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Regular Heptagon


RegularHeptagon

The regular heptagon is the seven-sided regular polygon illustrated above, which has Schläfli symbol {7}. 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).

HeptagonConstruction

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 X on a ruler AZ, and then build a square of side length AX. Then construct the perpendicular bisector at M to BC, and draw an arc centered at C of radius CE. Now place the marked ruler so that it passes through B, X lies on the arc, and A falls on the perpendicular bisector. Then 2theta=∠BAC=pi/7, and two such triangles give the vertex angle 2pi/7 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 360 degrees/7. While the angle subtended by a side is 360 degrees/7 approx 51.428571 degrees, Dixon gives constructions containing angles of 2sin^(-1)(sqrt(3)/4) approx 51.317813 degrees, tan^(-1)(5/4) approx 51.340192 degrees, and 30 degrees+sin^(-1)((sqrt(3)-1)/2) approx 51.470701 degrees.

HeptagonMidpoints

In the regular heptagon with unit circumradius and center O, construct the midpoint M_(AB) of AB and the mid-arc point X_(CB) of the arc CB, and let M_(OX) be the midpoint of OX_(CB). Then M_(OX)=M_(AB)=1/sqrt(2) (Bankoff and Garfunkel 1973).

HeptagonDiagonal

In the regular heptagon, construct the points X_(CB), M_(AB), and M_(OX) as above. Also construct the midpoint M_(OF) and construct J along the extension of M_(AB)B such that M_(AB)J=M_(AB)X_(CB). Note that the apothem OM_(AB) of the heptagon has length r=cos(pi/7). Then

1. The length x=M_(AB)M_(OF) is equal to sqrt(2)r=sqrt(2)cos(pi/7), and also to the largest root of

 8x^6-20x^4+12x^2-1=0,
(1)

2. M_(OJ)=sqrt(6)/2, and

3. M_(AB)M_(OX) is tangent to the circumcircle of DeltaM_(OF)OM_(AB)

(Bankoff and Garfunkel 1973).

HeptagonalTriangleQuad

Construct a heptagonal triangle DeltaABC in a regular heptagon with center O, and let BN and AM bisect ∠ABC and ∠BAC, respectively, with M and N both lying on the circumcircle. Also define the midpoints M_(MO), M_(NO), M_(MC), and M_(NC). Then

MN=1/2M_(MO)M_(NO)=1/2M_(MC)M_(NC)
(2)
=sqrt(2)M_(NO)M_(MC)
(3)
M_(MO)M_(MC)=M_(NO)M_(NC)=1/2
(4)
M_(MO)M_(NC)=1/2sqrt(2)
(5)

(Bankoff and Garfunkel 1973).


See also

Conchoid of Nicomedes, Edmonds' Map, Heptagon Theorem, Heptagonal Triangle, Neusis Construction, Klein Quartic, Polygon, Regular Polygon, Trigonometry Angles--Pi/7

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References

Aaboe, A. Episodes from the Early History of Mathematics. Washington, DC: Math. Assoc. Amer., 1964.Bankoff, L. and Demir, H. "Solution to Problem E 1154." Amer. Math. Monthly 62, 584-585, 1955.Bankoff, L. and Garfunkel, J. "The Heptagonal Triangle." Math. Mag. 46, 7-19, 1973.Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 59-60, 1982.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 194-200, 1996.Courant, R. and Robbins, H. "The Regular Heptagon." §3.3.4 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 138-139, 1996.Dixon, R. Mathographics. New York: Dover, pp. 35-40, 1991.Finlay, A. H. "Zig-Zag Paths." Math. Gaz. 43, 199, 1959.Heath, T. L. A Manual of Greek Mathematics. Oxford, England: Clarendon Press, pp. 340-342, 1931.Johnson, C. "A Construction for a Regular Heptagon." Math. Gaz. 59, 17-21, 1975.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 59-61, 1979.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991.

SeeAlso

Heptagon, Regular Polygon

Cite this as:

Weisstein, Eric W. "Regular Heptagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularHeptagon.html

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