A unified morphological method for generating subdivisions of periodic surfaces is presented. The surfaces are characterized by symmetry, frequency, and the topology of subdivision. An open-ended hyper-cubic lattice is suggested as the structure space within which the subdivided surfaces can transform continuously from one to another by varying different independent parameters. For the purposes of illustration, eight parameters are specified to demonstrate an 8-dimensional model for the structure space. Additional types of subdivisions, and other geometric or physical parameters encountered in building, extend this space to a higher-dimensional one which acts as a systematic framework for generating a large variety of surface structures and their transformations. Geodesic spheres are special cases within this continuum which includes plane surfaces and the hyperbolic tessellations. An infinite class of semi-regular hyperbolic tessellations analogous to the Archimedean polyhedra and plane tessellations are presented, and higher frequency subdivisions of hyperbolic tessellations are suggested.
ShubnikovA.V.KoptsikV.A., Symmetry in Science and Art, Plenum, 1974.
3.
FullerR.B., and MarksR., The Dymaxion World of Buckminster Fuller, Anchor, New York, 1973.
4.
CasparD.L.D., and KlugA., Cold Spring Harb. Symp. Quant. Biol., Cold Spring Harbor, 1962.
5.
GoldbergM., Tohoku Math. J., 1937.
6.
Brought to author's attention during a personal visit to Fuller's office, 1976.
7.
The models of all-triangulated skewed icosahedra first appeared in [4]; the author is indebted to the structural biologist S. Harrison (personal meeting, 1975) for the explanation of the ‘triangulation number’ method used by Caspar-Klug in [4].
8.
The author has been familiar with this aspect since 1975; Fuller didn't seem to be interested in this for some reason. The precise length and angle measurements have been worked out in detail by Tarnai (see [9]).
9.
TarnaiT., ed. Spherical Grid Structures, Hungarian Institute of Science, Budapest, 1987; see articles by TarnaiT. (p.125–159) and PavlovG.N. (p.9–64).
10.
LalvaniH., Multi-dimensional Periodic Arrangements of Transforming Space Stuctures, Ph.D., thesis, University of Pennsylvania, Philadelphia, 1981; published by University Microfilms, Ann Arbor, Michigan, 1982; see also [11].
11.
LalvaniH., Structures on Hyper-Structures, Lalvani, New York, 1982; based on [10].
12.
LalvaniH., Transpolyhedra, Lalvani, New York, 1977.
13.
CoxeterH.S.M., Introduction to Geometry, Wiley, New York, 1961, p.284–5.
14.
see Coxeter [17] for references to Mobius, SchwatzWythoff.
15.
FullerR.B., Synergetics, McMillan, New York, 1975; Fuller describes the fundamental region as the LCD (lowest common denominator) triangle.
16.
BurtM., Structural Topology, 6, 1982.
17.
CoxeterH.S.M., Regular Polytopes, Dover, New York, 1973.
TóthL. Fejes, Regular Figures, Pergamon, New York, 1964.
21.
Loeb [19], Ch. 9.
22.
BurtM., personal communication, 1978.
23.
Lalvani, [10,11], Fig. 10.
24.
Loeb [19], p. 56.
25.
CoxeterH.S.M.EmmerM.PenroseR., and TeuberM.L., eds. EscherM.C.: Art and Science, North-Holland, 1986; articles by RigbyJ.F. (p. 211–220) and DunhamD.J. (p. 241–248).
26.
MagnusW., Non-Euclidean Tessellations and Their Groups, Academic Press, New York, 1974.
27.
OppenheimerP.E., Transformation Groups and the Hyperbolic Plane, June 1978 (unpublished).
28.
LalvaniH., Coding and Generating Complex Periodic Patterns, The Visual Computer (1989) 5: 180–202.
29.
PopkoE., Geodesics, University of Detroit Press, Detroit, 1968.
30.
KennerH., Geodesic Math, Univ. of California Press, 1976.
31.
cited in [9]; independently suggested by the author.
32.
Fig. 11 is similar to the earlier figure of Coxeter [13], p. 387, though not the same (see also [28]).
33.
De BruijnN. (personal meeting, August 1985); subsequently generalized to “multi-grids” by De Bruijn.
34.
A slight variant of De Bruijn's multi-grids can be derived by subdividing a regular p-sided polygon; this can be done by drawing lines, parallel to p directions, from equally spaced points on all the edges of any regular polygon. Analogs of such poly-grids in higher Euclidean space, and in non-Euclidean spaces, are interesting possibilities. An example for p=5 based on the regular pentagon is show in Fig. A; the skewed pentagonal grid is obtained as shown with the pentagon in dotted lines.
35.
Lalvani [10,11], Fig. 72.
36.
Lalvani [10,11], Fig. 84.
37.
Lalvani [10,11], Figs. 53 and 60.
38.
TarnaiT., Buckling Patterns of Shells and Spherical Honeycomb Structures, Computers Math. Applic.17, No. 4–6, pp.639–652, 1989; see Fig. 5.
39.
Two dual sets of models showing a higher frequency skewed subdivision of the faces of a tetrahedron, octahedron and icosahedron were constructed by my students in the Morphology studio, School of Architecture, Pratt Institute, Fall 1982; each set contained seven subdivisions for each regular polyhedron.
40.
HanrahanP.LalvaniH., and McDermottR., Sketches of Polyhedra Transformations, a computer-animation video, N.Y.I.T., 1984; analogous subdivision of spherical, toroidal, and non-convex surfaces shown in Part 2.
41.
SenechalM., Escher Designs on Surfaces, In [25], p. 97–109.
42.
McDermottR. (personal communication, 1983): this dome was constructed in St. Johns, Newfoundland, 1978.
43.
CoxeterH.S.M., Twelve Geometric Essays, Southern Illinois University Press, Carbondale, 1968, p. 77.
44.
BurtM.KleinmannM., and WachmanA., Infinite Polyhedra, Technion, Haifa (Israel), 1974.
45.
MackayA.L., Two Dimensional Space Groups with Sevenfold Symmetry, Acta Cryst. (1986). A42, 55–56.
46.
Lalvani [10], p.206–211; also [11], p. 102–104.
47.
CoxeterH.S.M., Coloured Symmetry, In: [25], p. 15–33.
48.
LalvaniH., Continuous Transformations of Non-Periodic Tilings and Space-fillings, Symmetry Journal, VCH Publ., New York (in press).
49.
LalvaniH.Morphological Aspects of Space Structures, in: Studies in Space Structures, ed. NooshinH.. Multi-Science Publ., U.K. (in press); see Figs. 370 and 373, also Fig. 348a.
50.
LalvaniH., Structures and Meta-Structures, In: Symmetry of Structure, Abstracts 1, Eds. DarvasG.NagyD., The Hungarian Academy of Sciences, Budapest, 1989, p. 302–306.
51.
Architectural shape grammar studies with students, Technics studio, School of Architecture, Pratt Institute, 1987-present.
