This paper demonstrates the one-to-one correspondence between trivalent 3-polytopcs and fundamental architectural schemes. Enumeration results from the literature of combinatorics are given and are related to the problem of counting various classes of architectural plans and their adjacency structures. This work is related to recent architectural research by other authors: Mitchell et al (1976), Korf (1977), and Lynes (1977), The foundations are laid for a mathematical theory of architectural planning.
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References
1.
BeinekeL WPippertR E, 1972“A census of ball and disk dissections” in Graph Theory and Applications Eds AlaviYLickD RWhiteA T (Springer-Verlag, Heidelberg) pp 25–48.
2.
BondyJ AMurtyU S R, 1976Graph Theory and Applications (Macmillan, London).
3.
BowenHFiskS, 1967“Generation of triangulations of the sphere”Mathematics of Computation21250–252.
4.
BrownW G, 1965“Enumeration of triangulations of the disk”Proceedings of the London Mathematical Society14(3) 746–768.
5.
EberhardV, 1891Zur Morphologie der Polyeder (Teubner, Leipzig).
GrünbaumB, 1967Convex Polytopes (John Wiley, New York).
8.
GuyR K, 1958“Dissecting a polygon into triangles”Bulletin of the Malayan Mathematical Society557–60.
9.
GwiltJ, 1842Gwilt's Encyclopaedia of Architecture: Historical, Theoretical and Practical (Longmans Green, London).
10.
HaeckelE H, 1904Kunstformen der Natur (Verlag des Bibliographischen Instituts, Leipzig).
11.
HararyF, 1969Graph Theory (Addison-Wesley, Reading, Mass).
12.
KorfR E, 1977“A shape independent theory of space allocation”Environment and Planning B437–50.
13.
LynesJ A, 1977“Windows and floor plans”Environment and Planning B451–55.
14.
MacCormacR C, 1974“Froebel's kindergarten gifts and the early work of Frank Lloyd Wright”Environment and Planning B129–50.
15.
MitchellW JSteadmanJ PLiggetR S, 1976“Synthesis and optimization of small rectangular floor plans”Environment and Planning B337–70.
16.
MullinR C, 1965“On counting rooted triangular maps”Canadian Journal of Mathematics17373–382.
17.
PólyaC, 1937“Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen”Acta Mathematica68145–254.
18.
SteadmanJ P, 1976“A note on Combes's classification for rectangular dissections”Environment and Planning B333–36.
19.
StinyG, 1975Pictorial and Formal Aspects of Shape and Shape Grammars (Birkhauser Verlag, Basel, Switzerland).
20.
SullivanL H, 1924Autobiography of an Idea (Press of the American Institute of Architects, Washington, DC).
21.
TaylorT (Ed.), 1853Life of B.R. Haydon from his Autobiography and Journalsthree volumes, London (first edition, republished 1927; The World's Classics, London).
22.
ThompsonD' A W, 1917On Growth and Form (Cambridge University Press, Cambridge).
23.
TutteW T, 1962“A census of planar triangulations”Canadian Journal of Mathematics1421–38.
24.
TutteW T, 1963“A census of planar maps”Canadian Journal of Mathematics15249–271.
25.
TutteW T, 1977On the Enumeration of Convex Polyhedra Waterloo Research Report CORP 77/6, University of Waterloo, Ontario, Canada.
26.
UngarP, 1953“On diagrams representing maps”Journal of the London Mathematical Society28336–342.
27.
WrightF L, 1953The Future of Architecture (Horizon, New York).
28.
WrightF L, 1957A Testament (Horizon, New York).
29.
WrightI L, 1962“My father” in Architecture: Man in Possession of this Earth Ed. WrightF L (Doubleday, New York) pp 14–49.
