Pi Day: A researcher and architect proposes a ‘brother’ of Pi (π), named Psi (ψ) and equal to 3.140923 | Science | The USA Print

The number Pi (π), of which the international day is celebrated every March 14 due to the expression of this date in English, which precedes the number of the month to the number of the day (3.14), is about 4,000 years old, and its approximations they can be found in the Egyptian Rhind Papyrus, in the Bible, and in every culture since. This irrational number has crept into all facets of life, from engineering and architecture to statistics and quantum mechanics. But this ratio between the perimeter of a circle and its diameter also has its limitations. One was discovered by the architect and researcher at the University of Seville Joseph Cabeza-Lainez during the work on the Higher School of Engineering. The calculation of the area of ​​a roof executed with straight lines that rested on a semicircle was impossible with the existing knowledge. Pi wasn’t enough. After 30 years of research, two publications (science direct and other not reviewed) collect the bases of his new proposal: the number Psi (Ψ), with a value of 3.140923, close to its sister number Pi, but which allows its application in a geometric body that the researcher has baptized as Antisphere and also with uses in all areas. This is its formula:


Cabeza-Lainez, determined to unravel the greater efficiency of radiation and light —he has designed luminaires with greater light potential and lower consumption—, had begun to glimpse the solution to the problem in his book Radiant Light Transfer Fundamentals (Netbiblo, 2010), in which he develops the solution for radiation between a semicircle joined at the bottom to a rectangle. But it was the work of the School of Engineering that added the ultimate complexity with a surface that connected both shapes. How to find the area of ​​that surface formed by straight lines that start from the end of the rectangle to each of the points of the semicircle?

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One form of approximation was the length of the ellipse of the Indian mathematician Srinivasa Ramanujan, whose life inspired the film. The man who knew infinity. And with his formulas to determine the perimeter of an ellipse as a function of the two axes, he began to work to calculate the lateral surface of an antisphere divided into two hemispheres.

'Antisphera' divided into two hemispheres.
‘Antisphera’ divided into two hemispheres.J.CL.

For one antisphere with radius and height of the hemisphere equal to 1, the lateral area is extraordinarily similar to Pi squared, an algebraic expression that, according to the researcher, adds a new dimension: “Pi squared represents a three-dimensional surface that constitutes a new body, which it is a transition between a cube and the sphere. The plan is a square and the elevation a circle. It is the squaring of the circle”.

With Ramanujan’s approximations and years of research, during which he had to develop his own calculation software, comes an equation that opens the door for PSI:


“Psi is not Pi because there is a small difference between 3.14092 and 3.14159. Although it is approximate, it is also transcendent”, assures Cabeza-Lainez. And to prove it, he has started to apply PSI, the Antisphere and its sections in architectural works (houses, warehouses and tunnels) convinced that it presents unique optical, acoustic and thermal properties at half the cost because the surface area of ​​conventional shapes such as cylinders or rectangles is reduced. In the tunnels they have managed to concentrate and redistribute natural light from the new structures.

This last application is endorsed by the physicist and professor at the School of Civil Engineering of the University of Granada Antonio Manuel Peña García, who has not participated in the development of Psi, but has participated in the experimental application published by buildings and by Tunneling and Underground Space Technology.

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“The article reflects a revolutionary strategy for the use of light in tunnels,” says Peña. The researcher recalls that lighting is one of the factors with the greatest impact for sustainable development. “The energy consumption of the tunnels can cost hundreds of thousands of euros, close to a million a year,” he explains. To reduce it, Peña has worked on the conditioning of the accesses to these infrastructures, to reduce the reflection in the nearby areas with forest mass and gradually accommodate the vision to the greater darkness of the road under the mountain. And also with systems that, like a periscope, redirect sunlight inside. But this system, applied in China based on Peña’s research, “requires greater gauges (tunnel height) and costs a lot of money,” he explains.

Joseph Cabeza-Lainez shows a new vault geometry designed from the 'Antisphera' to reflect the light flow towards the road in a tunnel.
Joseph Cabeza-Lainez shows a new vault geometry designed from the ‘Antisphera’ to reflect the light flow towards the road in a tunnel.Alejandro Ruesga

“Joseph told me,” says the physicist, “that he had designed an absolutely marvelous and incredible surface and I remember asking him: does that surface allow you to direct the light coming from an angle where you want it to be? He said yes and I told him about my idea of ​​injecting light into the tunnels not from above, but from side periscopes that catch the light from outside, direct it and project it onto the pavement, 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 endorse any calculation that comes from Joseph Cabeza-Lainez”, concludes Peña.

The uniqueness of the antisphere is that “all its sections”, as explained by the architect who discovered Psi, “have exactly the same area, but none have the same shape”. The result is a unique body, sinuous, but made with straight lines. “It can be applied to a downspout, to a network of pipes, to an earthquake-resistant tower, to 50-meter-span buildings without pillars, even to biotechnological devices,” he specifies while showing 3D-printed reproductions of the developments. He also investigates the application of the new bodies to trains with better aerodynamic behavior and proposes the use of the antisphere for spaceships. “I don’t know the infinite applications it has. I come up with a new one every day, ”she says.

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