Euclid's Notion of Area and Scissors Congruence

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Ancient Greek mathematical found Euclid derived scissors congruence to calculate the areas of various shapes and the notion of geometric area. This inspired German mathematician David Hilbert to propose the third problem to cut and paste 3D objects back together without losing the original volume. Modern mathematicians use Euclid's notion of area to compute the area of shapes even today.

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In math class, you probably learned how to compute the area of lots of different shapes by memorizing algebraic formulas. Remember "base x height" for rectangles and "½ base x height" for triangles? Or "𝜋 x radius²" for circles? .

But if you were in math class in ancient Greece, you might have learned something very different. Ancient Greek mathematicians, such as Euclid, thought of area as something geometric, not algebraic. Euclid’s geometric perspective, recorded in his foundational work "Elements," has influenced research programs across centuries – even the work of mathematicians today, like the two of us.

Euclid's notion of area is used in modern mathematics

Modern mathematicians refer to Euclid’s concept of "having equal area" as "being scissors congruent." This idea, based on cutting up shapes and pasting them back together in different ways, has inspired interesting mathematics beyond just computing areas of triangles and squares. The story of scissors congruence demonstrates how classical problems in geometry can find new life in the strange world of abstract modern math.

David Hilbert proposed 23 mathematical problems of which the 3rd problem was related to Euclid's notion of area

Euclid’s notion of area .

Today, people think of the area of a shape as a single number that can be computed using algebraic formulas or calculus. So what does it mean to think of area as something geometric the way the ancient Greeks did? .

Imagine you’re back in math class and you have a pair of scissors, some tape and a piece of construction paper. Your teacher instructs you to make a new flat, two-dimensional shape using all of the construction paper and only straight-line cuts. Using your scissors, you cut the paper into a bunch of pieces. You start moving these pieces around – maybe you rotate them or flip them over – and you tape them back together to form a new shape.

Euclid derived certain formulas for computing the areas of different shapes like triangles and circles

Using your algebraic formulas for area, you could check that the area of your new shape is equal to the original area of the construction paper. No matter how a 2D shape is cut up – as long as all the pieces are taped back together without overlap – the area of the old and the new shape will always be equal.

For Euclid, area is the measurement that is preserved by this geometric "cutting-and-pasting." He would say that the new shape you made is "equal" to the original piece of construction paper – mathematicians today would say the two are "scissors congruent." .

The concept of scissors congruence is used to compute the area of polygons

What can your new shape look like? Because you’re only allowed to make straight-line cuts, it has to be a polygon, meaning none of the sides can be curved.

Could you have made any possible polygon with the same area as your original piece of paper? The answer, amazingly, is yes – there’s even a step-by-step guide from the 1800s that tells you exactly how to do it.

In other words, for polygons, Euclid’s notion of area is exactly the same as the modern one. In fact, you may have even used Euclid’s idea of area before in computations without knowing it.

The concept of scissors congruence can also be applied to cut a 3D object and reassemble it at the same volume

For example, you can use scissors congruence to compute the area of a pentagon. Since area is preserved if you cut the pentagon up into smaller triangles, you can instead find the area of these triangles (using "½ base x height") and add them up to get the answer.

Hilbert’s third problem .

Perhaps the most infa​mous example of scissors congruence comes from the third problem made famous by German mathematician David Hilbert. Hilbert wrote a list of 23 mathematical problems in 1900 with the goal of inspiring the next generation of mathematicians. Of these, the third problem asked: .

This idea of cutting and pasting shapes to preserve an area has been around since the ancient Greeks

"Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to form the second polyhedron?" .

Simply put, you are asked to figure out a way to cut up a 3D object with scissors and paste it back together to make a new 3D object. To solve it, you need to understand something very similar to Euclidean area – the idea that something geometric is preserved if you cut the 3D object up into pieces that can be reassembled.

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