Heron and His Shortest Distance Problem

During the Hellenistic times lived one very intelligent mathematician, engineer, and even physicist by the name of Heron of Alexandria.

​He was born in Alexandria, Egypt around 10 AD according to Evangelos Papadopoulos (2007).

The actual year of his birth is not fully known, with many accounts ranging from 5 AD to 30 AD. It is said on a website made for famous mathematicians that Heron’s “career started with teaching at the Musaeum. . .” (2012).

To add, the Musaeum was a place that was a part of the famous library of Alexandria. The site goes on to say “however his achievements as an inventor are most noteworthy” .
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He even has the honor to have the vending machine called his “brainchild” by this site, having the “idea of inserting a coin in a machine” to have something returned.

As you may can tell, Heron had many different famous works, especially since he was so diverse in what he did. In the mathematics he did a lot of work in geometry. One of his most famous and important works within geometry was Metrica.

This book, according to a final project done by Shannon Umberger of the University of Georgia, was the “greatest contribution to the mathematical world” and that it was “mainly of geometric nature, dealing with area and volume mensuration of various polygons and solids” (2000).

Heron also gave a way to lift heavy objects using “pulleys, levers and wedges” according to the site on famous mathematicians.

He wrote other works according to the Encyclopedia Britannica like Mensurae, Geodaesia, Liber Geëponicus, and many more (2016). In his other areas he did just as much, for example in mechanics he had some books writing, as explained in the encyclopedia.

Some of their names are “Pneumatica, Automatopietica, Belopoeica, and Chierobalistra.” This shows just how dedicated of a man Heron was and why he is still so influential today.

One of the most famous discovery he saw and proved is known as Heron’s Shortest Distance Problem. This problem is considered one of the first optimization problems.

According to wolfram, he tried “finding the relationship between the angle of incidence and the angle of reflection for the reflection of light off a flat surface” (“Heron's Problem”). It goes on to say that “the angles are equal, as can be seen by assuming that the path the light takes is the shortest possible and reflecting a straight path through the surface across the surface” (“Heron's Problem”).

He solved that given two points A and B on one side of a straight line XY, to find the point C on that straight line such that |AC| + |CB| is as small as possible.

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To solve this problem mathematically, Heron noticed that if B is reflected across the line XY, to B', then for any point C on the line, |CB| = |CB'|, and hence minimizing |AC| + |CB| is equivalent to minimizing |AC| + |CB'|. But since the shortest path from A to B' is a straight line, the point C that minimizes |AC| + |CB'| should be the point of intersection of the line XY with the line segment AB′. Any other path, such as A -> C' ->B', will clearly be longer (Sanchis, 2014).

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