## Heronian TrianglesJuly 25, 2014 The Greek philosophers contributed much to our understanding of the world. Archimedes (c. 287 BC - c. 212 BC) was one such philosopher who wasn't content with just understanding nature. As an inventor, he wanted to harness the laws of nature for the benefit of his fellow man. One of his inventions, still in use today, is the screw pump.
Pneumatica. This steam engine, although primitive, predates James Watt by seventeen centuries.
A of a triangle with sides a, b, and c, is simply,
Where the parameter, s, known as the semiperimeter, is half the perimeter; viz., The formula can be written in terms of the sides, only, as Mathematicians are creative individuals, and when they see something this simple, they immediately find ways to build something more complex out of it. So, the idea arises as to what integer sides will yield an integer area. Such an object is called an Heronian triangle, and these triangles are sometimes generalized to those for which the sides and areas are rational, not just integers. If the triangle is a right triangle whose sides are a Pythagorean triple, it's a Heronian triangle. This follows from the sides being integers and the formula for the area of a right triangle. The area must be an integer, since at least one of the non-hypotenuse sides of such a "3-4-5" triangle must be even. Sascha Kurz of the Department of Mathematics, Physics and Informatics, the University of Bayreuth, has posted an article about Heronian triangles on arXiv.[5] This is a version of a paper published in 2008 in the Serdica Journal of Computing.[6] Kurz' paper addresses the discovery of Heronian triangles by computation. The following table is a selection of just those Heronian triangles with greatest common divisor of one, and areas from 250-300. I don't know whether it's proven that there are an infinite number of such triangles, but it's likely that there are, and that this is a simple proof. Alas (or, happily), I'm not a mathematician, so these things don't keep me awake at night. Kurz gives a example of a very large Heronian triangle of area 75,954,096, for which (a,b,c) = (14962,13666,11700). The area and perimeter of the 354 "primitive" Heronian triangles (those whose sides have a greatest common divisor of one) with largest side up to 500 are plotted in the following graph. These data were calculated with my own C-language program, whose source code is found here.
## References:- Steam Engine Library, University of Rochester Collection at the Hopkin Thomas Project, himedo.net.
- Bennet Woodcroft, Translator, "The Pneumatics of Hero of Alexandria," Taylor, Walton and Maberly (London, 1851).
- Amelia Carolina Sparavigna, "Water, air and fire at work in Hero's machines," arXiv Preprint, January 18, 2011.
- Sascha Kurz, "On the generation of Heronian triangles," arXiv Preprint Server, January 11, 2014.
- Sascha Kurz, "On the generation of Heronian triangles," Serdica Journal of Computing, vol. 2, no. 2 (2008), pp. 181-196. A PDF copy is available, here, also.
Linked Keywords: Ancient Greek philosophy; Greek philosopher; Archimedes (c. 287 BC - c. 212 BC); nature; inventor; physical law; laws of nature; invention; screw pump; Loch Ness Monster; Tony Cragg; 's-Hertogenbosch; The Netherlands; Wikimedia Commons; Hero of Alexandria; Heron; steam engine; Aeolipile; James Watt; transliteration; Greek; manuscript; intellectual; mathematician; engineer; engineering drawing; physics; physical; mechanics; mechanical; Roman Empire; Vitruvius; Ctesibius (285-222 BC); lecture; course; science; engineering; mathematics; father; STEM fields; Heron's formula; area; triangle; high school; student; right angle; Euclidean geometry; right triangle; trapezoid; mathematical proof; acute; scalene; triangle; Inkscape; semiperimeter; perimeter; integer; Heronian triangle; rational number; Pythagorean triple; hypotenuse; parity; even; Sascha Kurz; Department of Mathematics, Physics and Informatics; University of Bayreuth; arXiv; paper; computation; greatest common divisor; infinite number; C programming language; computer program; source code; heron.c; Gnumeric; noise; cryptography. |
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