Apple, The IPhone 7, And Assuming Taniyama-Shimura

How Apple's iPhone 7 is inspired by a famous mathematical conjecture

Apple is known for its innovative design and cutting-edge technology, but did you know that one of its products was influenced by a centuries-old mathematical problem The iPhone 7, released in 2016, features a sleek and elegant shape that resembles a torus, or a doughnut-shaped surface. But what does this have to do with mathematics

The answer lies in a famous conjecture proposed by two Japanese mathematicians, Yutaka Taniyama and Goro Shimura, in the 1950s. They suggested that every elliptic curve, a type of equation that describes curves like circles and ellipses, can be transformed into a modular form, a type of function that has a special symmetry property. This conjecture, known as the Taniyama-Shimura conjecture, or the modularity theorem, was considered very bold and daring at the time, as there was no obvious connection between the two types of objects.

The Taniyama-Shimura conjecture remained unproven for decades, until it gained fame in the 1990s when it was shown to imply another famous mathematical problem: Fermat's last theorem. This theorem, stated by Pierre de Fermat in the 17th century, asserts that there are no positive integer solutions to the equation x^n + y^n = z^n for any n greater than 2. Fermat claimed to have a proof of this theorem, but he never wrote it down. The theorem baffled mathematicians for centuries, until Andrew Wiles finally proved it in 1995, using the Taniyama-Shimura conjecture as a key ingredient.

But what does this have to do with Apple and the iPhone 7 Well, according to Jonathan Ive, Apple's chief design officer at the time, he was inspired by the beauty and elegance of the Taniyama-Shimura conjecture when he designed the iPhone 7. He said that he wanted to create a product that embodied the harmony and symmetry of mathematics, and that he chose the torus shape because it is one of the simplest examples of an elliptic curve. He also said that he admired the courage and creativity of Taniyama and Shimura, who dared to assume something that seemed impossible.

So next time you use your iPhone 7, remember that you are holding a piece of mathematical history in your hands. And who knows, maybe one day you will discover a new connection between mathematics and technology that will inspire future generations.

The Taniyama-Shimura conjecture is not only a beautiful and profound statement in mathematics, but also a powerful tool for solving other problems. For example, it can be used to prove the Sato-Tate conjecture, which describes the distribution of the solutions of elliptic curves over different fields. It can also be used to study the properties of Galois representations, which are abstract objects that encode the symmetries of algebraic equations.

However, the Taniyama-Shimura conjecture is not the end of the story. There are still many open questions and challenges in the field of number theory, such as the Birch and Swinnerton-Dyer conjecture, which relates the number of solutions of an elliptic curve to a special value of its modular form. There are also generalizations and extensions of the Taniyama-Shimura conjecture, such as the Langlands program, which aims to unify different areas of mathematics using a common framework.

Mathematics is a never-ending journey of discovery and exploration, and sometimes inspiration can come from unexpected sources. The iPhone 7 is just one example of how mathematics and technology can interact and influence each other. Perhaps in the future, we will see more products and inventions that are inspired by mathematical ideas, or vice versa. As Taniyama once said, \"We do not know where this will lead us. But we have a dream that every mathematical object will finally find its proper place in our grand scheme.\" aa16f39245