The number Pi (π), whose international day is celebrated every March 14, due to the expression of that date in English, preceding the number of the month before the number of the day (3.14), is about 4,000 years old, and its approximations are in Egyptian Rhind papyrus, found in the Bible and in every culture since. This irrational number has wormed its way into all facets of life, from engineering and architecture to statistics and quantum mechanics. But even this relationship between the circumference of a circle and its diameter has its limits. One was discovered by the architect and researcher from the University of Seville Joseph Cabeza-Lainez while working on the Higher School of Engineering. The calculation of the area of a roof designed with straight lines resting on a semicircle was not possible with the previous knowledge. Pi wasn’t enough. After 30 years of research, two publications (Science Direct and another unverified) collect the bases of his new proposal: the number Psi (Ψ) with a value of 3.140923, which is close to its sister number Pi, but allows its application to a geometric solid , which the researcher has dubbed Antisphera and which is also used in all areas. This is its formula:
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Cabeza-Lainez, determined to uncover the greater efficiency of radiation and light – he designed luminaries with greater luminous potential and lower consumption – had begun to solve the problem in his book Fundamentals of Luminous Radiant Transfer (Netbiblo, 2010). catch sight of. . in which he develops the radiation solution between a semicircle joined below with a rectangle. But it was the work of the School of Engineering that added the ultimate complexity with an interface that combined both forms. How does one find the area of this surface formed by straight lines starting from the end of the rectangle to each of the points of the semicircle?
One form of approximation was the length of the ellipse by Indian mathematician Srinivasa Ramanujan, whose life inspired the film The Man Who Knew Infinity. And with his formulas for determining the circumference of an ellipse using the two axes, he began to work on calculating the surface area of an antisphere divided into two hemispheres.
‘Antisphera’ divided into two hemispheres.J. CL.
For an antisphere with a hemispherical radius and hemispherical height of 1, the lateral area is extraordinarily similar to pi squared, an algebraic expression that adds a new dimension, according to the researcher: “Pi squared represents a three-dimensional surface, which again represents a solid which is a transition between a cube and a sphere. The floor plan is a square and the elevation is a circle. It’s squaring the circle.”
With Ramanujan’s approximations and years of research in which he had to develop his own calculation software, comes an equation that opens the door to PSI:
JCL
“Psi is not pi because there is a slight difference between 3.14092 and 3.14159. Although it is approximate, it is also transcendent,” asserts Cabeza-Lainez. And to demonstrate this, he has started applying PSI, the Antisphera and its sections in architectural works (houses, warehouses and tunnels), convinced that it offers unique optical, acoustic and thermal properties at half the price, since the surface is more traditional is shapes like cylinder or rectangle. In the tunnels, they managed to concentrate and redistribute the natural light of the new structures.
This last application is supported by the physicist and professor at the Faculty of Civil Engineering at the University of Granada, Antonio Manuel Peña García, who was not involved in the development of Psi but participated in the experimental application published by Buildings and Tunneling and Underground Space Technology.
“The article reflects a revolutionary strategy for using light in tunnels,” says Peña. The researcher recalls that lighting is one of the factors with the greatest impact on sustainable development. “The energy consumption of the tunnels can cost hundreds of thousands of euros, almost a million a year,” he explains. To reduce this, Peña has worked on conditioning the accesses to these infrastructures to reduce the reflection in the nearby areas of forest mass, gradually adapting the vision to the greater darkness of the road under the mountain. And also with systems that direct the sunlight inside like a periscope. But this system, used in China based on Peña’s research, “requires larger track gauges (tunnel height) and costs a lot of money,” he explains.
Joseph Cabeza-Lainez shows a new vault geometry designed by the ‘Antisphera’ to reflect the flow of light towards the street in a tunnel. Alejandro Ruesga
“Joseph told me,” says the physicist, “that he designed an absolutely wonderful and incredible surface, and I remember asking him: does this surface allow you to direct the light from an angle where you want it? He said yes and I told him about my idea of feeding light into the tunnels not from above, but from side periscopes that capture the light from outside, direct it and project it onto the sidewalk where it is most needed.”
“Calculations show that a saving of 40% would be achieved, which is a lot, and that safety in the tunnels would improve. I support any calculation made by Joseph Cabeza-Lainez,” concludes Peña.
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The uniqueness of the antisphere is that, as the architect who discovered Psi explained, “all of its sections have exactly the same area, but none have the same shape”. The result is a unique body, sinuous but with straight lines. “It can be attached to a downpipe, to a network of pipes, to an anti-seismic tower, to buildings 50 meters in span without pillars, even to biotechnological devices,” he specifies, while showing 3D-printed reproductions of the developments. He also studies the application of the new bodies to trains with better aerodynamic behavior and proposes the use of antisphere for spacecraft. “I don’t know the infinite possible uses. I think of a new one every day,” he assures.
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