Constructing a Heptahedron – A new method using the Vesica Pisces

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.

heptahedron construction sequence

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

heptagon vesica pisces sequence

Squaring the Circle – A new method using the Vesica Pisces

Frank has discovered a new way to square the circle so that the perimeter of the square and the circumference of the circle are almost exactly equal. His method uses only a straight-edge and compass, and is the first to work from the inside out using the Vesica Pisces. The protocol is as follows (see the animated image below for visual reference):

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 a light line (m) connecting the center of circles #1 and #2, projecting it across most of the page.

4: Use where this line m meets the opposite side of circle #1 to be the center of a new circle, with the same radius as circle #2, this is circle #3.

5: Circles #2 and #3 form the Vesica Pisces, and intersect each other in two new points. Connect these points with a line (n), which is orthogonal to line m.

6: Where line n intersects circle #1 it creates two new points. Use these as the centers of two new circles with the same radius as Circles #2 and #3; let point Q be the center of circle #4 and point R be the center of circle #5. These new circles, #4 and #5, form another Vescia Pisces at 90 degrees to the first. The two Vescia Pisces cradle circle #1 where they overlap.

7: Circles #4 and #5 intersect in two points that lie on line m. Choose one of these points (P), and draw a line (s) orthogonal (at 90 degrees) to line m (and thus parallel to line n) at this point. From point P, extend line s on either side of line m so that its total length is just larger than the diameter of circle #1.

8: Note the center points of circles #4 and #5, Q and R. From each of these center ponts draw a line perpendicular to line n (parallel to line m), extending them to meet line s in two new points (S and T). Let S be the point made by extending from point Q and T be the point made from extending point R.

9: Open your compass to the length made between points S and R, and draw a circle (#6) of that radius with its center on point R.

10: Draw another circle (#7) of the same radius, with its center on point Q.

11: Circles #6 and #7 intersect line n in two new points (U and V) towards the outsides of the construction. Open your compass to the length made between the center of circle #1 and one of these new points, and draw a new circle (#8) with the same center as circle #1.

12: Circle #8 meets lines m and n in four points. Construct a total four lines (one on each point), two parallel to n and two parallel to m, such that the lines form a square in which circle #8 is circumscribed.

13: Open your compass to the original radius of circle #1. With the point of your compass on point U, mark a new point (Y) on line n towards the outside of the construction.

14: Draw a new circle (#9) with a center on point Y, with a radius equal to circle #1. Circle #9 is tangent to circle #8.

15: Open your compass to the distance between the centers of circles #1 and #9, and draw a final circle (#10) with the same center as circle #1. This circle has a perimeter almost exactly equal to the square constructed in step 12.

squaring the circle

 Click the image below for a bigger, static version of the full construction.

The mathematics of this construction are known exactly. Measurements of the circumference of the circle and the perimeter of the square agree to within 99.9+%, with the perimeter of the square being ever so slightly smaller than the circumference of the circle. It was proven in 1882 that it is impossible to perfectly square the circle using a finite number of steps with a straightedge and compass. This is a result of the Lindemann-Weierstrass theorem, which was involved in showing that the number pi is transcendental, and thus a non-algebraic number. Straightedge and compass constructions involving non-algebraic numbers can only be approximations.

Chestahedron Calculations

Dr. Karl Maret has kindly laid out the mathematical relationships that form the chestahedron and decatria (the dual of the chestahedron).

Click the image below to access the information (opens in a new window, via Microsoft’s SkyDrive).

Please note that there are four sheets in the Excel workbook (click the tabs at the bottom of the workbook to change which sheet you are viewing).

Lecture: Architecture and Form

The Sacramento Faust Branch of the Anthroposophical Society in America Presents

Architecture and Form:
The discovery of the archetypal form behind the building blocks of the universe.

Decka-in-Chesta-in-Edges

A Lecture by Frank Chester
Wednesday, May 16th, 2012 – 7:30-9pm

Stegmann Hall Teachers Education Room,
Rudolf Steiner College, Fair Oaks, CA

Goethe describes finding the archetypal plant “Urpflanze” in 1790, which he believes is the basis of all plants. Now there has been discovered a archetypal form, a basic model or a red line flowing through the building blocks of the universe (platonic forms) and their transformations. This discovery has been used to find the first form, the mother of form, the “Rosetta Stone” of form development. This basic model is really a single PLANE of generative forces underlying form fluctuations. In the lecture you will be shown how this Plane can be used in art and architecture.