Minimal Surface

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The Most Beautiful Mathematical Equations.

1. General Relativity

The equation above was formulated by Einstein as part of his groundbreaking general theory of relativity in 1915. The theory revolutionized how scientists understood gravity by describing the force as a warping of the fabric of space and time. The right-hand side of this equation describes the energy contents of our universe (including the ‘dark energy’ that propels the current cosmic acceleration). The left-hand side describes the geometry of space-time. The equality reflects the fact that in Einstein’s general relativity, mass and energy determine the geometry, and concomitantly the curvature, which is a manifestation of what we call gravity.

2. Standard Model
This equation describes the collection of fundamental particles currently thought to make up our universe. It has successfully described all elementary particles and forces that we’ve observed in the laboratory to date - except gravity, including recently discovered Higgs boson and phi in the formula. It is fully self-consistent with quantum mechanics and special relativity.

3. The Fundamental Theorem of Calculus 
This equation forms the backbone of the mathematical method known as calculus, and links its two main ideas, the concept of the integral and the concept of the derivative. It allows us to determine the net change over an interval based on the rate of change over the entire interval. The seeds of calculus began in ancient times, but much of it was put together in the 17th century by Isaac Newton, who used calculus to describe the motions of the planets around the sun.

4. 1 = 0.999999999….
This simple equation states that the quantity 0.999 followed by an infinite string of nines is equivalent to one, and is made by mathematician Steven Strogatz of Cornell University. Many people don’t believe it could be true. It’s also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity.

5. Special Relativity
Einstein makes the list again with his formulas for special relativity, which describes how time and space aren’t absolute concepts, but rather are relative depending on the speed of the observer. It shows how time dilates, or slows down, the faster a person is moving in any direction.

6. Euler’s Equation
This simple formula encapsulates something pure about the nature of spheres. It says that if you cut the surface of a sphere up into faces, edges and vertices, and let F be the number of faces, E the number of edges and V the number of vertices, you will always get V – E + F = 2. So, for example, take a tetrahedron, consisting of four triangles, six edges and four vertices. If you blew hard into a tetrahedron with flexible faces, you could round it off into a sphere, so in that sense, a sphere can be cut into four faces, six edges and four vertices. And we see that V – E + F = 2. Same holds for a pyramid with five faces - four triangular, and one square - eight edges and five vertices, and any other combination of faces, edges and vertices. The combinatorics of the vertices, edges and faces is capturing something very fundamental about the shape of a sphere.

7. Euler–Lagrange Equations and Noether’s Theorem
In this equation, L stands for the Lagrangian, which is a measure of energy in a physical system, such as springs, or levers or fundamental particles. Solving this equation tells you how the system will evolve with time. A spinoff of the Lagrangian equation is called Noether’s theorem. Informally, the theorem is that if your system has a symmetry, then there is a corresponding conservation law. For example, the idea that the fundamental laws of physics are the same today as tomorrow (time symmetry) implies that energy is conserved. The idea that the laws of physics are the same here as they are in outer space implies that momentum is conserved. 

8. The Callan-Symanzik Equation
Basic physics tells us that the gravitational force, and the electrical force, between two objects is proportional to the inverse of the distance between them squared. However, tiny quantum fluctuations can slightly alter a force’s dependence on distance, which has dramatic consequences for the strong nuclear force. What the Callan-Symanzik equation does is relate this dramatic and difficult-to-calculate effect, important when the distance is roughly the size of a proton, to more subtle but easier-to-calculate effects that can be measured when the distance is much smaller than a proton.

9. The Minimal Surface Equation
The minimal surface equation somehow encodes the beautiful soap films that form on wire boundaries when you dip them in soapy water. The fact that the equation is 'nonlinear,’ involving powers and products of derivatives, is the coded mathematical hint for the surprising behavior of soap films. 

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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.

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Spring Forest (5,3): embedded, unembedded, and cowl
12" x 11" x 9"
Knitted wool (Dream in Color Classy, in colors Happy Forest and Spring Tickle)
2009 and 2013
A (p,q) torus knot traverses the meridian cycle of a torus p times and the longitudinal cycle q times. Here are three instantiations of a (5,3) torus knot:
(a, middle) The knot embedded on a torus. A (p,q) torus knot may be drawn on a standard flat torus as a line of slope q/p. The challenge is to design a thickened line with constant slope on a curved surface.
(b, top) The knot projection knitted with a neighborhood of the embedding torus. The knitting proceeds meridianwise, as opposed to the embedded knot, which is knitted longitudinally. Here, one must form the knitting needle into a (5,3) torus knot prior to working rounds.
(c, bottom) The knot projection knitted into a cowl. The result looks like a skinny knotted torus.

Modern Striped Klein
2" x 14" x 7"
Knitted wool (Dream in Color Classy Firescorched in Aqua Jet with Sundown Orchid and Happy Forest)
2013
This Klein bottle was knitted from an intrinsic-twist Mobius band with the boundary self-identified. A Klein bottle can be viewed as the connected sum of two projective planes; here, the stripes highlight the two circles that generate the fundamental groups of the individual projective planes. In some positions, this coloring of the Klein bottle resembles an ouroboros (a snake eating its own tail). The design is more than 10 years old; I recently realized that I had no high-quality example of it (only worn classroom models) and thus created one. Dream in Color veil-dyed yarn was chosen to add a color depth to the seed-stitch texture. Images of this piece graced the cover of the March-April 2013 issue of American Scientist.

