [From Parallel Distributed Processing,chapter 16.] internal processing. Individual units update their activation level (a numerical value) as a function of the activation levels and connec- a major subset. They also represent a diverse er," to partition the patterns in such a way tion weights of other units that feed into the set of scientific disciplines, including cogni- that each unit responds to a unique feature system. Processing consists of a set of input tive psychology, computer science, physics, combination. The generalized delta rule units being activated (typically by an envi- mathematics, neuroscience, and molecular (chapter 8) is a procedure for adjusting the ronmental stimulus), which causes activa- biology. This collaboration in itself consti- weights on incoming connections as a func- tion patterns to propagate (across hidden tutes a major contribution to the interdisci- tion of the difference between a unit's ob- units if the network is multilayered), eventu- plinary field of cognitive science. Although tained activation value and the "correct" ally activating a set of response units. Learn- the work has implications for several fields, value. A "teacher"provides the correct acti- ing is based on mathematical algorithms that the central focus is human cognition. This vation values for the output units, and the adjust connection weights so that inputs emphasis reflects the fact that both McClel- algorithm then propagates weight adjust- produce responses that are more appropriate land and Rumelhart are cognitive psycholo- ments back to hidden and input units, in- by some specified criterion. gists; the third major contributor, Geoffrey cluding units that do not connect directly to The term "distributed" has two distinct Hinton, is a psychologically oriented com- output units. By exploiting hidden units, the meanings within connectionist models. All puter scientist. algorithm can learn to compute simple ver- such models perform distributedprocessing in The chapters fall into five major groups. sions of functions such as "exclusive or" and that each unit adjusts its activation levels and Part 1, in volume 1, includes four overview "odd versus even" that cannot be computed associated connection weights on the basis chapters by McClelland, Rumelhart, and with only input and output units. The latter of local computations. Such systems can Hinton. (The final chapter in volume 2 is limitation was highlighted by Minsky and exhibit massive parallelism and do not re- also an overview.) Chapters 1 through 3 in Papert in 1969 in a critique of "percep- quire oversight by a central processor. Some particular are essential reading, providing a trons," which were networks that lacked PDP models also use distributedrepresenta- general description of the motivation for hidden units. That critique effectively ended tions, in which meaningful elements are parallel distributed processing, a survey of an earlier generation's interest in adaptive represented by patterns of activation over a the core ideas and their variations, and an networks; the generalized delta rule is cru- number of units, rather than by single units. introduction to the notion of a "distributed cial in justifying the current revival. For example, a distributed representation of representation." Chapters 6 and 7 respectively describe the word cat might consist of two active The other sections of the books are much Smolensky's harmonytheory and Hinton and units, one representing the occurrence of ca moreitechnical. Part 2 includes four chapters Sejnowski's Boltzmann machine. These two in the first two letter positions and another on connectionist schemes for models of models, although couched in different ter- representing the occurrence of at in the final learning by adjusting connection weights, minology, are essentially identical. Each is two positions. Each of these units would the area that constitutes the core of recent based on a formal isomorphism between also form part of the representations of theoretical advances. Each chapter in this information processing and statistical ther- many other words. Distributed representa- section combines mathematical analysiswith modynamics. Processing consists of maxi- tions contribute to some of the more inter- computer simulation. Chapters 5 and 8, by mizing an optimization function based on esting properties of PDP networks as well as Rumelhart and colleagues, present two gen- the consistency of the activation pattern to some of their most difficult representa- eral algorithms for learning in networks. over a network of units that have symmetri- tional hurdles. Competitivelearning (chapter 5) is an algo- cal connection weights and binary ("on" or These two volumes consist of 26 chapters rithm whereby each unit in a mutually in- "off") activation levels. For example, if two by David Rumelhart, James McClelland, hibitory set learns to respond when patterns units have an excitatory connection, consist- and 14 other members of the "PDP Group" with a particular feature are presented. The ency is increased if they are both on or off that became active at the University of Cali- basic idea is that whichever unit responds rather than one on and one off. The updat- fornia at San Diego in 1982, with more most actively to an input pattern adjusts its ing process is probabilistic; the "noise" in recent offshoots at Carnegie-Mellon Univer- connection weights so as to respond slightly the process is at first large and then is sity and elsewhere. Although several notable more strongly to future occurrences of that progressively reduced. This "simulated an- figures in thle connectionist movement are input. The competing units eventually learn, nealing" is based on an analogy with the not represented, the contributors constitute without guidance from an external "teach- behavior of particles that are heated and

