The number pi (π) is celebrated internationally on March 14 because it starts with 3.14, or the 14th day of the third month. The mathematical constant was first calculated about 4,000 years ago, and its approximations can be found in numerous ancient cultures, including the Egyptian Rhind Mathematical Papyrus and the Bible. This irrational number has crept into every facet 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 limitations, such as the one discovered by Joseph Cabeza-Lainez, an architect and academic at the University of Seville (Spain).

During the construction of the university’s new engineering building, Cabeza-Lainez found that calculating the area of a roof with straight lines resting on a semicircle was impossible just by using pi. After 30 years of research, Cabeza-Lainez published a paper about his discovery in ScienceDirect and issued another unreviewed report. Both articles present his proposal of a number psi (Ψ), with a value of 3.140923, close to pi but which can be applied to a versatile geometric form that he calls an antisphere. This is the formula for calculating psi:

Cabeza-Lainez, determined to achieve greater efficiency of radiation and light, has designed light-pipes with better lighting capability and lower energy consumption. In the process, he glimpsed the solution to the problem, which he wrote about in a book about solar light transfer (Fundamentos de transferencia radiante luminosa; Netbiblo, 2010). Cabeza-Lainez’s solution addressed efficient solar light transfer between a semicircle joined at the bottom to a rectangle. But the construction of the university’s engineering building added more complexity with a design that called for a surface connecting both shapes. How to find the area of that surface formed by straight lines that run from the end of the rectangle to the points of the semicircle?

One approach was to use the length of an ellipse designed by Indian mathematician Srinivasa Ramanujan, whose life inspired the movie The Man Who Knew Infinity. With Ramanjuan’s formulas for determining the perimeter of an ellipse as a function of the two axes, Cabeza-Lainez began calculating the lateral surface of an antisphere divided into two hemispheres.

For an antisphere with a radius and hemisphere height equal to 1, the lateral area is remarkably similar to pi squared, an algebraic expression that involves a new dimension, says Cabeza-Lainez. “Pi squared represents a three-dimensional surface that constitutes a new form – a transition between a cube and a sphere. The base is a square, and the elevation is a circle. It is the squaring of the circle.”

Using Ramanujan’s estimating approach and his own extensive research that involved developing proprietary calculation software, Cabeza-Lainez created the equation that led to psi:

“Psi is not pi because there is a small difference between 3.14092 and 3.14159. Although the numbers are very close, the difference is transcendent,” said Cabeza-Lainez. He applied psi, the antisphere and its various components to architectural projects (houses, ships and tunnels) to demonstrate the significance. He became convinced that it has unique optical, acoustic and thermal properties that lower costs by 50% because it reduces the surface area of conventional shapes such as cylinders and rectangles. What’s more, he has successfully concentrated and transferred solar light from the new structures into tunnels.

The tunnel application was endorsed by Antonio Manuel Peña García, a physicist and engineering professor at the University of Granada (Spain). Although Peña was not involved in developing psi, he participated in the practical application experiment published in Buildings and Tunnelling and Underground Space Technology.

“The article reflects a revolutionary strategy for taking advantage of natural light in tunnels,” said Peña, who believes lighting can significantly impact sustainable development. “Energy consumption in tunnels can cost hundreds of thousands of euros, close to €1 million a year,” said Peña. To lower energy consumption, Peña has worked on reducing the solar reflection at tunnel access points by using forest mass, thereby requiring less visual adaptation to the tunnel darkness. He has also used periscope systems to redirect sunlight into tunnels. But Peña found that the periscope systems applied in China “required greater tunnel height and are very expensive.”

Peña said, “Joseph told me that he had designed an absolutely incredible new surface. I recall asking him, ‘Does that surface enable you to direct angled light to where you want it to go?’ He said yes, so I told him about my idea of transferring solar light into the tunnels – not from above – but from lateral periscopes that take exterior light and transfer and project it onto the pavement, where it’s most needed.”

“Calculations show that a savings of 40% would be achieved – which is a lot – and would also improve tunnel safety. I support any calculation that comes from Joseph Cabeza-Lainez,” Peña said.

The uniqueness of the antisphere form is that “every section has exactly the same area, but none have the same shape,” said Cabeza-Lainez. The result is a unique, sinuous shape made with straight lines. “It can be applied to a downspout, to a network of pipes, to an earthquake-resistant tower, to 150-foot [50-meter] ships without using columns, even to biotechnological devices,” he said, showing models made with 3D printers. He is also investigating applying these new geometric forms to endow trains with better aerodynamic properties, and says antispheres could also be used for spacecraft. “I don’t yet know all of the potential applications. I come up with a new one every day,” said Joseph Cabeza-Lainez.

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