7 Elegant Mathematical Proofs Without Words

Mathematics is a vast and ever expanding subject. It’s got numbers, symbols, formulas, proofs and diagrams. As one advances to higher mathematics, they are exposed to the rigor that underpins the foundation of modern math.

Despite the time mathematicians spend constructing proofs from first principles, there exists a unique and captivating form of demonstration known as a proof without words, or a visual proof 🎨

It really is an artistic illustration, a tableau that captures an identity or a mathematical statement in such a way that its truth is revealed as self-evident, needing no verbal accompaniment.

Here are 7 elegant proofs without words.

1. Area of A Circle

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The area of a circle can be approximated by cutting it up into a collection of thin segments, which are rearranged in a shape that’s roughly rectangular.

Each half of the circumference forms the width of the rectangle and the height is roughly the radius of the circle.

In the limiting case as the number of segments goes to infinity, the area of the circle becomes πr².

2. 2π > 6

2π > 6

Focus on the perimeter of the circle and the hexagon.

The radius of the circle is 1 and from the first proof we know that the circumference is 2π. The hexagon is made of 6 equilateral triangles, each of which has side length 1 and so the perimeter is 6.

Euclidean geometry tells us that the shortest distance between two points is a straight line. Consequently, the sum of each part of the circumference is greater than all 6 sides of the hexagon.

Therefore, 2π > 6.

3. Pythagoras Theorem

a² + b² = c²

This is one of the first theorem we all know and love. The Pythagoras theorem states that the sum of the squares of the two sides of a right-angled triangle is equal to the square of the hypotenuse.

The diagram on the left shows a square with area c² in the middle surrounded by 4 triangles with sides a, b and c.

By rearranging the shapes, we see that the area c² is equal to two areas a² and b², which is exactly what we want.

There are many more proofs of the Pythagoras theorem. Can you come up with a new one?

4. 1/2 + 1/4 + 1/8 + … = 1

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Starting with a square of side length 1, we cut the area in half and then cut one half in half. We then cut the quarter in half again, so on and so forth. The patter continues ad infinitum.

Eventually, the whole sum will represent the area of the square with side length 1 as 1/2 + 1/4 + 1/8 + … = ¹² = 1.

5. A Slightly More Complicated Infinite Sum

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This one is slight more involved than the previous one.

We start with an equilateral triangle with area 1. Our next step would be to divide into 4 equal equilateral triangles, each with 1/4 area. After that, we repeat the same process for the middle triangle with 1/4 area and get 4 triangles with 1/16 area.

In a similar fashion, the pattern continues on for an eternity, culminating in the sum 3/4 + 3/16 + 3/64 + … = 1.

6. The Sum of The First n Odd Natural Numbers is n²

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The diagram on the right starts with 1 dot at the top and increases by 3 dots on the next level, 5 dots on the next level, so on and so forth.

By rearranging the dots as a square, we see that 1 + 3 + 5 + … + (2n-1) = n².

7. Formula for The Infinite Geometric Series

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We will end this article with an elegant proof of the formula of the infinite geometric sum.

The triangle abc is similar to the triangle eda. Because of this, the ratio of their sides is equal.

ed/da = ab/bc

1 + r + r² + r³ + … = 1/(1-r)

And that’s it for proof without words!

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