22 MAY 1987 BOOK REVIEWS 993 then allowed to cool. Annealing serves to brain, but it is unknown and probably only gree of positive evidence for the second. The alleviate the "local minima" problem that all approximate. A single unit may correspond connection between two units can be direct- gradient-descent optimization procedures to a neuron, a cluster of neurons, or a ly translated into a simple rule of the form, must contend with (that is, the tendency for conceptual entity related in a complex way "If hypothesis 1 holds, then hypothesis 2 the search process to get trapped at some to actual neurons" (vol. 2, p. 329)-in holds," with a strength measure attached to solution that appears optimal relative to short, to virtually anything (perhaps includ- the rule. Summation of activation at each similar solutions but is inferior to some ing meaningful "symbols," the connection- unit serves to integrate multiple sources of other very different solution). Boltzmann ists' bUtenoire). Despite the admitted lack of converging or contradictory evidence re- machines also have a learning algorithm constraint on what units can represent, the garding a hypothesis. The various learning based on a comparison of the probability book adopts the rather misleading conven- algorithms allow revision of the inferential that any two connected units are on simulta- tion of terming their referents "microfea- relationships among hypotheses. As this de- neously when environmental stimuli are pre- tures." This seems reasonable enough for a scription suggests, many of the processing sented with the probability that both units unit that represents a low-level construct principles embodied in PDP models can be are on when the network is running freely in such as "a vowel preceded by a stop conso- readily incorporated into models that the absence of environmental inputs. Like nant and followed by a nasal" (chapter 18); choose to represent hypotheses as symbol the generalized delta rule, the Boltzmann on the other hand, a microfeature corre- structures rather than primitive units. learning rule can modify weights through- sponding to a "causal verb" (chapter 19) It is also apparent that the PDP approach out the entire network. seems like a conventional semantic feature; does not fundamentally contradict the func- Part 3, in volume 1, includes five chapters and the microfeature "has television" (chap- tionalist approach to cognition. A connec- dealing with technical aspects of PDP mod- ter 14) simply invites ridicule. tionist model of mind, like any cognitive els. Part 4, in volume 2, consists of six For many purposes it is useful to extract model, can be described at a level more chapters that describe applications of PDP the essential properties of connectionist abstractthan any particularimplementation. models to a rich body of experimental data models from their metaphorical neural trap- It also seems that the "computer metaphor" regarding such cognitive processes as word pings. In general terms, units represent for mind is not really being abandoned by recognition, speech perception, categoriza- hypotheses, and connections capture infer- the connectionists, but simply modified: the tion, the acquisition of verb morphology, ential dependencies among hypotheses. classical von Neumann serial processor is and sentence processing. From the point of Thus if one unit has an excitatory connec- replaced with a processor that performs view of , this section tion to another, this indicates that support some simple computation in parallel on provides the central evidence that the "brain for the first hypothesis provides some de- many data elements (for example, the "Con- metaphor" is in fact theoretically fruitful. Chapter 14, the most speculative chapter in this set, addresses an especially controversial . *u . ,LI.- ;u. Q**Q. , Qo * OWN. ; aspect of the PDP approach-its appropri- .0 0 . . ateness for describing high-level reasoning ...... * ...... 00. o- 0*O ..-...* and sequential thought processes. Finally, ceiling walls door window very-large part 5 includes five chapters that deal with , , , , . , , .. , -...., ~~~~. .. . .0 - .... E oo the biological relevance of PDP models 0.. .0. .m...... u* . " .... m . * . .. m...... *OE . E.. ...EE . . * (plus chapter 22, which more properly be- * ...... rnr.... . * .. *ErnErn*Ern .....*--o longs in part 3). Chapter 20, by Crick and Asanuma, provides a survey of current large medium smail very-smail desk telephone bed knowledge of cerebral anatomy and physiol- *o ...... * ..i 1000S . ogy; Sejnowski's chapter 21 is an intriguing .'