The Renaissance Mathematicus has completed yet another journey around the Sun and is, as of today, seventeen years old. Seventeen years of producing history of science blog posts to keep myself engaged and hopefully entertain and perhaps educate a few readers. Over the years I have penned either personal reflections or topics related to the number of years on my blogiversaries, and this year is no exception.
As regular readers may know, I have faced a challenging year in terms of health. Back in October, I had a lung collapse, an experience I wouldn’t recommend, which resulted in three very uncomfortable weeks in the hospital and several weeks recovering from the aftermath. However, I am pleased to report that in early May, I underwent a comprehensive lung examination, and my lungs are currently functioning well. Let’s hope that continues! This year, it was determined that what I had long believed to be degenerative orthopedic issues are largely neurological, caused by pinched nerves and compressed nerve roots. This condition has worsened, and for distances exceeding about fifty meters, I now depend on my electric wheelchair, with which I navigate the village where I reside. Interestingly, I have found that people are generally very kind and willing to help, offering to retrieve items from supermarket shelves that I can no longer reach. Aging is indeed, to say the least, fascinating.
17 is the 7th prime number. It is a Fermat number and the third of the five Fermat primes.
In number theory, it also qualifies as a Leyland number and a Leyland Prime of the format X^Y ± Y^X where X and Y are integers greater than 1, specifically using 2 and 3 (2^3 + 3^2) and using 3 and 4 (3^4 – 4^3). It is also one of the six Euler lucky numbers which are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k^2 − k + n yields a prime number.
However, it is in geometry that 17 plays a notable role.
Although all of Theodorus' work has been lost, Plato included Theodorus in his dialogue Theaetetus, which recounts his contributions. It is believed that Theodorus had demonstrated that all square roots of non-square integers from 3 to 17 are irrational through the Spiral of Theodorus.
The Spiral of Theodorus with the maximum right triangles laid edge-to-edge before completing one revolution. The largest triangle has a hypotenuse of √17.
The most intriguing instance of 17 in geometry arises from the work of Carl Friedrich Gauss (1777–1855). The ancient Greeks devised the mathematical challenge of determining which constructions and problems could be solved using only an ideal straightedge and an ideal compass.
The idealized ruler, known as a straightedge, is thought to be infinite in length, feature a single edge, and possess no markings. The compass is assumed to have no maximum or minimum radius and is considered to “collapse” when lifted from the page, preventing direct use for transferring distances.
One can, for example, bisect an angle, but not trisect it. The three classical problems discussed by the Greeks were the trisection of an angle, squaring a circle—i.e., constructing a square with the same area as a given circle—and doubling a cube—i.e., constructing a cube with double the volume of a given cube. All of these are unsolvable using a straightedge and compass.
A broader question is how many regular polygons can be constructed with only a straightedge and compass? All polygons with an even number of sides are constructible, but what about those with an odd number? In 1796, at just nineteen years of age, Gauss demonstrated that the heptadecagon, a regular seventeen-sided polygon, could indeed be constructed using straightedge and compass.
This accomplishment apparently inspired the young man to pursue a career in mathematics rather than philology. In his Disquisitiones Arithmeticae (Arithmetical Investigations), a treatise on number theory published in 1801, he presented the general theory for determining whether a regular polygon can be constructed with a straightedge and compass.
As is customary in my blogiversary post, I wish to express gratitude to all those who take the time to read my writings, especially those who comment, and most notably to those who correct my mistakes; it is always enlightening to learn something new.
So, off we head into the eighteenth year…