The regular polygon of 17 sides is called the heptadecagon, or sometimes the heptakaidecagon. Gauss proved in 1796 (when he was
19 years old) that the heptadecagon is constructible
with a compass and straightedge.
Gauss's proof appears in his monumental work Disquisitiones Arithmeticae.
The proof relies on the property of irreducible polynomial
equations that roots composed of a finite number of square
root extractions only exist when the order of the equation is a product of
the form 
,
where the 
are distinct primes of
the form
 |
(1)
|
known as Fermat primes. Constructions for the regular triangle (
), square (
), pentagon (
), hexagon (
), etc., had been given by Euclid, but constructions based
on the Fermat primes 
were unknown to the ancients. The first explicit construction
of a heptadecagon was given by Erchinger in about 1800.
The trigonometric functions 
and 
are both algebraic
numbers of degree 8 given respectively by
 |  |  |
(2)
|
 |  |  |
(3)
|
The heptadecagon with unit edge lengths has inradius and circumradius given by
 |  |  |
(4)
|
 |  |  |
(5)
|
both of which can be expressed in terms of finite root extractions. They can also be expressed as the largest roots of the algebraic equations
 |
(6)
|
 |
(7)
|
The area of the regular heptadecagon is given by
 |
(8)
|
where 
can be expressed as the largest root of
 |
(9)
|
The following elegant construction for the heptadecagon (Yates 1949, Coxeter 1969, Stewart 1977, Wells 1991) was first given by Richmond (1893).
1. Given an arbitrary point 
, draw a circle centered on 
and a diameter
drawn through 
.
2. Call the right end of the diameter dividing the circle into a semicircle 
.
3. Construct the diameter perpendicular to the original diameter by finding the perpendicular
bisector 
.
4. Construct 
a quarter of the way up 
.
5. Join 
and find 
so that 
is a quarter of 
.
6. Find 
so that 
is 
.
7. Construct the semicircle with diameter 
.
8. This semicircle cuts 
at 
.
9. Draw a semicircle with center 
and radius 
.
10. This cuts the line segment 
at 
.
11. Construct a line perpendicular to 
through 
.
12. This line meets the original semicircle at 
.
13. You now have points 
and 
of a heptadecagon.
14. Use 
and 
to get the remaining 15 points of the heptadecagon around the original circle
by constructing 
,
, 
, 
, 
, 
[filled circles], 
, 
, 
, 
, 
, 
[single-ringed filled circles], 
, 
, 
, 
, and 
[double-ringed filled circles].
15. Connect the adjacent points 
for 
to 17, forming the heptadecagon.
This construction, when suitably streamlined, has simplicity 53. The construction of Smith (1920) has a greater simplicity
of 58. Another construction due to Tietze (1965) and reproduced in Hall (1970) has
a simplicity of 50. However, neither Tietze (1965)
nor Hall (1970) provides a proof that this construction is correct. Both Richmond's
and Tietze's constructions require extensive calculations to prove their validity.
DeTemple (1991) gives an elegant construction involving the Carlyle
circles which has geometrography symbol 
and simplicity
45. The construction problem has now been automated to some extent (Bishop 1978).
à la Gauss."
Mathematica Educ. Res. 4, 31-36, 1995.Trott, M. "
à la Gauss."
§1.10.2 in 