. .30* a; ;::a; '. U...... r-... 1. discussion of the kinds of computations that ****@** ...... ~. ****.Q*1 ** 1... may be performed by the cerebral cortex; typewriter bookshelf carpet books deek-chair clock picture and chapters 23, 24, and 25 describe specific models of the coding of place information, neural plasticity, and human amnesia. r~~~~~~~~~~~~~~~~~~~~~~~~~~N-,,3 As the authors are careful to point out, U.... I.. I I...I I ....r.EI U...... O.,I I U ...... U ...rnr,I the attractiveness of the "brain metaphor" floor-lamp sofa easy-chair coffee-cup ash-fray fire-place drap.es underlying PDP models does not derive from the correspondence of the models to I * .-.1 aE E any recent breakthroughs in neuroscience. I I 000 I 00.Q.z .0C1 - 1OM-ol-.. 1 (Indeed, Crick and Asanuma's chapter ends stove sfnk refrigerator toaster cupboard coffee-p-ot dreeser with an interesting list of disanalogies be- ,1 E EU 1 *E EU**1 Ul. U.-Z I U.' *--.-E-. l i E . 13 mem.E.- U . - M. U No .. -. . E tween properties of real neurons and the 12 ".i ~rn>.r . -1 1 U- 13.- o . -13 ISONQ * "units" of the models described in other chapters.) The connectionist models pre- sented here resemble the brain only in ex- televieson bathtub toilet scale coat-hanger computer oven tremely general properties (there are many A representation of the connection weights between units in a parallel network. Each labeled box stands neurons, richly interconnected, that perform for a unit. "Within each unit, the small black and white squares represent the weights from that unit to simple computations in parallel on a rela- each of the other units in the system. The relative position of the small squares within each unit indicates which unit is connected." WVhite connections and black tively slow time scale). As the authors admit, the unit with that squares represent positive squares represent negative connections, with the size of the square representing the strength of the "The basic idea is that there is a mapping connection. The large, white squares in the "ceiling" and "walls" boxes show that there is a strong, between elements of the model and the positive weight between the two. [From Parallel Distributed Processing,chapter 14.s]

994- SCIENCE, VOL. 236 nection Machine" recently marketed by question whether any of the algorithms algorithms also have difficulty accounting Thinking Machines Corporation). The provides a model for some form of human for sequential behavior. In particular, none brain metaphor has influenced computers, learning. The chapters in part 4 use only the has yet demonstrated the capacity to learn yielding an updated computer metaphor for ungeneralized version of the delta rule (re- tasks involving sequences of actions in mind. stricted to networks without hidden units) which early components receive no direct The research reported in the book pro- in simulating detailed psychological data. In feedback yet are crucial to ultimate success vides a broad picture of the accomplish- chapter 8 the generalized version is used to (for example, a rat learning to negotiate a ments of the PDP approach and of its future generate solutions to several small-scale rela- maze to reach food, or a checkers player promise and problems. The accomplish- tional problems that require hidden units. setting up a triple-jump). These aspects of ments are substantial. The models provide Its solutions are often nonobvious and con- biological learning are not well captured by simple mechanisms for allowing fragments stitute impressive demonstrations of feature the PDP learning algorithms. of memory representations to complete extraction; however, the solutions generally The adequacy of the knowledge represen- themselves, for producing generalization to do not seem "humanlike." Each solution to tations proposed in the book is also open to similar patterns, and for "graceful degrada- a relational problem, such as deciding question. A general problem with PDP rep- tion" when the system is damaged. These whether a string of binary bits has an resentations is that a great deal of redundan- are all fundamental characteristicsof human odd or even number of l's, is obtained for cy is required. To process a word, for exam- cognition. At a more specific level, the indi- an input string of some particular fixed ple, the TRACE model of speech perception vidual psychological models, such as the length. For example, if strings of four bits assumes that the networks of feature, pho- word recognition model of McClelland and are presented, each labeled "even" or "odd," neme, and word units are reduplicated many Rumelhart and the speech perception model the network will in effect learn to count the times over the time segments into which the of McClelland and Elman, provide detailed number of I's and to associate the values 0, input signal is divided. In McClelland and accounts of fine-grained empirical results. It 2, and 4 with the response "even" and I and Rumeihart's model of visual word recogni- is hard to imagine any adequate account of 3 with "odd." No investigations of transfer tion, each letter position corresponds to a the impact of context effects on the interpre- are reported, but clearly this representation