An N-omino is a planar edge-connected set of N square cells. The terms polyomino (N-omino) and animal are equivalent, the first attributable to Golomb (1954) and the latter to Read (1962). This paper presents some results derived from a census of three different populations of N-ominoes for N = 6, 7, and 8. In all, 512 different polyominoes have been analysed in terms of properties of form which are of architectural interest. In this respect the paper extends our knowledge of the so-called âanimals of architectureâ (Frew et al., 1972). The paper concludes with some reflections on the nature of architectural solution spaces and design processes based on the experience of conducting this census, and the potential role of polyomino studies in architectural education.
Get full access to this article
View all access options for this article.
References
1.
AlexanderC., 1964, Notes on the Synthesis of Form (Harvard University Press, Cambridge, Mass.).
ArmourG. C.BuffaE. S., 1963, âA heuristic algorithm and simulation approach to relative location of facilitiesâ, Management Science, 9, 294â309.
5.
BermanG.FryerK. D., 1972, Introduction to Combinatorics (Academic Press, New York).
6.
BouwkampC., 1967, âCatalogue of solutions of the rectangular 3Ă4Ă5 solid pentomino problemâ, Department of Mathematics, Technische Hogeschool, Eindhoven, Netherlands.
7.
BouwkampC., 1969, âPacking a rectangular box with the twelve solid pentominoesâ, Journal of Combinatorial Theory, 7, 278â280.
8.
BouwkampC., 1971, âA new solid pentomino problemâ, Journal of Recreational Mathematics, 4(3), 179â186.
9.
BouwkampC.KlarnerD., 1970, âPacking a box with Y-pentacubesâ, Journal of Recreational Mathematics, 3, 10â26.
10.
deBruijnN., 1969, âFilling boxes with bricksâ, American Mathematical Monthly, 76, 37â40.
11.
CoxeterH. S. M.MoserW. O. J., 1965, Generators and Relations for Discrete Groups (Springer-Verlag, New York).
12.
DurandJ. N. L., 1823â1831, PrĂŠcis des Legons d'Architecture, 3 volumes, (L'Ăcole Royale Polytechnique, Paris).
13.
FawcettW., 1974, Measuring Adaptability, Dip. Arch. Thesis, School of Architecture, University of Cambridge, Cambridge, England.
14.
FeserV., 1968, âPentomino farmsâ, Journal of Recreational Mathematics, 1, 55â61.
15.
FletcherJ., 1965, âA program to solve the pentomino problem by the recursive use of macrosâ, Communications of the ACM, 8, 621â623.
16.
FloryP. J., 1969, Statistical Mechanics of Chain Molecules (John Wiley, New York).
17.
FrewR. S., 1973, Towards a Theory of Systems Architecture, Ph. D. Thesis, University of Waterloo, Canada.
18.
FrewR. S.RagadeR. K.RoeP. H., 1972, âThe animals of architectureâ, in Proceedings of the EDRA 3/AR 8 Conference, School of Architecture and Urban Planning, University of California, Los Angeles.
19.
GolombS., 1954, âCheckerboards and polyominoesâ, American Mathematical Monthly, 61, 672â682.
20.
GolombS., 1961a, âThe general theory of polyominoes. Part 1, dominoes, pentominoes, and checkerboardsâ, Recreational Mathematics Magazine, 4, 3â12.
21.
GolombS., 1961b, âThe general theory of polyominoes. Part 2, patterns and polyominoesâ, Recreational Mathematics Magazine, 5, 3â12.
22.
GolombS., 1961c, âThe general theory of polyominoes. Part 3, pentomino exclusion by monominoesâ, Recreational Mathematics Magazine, 6, 3â20.
23.
GolombS., 1962, âThe general theory of polyominoes. Part 4, extensions of polyominoesâ, Recreational Mathematics Magazine, 8, 7â16.
24.
GolombS., 1964, âReplicating figures in the planeâ, Mathematical Gazette, 48, 403â412.
25.
GolombS., 1965, Polyominoes (Charles Scribner, New York).
26.
GolombS., 1970, âTiling with polyominoesâ, Journal of Combinatorial Theory, 9, 60â71.
27.
HararyF., 1960, âUnsolved problems in the enumeration of graphsâ, Publications, Mathematical Institute, Hungarian Academy of Science, 5, 63â95.
28.
HararyF., 1972, Graph Theory (Addison-Wesley, Reading, Mass.).
29.
HararyF.PalmerE., 1973, Graphical Enumeration (Academic Press, New York).
30.
KellyJ. B., 1966, âPolynomials and polyominoesâ, American Mathematical Monthly, 73, 464â471.
31.
