Thales’ circles

“All is water”

624 notes


diatoms - california academy of sciences geology

1. Biddulphia deodora -  Miocene, ph#000058D,scale bar = 10 µm

2. Actinoptychus chenevierei - holotype, Cretaceous, ph#000677D, scale bar = 10 µm

3. Lithodesmium margaritaceum - Cretaceous, ph#000865D, scale bar = 10 µm

4. Aulacodiscus currus - holotype, Eocene, ph#001088D, scale bar = 10 µm

5. Triceratium diversum - holotype, Eocene, ph#001131D, scale bar = 10 µm

6. Triceratium swastika - Cretaceous, ph#000956D, scale bar = 10 µm

Going to look these up now.

(Source: Flickr / casgeology, via beautyandthemaths)

Filed under geometry algae

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Minimal Surface

In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having a mean curvature of zero. The term “minimal surface” is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints.

Art by Paul Nylander.

(via visualizingmath)

Filed under minimal surfaces math

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Gyroid & Geometry of interfaces: topological complexity in biology and materials:  A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970. The gyroid separates space into two identical labyrinths of passages. The gyroid has space group Ia3d. Channels run through the gyroid labyrinths in the (100) and (111) directions; passages emerge at 70.5 degree angles to any given channel as it is traversed, the direction at which they do so gyrating down the channel, giving rise to the name “gyroid”.  Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by the equation: sinX.cosY + sinY.cosZ + sinZ.cosX = 0

In nature, self-assembled gyroid structures are found in certain surfactant or lipid mesophases and block copolymers. In the polymer phase diagram, the gyroid phase is between the lamellar and cylindrical phases. Such self-assembled polymer structures have found applications in experimental supercapacitors, solar cells and nanoporous membranes. Gyroid membrane structures are occasionally found inside cells: biomimetic materials.

Article - Geometry of interfaces: topological complexity in biology and materials.

  • Figure 3: Topologically complex porous chitin structure in the wing scales of the butterfly Callophrys Rubi. Light microscopy of the ventral (upper) side of the wings illustrates the arrangement of individual scales on the wings (each several hundreds of micrometres in length). Electron microscopy reveals the structure of the wing-scales to consist of two layers of parallel ribs that cover a poly-crystalline porous chitin matrix structured according to a single srs net of symmetry I4132, commensurable with the lattice parameter of biological gyroid membranes.

  • The recognition of the chitin matrix as a consolidated three-dimensional cubic pattern rather than a two-dimensional film lining the gyroid is important. In particular, the spectacular optical features of these wing scales, certainly due in part to the cubic structure of the chitin, are dependent on this structural feature.  Also, the porosity of the chitin matrices in the wing scales of this species is tuned to optimize optical effects in vivo, possibly an impressive example of the efficacy of natural selection, rather than simply a coincidental by-product of the self-assembly process.
  • Figure 6:  Various views of the gyroid minimal surface. Within each of the two network domains bounded by the gyroid, there are two chiral elements of opposite rotation sense: threefold screw axes along the [111] and, of opposite rotation sense, fourfold screw axes along the [100] axes. The centred skeletal graph is the srs network with three-valent vertices.
  • See more at source.

Video: Gyroid by Daniel Piker on Vemeo.

Images: Green Hairstreak (Callophrys Rubi) by Ardeola - The Gyroid triply-periodic labyrinth - Biology and materials. - Method for constructing the graph in “Gyroids of Constant Mean Curvature” by Grobe-Brauckmann.

Filed under minimal surfaces gyroid biology

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Mathematics and Architecture

The link between math and architecture goes back to ancient times, when the two disciplines were virtually indistinguishable. Pyramids and temples were some of the earliest examples of mathematical principles at work. Today, math continues to feature prominently in building design. We’re not just talking about mere measurements — though elements like that are integral to architecture. Thanks to modern technology, architects can explore a variety of exciting design options based on complex mathematical languages, allowing them to build groundbreaking forms. Take a look at several structures past the break that were modelled after mathematics.

1. Mobius Strip Temple, China
2. Tetrahedral-Shaped Church, Colorado
3. Pentagonal, Phyllotactic Greenhouse and Education Center. The Eden Project, Cornwall
4. A Mathematically-Inclined Cucumber in the Sky, London
5. Experimental Math-Music Pavilion, 1958 World’s Fair
6. Modern Music-Math Home, Toronto
7. Solar Algorithm Wizardry, Barcelona
8. Cube Village, Holland
9. Magic Square Cathedral, Gaudi Cathederal, Barcelona
10. Fractal Gas Station Makeover, Los Angeles

How many of these have you been to?

(Personally, I’ve only seen the Gaudí Sagrada Família church, which isn’t a cathedral btw.)

Filed under math architecture