completely different set of feature detectors. tation of stimuli that will not embrace the provides no information about whether a (The implemented model can only recog- principles of interactive activation embodied novel input offive bits is even or odd. Unlike nize words up to four letters long.) This in these models. McClelland and Elman's the case of a person who has abstracted the representation, if taken seriously, implies TRACE theory provides an elegant account relational rule that "even" means "divisible that learning to recognize a particular letter of the phenomenon of categorical speech by 2," the delta rule's solution will never in the second position of words will have no perception in terms of the "canonicalizing" allow transfer to strings of unbounded impact on recognizing it in the third posi- effect of feedback from phoneme to feature length. tion, a prediction that seems dubious. Also, units. In addition, all the algorithms are ex- it is quite unclear how such architectures Not surprisingly, the work is still a long tremely slow, requiring thousands of repeti- could develop with experience, be extended way from providing a full account of human tions to reach solutions to even simple prob- to inputs of arbitrarylength, or (in the case cognition. The book offers a great deal of lems. The learning algorithms have so far of the TRACE model) be tuned to such intriguing and potentially fruitful specula- been shown to support a degree of parallel- crucial idiosyncrasies as speech rate. tion about how higher-level cognition ism more aptly characterizedas modest than The use of distributed representations to might be modeled in PDP terms. The dis- massive, since learning within networks con- represent sentences leads to additional archi- cussion of schemas and mental models in taining more than hundreds of units has yet tectural complexity. In a distributed repre- chapter 14 is of this nature, as is Smolensky's to be demonstrated. The difficulty of apply- sentation, the concept "boy" will be mapped description in chapter 6 of how physics ing the algorithms to very large networks onto many of the same units as is "girl." It is expertise might develop. The basic concep- may reflect inherent limitations of "bottom- therefore tricky to represent a sentence such tion of mental processing as a form of up" approaches to learning based solely on as "The boy kissed the girl" without mud- parallel constraint satisfaction constitutes a gradient-descent methods that adjust the dling the agent and patient roles. In chapter rich theoretical framework. In a few cases, weights of preexisting connections. If a unit 19 McClelland and Kawamoto, extending a however, speculation crosses the border into receives a large number of inputs, only a few proposal by Hinton, implement a possible flights of fancy, as when Hinton and Sej- of which actually correlate with the output solution to this type of problem that in- nowski identify the free-running stage of the the unit needs to produce in order to per- volves defining sets of units representing Boltzmann machine with dreaming (chapter form a new task, then most of the weight different semantic roles, such as agent or 7), or McClelland and Rumelhart postulate adjustments made on a given trial will fail to patient, and including units that respond to a mysterious "gamma" substance in the improve performance on the new task and conjunctions of features of the concepts brain and endow it with properties required may impair performance on tasks previously filling two different roles. This technique to reconcile PDP models with some of the learned. The PDP learning algorithms fail to exemplifies one approach to the general evidence concerning amnesia (chapter 25). exploit potential "top-down" types of infor- problem of representing "what goes with The greatest challenges facing connec- mation that could restrict the size of the what" in PDP networks, which has yet to be tionists concern the adequacy of their learn- search space in which the weight-adjustment definitively solved. ing mechanisms and of their knowledge procedures operate (for example, prior The representational complexities that the representations. Each of the three major knowledge of a specific domain, or general PDP approach must deal with severely exac- learning algorithms described in the book is heuristics for identifying plausible causal erbate the shortcomings of the available capable of extracting statistical regularities relations). Organisms from rats to humans learning procedures. The learning algo- present in patterned inputs, using combina- sometimes exhibit one-trial learning of rela- rithms are not inductively sufficient; that is, tions of hidden units to implicitly form tions consistent with their prior "theories" simply modifying connection weights with- novel feature detectors. These are significant and may fail utterly to detect statistical in a large homogeneous initial network on accomplishments; however, it i san open regularities that violate them. The learning the basis of environmental inputs would not

