Frank has also discovered a new straightedge and compass procedure to create a heptagon, a seven-sided polygon, using the Vesica Pisces. This is the first known construction that places the heptagon inside the center of the Vesica Pisces. As in the case of squaring the circle, a mathematically precise straightedge and compass construction is impossible, but this simple method comes very close. The construction process is as follows (refer to the animated diagram below):

1. Draw a circle (#1).

2: Open your compass to the diameter of circle #1, and draw a second circle (#2) (that has a radius exactly twice the original circle), whose center lies on the perimeter of circle #1.

3: Draw line m connecting the two crossing points of circles #1 and #2.

4: Connect the centers of circles #1 and #2 with line n, which is perpendicular to line m.

5: Place the tip of your compass on the point where lines n and m meet, and open it to the center of circle #1, and draw circle #3, which is exactly half the size of circles #1 and #2.

6: Circle #3 crosses line m in two places. Choose one crossing, point P, and place the tip of your compass there. This is one point of the heptagon. Open the compass to the nearby point where circles #1 and #2 meet, also on line m, and draw circle #4.

7: Circle #4 crosses circle #3 in two places. Choose one crossing, point Q, and place the tip of your compass there. Open your compass to the point where lines m and n meet in the center of the figure, and draw circle #5.

8: Circle #5 crosses circle #3 in two places, once near line m and once near line n. Place the tip of your compass on the place where circle #5 crosses circle #3 near line n. This is another point on the heptagon. Open the compass to point P and draw circle #6.

9: Circle #6 hits circle #3 in a new point on the heptagon. Draw a new circle, #7, with the same radius as circle #6 from this new point.

10: Circle #7 hits circle #3 in a new point on the heptagon. Draw a new circle, #8, continuing with the same radius.

11-14: Continue in this way until you have created circles #9 – #12.

15: Connect the seven points on circle #3 into the heptagon.

The construction is made up almost entirely of the Vesica Pisces form, repeated nine times:

hmm.

According to my calculations, this is pretty darn accurate ! I used some computer scripting to come up with the following results:

Degrees 2/7 of a circle estimated 102.941402865

Degrees 2/7 of a circle actual 102.857142857

difference ratio %0.0819194519666

According to wikipedia http://en.wikipedia.org/wiki/Heptagon#Construction, there are more accurate methods for approximating a heptagon but they require a ruler or grid paper to perform and thus are not as elegant as this 🙂