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Crystals: animal, vegetable or mineral?

Hyde ST - Interface Focus (2015)

Bottom Line: The idea that there is a clear distinction between these two classes of matter has waxed and waned in popularity through past centuries.The older picture of disjoint universes of forms is better understood as a continuum of forms, with significant overlap and common features unifying biological and inorganic matter.In addition to the philosophical relevance of this perspective, there are important ramifications for science.

View Article: PubMed Central - PubMed

Affiliation: Department of Applied Mathematics, Research School of Physics and Engineering , The Australian National University , Canberra, Australian Capital Territory 0200 , Australia.

ABSTRACT
The morphologies of biological materials, from body shapes to membranes within cells, are typically curvaceous and flexible, in contrast to the angular, facetted shapes of inorganic matter. An alternative dichotomy has it that biomolecules typically assemble into aperiodic structures in vivo, in contrast to inorganic crystals. This paper explores the evolution of our understanding of structures across the spectrum of materials, from living to inanimate, driven by those naive beliefs, with particular focus on the development of crystallography in materials science and biology. The idea that there is a clear distinction between these two classes of matter has waxed and waned in popularity through past centuries. Our current understanding, driven largely by detailed exploration of biomolecular structures at the sub-cellular level initiated by Bernal and Astbury in the 1930s, and more recent explorations of sterile soft matter, makes it clear that this is a false dichotomy. For example, liquid crystals and other soft materials are common to both living and inanimate materials. The older picture of disjoint universes of forms is better understood as a continuum of forms, with significant overlap and common features unifying biological and inorganic matter. In addition to the philosophical relevance of this perspective, there are important ramifications for science. For example, the debates surrounding extra-terrestrial life, the oldest terrestrial fossils and consequent dating of the emergence of life on the Earth rests to some degree on prejudices inferred from the supposed dichotomy between life-forms and the rest.

No MeSH data available.


Related in: MedlinePlus

(a) M.C. Escher's drawing of a tessellation of the Euclidean plane (), Angels and Devils (with orbifold 4*2); (b) Escher's hyperbolic Angels and Devils tessellation, drawn in the Poincaré disc model of ℍ2 (with orbifold 4*3); (c) A fragment of the hyperbolic tiling in (b), excised from ℍ2and mapped into a single (rhombohedral) unit cell of the P-surface (a three-periodic minimal surface) embedded in ; (d) A larger fragment of the P-surface, made of many unit cells, tiled with Angels and Devils. (Images (b–d) courtesy of Stuart Ramsden).
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RSFS20150027F5: (a) M.C. Escher's drawing of a tessellation of the Euclidean plane (), Angels and Devils (with orbifold 4*2); (b) Escher's hyperbolic Angels and Devils tessellation, drawn in the Poincaré disc model of ℍ2 (with orbifold 4*3); (c) A fragment of the hyperbolic tiling in (b), excised from ℍ2and mapped into a single (rhombohedral) unit cell of the P-surface (a three-periodic minimal surface) embedded in ; (d) A larger fragment of the P-surface, made of many unit cells, tiled with Angels and Devils. (Images (b–d) courtesy of Stuart Ramsden).

Mentions: We can move between patterns in and related patterns in ℍ2 with ease, by editing the order of symmetry elements [28]. Consider, for example, a flat two-dimensional crystal, such as the pattern of Angels and Devils by Escher, drawn in . This planar crystalline pattern can be negatively curved, by the addition of disclinations, to form a generalized hyperbolic (two-dimensional) crystal in ℍ2, as illustrated in figure 5a. Note that in this image the hyperbolic crystal is portrayed within the Poincaé disc model, so that the various angels and devils appear to shrink as they approach the disc edge. In fact, they do not in ℍ2, and this effect is an artefact of the map. In the true crystal (i.e. in ℍ2), all angels and all devils are identical. The image in figure 5b therefore portrays a regular tessellation—and example of Bernal's ‘repeat organization’—of ℍ2 with just two tiles, one angelic, the other diabolic. A map of the Angels and Devils tiling onto the P surface is illustrated in figure 5c. Structures that form these surfaces can therefore be viewed through two distinct perspectives. They can be comprehended as patterns embedded within , and their triplet of lattice vectors makes them members of Bernal's ‘small branch of crystallography, three-dimensional lattice crystallography’. That view corresponds to the image in figure 5d. Alternatively, they can be perceived within the confines of two-dimensional ℍ2 alone, rather like the view of a very thin, two-dimensional ant living in the hyperbolic surface. From that perspective, they are two-dimensional, hyperbolically curved crystals, arranged according to the *246 isometries of ℍ2. That view corresponds to the tiling of ℍ2 itself, shown in figure 5b.Figure 5.


