HAWAIIAN FLOWERS WITH FIVEFOLD SYMMETRY

THE DISCOVERY OF SPACE FRAMES WITH FIVEFOLD SYMMETRY

Fivefold Symmetry ◽

10.1142/9789814439497_0013 ◽

1992 ◽

pp. 205-219

Author(s):

Stephen C. Baer

Keyword(s):

Fivefold Symmetry ◽

Space Frames

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Observation of fivefold symmetry structures in computer models of dense packing of hard spheres

Journal of Non-Crystalline Solids ◽

10.1016/j.jnoncrysol.2007.05.113 ◽

2007 ◽

Vol 353 (32-40) ◽

pp. 3545-3549 ◽

Cited By ~ 13

Author(s):

A.V. Anikeenko ◽

N.N. Medvedev ◽

A. Bezrukov ◽

D. Stoyan

Keyword(s):

Hard Spheres ◽

Computer Models ◽

Dense Packing ◽

Fivefold Symmetry

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Ontogeny in the fossil record: diversification of body plans and the evolution of “aberrant” symmetry in Paleozoic echinoderms

Paleobiology ◽

10.1666/06053.1 ◽

2007 ◽

Vol 33 (1) ◽

pp. 149-163 ◽

Cited By ~ 66

Author(s):

Colin D. Sumrall ◽

Gregory A. Wray

Keyword(s):

Great Reduction ◽

Fossil Record ◽

Evolutionary History ◽

Total Loss ◽

Cambrian Explosion ◽

Growth Stages ◽

Ecological Constraints ◽

Fivefold Symmetry ◽

Primary Body ◽

History Of

Echinoderms have long been characterized by the presence of ambulacra that exhibit pentaradiate symmetry and define five primary body axes. In reality, truly pentaradial ambulacral symmetry is a condition derived only once in the evolutionary history of echinoderms and is restricted to eleutherozoans, the clade that contains most living echinoderm species. In contrast, early echinoderms have a bilaterally symmetrical 2-1-2 arrangement, with three ambulacra radiating from the mouth. Branching of the two side ambulacra during ontogeny produces the five adult rays. During the Cambrian Explosion and Ordovician Radiation, some 30 clades of echinoderms evolved, many of which have aberrant ambulacral systems with one to four rays. Unfortunately, no underlying model has emerged that explains ambulacral homologies among disparate forms. Here we show that most Paleozoic echinoderms are characterized by uniquely identifiable ambulacra that develop in three distinct postlarval stages. Nearly all “aberrant” echinoderm morphologies can be explained by the paedomorphic ambulacra reduction (PAR) model through the loss of some combination of these growth stages during ontogeny. Superficially similar patterns of ambulacral reduction in distantly related clades have resulted from the parallel loss of homologous ambulacra during ontogeny. Pseudo-fivefold symmetry seen in Blastoidea and the true fivefold symmetry seen in Eleutherozoa result from great reduction and total loss, respectively, of the 2–1–2 symmetry early in ontogeny. These ambulacral variations suggest that both developmental and ecological constraints affect the evolution of novel echinoderm body plans.

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Fivefold symmetry as an inhibitor to hard-sphere crystallization

Physical Review E ◽

10.1103/physreve.83.061505 ◽

2011 ◽

Vol 83 (6) ◽

Cited By ~ 32

Author(s):

Nikos Ch. Karayiannis ◽

Rohit Malshe ◽

Juan J. de Pablo ◽

Manuel Laso

Keyword(s):

Hard Sphere ◽

Fivefold Symmetry

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PLANAR PATTERNS WITH FIVEFOLD SYMMETRY AS SECTIONS OF PERIODIC STRUCTURES IN 4-SPACE

International Journal of Modern Physics B ◽

10.1142/s0217979290001054 ◽

1990 ◽

Vol 04 (15n16) ◽

pp. 2217-2268 ◽

Cited By ~ 108

Author(s):

M. BAAKE ◽

P. KRAMER ◽

M. SCHLOTTMANN ◽

D. ZEIDLER

Keyword(s):

Detailed Analysis ◽

Periodic Structures ◽

Root Lattice ◽

Fivefold Symmetry ◽

Dual Complex

Two quasiperiodic planar patterns with fivefold symmetry are derived from the root lattice A4 in 4-space. A detailed analysis of the geometry of the A4 Voronoi complex and its dual complex is presented with special emphasis on fivefold symmetry. By means of the general dualization method, 2D patterns are obtained, one with triangular tiles and a second which turns out to be the well-known Penrose pattern. The vertex configurations and their relative frequencies, the deflation rules, and the Fourier properties of these patterns are worked out in the framework of the dualization method and Klotz construction.

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