http://www.nexusjournal.com/96/emmer.html

from that page:

Architecture and Mathematics:

Soap Bubbles and Soap Films

Michele Emmer, Professor of Mathematics

Universit? di Roma "La Sapienza," Rome

and Universit? "Ca' Foscari," Venice

"I hope that none of you are yet tired of playing with bubbles, because, as I hope we shall see, there is more in a common bubble than those who have only played with them generally imagine." Charles V. Boys

When Joseph Plateau published his treatise on soap bubbles and film in 1873, soap bubbles already had their own place in literature and art. Plateau's problem consists in taking a generic curve in three-space and finding a surface with the least possible area bounded by that curve. The empirical solution may be obtained by dipping a tridimensional model of the curve into soapy water, resulting in a form called a minimal surface. When a soap bubble is blown, the soapy surface stretches; when blowing ceases, the film tends toward equilibrium. The sphere presents the least exterior surface area of all surfaces containing the same volume of air.

The isoperimetric property refers to the fact that the circumference encloses the largest surface area. It is reasonable to suppose that people of ancient times in charge of founding a town were aware of the isoperimetric property, at least empirically: A town wall of the least possible length containing the largest area had to be circular. The circular plan is more prevalent in some periods of history than in others. C.N. Ledoux presented a circular plan for a town, the form as "pure as the one the Sun describes in its movement."

Like the circle, the sphere also appears in architecture. Ledoux planned a spherical house. His contemporary, Boull?I, used the sphere in the cenotaph of Newton. The hemispherical igloos of the Eskimos solve the problem of a structure based on a plane with the greatest possible volume for the same external surface.

Frei Otto used soap film models to design his tensile-structures. He developed a technique to obtain a precise photogrammetric evaluation of soap film models and another method to simulate peaks in a membrane of soap films. Otto's Institute of Architecture, Stuttgart, was built along the lines of such a model.

H.A. Schwarz solved Plateau's problem for a non-plane boundary by developing the periodic minimal surface. Infinite periodic minimal surfaces, combinations of saddle polygons or surfaces, are more stable. Such a surface has been adapted as a play sculpture in the Brooklyn Museum, where children can actually enter into the labyrinthic structure of periodic minimal surfaces.

--------------------------------------------------------------------------------

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MYNOTE:

"The empirical solution may be obtained by dipping a tridimensional model of the curve into soapy water, resulting in a form called a minimal surface. When a soap bubble is blown, the soapy surface stretches; when blowing ceases, the film tends toward equilibrium. The sphere presents the least exterior surface area of all surfaces containing the same volume of air."

is this the key to get centroidal cells?

or is it with a formula to get center of mass using density function?

from that page:

Architecture and Mathematics:

Soap Bubbles and Soap Films

Michele Emmer, Professor of Mathematics

Universit? di Roma "La Sapienza," Rome

and Universit? "Ca' Foscari," Venice

"I hope that none of you are yet tired of playing with bubbles, because, as I hope we shall see, there is more in a common bubble than those who have only played with them generally imagine." Charles V. Boys

When Joseph Plateau published his treatise on soap bubbles and film in 1873, soap bubbles already had their own place in literature and art. Plateau's problem consists in taking a generic curve in three-space and finding a surface with the least possible area bounded by that curve. The empirical solution may be obtained by dipping a tridimensional model of the curve into soapy water, resulting in a form called a minimal surface. When a soap bubble is blown, the soapy surface stretches; when blowing ceases, the film tends toward equilibrium. The sphere presents the least exterior surface area of all surfaces containing the same volume of air.

The isoperimetric property refers to the fact that the circumference encloses the largest surface area. It is reasonable to suppose that people of ancient times in charge of founding a town were aware of the isoperimetric property, at least empirically: A town wall of the least possible length containing the largest area had to be circular. The circular plan is more prevalent in some periods of history than in others. C.N. Ledoux presented a circular plan for a town, the form as "pure as the one the Sun describes in its movement."

Like the circle, the sphere also appears in architecture. Ledoux planned a spherical house. His contemporary, Boull?I, used the sphere in the cenotaph of Newton. The hemispherical igloos of the Eskimos solve the problem of a structure based on a plane with the greatest possible volume for the same external surface.

Frei Otto used soap film models to design his tensile-structures. He developed a technique to obtain a precise photogrammetric evaluation of soap film models and another method to simulate peaks in a membrane of soap films. Otto's Institute of Architecture, Stuttgart, was built along the lines of such a model.

H.A. Schwarz solved Plateau's problem for a non-plane boundary by developing the periodic minimal surface. Infinite periodic minimal surfaces, combinations of saddle polygons or surfaces, are more stable. Such a surface has been adapted as a play sculpture in the Brooklyn Museum, where children can actually enter into the labyrinthic structure of periodic minimal surfaces.

--------------------------------------------------------------------------------

Order Nexus: Architecture and Mathematics

abstract index | Previous abstract | Next abstract | E-mail Michele Emmer | Nexus Network Journal Homepage

MYNOTE:

"The empirical solution may be obtained by dipping a tridimensional model of the curve into soapy water, resulting in a form called a minimal surface. When a soap bubble is blown, the soapy surface stretches; when blowing ceases, the film tends toward equilibrium. The sphere presents the least exterior surface area of all surfaces containing the same volume of air."

is this the key to get centroidal cells?

or is it with a formula to get center of mass using density function?