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Pi is an irrational number whose decimal part is infinitely non-cyclic, that is, it cannot be expressed by the ratio of two integers. We often use 3.14 or 22 stroke 7 to approximate π, but these are only approximate values, not exact values. So, in other universes, does π have the same value? If in some universe, pi is not 3.14... So what is the difference between this universe and our present universe?

To answer this question, we must first make clear what pi is. Pi is defined as the ratio of the circumference to the diameter of a circle. That is to say, in a flat space, any circle satisfies C = 2 π r (C is the perimeter and r is the radius). This definition looks simple, but it actually hides an important assumption: space is Euclidean space.

Euclidean space refers to the space that satisfies the Euclidean axiom system. The Euclidean axiom system includes five basic axioms and some corollary theorems. One of the most important axioms is the parallel axiom: after a point outside the straight line, there is and only one straight line is parallel to it.

Euclidean space can be regarded as a spatial model in our daily life. In Euclidean space, the sum of the interior angles of triangles is 180 degrees, the diagonal of a square is equal, the diagonal of quadrilaterals in a circle is 180 degrees, and so on. And the pi is exactly in line with our expectations of the nature of the circle: no matter how big or small the circle is, where it is placed, how it rotates or changes its position, the ratio of its perimeter to diameter remains constant.

However, not all possible spaces are Euclidean spaces. In fact, before the 19th century, it was thought that only Euclidean space could conform to logical and natural laws. However, with the development of mathematics and physics, people gradually discovered non-Euclidean space.

Non-Euclidean space refers to a space that does not satisfy some axioms (especially parallel axioms) in the Euclidean axiom system. For example, when you draw a figure on a sphere, you will find a lot of strange phenomena: the sum of the interior angles of triangles is greater than 180 degrees, the shortest path no longer follows a straight line, and so on.

So, will pi remain constant in non-Euclidean space? The answer is no. In non-Euclidean space, pi is no longer a constant, but a variable, which depends on the size and position of the circle. What causes it? The reason is that in non-Euclidean space, space itself is curved. This means that in different places, distance, angle, area and so on have different measurement methods and results.

So what is the pi on the sphere? The answer is that there is no definite value. Because on a sphere, pi depends on the size of the circle. If we draw a small circle (relative to the radius of the sphere), it looks like a circle on a plane, with a ratio of perimeter to diameter close to 3.14. But if we draw a large circle (close to half a sphere), it looks like a straight line with a ratio of perimeter to diameter close to 1.

More generally, if we draw a small circle (relative to the curvature of space) in any non-Euclidean space, it looks like a circle on a plane, with a ratio of perimeter to diameter close to π; but if we draw a large circle (relative to the curvature of space), then it does not look like a circle on a plane, and the ratio of its perimeter to diameter will deviate from π.

So, how is pi calculated in non-Euclidean space? One way is to use the so-called Gauss-Bonnet theorem. This theorem tells us that on any surface, the Gaussian curvature of a small region is proportional to the difference between the sum of the interior angles of the triangles in the region and 180 degrees. In other words, if we draw a small circle on a surface and divide several triangles within the circle, then the difference between the sum of the triangles and 180 degrees can reflect the difference between pi and pi.

This article comes from the official account of Wechat: Vientiane experience (ID:UR4351), author: Eugene Wang

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