Repeat this with the compass point at B using the same radius. The intersection
point of these two arcs is at P. As AP = AB and BP = AB all three sides of the
triangle APB are equal. The actual triangle is completed by joining AB, AP and BP
using the straight edge.
If you think having a straight edge seems rather a luxury, you’re not alone –
the Dane Georg Mohr thought so too. The equilateral triangle is constructed by
finding the point P and for this only the compasses are required – the straight
edge was only used to physically join the points together. Mohr showed that any
construction achievable by straight edge and compasses can be achieved with the
compasses alone. The Italian Lorenzo Mascheroni proved the same results 125
years later. A novel feature of his 1797 book Geometria del Compasso, dedicated
to Napoleon, is that he wrote it in verse.
A prince is born
Carl Friedrich Gauss was so impressed by his result showing a 17-sided polygon could
be constructed that he decided to put away his planned study of languages and become
a mathematician. The rest is history – and he became known as the ‘prince of
mathematicians’. The 17-sided polygon is the shape of the base of his memorial at
Göttingen, Germany, and is a fitting tribute to his genius.
For the general problem, the polygons with p sides where p is a prime
number are especially important. We have already constructed the 3-sided
polygon, and Euclid constructed the 5-sided polygon but he could not construct
the 7-sided polygon (the heptagon). Investigating this problem as a 17 year old,
a certain Carl Friederich Gauss proved a negative. He deduced that it is not
possible to construct a p-sided polygon for p = 7, 11 or 13.
But Gauss also proved a positive, and he concluded that it is possible to
construct a 17-sided polygon. Gauss actually went further and proved that a p-
sided polygon is constructable if and only if the prime number p is of the form
Numbers of this form are called Fermat numbers. If we evaluate them for n =
0, 1, 2, 3 and 4, we find they are the prime numbers p = 3, 5, 17, 257 and
65,537, and these correspond to a constructible polygon with p sides.
When we try n = 5, the Fermat number is p = 2^32 + 1 = 4,294,967,297.
Pierre de Fermat conjectured that they were all prime numbers, but unfortunately