The regular heptagon is the seven-sided regular polygon illustrated above, which has Schläfli symbol 
. According to Bankoff and Garfunkel
(1973), "since the earliest days of recorded mathematics, the regular heptagon
has been virtually relegated to limbo." Nevertheless, Thébault (1913)
discovered many beautiful properties of the heptagon, some of which are discussed
by Bankoff and Garfunkel (1973).
Although the regular heptagon is not a constructible polygon using the classical rules of Greek geometric
construction, it is constructible using a Neusis
construction (Johnson 1975; left figure above). To implement the construction,
place a mark 
on a ruler 
,
and then build a square of side length 
. Then construct the perpendicular bisector at 
to 
, and draw an arc centered at 
of radius 
. Now place the marked ruler so that it passes through 
, 
lies on the arc, and 
falls on the perpendicular bisector. Then 
, and two such triangles give the vertex
angle 
of a regular heptagon. Conway and Guy (1996) give a Neusis
construction for the heptagon. In addition, the regular heptagon can be constructed
using seven identical toothpicks to form 1:3:3 triangles (Finlay 1959, Johnson 1975,
Wells 1991; right figure above). Bankoff and Garfunkel (1973) discuss the heptagon,
including a purported discovery of the Neusis
construction by Archimedes (Heath 1931). Madachy (1979) illustrates how to construct
a heptagon by folding and knotting a strip of paper, and the regular heptagon can
also be constructed using a conchoid of Nicomedes.
Although the regular heptagon is not constructible using classical techniques, Dixon (1991) gives constructions for several angles very
close to 
.
While the angle subtended by a side is 
, Dixon gives constructions
containing angles of 
,
,
and 
.
In the regular heptagon with unit circumradius and center 
,
construct the midpoint 
of 
and the mid-arc point 
of the arc 
, and let 
be the midpoint of 
. Then 
(Bankoff and Garfunkel 1973).
In the regular heptagon, construct the points 
, 
, and 
as above. Also construct the midpoint 
and construct 
along the extension of 
such that 
. Note that the apothem 
of the heptagon has length 
. Then
1. The length 
is equal to 
,
and also to the largest root of
 |
(1)
|
2. 
,
and
3. 
is tangent to the circumcircle of 
(Bankoff and Garfunkel 1973).
Construct a heptagonal triangle 
in a regular heptagon with center 
, and let 
and 
bisect 
and 
, respectively, with 
and 
both lying on the circumcircle. Also define the midpoints
, 
, 
, and 
. Then
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
(Bankoff and Garfunkel 1973).