22 MAY 1987 BOOK REVIEWS 995 suffice to construct the specialized represen- articulated in this book will have a major crystallographers, and physicists. In doing tations required for different tasks. The per- impact on cognitive science. Researchers in so they were forced to take a fresh look at formance of a weight-adjustment algorithm cognitive psychology, , what was known, what was thought to be generally depends on the prespecified archi- parallel computation, neuroscience, linguis- known, and what had yet to be investigated, tecture of the network within which it oper- tics, and other fields as well will find the and to provide a framework in which all of ates (for example, the number of layers, the work immensely stimulating. The greatest this information could be succinctly put number of units within each layer, and the value of the book is that it clearly lays out a down. The project, which was started about overall pattern of connectivity). In all the paradigm and applies it to concrete and 10 years ago and has only now been brought simulations reported in the book, the net- interesting examples. The book opens a to (partial) completion, is well worth the work is crafted by the researcherto perform door that cognitive scientists can enter: wait. some specific task. As yet there seem to be some will stay and join the movement, To present this material Gruinbaum and no principles that constrain network archi- others will steal a few ideas and leave, and Shephard needed to develop a standardized tecture and no proposed learning techniques others yet will learn why they prefer to stay terminology. They have chosen terminology that might allow induction of modular sub- outside. All will want to take a peek. that is clear yet flexible and thus well suited nets or other specialized architectures. KEITH J. HOLYOAK to the rich range of phenomena encoun- Although major hurdles loom ahead, it is Department of Psychology,University of tered. Since this book begins with the sim- clear that the PDP approach so forcefully Calzforrniq,Los Angeles, CA 90024 plest of concepts, it can be started and read with enjoyment by a high-school student. However, the reader needs staying power. As the book progresses, relatively elemen- tary concepts and definitions are developed, Shapes in the Plane but in the building-block style typical of a mathematical theory. Furthermore, the au- thors have felt free to use elementary ideas patterns was not developed until that pre- from topology, group theory, and number Tilings and Patterns. BRANKO GRtNBAUM and sented in this book. theory, defining the necessary terminology G. C. SHEPHARD. Freeman,New York,1986. xii, 700 pp., illus. $59.95. What Grunbaum and Shephard have as they go along. Thus, although the treat- done, in a dazzling display of scholarship, ment is elementary in the sense that all Throughout history people have filled erudition, and research, is collect in one concepts and ideas that are not well known floors, stained-glass windows, and fabrics volume a compendium of the accumulated by persons with modest mathematics back- with shapes that occupied the plane without knowledge about tilings and patterns devel- grounds are fully, clearly, and carefully ex- holes or overlaps. These shapes, or tiles, oped by a wide range of individuals includ- plained, one cannot hope to understand the often endow plane surfaces with patterns of ing artisans and craftsmen, mathematicians, statements in the middle of the book with- remarkablesymmetry and beauty. Although tilings of the plane have been found in varied contexts and cultures, the systematic 10 study of their types and properties is surpris- II0 () 0 0I () ingly new. Except for a modest beginning in the 17th century by Johannes Kepler, there were few studies of tilings until the 20th 0~~~~~~~~~~~~~~~~~ o2 of 2 2% century, and not until the research of this book's authors did the study of tilings be- come a full-fledged subbranch of geometry. To develop a theory of tilings, one must adhere to the rules concerning which shapes are allowed for tiles and the rules about how shapes can be placed next to one another. For example, if one starts with a supply of identical rectangular tiles, one may place the "A modification by Pen- tiles so that the edges of the rectangles rose of his set P1 of aper- match, producing the familiar tiling of floor- iodic prototiles. Each ings, where four tiles meet at a point. But edge is replaced by a cir- cular arc whose center is other tilings by rectangles can occur if we the 'point' of a pentacle allow the tiles to adjoin in additional ways. or half-pentacle. A small Infinitely many types of tiling are possible portion of the original when the tiles are not laid down edge to P1 tiling is reproduced at the top of the diagram edge. Patterns, the other word sharing the to show the relationship book's title with tilings, involve symmetry between the tilings. considerations that arise when motifs of Three of the prototiles various kinds are systematically located in have been colored black the plane. Although beautiful patterns have and three are white. It is conjectured that these been created and examined by mathemati- ,curvilinear'tiles are also cians in addition to craftsmen and artists for aperiodic." [From Til- thousands of years, a coherent theory of ings and Patterns]

996 SCIENCE, VOL. 236