Crystals: animal, vegetable or mineral?

Hyde ST - Interface Focus (2015)

(a) M.C. Escher's drawing of a tessellation of the Euclidean plane (), Angels and Devils (with orbifold 4*2); (b) Escher's hyperbolic Angels and Devils tessellation, drawn in the Poincaré disc model of ℍ2 (with orbifold 4*3); (c) A fragment of the hyperbolic tiling in (b), excised from ℍ2and mapped into a single (rhombohedral) unit cell of the P-surface (a three-periodic minimal surface) embedded in ; (d) A larger fragment of the P-surface, made of many unit cells, tiled with Angels and Devils. (Images (b–d) courtesy of Stuart Ramsden).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4590423&req=5

RSFS20150027F5: (a) M.C. Escher's drawing of a tessellation of the Euclidean plane (), Angels and Devils (with orbifold 4*2); (b) Escher's hyperbolic Angels and Devils tessellation, drawn in the Poincaré disc model of ℍ2 (with orbifold 4*3); (c) A fragment of the hyperbolic tiling in (b), excised from ℍ2and mapped into a single (rhombohedral) unit cell of the P-surface (a three-periodic minimal surface) embedded in ; (d) A larger fragment of the P-surface, made of many unit cells, tiled with Angels and Devils. (Images (b–d) courtesy of Stuart Ramsden).
Mentions: We can move between patterns in and related patterns in ℍ2 with ease, by editing the order of symmetry elements [28]. Consider, for example, a flat two-dimensional crystal, such as the pattern of Angels and Devils by Escher, drawn in . This planar crystalline pattern can be negatively curved, by the addition of disclinations, to form a generalized hyperbolic (two-dimensional) crystal in ℍ2, as illustrated in figure 5a. Note that in this image the hyperbolic crystal is portrayed within the Poincaé disc model, so that the various angels and devils appear to shrink as they approach the disc edge. In fact, they do not in ℍ2, and this effect is an artefact of the map. In the true crystal (i.e. in ℍ2), all angels and all devils are identical. The image in figure 5b therefore portrays a regular tessellation—and example of Bernal's ‘repeat organization’—of ℍ2 with just two tiles, one angelic, the other diabolic. A map of the Angels and Devils tiling onto the P surface is illustrated in figure 5c. Structures that form these surfaces can therefore be viewed through two distinct perspectives. They can be comprehended as patterns embedded within , and their triplet of lattice vectors makes them members of Bernal's ‘small branch of crystallography, three-dimensional lattice crystallography’. That view corresponds to the image in figure 5d. Alternatively, they can be perceived within the confines of two-dimensional ℍ2 alone, rather like the view of a very thin, two-dimensional ant living in the hyperbolic surface. From that perspective, they are two-dimensional, hyperbolically curved crystals, arranged according to the *246 isometries of ℍ2. That view corresponds to the tiling of ℍ2 itself, shown in figure 5b.Figure 5.

Bottom Line: The idea that there is a clear distinction between these two classes of matter has waxed and waned in popularity through past centuries.The older picture of disjoint universes of forms is better understood as a continuum of forms, with significant overlap and common features unifying biological and inorganic matter.In addition to the philosophical relevance of this perspective, there are important ramifications for science.

View Article: PubMed Central - PubMed

Affiliation: Department of Applied Mathematics, Research School of Physics and Engineering , The Australian National University , Canberra, Australian Capital Territory 0200 , Australia.

ABSTRACT
The morphologies of biological materials, from body shapes to membranes within cells, are typically curvaceous and flexible, in contrast to the angular, facetted shapes of inorganic matter. An alternative dichotomy has it that biomolecules typically assemble into aperiodic structures in vivo, in contrast to inorganic crystals. This paper explores the evolution of our understanding of structures across the spectrum of materials, from living to inanimate, driven by those naive beliefs, with particular focus on the development of crystallography in materials science and biology. The idea that there is a clear distinction between these two classes of matter has waxed and waned in popularity through past centuries. Our current understanding, driven largely by detailed exploration of biomolecular structures at the sub-cellular level initiated by Bernal and Astbury in the 1930s, and more recent explorations of sterile soft matter, makes it clear that this is a false dichotomy. For example, liquid crystals and other soft materials are common to both living and inanimate materials. The older picture of disjoint universes of forms is better understood as a continuum of forms, with significant overlap and common features unifying biological and inorganic matter. In addition to the philosophical relevance of this perspective, there are important ramifications for science. For example, the debates surrounding extra-terrestrial life, the oldest terrestrial fossils and consequent dating of the emergence of life on the Earth rests to some degree on prejudices inferred from the supposed dichotomy between life-forms and the rest.

No MeSH data available.


Related in: MedlinePlus