Free-Range Mathematician
Sarah Lawrence College / Smith College
Hadley, MA
http://www.toroidalsnark.net

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Commenting on the work of Terasaki et al. revealing ER’s helicoidal structure, Wallace Marshall writes in the same issue of Cell Leading Edge report suggesting this sort of study has been neglected in the ‘omics era. Highlighting the work of pioneering biomathematician D’Arcy Wentworth Thompson, he suggests the author’s field has been neglected with the discovery of the genetic code,

...and with it the desire to explain away questions of cellular structure by telling ourselves that geometry is encoded in the genome. Although the genome is not a blueprint that explicitly encodes shape, the genome does encode proteins that sculpt cellular structures, for example by dictating membrane curvature. The existence of such proteins goes against the concepts of D’Arcy Thompson and appeared to be a final nail in the coffin of his Pythagorean approach to cell biology.

On the differential equations that helped resolve the ER’s structure, Marshall notes:

The fact that the helicoid shape is predicted without having to know the value of any physical constants is one of the most beautiful and surprising results of this paper. It is not very shocking that physical properties of cellular components contribute to their shape... but usually when one talks about modeling the relation between physical forces and biological structures, one ends up having to know, or at least estimate, the value of various physical constants like elastic moduli, rate constants, and so on.

In this sense, it may be said that the shape of the ER connectors comes from mathematics rather than physics.

This simple mathematics emerges due to the helicoid's minimal surface: i.e. average curvature of the sheets is zero (as they are saddle-shaped).

Whereas helicoids are not seen that often, another minimal surface, the gyroid, arises in a huge number of contexts. The gyroid contains helical twists similar to the helicoid but, whereas the helicoid is periodic along one axis, the gyroid is periodic in three perpendicular axes.

This gives it the abstract (cubic) forms pictured above in diagrams of gyroid formations observed in butterfly scales, which “divide the cubic structure into two regions forming two continuous interpenetrating networks in the system” (maximising the band gap of the scales, related to their interaction with light).

The scales are formed of chitin (a ubiquitous structural biomolecule found across life’s kingdoms), but this is deposited into gaps formed by folds in the plasma membrane separated by smooth ER tubules,

suggesting that the elastic properties of biological membranes may drive the formation of complex-looking minimal surfaces. Under conditions of stress or viral infection, ER can form periodic structures, some of which represent triply periodic minimal surfaces.

The fact that Terasaki et al. needed advanced microscopy methods to visualize their structure suggests that we might not see recognizable mathematical forms because we don’t yet know how to look for them.

Read Thompson’s 1917 masterpiece of biophysics, On Growth and Form (extract pictured above) in full at archive.org.

Merete Rasmussen :

“I am Danish and have lived in London since 2005. My studio is in Camberwell, South London

For galleries that stock my work and current/upcoming exhibitions, please see ‘news’, or email me for more information. We also have open studios twice a year.

About my work

I work with abstract sculptural form

I am interested in the way one defines and comprehends space through physical form. My shapes can represent an idea of a captured movement, as a flowing form stretching or curling around itself, or the idea can derive from repeated natural forms or even complex mathematical constructions. Different form expressions appeal to me and results in my continuous exploration with many different variations: soft but precise curves, sharp edges, concave surfaces shifting to convex; the discovery and strength of an inner or negative space. I am intrigued by the idea of a continuous surface, for example with one connected edge running through an entire form.

I work with the idea of a composition in three dimensions, seeking balance and harmony. The finished form should have energy, enthusiasm, and a sense of purpose.

The form is emphasized by a monochrome matt surface and I find that strong colour builds further importance, strength and energy. My work is hand built in coiling technique in stoneware clay, and mainly unsuitable for outdoor display.”

w: http://www.mereterasmussen.com/

tumblr source : http://theklob.com/

via John Baez

The gyroid is a surface that chops space into two parts.  This is a portion of it, made into an elegant sculpture by Bathsheba Grossman. 

A minimal surface is a surface in ordinary 3d space that can’t reduce its area by changing shape slightly. You can create a minimal surface by building a wire frame and then creating a soap film on it. As long as the soap film doesn’t actually enclose any air, it will try to minimize its area - so it will end up being a minimal surface.

If you make a minimal surface this way, it will have edges along the wire frame. A minimal surface without edges is called complete. For a long time, the only known complete minimal surfaces that didn’t intersect themselves were the plane, the catenoid, and the helicoid. You get a catenoid by taking an infinitely long chain and letting it hang to form a curve called a catenary, and then turning the curve around to form a surface of revolution with equation like this:

sqrt(x^2 + y^2) = c cosh(x/c)

for your favorite constant c.  A helicoid is like a spiral staircase; in cylindrical coordinates it’s given by the equation

z = c θ

for some constant c. You can see a helicoid here - and see how it can continuously deform into a catenoid:

• Eric Weisstein, Helicoid, http://mathworld.wolfram.com/Helicoid.html

In 1970, A. H. Schoen discovered another minimal surface: the gyroid!  This one is very different.  It’s triply periodic, meaning that it repeats itself over and over as we move in 3 different directions in space, like a crystal.  He was working for NASA, and his idea was to use it for building ultra-light, super-strong structures:

• Eric Weisstein, Gyroid, from Mathworld, http://mathworld.wolfram.com/Gyroid.html

Starting around 1987, people started using computers to find lots minimal surfaces.  You can see a bunch of triply periodic ones here:

• Ken Brakke, Triply periodic minimal surfaces, http://www.susqu.edu/brakke/evolver/examples/periodic/periodic.html

How do you find a minimal surface?  For starters, a minimal surface needs to be locally saddle-shaped. More precisely, it has zero mean curvature: at any point, if it curves one way along one principal axis of curvature, it has to curve an equal and opposite amount along the perpendicular axis. Supposedly this was proved by Euler. If we write this requirement as an equation, we get a second-order nonlinear differential equation called Lagrange’s equation. So, finding new minimal surfaces amounts to finding new solutions of this equation.