KlarnerD., 1963, âCovering a rectangle with L-tetrominoesâ, American Mathematical Monthly, 70, 760.
KlarnerD., 1967, âCell growth problemsâ, Canadian Journal of Mathematics, 19, 851â863.
34.
KlarnerD., 1969, âPacking a rectangle with congruent N-ominoesâ, Journal of Combinational Theory, 7, 107â115.
35.
LunnonW. F., 1969, Generalized Polyominoes, Ph.D.Thesis, University of Manchester, England.
36.
LunnonW. F., 1971, âCounting polyominoesâ, in AtkinA. O. L. and BirchB. J., (eds.), Computers in Number Theory (Academic Press, London).
37.
MacCormacR. C., 1974, âFroebel's kindergarten gifts and the early work of Frank Lloyd Wrightâ, Environment and Planning B, 1, 29â50.
38.
MadachyJ. S., 1969, âPentominoesâsome solved and unsolved problemsâ, Journal of Recreational Mathematics, 2, 181â188.
39.
MarchL., 1972, âModern movement to Vitruvius: Themes of education and researchâ, Royal Institute of British Architects Journal, 78, 101â109.
40.
MarchL., 1975, âA boolean description of a class of built formâ, in MarchL., (ed.), The Architecture of Form (Cambridge University Press, London), forthcoming.
41.
MarchL.SteadmanP., 1971, The Geometry of the Environment (RIBA Publications, London).
42.
MatelaR. J., 1974, An Analysis of the Animals of Architecture: A Complete Enumeration of Polyominoes by Some of Their Architectural Properties, M.E.D. Thesis, Yale University, New Haven, Conn., USA.
43.
MitchellW. J., 1973, âVitruvius computatusâ, in Proceedings, LUBFS Conference on Models and Systems in Architecture and Building (Medical and Technical Press, Lancaster), forthcoming.
44.
MitchellW. J., 1974, âTechniques of automated design in architecture: A survey and evaluationâ, Computers and Urban Society, 1 (1), in press.
45.
MitchellW. J.DillonR. L., 1972, âA polyomino assembly procedure for architectural floor planningâ, in Proceedings of the EDRA 3/AR 8 Conference, School of Architecture and Urban Planning, University of California, Los Angeles.
46.
ParkinT. R., 1967, âPolyomino enumeration resultsâ, presented at the SIAM Fall Meeting, Santa Barbara.
47.
PriharZ., 1956, âTopological properties of telecommunication networksâ, Proceedings of the Institute of Radio Engineers, 44, 929â933.
48.
RashevskyN., 1954, âTopology and life. In search of general mathematical principles in biology and societyâ, Bulletin of Mathematical Biophysics, 16, 317â318.
49.
RashevskyN., 1955a, âNote on a combinatorial problem in topological biologyâ, Bulletin of Mathematical Biophysics, 17, 45â54.
50.
RashevskyN., 1955b, âSome theorems in topology and a possible biological implicationâ, Bulletin of Mathematical Biophysics, 17, 111â126.
51.
RashevskyN., 1955c, âSome remarks on topological biologyâ, Bulletin of Mathematical Biophysics, 17, 207â218.
52.
RashevskyN., 1955d, âLife, information theory, and topologyâ, Bulletin of Mathematical Biophysics, 17, 229â235.
53.
ReadR. C., 1962, âContributions to the cell growth problemâ, Canadian Journal of Mathematics, 15, 1â20.
54.
RittleH., 1968, âTheories of cell configurationsâ, in MooreG. (ed.), Emerging Methods in Environmental Design and Planning (MIT Press, Cambridge, Mass.).
55.
ShapiraH., 1974, A Procedure for Generating Floor PlansâComputer Aided Design, M.E.D. Thesis, Yale University, New Haven, Conn., USA.
56.
SouderJ. J.ClarkW. E.ElkindJ. I.BrownM. B., 1964, Planning for Hospitals; A System Approach using Computer-Aided Techniques (American Hospital Association, Chicago).
57.
SteadmanP., 1973, âGraph theoretic representation of architectural arrangementâ, Architectural Research and Teaching, 2, 161â172.
58.
TaborP., 1975, âAnalyzing route patternsâ, in MarchL., (ed.), The Architecture of Form (Cambridge University Press, London).
59.
VilenkinN. Ya., 1971, Combinatorics (Academic Press, New York).
60.
WaddingtonC.H., (ed.), 1972, Towards a Theoretical BiologyâEssays (Edinburgh University Press, Edinburgh).
61.
WalkupD., 1965, âCovering a rectangle with T-tetrominoesâ, American Mathematical Monthly, 72, 986â988.
62.
WhyteL. L., 1965, Internal Factors in Evolution (Trinity Press, London).
