Pythagorean Theorem and the Nature of Proofs

Has anyone ever claimed to disprove the Pythagorean Theorem? The answer is a clear and definitive no. The statement is not just that no such proof exists - it is that for it to be disproven, our fundamental theory of planar geometry would have to be inconsistent. We are in no imminent danger of this happening.

The Pythagorean Theorem, a cornerstone of Euclidean geometry, asserts that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. This concept has been thoroughly examined, with hundreds of elegant and varied proofs. A simple disproof of this theorem would introduce a contradiction into the very fabric of geometry, which is currently not deemed possible.

Geometric Contexts Beyond Euclid

While the Pythagorean Theorem holds true in Euclidean geometry, it does not necessarily extend to other geometries. Specifically, in spherical geometry, which studies shapes on the surface of a sphere, and in elliptic geometry, which models a space where Euclid's fifth postulate is replaced, the theorem takes on a different form.

Spherical Geometry and the Generalization

In spherical geometry, a branch of non-Euclidean geometry, right triangles can indeed exist, but their properties differ from those in Euclidean space. A right spherical triangle can have up to three right angles, and the Pythagorean Theorem as we know it does not hold. The traditional s^2 t^2 u^2 relationship is altered.

Transitioning to Elliptic Geometry

Elliptic Geometry, another type of non-Euclidean geometry, transforms the familiar concepts of length and angle. Here, a point is represented by a pair of antipodal points on a sphere. It's a complex but fascinating adjustment where traditional Euclidean concepts need to be redefined.

Redefines of Quadrance and Spread

Ultimately, in elliptic geometry, a concept known as quadrance replaces Euclidean length, and spread replaces the angle. Quadrance is defined in a way that any two collinear points yield a quadrance of 0, and any two perpendicular lines give a quadrance of 1. This set-up leads to a new formulation that generalizes the Pythagorean relationship:

Elliptic Pythagorean Theorem:
[1 - c (1 - a) (1 - b)]

When applied to a spherical right triangle with a right angle at C, this theorem simplifies because the spread at C is 1. This gives:

[1 - c 1 - a 1 - b]

Which means:

[c ab - ab]

This form is distinctively different from the Euclidean version but remains true under certain geometric conditions.

Connecting Quadrance and Spread

A deep dive into the law of trigonometry known as the Cross Law, we can derive the relationships that govern the quadrance and spread of the sides of an ellipse. The formula, jaw-dropping in its complexity, becomes:

Cross Law:
[abC - a - b - c^2 4(1 - a)(1 - b)(1 - c)]

And its dual form:

Dual Cross Law:
[ABc - A - B - C^2 4(1 - A)(1 - B)(1 - C)]

By exploring these laws, the unique nature of elliptic geometry is brought into focus, showcasing how even fundamental concepts like the Pythagorean Theorem adapt to new geometric environments.

Conclusion

The Pythagorean theorem, while fundamental in Euclidean geometry, reveals its limitations and transforms in non-Euclidean geometries such as spherical and elliptic. The exploration of these theorems opens up new vistas in geometry and deepens our understanding of mathematical principles across different dimensions and shapes.